1226437

Renormalisation perturbative et T-dualite - Nouvelles
metriques d’Einstein et super-espace harmonique
Pierre-Yves Casteill
To cite this version:
Pierre-Yves Casteill. Renormalisation perturbative et T-dualite - Nouvelles metriques d’Einstein et
super-espace harmonique. Physique mathématique [math-ph]. Université Paris-Diderot - Paris VII,
2002. Français. �tel-00002166�
HAL Id: tel-00002166
https://tel.archives-ouvertes.fr/tel-00002166
Submitted on 17 Dec 2002
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µ
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R .
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=
2
Z
d2 x Gij (η µν + µν ) ∂µ φi ∂ν φj
ij
ij
k
k
gij = G(ij) ,
hij = G[ij] .
äw wutz z } xvut f = f B
äw tu zv®tz t }tx x{uzvtx |tx wu v|v xxv vxz zx }vxuvw
ij,k
ij
s
ks
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t |}t ρ §T = ρ © tu |{
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zvxxtwu {vwutwwu t w ww uzvv}t wx } wt}}t uzvt tx t uzvtx
wt zzwu £uzt v}twutx zt{vtz zzt wuvt t xv }tx w|uvwx χ xwu
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§wu ric© È
p
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Ric = −(G.ric.G) + D v ,
v = −2 G f − ∂ ln( det g) .
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tu xt{}tz uzwt È t}}t w~txu vt{{twu xxv}t t z|t t } {uzv|t ric wt
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wx }~}vu §ÆÖ© È
¤ zuvz t G
Ricij = −χs (G.
−1
= B + A.φ
∂B
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∂ρs
w tu xxv |zvzt
∂G
∂B
= −G.
.G .
∂ρs
∂ρs
tx t zt}uvwx tz{tuutwu }zx ~utwvz } z{t §ÆÎ© zt|tz|t È
Ricij = χs
p
vi = −2 Git fst s − ∂i ln( det g) .
∂Gij
+ Dj v i ,
∂ρs
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s
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ij
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t xw twut}}t

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z }tx }ztx xt{v¦xv{}tx §ÆÔ© w~txu wz}}t{twu x zv®t ~txu |t v xt
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3
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t xuz|uzt x~|zvu f = −2a δ tx }ztx t vt y uzvx v{twxvwx xwu vwxv
xztx tw t |}xxtx xvwu } w}}vu t a
z } |}xxt ¤ §a = 0© } zzvu v}uwvt txu zv®t
z } |}xxt §a 6= 0© } zvxzt tx x{uzvtx zvutx |wvuvwwtz }
zv®|uvw t } zzvu v}uwvt ¤vwxvxv w ztw } {uzv|t B } }x wz}t
st
1
s
t1
ij
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s2 s1 r3 ij
} zt}uvw §ÆÔ© txu }zx v}twut xxu{t xvwu È
§ÆÅÕ©
ν ≡ n2 r2 + n3 r3 = 0 ,
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t }t xxu{t §ÆÅÕ© xvu zv® t|tu wx }t |x t }~}zt vw|v §n =
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det B
2
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¡ztz¦zuw
2
1
2
2
2
3
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i
dσi = ε
1
ijk σj ∧ σk ,
2
ε = ±1 .
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}t uzv}tu ~σ z
L
⊕ su(2)R
vu xz
δ~σ = ~R ∧ ~σ .
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w|uvwx β(t) tu γ(t) xwu vztwutx } x{uzvt SU (2) txu zvxt tw w U (1) tu
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t x{uzvt ®w} xtz SU (2) × U (1) vxt }tx uzwxz{uvwx vw®wvuxv{}tx t
su(2) ⊕ su(2) wwtzwu }zx
L
R
R
L
R
L
R
δ~σ = ~L ∧ ~σ .
wx ux }tx |x xv w = γ(t) }t zt t x{uzvt txu }zv y SU (2) ×SU (2)
tu } {uzvt g txu }zxβ(t)
|wz{{twu }ut
äw tu }zx y zuvz t } {uzvt g ®wvz }t {}t xv{ |zztxwwu È
R
1
S=
T
Z
d2 x gij ∂+ φi ∂− φj .
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L
Ø ÛÜ! ># z{&(|*ÜÚ'Ü ) &ÚÜ .Ù&'*Ü
z |{{vu w x {φ = t, φ = θ, φ = ϕ, φ = ψ} tu g = g dφ dφ
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0
1
2
3
ij
i
j
g3 = β(t) (σ1 2 + σ2 2 ) + γ(t) σ3 2 ,
tw }vwu {{twuw{twu }t utz{t α(t) dt t }~w zutz y } ®w tx Å¦z{tx
©
©
J
σ twu }zx }t z }t tx |zwux J §ε = 1 K §ε = −1 t }~w vu wx
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wu }zx £uzt zt{}|x wx } uzvt }t z t wt}}tx |zwwtx {r, y, z}
äw uvtwu z } {uzvt ®w}t
2
i
µ
i
r2 + β(t)2
ĝ = α(t) dt +
∆
2
<
i
µ
ry
dy
dr + 2
r + β(t)2
2
+
β(t)
y 2 β(t) γ(t) 2
2
dz
dy
+
r2 + β(t)2
∆
§ÆÅÆ©
∆ = y 2 β(t) + r2 + β(t)2 γ(t) .
Æ¦z{t xx|vt utwuvt} t uzxvw x~|zvu
§ÆÅÉ©
{uzvt ĝ tz x x{uzvt SU (2) tu wt xxt }x ~w U (1) xx|v
îv}}vw ∂
äw ®wvu }tx {uzv|tx ĝ ĥ tu Ĝ xt}}tx z H = ĥ dφ̂ ∧ dφ̂ ĝ =
ä
ĝ dφ̂ dφ̂ tu Ĝ = ĝ + ĥ t| {φ̂ = t, φ̂ = r, φ̂ = y, φ̂ = z} w wutz ric }t
utwxtz t ¥v||v t } uzvt vwvuv}t tu R̂ic |t}v t } uzvt }t
H=−
y 2 β(t)
r y γ(t)
dr ∧ dz +
dy ∧ dz .
∆
∆
z
ij
ij
i
j
ij
ij
ij
ij
0
ij
1
1
2
2
3
ij
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t w{zt z{uztx ttwu £uzt ||x wx }tx w|uvwx α(t) β(t) tu γ(t)
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}t xt} |}t ztwz{}vx wx wt ztwzwx tw |{ut t tx {uzvtx xv¦
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t |tuut tzwvzt zzvu v ®wvzwu }tx uzvtx vwvuv}tx w xtw|t t uzxvw
} zzvu xv¦vwxutvw tx uzvtx vwvuv}tx x~|zvu xv{}t{twu
ricij = λ gij + D(i vj) .
{r, y, z}
¡
¡
ÆÖ
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tx vx{uzvtx t } uzvt vwvuv}t z|twu îv}}vw ∂ zx } z{t t v y v =
δ f (t)
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}~v}tw|t
§ÆÅÍ©
ric = λ g + D v ⇐⇒ R̂ic = λ̂ Ĝ + D v̂ + ∂ ŵ ,
<

 λ̂ = λ ,
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v̂ = −2 λ ĝ X + ∂ Log∆ + v ,
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uwu ®wv z X = r ∂ + y ∂
u ~z }~}vu λ̂ = λ {wuzt t } |wxuwut t |}t T xt ztwz¦
{}vxt t|ut{twu t } {£{t w z } uzvt vwvuv}t tu } uzvt }t |t t
wx wx uvwx zx t {wuztz
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t } x}uvw
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wx }tx |wuzt¦utz{tx y wt |}t |{ t } uzvt }t |t t } uzvt
vwvuv}t
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} |wxtzuvw t }λzzvu
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x~|zvzt xx } z{t ~w zvtwu wx xÕÆ v} txu {wuz t |t} w~txu {vx
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z{t ~w{}vt v}uwvt ww }vt |tuut vx y wt uz|t ww w}}t xz }tx |wxuwutx
t xuz|uzt t }~}zt }vxt
w vu }~zvuvw utz{t −2 λ ĝ X wx v̂ txu }v vu t wx |wxv¦
zwx t } |wxuwut t |}t T txu tuzvtzt y } {uzvt g wx } uzvt
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y wt |}t t } uzvt {vx } zzvu {uzvt v|v wwt z } uzvt
}t
is
s
R̂icij = χT
i
∂
Ĝij (T ) + Dj v̂i + ∂[i ŵj] .
∂T
z uut {uzvt vwxutvw xv¦vwxutvw y |wxuwut |x{}vt ww
w}}tv} txu uzx xxv}t t vzt w |wt{twu t |zwwtx tz{tuuwu ~|zvzt
} {uzvt xx } z{t
gij (λ) =
tij
,
λ
< t txu wt {uzvt t |wxuwut |x{}vt }t y }~wvu äw vu vwxv t|
w w xxu{t t |zwwtx ~v} txu xxtª xv{}t t ztwuztz }t |}t wx }
{uzvt vwvuv}t È
ij
1
1
gij (λ) =
tij = gij (λ T ) .
T
λT
z vz |wuvwtz y uv}vxtz }tx zx}uux t } xt|uvw z|twut v} xu w|
~}vtz }t |utz }} tu t vzt }t zt{}|t{twu λ −→ λ T wx §ÆÅÍ© tu
§ÆÅÎ© äw tu z v}}tzx xtz wx §ÆÅÅ©
1
T
α(t) =
a(t)
,
λT
b(t)
,
λT
β(t) =
γ(t) =
c(t)
,
λT
< }tx w|uvwx a(t) b(t) tu c(t) xwu vwtwwutx t λ tu t T äw w| utwvz
wt {uzvt }t Ĝ (T ) v tw |}t } txu }zx xxv}t t {wuztz
}~}vu
ij
Ĝij − 2 Dj (ĝis X s ) − 2 ∂[i X s Ĝsj] = −T
|t v wwt z }t utwxtz t ¥v||v t } uzvt }t
R̂icij = χT
∂
Ĝij + Dj (vi + ∂i Log∆) ,
∂T
∂
Ĝij ,
∂T
χT = −λ T 2 .
tuut tzwvzt zt}uvw {wuzt y } vx }~v}tw|t y wt |}t tu } |wxtzuvw
t } zzvu v}uwvt uwx t z } uzvt vwvuv}t w vtw
∂
ricij = λ T gij + Dj vi = tij + Dj vi = −λ T
∂T
2
tij
λT
+ Dj v i = χ T
∂
gij + Dj vi .
∂T
wx |tuut tzxt|uvt < }t |}t txu vw|} wx } {uzvt vwvuv}t } z¦
zvu v}uwvt txu vtw zv®t È }tx vtztw|tx x}{twuvztx |{ t }
uzvt }t xwu }v{vwtx z } zt®wvuvw v}uw
Ψ̂ = Ψ + Log ∆ .
Æð
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vwvw ®5 WH312/N54KH3 Q21 14/LW4L/21 WHI.J2¯21
wx w |zt ~}v|uvwx xtz¦x{uzvtx ¢x tu tuxx wu |zvu |{¦
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îï}tz ÔÎ x }}wx twvxtz v|v |tzuvwx |x < } {uzvt vwvuv}t SU (2)×U (1)
txu îï}tz tu {wuztz t |tuut zzvu tvxut tw|zt z }tz {uzvt }t
x xxtzwx ~v} tvxut w |v t |zwwtx }{ztx z }tx¦
t}}tx }tx x{uzvtx SU (2) × U (1) vxxtwu }vwvzt{twu Çwt ut}}t uxt v{}vt
}~vwuzv}vu tx xuz|uztx |{}ttx ¤®w t ®wvz |t}}tx¦|v wx ®wvxxwx z
} {uzvt vwvuv}t §ÆÅÅ© }t vtztvw
e0 =
e2 =
p
p
α(t) dt , e1 =
β(t) σ2 , e3 =
p
p
β(t) σ1 ,
γ(t) σ3 .
zxt β(t) w~txu x wt |wxuwut v} txu uzx xxv}t t zt®wvz wx
©
§ÆÅÅ } |zwwt t t w y |t t }~w vu β(t) = t Çwt |wvuvw xxwut z
t } uzvt t zu xvu îï}tz txu }zx wwt z }~vtwuvu
1
§ÆÅÖ©
γ(t) =
.
α(t)
tuut |wvuvw zwvu tw ttu } z{t t îï}tz xvwut È
ρ1 = e0 ∧ e3 + e1 ∧ e2 = dt ∧ σ3 + t dσ3 = d(t σ3 ) .
ú~vtwu¦v} }zx t } xuz|uzt |{}tt xx|vt y ρ zx }vxuvw £ v w xt
1
σ̂i = −Ĝsi dφ̂s ,
w tu |zvzt } {uzvt }t xx } z{t
ĝ =
tu zv®tz t } Å¦z{t
i, s ∈ J1, 3K,
1
dt2 + t (σˆ1 2 + σˆ2 2 ) + γ(t) σˆ3 2 ,
γ(t)
1
ρ̂1 = dt ∧ σ̂3 + t σ̂1 ∧ σ̂2 = Jˆ1ij dφ̂i ∧ dφ̂j
2
txu vtw wt z{t t îï}tz ¿ÒÄ» ¶¸·á¼¸µ äw tw ttu z } xuz|uzt ztxt
|{}tt Jˆ xx|vt }tx zzvux
1


Jˆ J sj = −δi j ,

 1is 1
Jˆ1(ij) = 0 ,



Di Jˆ1jk = 0 ,
ÆÔ
Ø ÛÜ! ># z{&(|*ÜÚ'Ü ) &ÚÜ .Ù&'*Ü
< D txu } tzvt |zvwut t| uzxvw uwx t } zxtw|t t uzxvw vu t
}~w w~ }x } tz{tuzt t } z{t t îï}tz {vx } zt}uvw Ô×
H
uwu ®wv z §ÆÅÉ©
dρ̂1 = (? dH) ∧ ρ̂1 ,
z v}}tzx }zxt ~tw }x t §ÆÅÖ© w
γ(t) = t +
a
,
t
} {uzvt vwvuv}t §ÆÅÅ© x~vtwuv®t }zx y } {uzvt ~|v¦ wxw ×ð tu
tvtwu w| tz¦îï}tz tx xuz|uztx |{}ttx u¦}tx t } uzvt vwvuv}t
zwvxxtwu }zx z xv{}t |wt{twu σ −→ σ̂ |t}}tx t } uzvt }t È
i
i

q
p
t
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q
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wvwx È3 IHQJ2 XÉ.2/PÊËXJ2/
äw |wxvzt t {}uv}tux t |{x |{}ttx ϕ tu ϕ y (n+1) |{xwutx
v xt uzwxz{twu xt}w w }tu t SU (2) È
1
2
H
φ=
ϕ1
ϕ2
,
δφ = ~ε ·
~τ
φ,
2i
tu xt}w } ztzxtwuuvw w{twu}t t SU (n + 1) È
T
δϕ1 = ~ε ·
~λ
ϕ1 ,
2i
δϕ2 = ~ε ·
~λ
ϕ2 ,
2i
< }tx ~λ xwu }tx {uzv|tx t t}}¦¡ww (n + 1) × (n + 1) tz{vuvtx t uz|t w}}t
zuzvu tu ztt{w zutwu t } {uzvt } }x xv{}t |t}}t t }~tx|t
}u È }t xt|utz xwvt }zwvtw x~|zvu }zx ∂ φ . ∂ φ |wxuz|uvw
{}t tz¦îï}tz xt vu }zx tw t ut{x È
Å äw t uu ~z }t U (1) t |{ t t A ®wv z }tx uzwxz{¦
uvwx vw®wvuxv{}tx È
+
µ
µ
µ
δϕ1 = i ε(x) ϕ1 ,
δϕ2 = i ε(x) ϕ2 ,
δAµ = ∂µ ε(x) .
tx zvtx zuvt}}tx ∂ xwu }zx zt{}|tx z }tx zvtx |zvwutx
®

D = ∂ − i A tu zx uvw t } t }t xt|utz xwvt }zwvtw
x~|zvu
(ϕ̄ ∂ ϕ − ϕ ∂ ϕ̄ )
§ÍÅ©
L = ∂ ϕ̄ ∂ ϕ +
.
4 ϕ̄ ϕ
t }zwvtw wwt } vxuw|t z xv{}t xxuvuuvw ∂ −→ d
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φ ~τ φ = ~b .
~zvvwt xtz¦x{uzvt t |tuut |wuzvwut xtz t}v|vut }x }vw wx }t
|zt xtz¦tx|t z{wvt ¤t| }t |v ~b = (0, 0, 1) |t}}t¦|v xt uzvu
z }tx {}uv}tux z
µ
µ
µ
µ
i µ
µ
i
µ
i
i µ
i
2
i
i
i
µ
+
ϕ̄1 ϕ2 = 0 ,
ϕ̄1 ϕ1 − ϕ̄2 ϕ2 = 1 .
zt{vzt }vu zwvu vwxv t |wuzvwutx zt}}tx uwvx t } tv{t
w~tw zwvu ~wt
ßÄà¿·ÑºÄá ì

• x x{{tx zuvx ~wt uzvt xxwu 4(n + 1) z{uztx zt}x w
twu w îv}}vw tu tw ®wu uzvx |wuzvwutx zt}}tx w uvu w| y wt
uzvt ®w}t y 4n |zwwtx
ÍÅ
Ø ÛÜ! T # U Ý& Ü!,ÝÁA-Û!(Ü C &Ú Ù&Û(* E-ÙA-Û!({&Ü
Í Æ© zvxt }t SU (2) tw w U (1) tzv{ z }tx zuuvwx
• |wuzvwut §
uz t ~b tx vx{uzvtx ®w}tx t } uzvt xtzwu w| U (1) ×SU (n+1)

• zuzvu tu ztt{w w~wu x utw } {uzvt t}v|vut t |t {}t
w ttu wt wwt zuvt uzv} ztxut y vzt vx~v} u tw|zt uztz
wt zt®wvuvw tx {}uv}tux ϕ tu ϕ ®w ~utwvz tx |zwwtx vw¦
twwutx X v tzwu xuvxvzt } |wuzvwut §ÍÆ© È |~txu |t ~w t}}t
} ·Âá¸¾º¶¼¸µ ¹Äá »¸µ¶·¿¼µ¶Äá tuut zt®wvuvw vut tu zx ztzu wx
§ÍÅ© } vxuw|t xtz utwt tw zt{}wu }tx utz{tx ∂ X z dX
• t ut t |wxuz|uvw t t xw zvvwt xtz¦x{uzvt u z¦
{} t w {u{uvt wx } |wxuz|uvw vut Ê Ñº¸¶¼Äµ¶ óëêÄ·Á
ÌÍó¾Ä·Ì î¥ð× }}t |wxvxut tw zuwu ~wt {uzvt xv{}t §v|v }~tx¦
|t }u© y v{vwtz tuvu y tuvu }t w{zt t v{twxvwx v tx ®uvwx
t t tu tx |wuzvwutx {{t }tx {uzvtx tz¦îï}tz xwu z|{twu
t v{twxvw 4n }t w{zt uu} t îv}}vwx tu t |wuzvwutx zt}}tx vu
£uzt w {}uv}t t 4
H
H
1
T
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i
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i
i
wvwv È3 IHQJ2 TL542/3KH3KTL2
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t t ww¦t}vtwwt SU (2) t |{ t t V~ ®wvt z }tx uzwxz{uvwx
vw®wvuxv{}t È
µ
+
µ
µ
δφ = ~ε(x).
~τ
φ,
2i
δ V~µ = ∂µ ~ε(x) + ~ε(x) ∧ V~µ .
tx zvtx zuvt}}tx ∂ xwu vwxv zt{}|tx z }tx zvtx |zvwutx
¤

~τ
D = ∂ − i · V~ zx }v{vwuvw tx uzvx tzx t }v tzu t V~ z=|t
2
uvwx |{
w uvtwu } vxuw|t
(φ ~τ dφ − dφ ~τ φ)
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dφ . dφ +
.
µ
µ
µ
µ
µ
+
+
+
2
4 φ+ . φ
xtz¦x{uzvt v{xt |tuut vx } |wuzvwut x|}vzt
φ+ . φ = 1 .
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ßÄà¿·ÑºÄá ì

|wuzvwut zt}}t È tw|zt
• |v w t uzvx tzx t }v tzu tu w v{xt wt ¤
®
wt vx } {uzvt w}t xtz t v{twxvw 4n } ®w t }tz zuv|}t
zuzvu tu ztt{w xztwu t |{vwtz }tx t z|txxx z|twux
tw zuwu ~wt |uvw y 4(n + 2) z{uztx ®w ~utwvz zx }tx t
®uvwx t t tu }tx uztx |wuzvwutx wt {uzvt y 4n v{twxvwx
vtw t ww t}zt w twxt t |tuut vt tz{tu ~uvz y }~tutwxvw
utzwvwvt t } {uzvt ~|v¦ wxw
ÍÆ
T #³ # UÜ !Ù.*DAÜ %Ü * !-ÝÙ*&Û(ÙÚ %ÜÝ 'ÙÚÛ!(ÚÛÜÝ
tx vx{uzvtx t } uzvt ®w}t xwu {vwx SU (2) × SU (n + 1) w
vu w tu }tx utwzt y Sp(n + 1) |{{t w }t tzz }x }vw
• t ut t |wxuz|uvw }v xxv u z{} t w {u{uvt wx
} |wxuz|uvw vut Ê Ñº¸¶¼Äµ¶ Ñº¿¶Ä·µ¼¸µ¼ÑºÄÌ }ð× t w{zt
tt{}tx utzwvwvtx utwx z |tuut {ut xwu wwx wx }ð×
}ÔÆ {vx }y tw|zt }tx {uzvtx t}v|vutx w~wu x u wwtx
•
H
T
7X MN VK
} ztxuvu z }tx t {}tx z|twux y zt®wvz }tx 4n z{uztx ztxuwux
wx }t xt|utz xwvt t w y zxzt }tx |wuzvwutx t uzv} txu }zx
zt{twu ¿¾íÂÃ·¼ÑºÄ z tt{}t xv w zu }zwvtw
L = ∂µ ΦT ∂µ Φ
<


ϕ1
Φ =  ϕ2  ,
ϕ3
}tx ϕ uwu zt}x tu t }~w v{xt } |wuzvwut Φ Φ = 1 , w vu vtw ~w |v
t |zwwtx ztxut y vzt äw tu vwxv y zuvz t } |wuzvwut |vtz ~tzv{tz
ϕ tw w|uvw t ϕ tu ϕ t }~w ztzut }zx wx }t }zwvtw È
T
i
3
1
2
L = ∂µ ϕ i ∂µ ϕ i +
(ϕi ∂µ ϕi )2
,
1 − ϕi ϕi
i = 1, 2 .
xxuvuuvw z{t}}t ∂ −→ d tz{tu vwxv ~utwvz wt vxuw|t xz } xzt S
t| }tx |zwwtx tu tx |x < } zx}uvw tx |wuzvwutx txu xxv xv{}t
xwu {}tztxt{twuϕ zztxϕ
z }t {}t tz¦îï}tz §ÍÅ© ¤}ztª¦{ tu ztt{w wu utw
w xxu{t t |zwwtx xuvxvxwu } |wuzvwut §ÍÆ© z ~b = (0, 2, 0) wx
¤ðÕ t}v¦|v txu ww z
2
µ
1
2

ϕ1 = f (1 + ū.u)−1/2 (u, 1) ,




 ϕ = f (1 + v̄.v)−1/2 (v̄, 1) ,
2
1/4


(1 + ū.u)(1 + v̄.v)


 f =
,
(1 + u.v)2
< u tu v xwu tx t|utzx y n |{xwutx |{}ttx wx |tx |zwwtx }
{uzvt
txu t}v|vut{twu îï}tz t| z utwuvt} t îï}tz
(1 + ū.u)(1 + v̄.v)
K=
(1 + u.v)(1 + ū.v̄)
1/2
.
}x wu zv®tz } w}}vu utwxtz t ¥v||v {vx w~wu {wuztz } zzvu
tz¦îï}tz t } {uzvt ttwwu vx z vtzxtx |wxvzuvwx v}x wu
ÍÉ
Ø ÛÜ! T # U Ý& Ü!,ÝÁA-Û!(Ü C &Ú Ù&Û(* E-ÙA-Û!({&Ü
|wt|uz t |t}}t¦|v uvu } {uzvt t }v }×Ô tuut |wt|uzt w~ u
{wuzt t uzªt wx }x uz z }twu wx }ÔÍ ¤zwu }t |x
ðÅ
n = 1 vu u zx} wx < v} uvu {wuz t |tuut {uzvt ~vx{uzvtx
®

×ð |x zuv|}tz tx {uzvtx t }v
U (2) x~vtwuv vu y |t}}t ~ |v¦ wxw
z } v{twxvw Í
¥t}wx t } {uzvt t }v uvu ®wvt y zuvz tx |zwwtx z
|{}ttx §α = 1, · · · , n© xz CP z }txt}}tx } {uzvt t vwv¦u x~|zvu
α
n
gαβ̄ =
∂2K
,
∂zα ∂zβ̄
K = ln(1 + z̄.z) .
äw |{}uvu }zx |t}}tx¦|v z }tx |zwwtx ζ |{}ttx §α = 1, · · · , n© t|
α
t = g αβ̄ (z, z̄) ζα ζ̄β .
{uzvt tz¦îï}tz t }v |}vu }zx utwuvt} t îï}tz K tw |¦
zwwtx }{ztx (z , ζ ) ww z
α
α
K = ln(1 + z̄.z) − ln(1 +
√
1 + 4 t) .
~vtwuv®|uvw t| §ÍÅ© tu t| }t t|utz ~b = (0, 0, 1) txu ww wx }ÔÍ È
1
ϕ1 = p
(z, 1) ,
(1 + z̄.z) h
p
(1 + z̄.z) h (ζ̄, −z̄ ζ̄) ,
ϕ2 =
h=
2
√
.
1 + 1 + 4t
{{t }t {wuzt |tu tt{}t ww xt}t{twu v} w~txu x vx t uztz t ÊwwtxÌ
|zwwtx v xv{}v®tzwu } zx}uvw tx |wuzvwutx {vx t xz|z/u |t}}tx¦|v wt
xtzwu wz}t{twu x }tx ÊwwtxÌ |zwwtx v wwtzwu ®w} wt {uzvt
t}v|vut xv{}t vzt zt|wwvxx}t
w |t v |w|tzwt }t {}t utzwvwvt uz z zuzvu tu ztt{w
}~vtwuv®|uvw t } {uzvt utwt wt xt{}t x vz u zt wx } }vuuz¦
uzt } x~vu tw vu {}t IH {}t utzwvwvt |{|u |zztxwwu
uvtwu Sp(n + 1) äw }t {wuzt tw xwu
n
Sp(1) × Sp(n)
ϕα1 = q0α − i q3α ,
ϕα2 = q2α − i q1α ,
α = 1, · · · , n + 1 ,
< q tu q §i = 1, 2, 3© xwu zt}x äw tu }zx ®wvz w t|utz y n + 1 |{xwutx
utzwvwvtx È
α
0
α
i
α
ζ =
q0α e0
+
3
X
qrα er
,
|wuzvwut §ÍÍ© tvtwu }zx
r=1
α
α
ζ̂ ζ = 1 ,
<
α
e0 er = er e0 = er
.
er es = −δrs e0 + rst et
ζ̂ =
ÍÍ
q0α e0
−
3
X
r=1
qrα er ,
T #T # Î-Û!({&ÜÝ Á Ü!,ÏÐ*Ü! ÜÛ Ý& Ü!,ÜÝ 'Ü !AÙÚ({&Ü
uwvx t } vxuw|t §ÍÉ© x~vtwuv®t y
dζ̂ α dζ α − (ζ̂ α dζ α )(dζ̂ β ζ β ) .
|wuzvwut tu }zx x~}v{vwtz tw xwu
ζα = √
z α = 1, · · · , n ,
χα
1+ρ
ζ n+1 = √
1
,
1+ρ
ρ = χ̂α χα .
äw ztuzt }zx z } {uzvt }~|zvuzt t ut vwv¦u |zztxwwut y
IH È
n
ds2 =
dχ̂α dχα (χ̂α dχα )(dχ̂β χβ )
−
.
1+ρ
(1 + ρ)2
z n = 1 w uvtwu } xzt S ~vx{uzvtx Sp(2) ∼ SO(5)
4
VV
ÑMÀN L¿7WÒÓL N7W7 LW
ÀN
|tzut t wt}}tx {uzvtx wx }t |zt xtz¦x{uzvt w~ £uzt
z}vxt |w|zut{twu t z=|t y }~vwuz|uvw ~w z{}vx{t xv{}t tu vxxwu
|t}v xtz¦tx|t z{wvt wx }tt} } xtz¦x{uzvt N = 2 txu u{uv¦
t{twu z}vxt t}v¦|v vu xw zvuvw wx î ðÍ äuzt t w{ztxtx
}v|uvwx |t z{}vx{t t xtz¦|{x tz{vx t {tuuzt tw tzt } |wxuz|¦
uvw t}v|vut t wt}}tx {uzvtx tz¦îï}tz y zuvz xt|utz xwvt t
{}tx xv{ wx }t |zt t } xtz¦x{uzvt N = 2 u tw zwvxxwu wt {¦
ut xxu{uvt z |wxuzvzt t ut}}tx {uzvtx v} tz{tu t }vzt vzt|ut{twu
xz }t }zwvtw }tx x{uzvtx t xxtz } {uzvt ®w}t
wx }t xt|utz w}uvt ^303 xu}t xx }~|uvw t } xtz¦x{uzvt N = 2
}tx |zwwtx xtz¦tx|t z{wvt xwu
§ÍÎ©
(ζ) = {x = x − 2 i θ σ θ̄ , θ , θ̄ . , u } .
+
m
A
m
+
m
+
+
α
+
α
±
i
t xtz¦tx|t z{wvt uvzt xw w{ tx |zwwtx u v xwu tw vu }tx
z{wvtx xzvtx xz } xzt S È
±
i
2
u+
u−
1
1
+
u2 u−
2
=
cos θ
i sin θ e−i φ
i sin θ ei φ
cos θ
,
t|
u+i u−
i = 1
+ −
+
ui uj − u−
i uj = ij
.
tx vwv|tx i, j ∈ J1, 2K xwu {wux tu vxxx y }~vt utwxtz wuv¦x{uzvt
t| } |wtwuvw = = 1 } txu |v}t t zv®tz t uu utz{t t } z{t X
xt ztuut xz }tx z{wvtx u xt}w
12
ij
+i
21
±i
+s
+i
+s
−i
X +i = (u−
− (u+
.
s X )u
s X )u
ÍÎ
tx θ tu θ̄
+
+
Ø ÛÜ! T # U Ý& Ü!,ÝÁA-Û!(Ü C &Ú Ù&Û(* E-ÙA-Û!({&Ü
xwu tx zv}tx zxx{wvtwwtx tu }~w
+
+
i
θα+ = θαi u+
i , θ̄ α. = θ̄ α. ui .
äw tu {vwutwwu |zvzt } z{t wz}t tx xtz¦|{x wx }t xt|utz
w}uvt È
Q (x , θ , θ̄ , u ) = F + θ M + θ̄ N + i θ σ θ̄ A + θ θ̄ D + tz{vwx .
tx |{xwutx F M N A tu D xwu tx xzvtx wz}t{twu vw®wvtx tw u
äw w| wz}t{twu wt vw®wvu t |{x v}vvztx {{t w wt x~vwuztxxt
~ xt|utz xwvt }zwvtw w wt uvtwz {vx |{ut tx |{xwutx
tz{vwvtx tx xtz¦|{x äw ®wvu z v}}tzx } zvt |zvwut
< ∂ = u ∂ .
D = ∂ − 2 i θ σ θ̄ ∂ ,
∂u
wxvut wuwx t }t xtz¦tx|t §ÍÎ© txu zt} vx y vx t } |wvxw wz}v¦
xt e zvu t } |wvxw |{}tt tu t } zt|uvw wuv}t xz S ¤vwxv
w
+
A
+
+
+
±
i
+
−
++
+2
−
−
−
m
++
+
+2
+
−
m +
+2
−
m
+2
−3
±
i
−3
m
+
±
i
±±
m
∓i
2
±
±i
uf
,
i = u
+ = −θ + ,
θf̄
+ = θ̄ + ,
θf
++ = D ++ .
]
D
w®w tw }x SU (2) xtz¦x{uzvt vxxwu xz }tx vwv|tx i, j w vw¦
uzvu w SU (2) t }v¦Ôzxt xz tx vwv|tx a, b tw ®wvxxwu }t }tu Q
z
tu Q = −Qf .
Q
=Q
|v xxv }tx vwv|tx a tu b xwu {wux tu vxxx y }~vt t
} txu }zx xxv}t ~ww|tz }t zx}uu t î ðÍ È
éÄ ¾¿í·¿µí¼Äµ ¾Ä ê¾ºá íÂµÂ·¿¾ ¹õºµ à¸¹ã¾Ä á¼íà¿ µ¸µÁ¾¼µÂ¿¼·Ä ½ áºêÄ·Á
áëàÂ¶·¼Ä í¾¸Ã¿¾Ä ÕÖÏ× Äµ ¹¼àÄµá¼¸µ è× Äá¶ ¹¸µµÂ ê¿·
Z
1
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L
=
[d θ d θ̄ ][du] L (Q , u ) D Q + L (Q , u ) . §
S
+
a
PG
+
a=1
+
+
a=2
+
ab
+
2 + 2 +
HK
+
a
2
+
±
++
+a
+4
+
±
~vwuzuvw xz [d θ d θ̄ ] tuzvu }tx utz{tx tw θ θ̄ uwvx t }~vwuzuvw
xz [du] wt zt t
}t xvw}tu xz S tx xzvtx tw u w zuvt wx ztwzwx
uzx
2 + 2 +
+2
+2
±
i
2
+
±
+
L+
a (Q , u ) = Qa .
{uzvt |zztxwwut y §ÍÖ© wt vu twzt t t uztx Ê|zwwtxÌ
t}}tx¦|v xwu }t }x xtwu wwtx z }t utz{t ~zzt w tw t
t
}~w wut F z utwvz } vxuw|t v} u w| ~z |}|}tzu ux Q}tx ||{x
v}vvztx §M N A D tu }tx utz{tx ~zzt xzvtz y w tw u t F © wt
ztz t }t xvw}tu xz S utz{t tw θ θ̄ vx tt|utz } xxuvuuvw ∂ −→ d
tx {uzvtx utwtx xtzwu }zx uutx tz¦îï}tz
ÍÖ
i
a
+
i
−
a
−
a
−
a,m
2
−3
a
+2
+2
+
a θ=0
+
i
+
a
m
T #Ø# Ä!-, ÙÛÜÚÛ(Ü*Ý Å Ü@ÜA *ÜÝ
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t utz{t L (Q , u ) t §ÍÖ© txu t} ê·ÂÁê¸¶Äµ¶¼Ä¾ |z |tu tu |wuvtwu
uutx }tx vwz{uvwx xz } {uzvt ®w}t |zztxww|t twuzt }t z¦utwuvt}
tu } {uzvt ®w}t w~txu x |wwt y zvzv |t}}t¦|v w~uwu t}v|vut ~wt vx }tx
|}|}x §}v{vwuvw tx |{x v}vvztx }x twut}}t{twu ®uvw tx tx tu
zx}uvw tx |wuzvwutx© tt|ux } x~vu w| t |wxuzvzt wt |zztxww|t
twuzt z¦utwuvt}x tu {uzvtx ®w}tx ttwwu |{{t wx }t tzzwx z }
xvut }tx vx{uzvtx t } {uzvt ®w}t zvxxtwu |}vzt{twu wx }t z¦utwuvt}
~txu wx |t tzwvtz t xtzwu vwuzvu tx utz{tx zvxwu t}v|vut{twu }t SU (2)
}t SU (2) wx ä¥ðð v} u {wuz t uu îv}}vw t SU (2) uvu
}{zt uwvx t uu îv}}vw t SU (2) uvu uzv¦}{zt ¤vwxv uut
{uzvt utwt y zuvz }zwvtw §ÍÖ© v wt zvxt x uu}t{twu }t SU (2)
wwtz wt {uzvt {}uv|twuzt §2S 3 ¤wwtt ©
+4
+
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PG
S
S
PG
PG
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zt{vzt |zztxww|t twuzt }t z¦utwuvt}
, u ) tu wt {uzvt
tz¦îï}tz tw v{twxvw Í u wwt wx äðÖ Ltu(Qzuv|}t
wutz {zt

}t u t } zt|tz|t tx |uvwx xtz¦tx|t z{wvt v xwu xx|vtx
{uzvtx {}uv|twuztx tx utzx wu u}v t } twxvu }zwvtwwt
§Í×©
fQ
fD Q +λ Q
,
L =Q
+4
TN
+
++
+
+
+
±
2
+
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VT N = λ +
1
.
~
|X|
äw |{ztw {vt }tx vx{uzvtx t §Í×© xv w ®wvu }t uzv}tu a ut} t
(ab)
a
11
=a
22
= 0,
a
12
√
i λ
=
.
2
äw tu }zx z|zvzt y w |utz §¦Æ© zx §Í×© xx } z{t
+
++ +a
Q+
Q + aab Q+
a D
a Qb
2
.
t}}t¦|v {wuzt t}v|vut{twu } zvxzt SU (2) tw w U (1) z }t uzv}tu a
{{t |t tzwvtz }zwvtw wt |{zut |wt tww|t t}v|vut tw u }t
®
SU (2) txu zxtz tx vx{uzvtx w}tx xwu w| SU (2) × U (1) t xtz¦
tx|t z{wvt tz{tu t vz vzt|ut{twu xz }t }zwvtw }tx vx{uzvtx t }
{uzvt ®w}t
Í×
PG
ab
PG
±
i
S
S
PG
Ø ÛÜ! T # U Ý& Ü!,ÝÁA-Û!(Ü C &Ú Ù&Û(* E-ÙA-Û!({&Ü
wÚwv VLWXKPÛ531H3
t z¦utwuvt} wwwu |v¦ wxw u uz wx äðÖ tw uv}v¦
xwu wx }t xtz¦tx|t z{wvt }~z|t uvtwu tz¦îï}tz uv}vx z
zuzvu tu zvt{w z |t} w |wxuzvu ~z wt uzvt t t t| w
}tu t SO(2) x}{twuvzt ~vwv|tx A, B = 1, 2 È }tx xtz¦|{x x~|zvtwu
{vwutwwu Q t }zwvtw t zu txu }zx
Z
1
§Íð©
Q Q +c
,
L
=
[d θ d θ̄ ][du] Q D Q + W
2
t| D c = 0
tuut vx¦|v }t SU (2) txu |wxtz uwvx t }t utz{t c = c u u zvxt
t}v|vut{twu }t SU (2) tw w U (1) ~vwzvw|t t t }|}t txu wwt z
+a
A
+
aA
2 + 2 +
EH
+a
A
++
++
+a
A
AB
+
aB
++
++ ++
++
PG
S
(ij) + +
i j
S
+a
δQ+a
A = ε AB QB ,
δW ++ = D++ ε .
t |{ t t W t }t zJ}t ~w {}uv}v|utz t zwt v vwuzvu wt
|wuzvwut xz }tx |{x Q
äw ®wvu }tx |{xwutx tx xtz¦|{x z
(
Q = F + i θ σ θ̄ A
+ θ θ̄ D + tz{vwx ,
W
= i θ σ θ̄ W + θ θ̄ q + tz{vwx .
äw tu tw ttu {wuztz t }tx utz{tx tw θ tu θ̄ vxzvxxtwu y } ®w tx
|}|}x z v}}tzx w uv}vx } t t txx¦«{vw z }t xtz¦t|utz W È
w }zx W | = 0
¤zx t}t{twu tu tuz|uvw tx utz{tx tw θ θ̄ w uvtwu }~|zvuzt
}zwvtw §Íð© xx } z{t
++
+a
A
+
aA
+
aA
++
+
+
m +
m +
m
+2
−
m,aA
+2
+2
+2
−3
aA
−2
+2
+2
++
++
θ=0
+2
LEH
1
=
2
Z
[du]
−3 ++ +a
+a
2DaA
∂ FA + 2 A−
m,aA ∂m FA −
+2
1 −
A
∂ ++ A−a
m,A
2 m,aA
+
+a
−2 ++
+ AB q −2 FA+a FaB
+ AB Wm A−
c
m,aA FB + q
.
} u {vwutwwu }v{vwtz }tx |{x v}vztx utwvz }tx |wuzvwutx tu ®tz }
t xz SO(2) u |t|v xt vu z=|t uvwx {t{twu tx tzwvztx
x~|zvtwu È
• z zzu y D È
−3
aA
∂ ++ FA+a = 0
=⇒
FA+a ≡ FAia u+
i .
¤vwxv xt} }t zt{vtz utz{t tw u wx }t t}t{twu t F w~txu x w} |t
v tz{tu ~}v{vwtz wt vw®wvu t |{x v}vvztx ¨ tx F xtzwu }tx uztx
|zwwtx |wuzvwutx t } {uzvt zt|tz|t
Íð
+
i
+a
A
ia
A
T #Ø# Ä!-, ÙÛÜÚÛ(Ü*Ý Å Ü@ÜA *ÜÝ
•
z zzu y q È
−2
+
AB FA+a FaB
+ c++ = 0 .
} x~vu ~w uzv}tu t |wuzvwutx xz }tx È }tx |wuzvwutx xwu w| wwtx z
}t |utz |{ t t W wx }tF}zwvtw vwvuv} §Íð©
È
• z zzu y A
ia
A
++
−
m,aA
§ÍÔ©
txu }v{vu y xw wvt utz{t tw u |tuut zt}uvw x~vwuzt vx{twu tu
a
{{t
zwvu F
+a
A
+a
+a
∂ ++ A−
m, A = 2 ∂m FA + Wm AB FB .
+
i
a
FA−a = FAia u−
i .
−a
−a
,
A−
m, A = 2 ∂m FA + Wm AB FB
¤wu t ®tz } t z=|t uvwx {t{twu tw W v} txu }x
xv{}t t ztzutz ~z |tuut tzwvzt tztxxvw wx }t }zwvtw tu ~vwuztz
xz }tx z{wvtx u± äw uvtwu }zx
m
i
1
LEH = ∂m FAia ∂m FiaA + Km Wm + α Wm 2 ,
2
<
1
α = F2 ,
8
1
Km = − AB FAia ∂m FiaB .
2
F 2 = FAia FiaA ,
äw tu {vwutwwu ®tz } t SO(2) }tx uvwx {t{twu tw W wwwu
m
Wm = −
äw ztxut }zx t| } vxuw|t
LEH = L0 + L
t| ,



<
Km
.
2α

 L
tu }t uzv}tu t |wuzvwutx
1
∂m FAia ∂m FiaA
2
,
Km 2
= −
4α
L0 =
t|
j
AB FAia FaB
+ cij = 0 .
v w vtwuv®t F ≡ 2 i ϕ tu F ≡ 2 i ϕ w ztuzt y w |utz zx }t }zwvtw
§ÍÅ© tu } |wuzvwut §ÍÆ© wwx z zuzvu tu zvt{w wx }t |x n = 1 È v}
x~vu vtw t } {uzvt ~|v¦ wxw
ÍÔ
ia
1
1
ia
2
2
Ø ÛÜ! T # U Ý& Ü!,ÝÁA-Û!(Ü C &Ú Ù&Û(* E-ÙA-Û!({&Ü
wÚwæ ÜHLMJ2 O5LMPSÈO 5N2W I51121 QKÝ0/23421
wx äðÖ }tx utzx wu zx t {}wtz }tx t z¦utwuvt}x z¦
|twux È
+
+ 2
++ +a
++
LdT N = Q+
QA + W ++ AB Q+a
+ aab Q+
.
aA D
A QaB + c
aA QbA
tx vx{uzvtx ®w}tx xwu }zx U (1) × U (1) y |xt t } }t zvxzt c tu
ä¥ðð t }~w uvtwu vwxv } {uzvt }t
a } u {wuz wx ¤
¦Ç t utwuvt} §2S 3 wwtt © È
S
PG
++
ab
V =λ+
1
1
+
.
~
~
~
~
|X − ξ| |X + ξ|
|wxuwut λ z/u wx }t z¦utwuvt} v }t uzv}tu a }t t|utz ξ~ v }t uzv}tu
c
} {wvu tw|zt }t z{uzt tz{tuuwu ~xx|vtz wt {xxt vztwut y
||w tx |twuztx §}t z{uzt ρ wx §Ö©© |t|v ®w ~utwvz } {uzvt y
t |twuztx } }x wz}t x wx zx} |t z}{t wx }~zuv|}t Ê U (1) ×
Î© Ç
1 _ ^ ^2 1_^2 S_ i _ ^2 _ 2 Ì 2S ¤
U (1) Þa] 0 Z cZ b0 \ cb ] bcZ \ap0 \p] 0 § 3 wwtt wt w t
z|tz txu t zuvz z¦utwuvt}
ab
++
+
+
+ 2
++
ab +
++
L+4
AB Q+a
+ aab Q+
,
A QaB − β0 a QaA QbA + c
aA QbA
dT N + = W
t| wt vwzvw|t t t SO(2) }|}t {v®t È
+a
ab +
δQ+a
=
ε
Q
+
β
a
Q
AB
0
A
B
bA ,
δW ++ = D++ ε .
~txu } |wxuwut β v wwtz wt {xxt vztwut |twuztx +ξ~ tu −ξ~ ttwwu
|tuut zxtwuuvw txu tw vu t xuvxvxwut |z w wut wt vxx{uzvt zwt
twuzt xw utz{t Ê|v¦ wxwÌ zuvt tw Q tu xw utz{t Ê¦ÇÌ z¦
uvt tw Q z v}}tzx w vu t } {uzvt ~|v¦ wxw x~uvtwu z wt
uzvt t t |wuzvzt{twu y |t}}t t ¦Ç } txu xxv}t ~wv®tz |tx t
{uzvtx vwxv ~v} u {wuz wx ÕÕ t w y w~vz t tx utz{tx
zuvtx wx }tx xtz¦|{x z |t} w zut w xtz¦|{ g t w
y zuvz t } twxvu }zwvtwwt
0
+a
A
+a
A
+
r
+
+
++ +a
ab +
++
LdT N ++ = Q+
QA + gr+ D++ g +r + W ++ AB Q+a
aA D
A QaB − β0 a QaA QbA + c
+
+r
ab +
+ V ++ 2 u+
g
−
a
Q
Q
r
aA bA .
} u }zx tz }tx t U (1) |{{uwux xvwux È
Å δ Q = ε Q + β a Q , δ g = 0 , δ W = D ε , δ V
Æ δ Q = ϕ a Q , δ g = ϕ u , δ W = 0 , δ V = D ϕ .
ÎÕ
ε
+a
A
ϕ
+a
A
AB
ab
+a
B
+
bA
0
+
ϕ r
ab
+
bA
+
ε r
+
r
ϕ
ε
++
++
ϕ
++
++
ε
++
§ÍÅÕ©
++
=0.
T #Ø# Ä!-, ÙÛÜÚÛ(Ü*Ý Å Ü@ÜA *ÜÝ
uzwxz{uvw t t §Æ© tz{tu t ztxuztvwzt g z } |wvuvw u
xvu
+
r
−
r

§ÍÅÅ©
g +r = 0
+s
u−r .
g +r = − u+
s g
zvuvw z zzu y V zwvu } |wuzvwut
++
1
+
+r
u+
= aab Q+
r g
aA QbA ,
2
v wwt }zx z }t utz{t |vwuvt
+r
gr+ D++ g +r = u+
r g
2
=
1 ab + + 2
a QaA QbA .
4
äw ztuzt vtw y w |utz zx }t utz{t zuvt z¦utwuvt} t u¦Ç
~txu z |tuut {ut t }t t t } {uzvt }t ¦Ç
t| tx {xxtx vztwutx u utwt wx }~zuv|}t ÕÆ Y t}}t zvu tw uwu
t }v{vut tz¦îï}tz ~wt wt}}t {uzvt ~vwxutvw y t} u¦} wu
} twxvu }zwvtwwt xt zvu vtw y §ÍÅÕ© }zxt }~w vu utwzt } |wxuwut
~vwxutvw tzx ªz t|v xtz t} }x tw uv} wx }t vuzt Î
v w xt g = g u + · · · }tx |wuzvwutx t }~w uvtwu xt }vxtwu xz }t
}zwvtw È
+r
ri
+
i
a(i
j)
j
i
AB FA FaB − β0 aab FaA
FbA
+ c(ij) = 0
1
j
i
g (ij) − aab FaA
FbA
=0
2
g ij
+
FaA
uwx v|v t }t |v t t §ÍÅÅ© tz{vx t zvzt y w uzv}tu g ¤vwxv
v} u zuvz ~wt {uzvt y ÅÆ v{twxvwx §ð z }tx tu Í z g © ¤zx vz
t îv}}vw tu utw ÉRÉßÖ |wuzvwutx zt}}tx w uvtwu vtw wt {uzvt
y Í v{twxvwx tx vx{uzvtx ®w}tx xwu U (1) × U (1) t}}tx¦|v ttwu £uzt
{twutx wx t tu xt}t{twu t |x È
Å a = 0 È w uvtwu }zx ¦Ç
Æ c = 0 tu β = 0 È w uvtwu }zx |v¦ wxw
t w }x wz}t }t z¦utwuvt} xx|v y w {}uv|twuzt t}|wt u ww
wx ä¥ðð Çwt zvuvw }x uzwxztwut tu }x xv{}t xt uzt wx ÕÅ
uwx tw®w t } {uzvt ~¤uv¦ vu|vw v wt xxt |w îv}}vw uzv¦
}{zt §2S 3 ¤wwtt © w~ x tw|zt £uzt vut xtz¦tx|t z{wvt
{£{t xv xw z¦utwuvt} txu |ww w ttu xv wx |wxvzwx
µ f
§ÍÅÆ©
fQ
fD Q +λ Q
,
Q +Q
L =Q
+
2
2

w vu t }t |}t tw µ zvxt }t U (1) v xxvxut t| }t |}t tw λ }
wt ztxut }zx }x |{{t vx{uzvtx t |t}}tx uvtx t } {uzvt ~¤uv¦
vu|vw SU (2) ¡}tztxt{twu y |xt tx utz{tx zuvtx }zwvtw
ÎÅ
S
(ij)
+
r
PG
(ab)
(ij)
0
+
AH
+
++
+
+
+
PG
S
2
+
2
+2
2
Ø ÛÜ! T # U Ý& Ü!,ÝÁA-Û!(Ü C &Ú Ù&Û(* E-ÙA-Û!({&Ü
}tx uvwx xz } xzt S }zx |}|} tx |{x v}vvztx xwu ~wt ut}}t
|{}tvu t tzxwwt w~ x~v|v tx|twzt y } {uzvt t}v|vut w ttu
}tx |{x v}vvztx xwu ®x z }tx uvwx {t{twu v vxxtwu ~w
zzt }t tz }zwvtw wx }tx xtz¦|{x ¤vwxv xv |t}v¦|v txu }x t
tz t w tu £uzt xz t }~w wt zz vz t tx uvwx vztwuvt}}tx
¾¼µÂ¿¼·Äá y zxzt ~txu t|ut{twu |t v xt xxvu wx } xt|uvw z|twut §2S 3
uvw §ÍÔ©© uwx t wx }t |x z¦utwuvt} zuvt t ¦Ç §Í×©
} zxtw|t U (1) ztxuwu tz{tu t ztuztz |tuut }vwzvxv}vu tu ~uvz y
} {uzvt
2
PG
ÎÆ
¯ýqkm à
áqmotkr týqkmpo°potkr kq
r km krýnk ýml°potk
tx {uzvtx tz¦îï}tz tw v{twxvw Í t| {vwx w îv}}vw uzv¦
}{zt uwu |wwtx vwxv ~v} txu zt} wx }~¤wwtt |t |vuzt {wuzt
|{{twu v} txu xxv}t ~tw utwvz tx tutwxvwx utzwvwvtx uzx wx }t
|zt xtz¦tx|t z{wvt w ttu y {vwx t zvxtz uu}t{twu }t SU (2)
w txu |tzuvw t ztuztz wt {uzvt {}uv|twuzt wx }t |zt tx xtz¦x{uzvtx
}}tx N = 2 tw v{twxvw Í
PG
Ù
MM
w tu }twu wu tuzvzt wx ÕÕ }t xt|utz xwvt tx
xtzzvux utwtx wx äð× tz zx}uu txu ~ uut {uzvt tz¦
îï}tz t }zwvtw §ÍÖ© |zztxw wt {uzvt utzwvwvt wu }t }zwvtw
txu ww z
LQ
<
1
=
2
Z
+ ++ +a
+i 2
[d2 θ+ d2 θ̄+ ][du] −qi+ D++ q +i + κ2 (u−
Q + L+4 (Q+ , v ± )
i q ) Qa D
1
(ij)
+ κ2 Vm
Vm(ij) ,
2
(ij)
Vm
=3
Z
§ÎÅ©
+k 2 +a
+
f
)
F
∂
F
.
[du] u−i u−j f +k ∂m fk+ − κ2 (u−
m
a
k
|wxuwut κ txu zzuvwwt}}t y } |wxuwut ~vwxutvw tx xtz¦|{x x¦
}{twuvztx q xwu t}x Ê|{twxutzxÌ äw x
tu F = Q | .
f =q |
2
+
i
+
i
+
i θ=0
+
a
+
a θ=0
tx u
±
i
Ø ÛÜ! Ø# Î-Û!({&ÜÝ {&ÛÜ!Ú(ÙÚ({&ÜÝ ÜÛ Ý& Ü!,ÜÝ 'Ü !AÙÚ({&Ü
xwu zt{}|x wx L z t wt}}tx z{wvtx v ®wvtx z
±
i
+4
tu v = u .
z v}}tzx } zuvt xwvt f |{twxutz txu |wuzvwut z } zt}uvw
Z
1
§ÎÆ©
.
[du] f ∂ f − κ (u f ) F ∂ F
=
v +i =
+s
(u+
q +i
s q ) −i
+i
=
u
−
u
+t
+t
(u−
(u−
t q )
t q )
−i
−i
+i
+i −− +
i
2
−
i
+i 2
+a −−
}v{vut tz¦îï}tz |~txu y vzt } }v{vut κ
xutzx z
2
q +i −→
wx |tuut }v{vut w }zx
−→ 0
+
a
κ2
xt uzvu xz }tx |{tw¦
u+i
.
|κ|
tu (u q ) −→ 0 =⇒ v −→ u ,
|t v twuz/wt } |wtztw|t }zwvtw utzwvwvt §ÎÅ© tzx }t }zwvtw
tz¦îï}tz §ÍÖ© }zxt κ −→ 0
w zuvt ux }tx |}|}x x~ztwu wx } t
t| f = ω δ ,
f =f u
|t v zwvu }tx xv{}v®|uvwx xvwutx È
D++ qi+ −→ 0
− +i 2
+i 2
κ2 (u−
i q ) −→ (ui u ) = 1 ,
+
i
+i
±i
±i
2
+i
j
+
j
i
j
j
i
i
 + +i
u f =0,

 i
+i
u−
=ω ,
i f

 ij
f fij = 2 ω 2 .
t |v tz{tu t z|zvzt } |wuzvwut §ÎÆ© xx } z{t
1
< F = F F .
κ ω =
,
1− F
¤vwxv y w xvwt zx v wt |wt x }t }zwvtw ω txu ® È }tx |{twxutzx
w~vwuzvxtwu x t tzx t }vtzu x}{twuvztx t |v t t wwt z
}t t|utz V }~|zvuzt
2
2
2
κ2
2
ia
A
2
(ij)
m
(i
iaA
j)a
(ij)
Vm
= −κ2 ω 2 FaA ∂m FA .
w®w }zxt L (Q , v ) txu wt z{t v¦}vwvzt wx }tx xtz¦|{x tu
wx }tx z{wvtx v |{{t z tt{}t Q Q + 2 v g + c v v wt
xv{}v®|uvw uzx z|v}t z/u xv w xt z }tx xtz¦|{x È
§ÎÉ©
Q̂ = |κ| (u q ) Q .
twxvu }zwvtwwt t §ÎÅ© x~|zvu }zx È
+4
+
±
+
AB
+
−
i
+i
+a
A
+
aB
+
r
+r
(ij)
+
++ +a
Q̂ + L+4 (Q̂+ , |κ| q + ) ,
LQ = −qi+ D++ q +i + Q̂+
a D
< }tx z{wvtx v zxtwutx wx L wu u zt{}|tx z |κ| q
ÎÍ
+
i
+4
+
i
+
i
+
j
Ø#># [email protected]ÛÜÚÝ(ÙÚ {&ÛÜ!Ú(ÙÚ({&Ü %âzE&'(,ãÚÝÙÚ
Ù 4 6 ÀNÀN µNLW
¤ zuvz z¦utwuvt} §Íð© ww |vuzt Í t } {uzvt ~|v¦
wxw tu ®w ~utwvz }~tutwxvw utzwvwvt t |tuut tzwvzt w xvu w|
~v} u {vwutwwu uv}vxtz } twxvu }zwvtwwt
+ ++ +a
+
++
+i 2
,
QA + W ++ AB Q+a
LQEH = −qi+ D++ q +i + κ2 (u−
i q ) QaA D
A QaB + c
t| |tuut vx
c++ = cij vi+ vj+ ,
D++ c++ = 0 .
w uv}vxwu } zt®wvuvw §ÎÉ© |tuut twxvu xt xv{}v®t tw
LQEH =
−qi+
D
++ +i
q
+
Q̂+
aA
D
++
Q̂+a
A
+W
++
AB Q̂+a
A
Q̂+
aB
2
+κ
cij qi+ qj+
.
äw tu }zx zv®tz }~vwzvw|t t |tuut tzwvzt tztxxvw xx }~|uvw SO(2)
}|} ut} t
+
δ Q̂+
aA = ε AB Q̂aB ,
δq +i = ε κ2 cij qj+ ,
δW ++ = D++ ε .
uwx t z zzu |x tz¦îï}tz } t vu £uzt ®wvt t w v¦
ztwut È y |xt vu t c txu {vwutwwu {}uv}v z w utz{t wt |wutwwu
}t |{twxutz |t}v¦|v vu }vuvzt{twu xvz wt z{uvw vw®wvuxv{}t xv
w tu {vwutwvz W = D ε
tx|twut tzx } {uzvt t }~tutwxvw utzwvwvt ~|v¦ wxw
wwt }zx }vt y tx |}|}x uu y vu |{z}tx y |t t } xt|uvw §ÍÎÆ©
t¦|v tz{tuutwu ~uvz }zwvtw
++
++
<
κ2 ω 2
L=
2
++
κ2 2
κ2 (ij)
2
ia
ĉ Wm + Vm Vm(ij) ,
∇m FA ∇m FiaA −
4
2
tu ∇ F = ∂ F + 1 W F .
tx uvwx t |{ z zzu |{ t t2W wwtwu }zx
ĉ2 = cij cij
m
ia
A
m
ia
A
m AB
ia
B
m
Wm = 2
AB FAia ∂m F̂iaB
F̂ 2 − ĉ2 κ2
.
tuut {uzvt txu |{}ut z }t bäb0 uzv}tu t |wuzvwutx t z |v¦
wxw È
a(i
j)
AB FA FaB + c(ij) = 0 .
x w~wx zxtwu v|v t } {vuv uzv} È v} u tw|zt zxzt }tx
|wuzvwutx tu wwtz t ÊzvtxÌ |zwwtx z } {uzvt ®w}t t v{twxvw Í
Çwt wwt |zvuzt t |t}}t¦|v txu
1
g=
4 (1 − κ2 s)2
s − κ2 c2 2
s2 − c2 2
2 2
2
2
ds
+
(s
−
κ
c
)(σ
+
σ
)
+
σ3
1
2
s2 − c2
s − κ2 c2
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,
Ø ÛÜ! Ø# Î-Û!({&ÜÝ {&ÛÜ!Ú(ÙÚ({&ÜÝ ÜÛ Ý& Ü!,ÜÝ 'Ü !AÙÚ({&Ü
< }tx Å¦z{tx σ zv®twu dσ = σ ∧ σ t}}tx¦|v xwu vwzvwutx xx }~|uvw t
¤
SU (2) tu uzwtwu |{{t w t|utz xx }~|uvw t SU (2) vwxv }tx vx{uzvtx

xwu vtw SU (2) × U (1) }t z{uzt c |zvwu } zvxzt t SU (2) uwx t
}zx t wx }t |x tz¦îï}tz } }v{vut c −→ 0 ztwwt }~tx|t }u w ztuzt
v|v } xzt S
i
i
1
2 ijk
j
k
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S
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S
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4
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6 ÀNÀN N NX NXWå5
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t w uu y vu xv{v}vzt wx } }tuuzt Ê `a]10_Z^cZ^2 0æ10Z\^cZ cS 1i0 fcakg0 d]ake
çèd b01_^2é §2S 3 ¤wwtt Í© äw uvtwu }zx wt {uzvt ~vwxutvw y t} u¦
} vwzvwut xx }~|uvw tx x{uzvtx U (1) × U (1) tuut {uzvt tw
t t z{uztx a tu c v wx } }v{vut tz¦îï}tz wwtwu ztxt|uvt{twu }t
utwuvt} y }~vw®wv tu } vxuw|t twuzt }tx t |twuztx t } {uzvt }t ¦
Ç §2S 3 ¤wwtt © wx }~zuv|}t ÕÆ }x uv}} v} u zu }t z{uzt

Í Î É©
β t } xt|uvw § v wwt wx } }v{vut κ −→ 0 tx {xxtx v ztwutx
t |twuztx } xxv u xxv}t t zutz w uzv{t z{uzt α v vxz/u
wx } }v{vut tz¦îï}tz ¤vwxv tx {uzvtx utzwvwvtx vztwutx ttwu
vz } {£{t }v{vut tz¦îï}tz
twxvu }zwvtwwt t zu xt vu §ztxt© vzt|ut{twu t |t}}t v
|zztxw |x tz¦îï}tz §ÍÅÕ© ¥t}wx ~v} u vwuzvzt w xtz¦|{
x}{twuvzt g ®w t zvxtz }t SU (2) xwx utz{t zuvt tw Q x |zvwx
v|v |tuut twxvu xx x z{t xv{}v®t tw Q̂ ĝ È
PG
S
0
0
+
r
+
PG
+
+
++ +a
Q̂A + ĝr+ D++ ĝ +r
LQdT N + = −qi+ D++ q +i + Q̂+
aA D
h
i
+ +
+ +
+
+
2 (ij)
(ab) +
q
q
−
ĝ
ĝ
Q̂
Q̂
+
κ
c
+ W ++ AB Q̂+a
Q̂
−
β
a
0
i j
i
j
aA bA
A
aB
h
i
+ +
+ +
+
2 (ij)
(ab) +
++
+ +r
qi qj − ĝi ĝj .
− a Q̂aA Q̂bA + α0 κ c
+V
2|κ| qr ĝ
§ÎÍ©
t }zwvtw txu vwzvwu xx }tx t uzwxz{uvwx tt U (1) v |{{utwu È


δε Q̂+a

A





δε ĝ +r



δε q +i





δε W ++





δε V ++
(ab) +
Q̂b A
= ε AB Q̂+a
B + ε β0 a
= ε κ2 c(rs) ĝs+
= ε κ2 c(ij) qj+
tu
= D++ ε
=0


δϕ Q̂+a

A





δϕ ĝ +r



δϕ q +i





δε W ++





δε V ++
= ϕ a(ab) Q̂+
bA
= ϕ |κ| q +r + ϕ α0 κ2 c(rs) ĝs+
= ϕ |κ| ĝ +i + ϕ α0 κ2 c(ij) qj+
=0
= D++ ϕ
uwx t } zxtw|t ~w utz{t tw ĝ ĝ wx }t |utz t W xtwut wx }t
|x tz¦îï}tz txu v|v ztw w|txxvzt z t |t |utz xvu vtw vwzvwu xx }
uzwxz{uvw t t t z{uzt ϕ wx }t |x |wuzvzt }tx t U (1) x wt
|{{utzvtwu x |t v twuzvwtzvu z } xvut tx |wuzvwutx x}{twuvztx v
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+
i
+
j
++
.
Ø#³ # [email protected]ÛÜÚÝ(ÙÚ {&ÛÜ!Ú(ÙÚ({&Ü %& %Ù&.*Ü +&.,ê?+
uzvv}vxtzvtwu } {uzvt t utz{t wt z zzu y §ÍÅÕ© twuz/wt y xw uz
wt zt®wvuvw tx uzwxz{uvwx t t t z{uztx ε È w w~ }x δ g = 0
|{{t wx }t |x tz¦îï}tz
tx z{uztx x}{twuvztx α tu β xwu ztwx xxv}tx z }t vu ~v}
tvxut tx }vtzux wu y } w t tz t U (1) vwtwwux xz }t }zwvtw
t z{uzt α wt tu x £uzt vwuzvu wx }t |x tz¦îï}tz |z
L
v} ztxt xz }~tvxutw|t ~w U (1) x}{twuvzt z }t }zwvtw utzwvwvt
§ÎÍ© z zzu }zwvtw tz¦îï}tz |zztxwwu §ÍÅÕ© w ttu w vu
uzx vtw wx §ÎÍ© } xv{v}zvu twuzt }tx utz{tx tw q tu |t tw ĝ È }t }zwvtw
utzwvwvt txu vwzvwu xx } zuuvw tz}vt t q tu ĝ
ε
0
QdT N +
+r
0
0
+i
+r
+i
δĝ +r = ϕ |κ| q +r ,
+r
δq +i = ϕ |κ| ĝ +r .
w|zt wt vx }tx t uzv}tux t |wuzvwutx xt }vxtwu xz } twxvu }zw¦
vtwwt §ÎÍ© È
a(i
j)
j
i
AB FA FaB − β0 a(ab) FaA
FbA
− κ2 c(kl) g (ki) g (lj) + c(ij) = 0 ,
tu
j
i
2 g (ij) − a(ab) FaA
FbA
+ α0 c(ij) − κ2 c(kl) g (ki) g (lj) = 0 .
¤ zuvz t L
wx wx tx|twzt tw |{xwutx zxzt }tx |wuzvwutx
tu |}|}tz } {uzvt ®w}t t}}t¦|v tu x~|zvzt xx } z{t |wuz|ut
QdT N +
P
4D g =
A
2
<
Q
dφ +
dα
4P
g0 =
2
1 + a2 λ ρ 2 2 2
X dα
+ A g0 +
P
dY 2 + dX 2 + a2 λ (X dY − Y dX)2
1 + a2 λ ρ 2
txu } {uzvt xz } xzt S §a λ < 0© }~tx|t }u §a λ = 0© }t }w tz¦
}vt §a λ > 0© äw x ρ = X + Y |wxuwut ~vwxutvw u Λ = −16 λ
t| λ = tx w|uvwx D A P tu Q wt twtwu t tx |zwwtx X tu Y
vwxv t tx z{uztx a c α β tu t } |wxuwut ~vwxutvw tx t îv}}vw ∂
tu ∂ ztwtwu }zx }~vx{uzvt U (1) × U (1) uzwxztwut wx }~|zvuzt t g
tx uztx w|uvwx v zvxxtwu wx g xwu uzx |{}vtx |t v vu
~v} w~ x £uzt xxv}t t zv®tz z w |}|} vzt|u z{vx }tx |x α = β = 0
t |tuut {uzvt uvu vtw vwxutvw tu t xw utwxtz t t} uvu u¦} z
|t} wx wx uv}vx wt z|t t y zªwûx¢v zªÔÅ tu ÔÎ tx
utzx wu {wuz t z uut {uzvt ~vwxutvw y t} u¦} v xxt
w îv}}vw ∂ v} tvxut w xxu{t t |zwwtx wx }tt} } {uzvt tu x~|zvzt
xx } z{t
1
1
§ÎÎ©
(dφ + Θ) + W(e (dν + dµ ) + dw ) .
g=
2
2
2
2
2
2
2
κ2
4
0
0
α
φ
0
φ
2
w2 W
v
Î×
2
2
2
0
Ø ÛÜ! Ø# Î-Û!({&ÜÝ {&ÛÜ!Ú(ÙÚ({&ÜÝ ÜÛ Ý& Ü!,ÜÝ 'Ü !AÙÚ({&Ü
{uzvt g txu }zx vwxutvw y t} u¦} xv tu xt}t{twu xv

Λ


(a)
−2 W = 2 − w ∂w v ,


3


(b)
∂ν2 + ∂µ2 v + ∂w2 (ev ) = 0 ,





 (c) −dΘ = ∂ W dµ ∧ dw + ∂ W dw ∧ dν + ∂ (W ev ) dν ∧ dµ .
ν
µ
w
} u xxv}t ~|zvzt g xx } z{t ww z zªwûx¢v tu tu t zv®tz
|tu twxt{}t t uzvx |wvuvwx {uzvt t wx wx utwt txu vtw wt
{uzvt ~vwxutvw y t} u¦}
z v}}tzx v} u xxv}t t {wuztz t g uvu |utz |wz{t w
t §ÎÎ© zx wt wt}}t x}uvw uvwx |}tx ~vwxutvw¦¡ût}} È
2

 Ric µν =

1
2
dF − = 0 ,
Fµρ g ρσ Fνσ − 14 gµν Fρσ F ρσ .
dF + = 0 .
w ttu wt v}tw|t t y }tzu }×ð xxzt t uut {uzvt y t}
u¦} tu x}uvw tx uvwx |}tx vwxutvw¦¡ût}} txu wt {uzvt îï}tz
y |zzt x|}vzt w}}t tu z|vzt{twu t }x uut {uzvt vwxutvw y t}
u¦} t| {vwx w îv}}vw txu |wz{t y wt {uzvt îï}tz y |zzt
txu wt x}uvw
x|}vzt w}}t }t |utz |wz{t uwu } tw zx}ut t
tx uvwx |}tx vwxutvw¦¡ût}}w} ztxuvu y |{ztz w|t}}tg ¦|v x}uvwx
|wwtx wx } }vuuzuzt È }tx {uzvtx t tzx¦xzt}¦v}xw tz×Å ×Æ tu }tx
{uzvtx t }twx¢v¦ t{vwx¢v ×Ö x wx zv® t w g w~zuvtwu y
|wt t |tx t |}xxtx
tx vztwutx }v{vutx t g xwu uv}}tx xz } ®zt §ÎÅ©
2
2
2
ÙV
N 77
ztt ~v} uvu xxv}t t uztz t}v|vut{twu t wt}}tx {uzvtx

~ vwxutvw z=|t xtz¦tx|t z{wvt u wwt w v{twxvw Í }tx {¦
uzvtx utwtx xwu t }x |wz{tx y tx x}uvwx tx uvwx |}tx vwxutvw¦
¡ût}}
t w wz}t } zx}uvw tx uvwx ~vwxutvw vu z/uzt tx ¦
uvwx vztwuvt}}tx |}tx ww¦}vwvztx tv{t zzt z=|t xtz¦tx|t
z{wvt v} txu xxv}t ~utwvz tx |}xxtx t x}uvwx utzwvwvtx wt w|tx¦
xvuwu t } zx}uvw ~uvwx vztwuvt}}tx ê·Äà¼Ä· ¸·¹·Ä t}}tx¦|v ¦
zvxxtwu }zx |}|} tx |{x v}vvztx }zu ut{x t}}tx xwu t }x
¾¼µÂ¿¼·Äá È v} u z |t} t }t }zwvtw t zu xvu }x zuvt wx
}tx xtz¦|{x w |wuzt¦zuvt } zt}}t v|}u tvtwu ¿¾íÂÃ·¼ÑºÄ È v} x~vu t
Îð
Ø#T # ØÙÚ'*&Ý(ÙÚ ÜÛ Ü!Ý Ü'Û(|ÜÝ
double
Taub−NUT ++
ñì íî ï
ëì íî ï
double
Taub−NUT
òóôôõô ÷ö
ëì íî ï
ñì íî ï
ð íî ï
òóôôõô ÷ö
Eguchi−Hanson
double
Taub−NUT+
double
Taub−NUT
ø íî ï
Taub−NUT
ð íî ï
ñì íî ï
Eguchi−Hanson
ùúôôûõüõ ýþõ ÿ ò õ
íî ï
vzt ÎÅÈ x }v{vutx
} zx}uvw tx |wuzvwutx ~wt zu tu t } {vxt tw z{t t } {uzvt ~uzt
zu ®w t ztztz }tx }{twux {uzv|vt}x g tuut {vxt tw z{t w|txxvut x
~v} t}tx z{uztx }~uv}vxuvw t }~uv} vwz{uvt ¤vwxv } z{t zut
t } {uzvt g zx zx}uvw tx |wuzvwutx uvtwu xz |vw tx ~v{ztxxvw x
z{t t}t xz }x t |twu tx ¨ zvx |wt{twux t |zwwtx x||txxvx
wu u w|txxvztx ®w t vz |zvzt } {uzvt xx wt z{t }x xv{}t } w~tw
ztxut x {vwx t }tx vtzx utz{tx v zvxxtwu tu tw zuv|}vtz } w|uvw Q
xwu tw|zt zt}uvt{twu |{}vx } {wt xtz¦tx|t z{wvt wt {¦
ut v }v tz{tuzvu t zwvz tx |zwwtx }x Êwuzt}}txÌ wx }txt}}tx }
{uzvt utwt xtzvu }x xv{}t
¤}zx t wx |zvvwx }~zuv|}t ÕÆ }tzw¢ tu ttzxtw wu uz
ÕÅ
} }vwzvxuvw |zztxwwut {uzvtx ~vwxutvw y t} u¦} tu
~vx{uzvtx U (1) × U (1) y Í v{twxvwx È v}x wu |zvzt |t}}tx¦|v xx wt z{t
|{|ut v wt tw t ~wt w|uvw tuut w|uvw |{z}t utwuvt}

g^ ^_
£
V tx {}uv|twuztx vu uzt x}uvw ~wt uvw v ztwuvt}}t Z[] 0 tv{t
zzt wx }tx |zwwtx ttwwu } {ut ~v}x wu t{}t wx }tz zt¦
|tz|t wt }tz tz{tu x ~}}tz }x }vw tzx tx {uzvtx ~vx{uzvtx }x v}tx
t xtz¦tx|t z{wvt wt xxt x wt ut}}t }v{vuuvw tu v} xt{}t xxv}t
t zutz tx z{uztx v z tt{}t zvxtzvtwu }t U (1) ztxuwu tw zuwu
wt uzvxv{t Ê{xxtÌ wx } }v{vut tz¦îï}tz z |t} uzvx z{uztx tu }
|wxuwut ~vwxutvw xxtwu
~uztx vtx z{tuutxtx ztxutwu y t}ztz } x~vu tx tutwxvwx utzwv¦
ÎÔ
ij
S
Ø ÛÜ! Ø# Î-Û!({&ÜÝ {&ÛÜ!Ú(ÙÚ({&ÜÝ ÜÛ Ý& Ü!,ÜÝ 'Ü !AÙÚ({&Ü
wvtx tx {uzvtx t }v tu tx tutwxvwx tz¦îï}tz utzwvwvtx t }
{uzvt t ¦Ç tzx tx tx|tx t v{twxvw 4n z v}}tzx x~v} txu xxv}t
t zvxtz uu}t{twu }t SU (2) w tu txztz utwvz wx }t |x tz¦îï}tz t
wt}}tx {uzvtx xwx îv}}vw uzv¦}{zt
PG
ÖÕ
kpo±
jkp°mlýorýqo°p pk °nk ¹p
l° ²k ro¾lý p°p opýomk
äw |wxvzt w {}t xv{ ww¦}vwvzt ~|uvw
1
S=
2T
Z
d2 x [gij (φ) η µν + hij (φ) µν ] ∂µ φi ∂ν φj .
§¤Å©
tx zv}tx φ xwu tx |{x vxxwu xz w tx|t v¦v{twxvwwt} |}vvtw
¡vw¢ûx¢vtw tu y }tzx xz wt zvu zvt{{wwvtwwt t }~w t}}t tx|t
|v}t tu tx|t |v}t txu {wv ~wt {uzvt g (φ) wxM}tx |zwwtx }|}tx φ
t utwxtz txu x{uzvt tu xw vwtzxt txu wu g vxuw|t xz M txu }zx
wwt z g
i
i
ij
ij
ij
ds2 = gij dφi dφj .
z v}}tzx wuzt {}t xv{ xxt w utz{t t |}t t txx¦«{vw¦vuutw
§«© t z } zxtw|t utwuvt} t uzxvw h t utwxtz h txu wuv¦x{uzvt
tu tz{tu ~vwutzzutz } {uzvt xz }~tx|t |v}t |{{t wt {tuzvt t| uz¦
xvw t}}t¦|v u{uvt{twu tz{t txu ®wvt z
ij
Tijk =
ij
3
1
∂[i hjk] = (∂i hjk + ∂j hki + ∂k hij ) .
2
2
t utwuvt} t uzxvw w~txu ®wv ~y w zuuvwwt} zx w ttu v} txu vtwu t
uu utz{t t } z{t ∂ Φ wx h wt |wt x } uzxvw Çwt vwuzuvw z
zuvt }t vu t {£{t vxz/uzt t }~|uvw S
[i
j]
ij
MM wx |t |zt t {uzvt t| uzxvw wx ®wvzwx }tx x{}tx t zvx¦
ut} z
i
Γijk = γjk
+ T i jk ,
< γ
xvw È
i
jk
ÜÚ%(@ # ´ÜÚÙ!A*(ÝÛ(ÙÚ ) &ÚÜ .Ù&'*Ü %â&Ú AÙ%D*Ü Ý(EA ÚÙÚ,*(Ú-(!Ü
txu } |wwtvw x{uzvt t tv¦vvu vut}}t tx {uzvtx xwx uz¦
i
γjk
=
1 is
g (∂j gks + ∂k gjs − ∂s gjk ) .
2
j↔k
|wwtvw Γ w~txu z |wxtwu }x x{uzvt §
i
jk
© v w wut
Gij = gij + hij ,
}zx } |wwtvw t| uzxvw x~|zvu xxv
Γijk =
1 is
g (∂j Gks + ∂k Gsj − ∂s Gkj ) .
2
tx zvtx |zvwutx t| uzxvw xwu }zx wwtx z
(
Di Aj = ∂i Aj + Γjis As = ∇i Aj + T j is As ,
Di Aj = ∂i Aj − Γsij As = ∇i Aj − T s ij As ,
< ∇ xvwt } zvt |zvwut xwx uzxvw äw ®wvu twxvut }t utwxtz t ¥vt{ww
t| uzxvw z
[Dk , Dl ] v i = Ri j,kl v j − 2 T s kl Ds v i ,
|t v wwt È
Ri j,kl = ∂k Γilj + Γiks Γslj − (k ↔ l) .
w®w }t utwxtz t ¥v||v tu } |zzt x~|zvtwu
R = Rics s .
Ricij = Rs i,sj ,
t utwxtz t ¥vt{ww t| uzxvw R w~txu }x x{uzvt z }~|wt tx
|}tx ~vwv|tx (ij) ↔ (kl) {vx ztxut wuvx{uzvt xz }tx vwv|tx §i ↔ j © tu
§k ↔ l © t utwxtz t ¥v||v t| uzxvw w~txu w| }x x{uzvt È
ij,kl
Ric[ij] = −Ds T s ij = −∇s T s ij .
x zwx xxv txvw t } zvt t vt L z uu utwxtz S ®wv xz
~ −→ φ
~
M w wx w |wt{twu t |zwwtx φ
ij
o
~ o ) ∂µ φoi ∂ν φoj = Sij (φ)
~ ∂µ φi ∂ν φj .
Sijo (φ
v φ~
o
~ + η ~v
=φ
tu }~w }zx
} zvt t vt t S z zzu y ~v txu ®wvt z
o
ij
~ = S o (φ)
~ − η L S o (φ)
~ + O(η 2 ) ,
Sij (φ)
ij
ij
~v
s
s
s
L (Sij ) = v ∇s Sij + Ssj ∇i v + Sis ∇j v .
~v
ÖÆ
§¤Æ©
#># ´ÜÚÙ!A*(Ý.(*(Û- ) &ÚÜ .Ù&'*Ü
4 XM ¶ N XN
tx vtztw|tx y wt |}t {}t xv{ ww }vwvzt §¤Å© wu u wwtx
z zvtw zvðÎ tu }~|uvw ztwz{}vxt y wt |}t tw v{twxvw d = 2 − ε x~|zvu
SR1
1
=
2T
Z
~T
Ricij (η µν + µν ) ∂µ φi ∂ν φj .
d x Gij +
2πε
2
tx |wuzt¦utz{tx xxv}tx |}t tu |{x xwu wwx z }t t}¦
t{twu tw ~ t }~|uvw wt wx }t x|{ v{twxvwwt} {vwv{} xv w ®wvu }
|wxuwut λ tu }t t|utz ~v z

1
1
~T

2

λ + O(~ ) ,
1+
 o =
T
T
2πε


~o = φ
~ + ~ T ~v + O(~2 ) ,
 φ
4πε
< T tu φ~ xwu ztxt|uvt{twu }t |}t w tu }tx |{x wx w uvtwu wt
tv{t |zvuzt t }~|uvw ztwz{}vxt y wt |}t È
o
0
SR1
o
~ T vi
~ T vj
j
dx
Gij (η + ) ∂µ φ +
∂ν φ +
4πε
4πε
Z
1
~T
1
(η µν + µν ) ∂µ φi ∂ν φj .
=
d2 x Gij +
λ Gij + L (Gij )
2T
2πε
2 ~v
1
=
2T
Z
2
~T λ
1+
2πε
µν
µν
i
} ztxut y |{ztz S y S ®w ~tw vzt } |wvuvw t ztwz{}vxv}vu y wt
|}t t } uzvt §¤Å© w uv}vxwu ∇ g = 0 tu tw vxwu }~vtwuv®|uvw S ≡ G
wx §¤Æ© w {wuzt }~}vu
1
R
10
R
i jk
L(Gij ) = 2Dj vi + ∂[i ζj] ,
~v
ij
ij
ζi = 2 k s Gsi .
u utz{t t } z{t ∂ ζ vxz/u t }~|uvw z=|t y wt vwuzuvw z zuvtx
w utwwu |{ut t |tuut }vtzu w uvtwu } |wvuvw t ztwz{}vxv}vu y wt
|}t {}t xv{ ww¦}vwvzt È
[i j]
Ricij = λ Gij + Dj vi + ∂[i wj] .
ztwz{}vxv}vu y wt |}t t } uzvt §¤Å© xt uzvu w| z wt |wuzvwut
{uzvt È ¾¿ àÂ¶·¼ÑºÄ ¹¸¼¶ ¶·Ä Ñº¿á¼ÁÐ¼µá¶Ä¼µ ¿ÒÄ» ¶¸·á¼¸µ w xzwu
zuvt x{uzvt tu zuvt wuv¦x{uzvt |tuut |wvuvw xt z|zvu
(
Ric(ij) = λ gij + D(i vj) ,
Ric[ij] = λ hij + vs T s ij + ∂[i (w − v)j] .
ÖÉ
ÜÚ%(@ # ´ÜÚÙ!A*(ÝÛ(ÙÚ ) &ÚÜ .Ù&'*Ü %â&Ú AÙ%D*Ü Ý(EA ÚÙÚ,*(Ú-(!Ü
ÖÍ
kpo±
áqmotkr rkkr kp olkpro°p º
ÑMÀN ÀNW
SU (2) × U (1)
x |wxvzwx v|v }tx {uzvtx y Í v{twxvwx t |¦{wvu w xx }tx
vx{uzvtx SU (2) × U (1) t}}tx¦|v ttwu x~|zvzt xx } z{t
g = α(t) dt2 + β(t) (σ1 2 + σ2 2 ) + γ(t) σ3 2 ,
< }tx σ xwu }tx Å¦z{tx vwzvwutx t SU (2) v twtwu tx w}tx ~}tz
{θ, ϕ, ψ} È
i
σ1 2 + σ2 2 = dθ2 + sin2 θ dϕ2 ,
tx vtwuvux t ¡ztz¦zuw x~|zvtwu
dσi = ε
1
ijk σj ∧ σk ,
2
zzvu xv¦vwxutvw txu }zx wwt z
ricij = λ gij + D(i vj) ,
σ3 = dψ + cos θ dϕ .
ε = ±1 .
v = v0 (t) dt .
z{t zuv|}vzt t v txu v{xt z }tx x{uzvtx t}v¦|v w~txu tw vu ®wv
~ îv}}vw ∂ zx wu } Å¦z{t |zztxwwut x~|zvu γ(t) σ
wx }t |x zuv|}vtz < β(t) = cste wt zt®wvuvw t } |zwwt t
tz{tu uzx t ztwzt β(t) = t |wvuvw xv¦vwxutvw txu }zx v}twut
y uzvx uvwx vztwuvt}}tx ww¦}vwvztx zuwu xz α(t) γ(t) tu v (t) È
ψ













3
0
1
+
t2
α0 (t)
γ 00 (t)
γ 0 (t)2
v0 (t)
−
+
= 2 λ α(t) + 2 v00 (t) −
2
α(t)
γ(t)
α(t)
2 γ(t)
γ(t)
α0 (t) γ 0 (t)
2 2−
α(t) +
−
= 4 λ t α(t) + 2 v0 (t)
t
α(t)
γ(t)
1
γ 0 (t)
+
t 2 γ(t)









2 γ(t)2
α0 (t) γ 0 (t) 2 γ 00 (t)
γ(t)
2


α(t) +
+
− 0
= 4 λ 0 α(t) + 2 v0 (t)
 − + 2 0
t t γ (t)
α(t)
γ(t)
γ (t)
γ (t)
§Å©
ÜÚ%(@ # Î-Û!({&ÜÝ &Ý&Ü**ÜÝ ÜÚ %(AÜÚÝ(ÙÚ T
} txu xxv}t ~}v{vwtz α(t) tu v (t) t |t xxu{t {vx w uvu }zx y wt uvw
vztwuvt}}t ut{twu ww¦}vwvzt ~zzt uzt tw γ(t) z }~vwxuwu ww zx}t
t}}t¦|v {wuzt |ttwwu t } {uzvt wz}t xv¦vwxutvw y uzt v{twxvwx
SU (2) × U (1) vu twzt t uzt z{uztx vwtwwux zx}uvw txu
|ttwwu xxv}t xv }~w v{xt |tzuvwtx |wvuvwx ®w t ztxuztvwzt } zt|tz|t
|x tx {uzvtx ~vwxutvw |x t |tzuvwtx {uzvtx îï}tz
0
04/KTL21 QUVK3142K3
wxwx
v }~w v{xt } |wvuvw v (t) = 0 }tx xxu{t §Å© txu uu}t{twu zx} tu
} x}uvw v tw t t z{uztx A tu B txu wwt z
0

1
1


·
,
α(t)
=


1 + A t γ(t)







 γ(t) =
√
4t
4 λ t2 3 + 1 + A t
B√
1 + At .
2 −
3 +
√
√
3 1 + 1 + At
t
1 + 1 + At
v A = 0 g x~vtwuv®t y }~tutwxvw îï}tz¦vwxutvw t } {uzvt ~|v¦ wxw t
|wt{twu t |zwwtx t ≡ s tu B = −a tz{tu t ztuztz x z{t xt}}t È

• utwxvw îï}tz¦ vwxutvw t |v¦ wxw È
2
4
§Æ©
v A 6= 0 } {uzvt x~vtwuv®t y } }zt |}xxt t {uzvtx ~vwxutvw |tzut
z zutz zÖð w tt|uwu }t |wt{twu t |zwwtx
t| A = 1 tu B = −8(M − n) n ,
t −→ t − n
n
w tw uvtwu }~|zvuzt xvwut }x xv{}t
g=
4 2
ds + s2 σ1 2 + σ2 2 + F σ3 2 ,
F
2
F =1−
2
a4 2 λ 2
−
s .
s4
3
3
2





4 n2
t2 − n2 2
2
2
2
2
f (t) σ3 2 ,
dt + (t − n ) (σ1 + σ2 ) + 2
g =
2
f (t)
t −n



 f (t) = t2 − 2 M t + n2 − λ (t − n)3 (t + 3 n) .
3
§É©
tx |wxuwutx A tu B uwu zt}}tx M n tu t vtwu £uzt xv{}uw{twu zt}x
v{vwvztx zx tuut |}xxt t {uzvtx t zutz |wuvtwu twuzt uztx
Ç §M = n tu λ = 0© È
• ¦
g=
t+n 2
t−n 2
dt + (t2 − n2 )(σ1 2 + σ2 2 ) + 4 n2
σ3 ,
t−n
t+n
ÖÖ
#"# Î-Û!({&ÜÝ {&Ý(,z(ÚÝÛÜ(Ú SU (2) × U (1)
•
|ûzªx|v} t| |wxuwut |x{}vt §dψ ≡ dΨ vx n −→ 0© È
2n
g=
•
1 2
dt + t2 (dθ2 + sin2 θ dϕ2 ) + G dΨ2 ,
G
t xz P (C)#P (C) §
2
§Í©
2M
λ
− t2 ,
t
3
s
√
4λ 3
1 + ν2
t = nν s M = − n n = i 3
3
λ (3 + 6 ν 2 − ν 4 )
2
G=1−

4 ν2
3
1 − ν 2 s2
2
2
2
2
2
(σ1 + σ2 ) +
σ3
1+ν
g=
,
H ds +
λ
3 + 6 ν2 − ν4
H (3 + 6 ν 2 − ν 4 )2
©È
§Î©
1−ν s
< H =
tu < ν txu }~wvt x}uvw wx [0, 1] t
− s) (3 − ν − ν (1 + ν ) s )
}~uvw(1}
zvt 4 ν (3 + ν ) = 3 + 6 ν − ν
2
2
2
2
2
2
2
wxwv
2
2
4
04/KTL21 ÊËXJ2/ TL51KPVK3142K3
|wvuvw α(t) = 1 tz{tu xxv t zxzt }t xxu{t §Å© tu }~w
γ(t)
ä
uvtwu }zx |tzuvwtx {uzvtx
SU (2) × U (1) v xwu îï}tz w xxt v|v ~v}
tvxut w |v t |zwwtx }{ztx z }txt}}tx }tx vx{uzvtx vxxtwu
}vwvzt{twu |t v v{xt }~vwuzv}vu tx xuz|uztx |{}ttx z{t t îï}tz
txu wwt z
K = d(t σ3 ) .
wt}}t x}uvw tw xxv t t z{uztx C tu D tu x~|zvu È
g=
<
1
dt2 + t σ1 2 + σ2 2 + γ(t) σ3 2 ,
γ(t)
D eC t
2
γ(t) =
+t+ 2
t
C t
2λ
1−
C
Ct
e
v0 (t) = −C ,
1
− 1 − C t − C 2 t2
2
.
tuut |}xxt t {uzvtx |tzut z t tu }twu ÔÖ txu y wuzt
|wwvxxw|t }t xt} tt{}t t {uzvtx ~vx{uzvtx U (2) xv¦vwxutvw tw D = 4
t } }vuuzuzt wx } }v{vut C −→ 0 w v = 0 tu w ztuzt w| } {uzvt
vwxutvw t| }t utwuvt} t îï}tz K } x~vu t }~tutwxvw îï}tz¦vwxutvw t }
{uzvt ~|v¦ wxw z ztuztz §Æ© v} u tt|utz }t |wt{twu t
|zwwtx t ≡ s |zztxww|t twuzt }tx z{uztx txu D = B = −a t}}t¦|v
{wuzt t }tx
uztx z{uztx t } x}uvw m wz}t t §Å© wt ttwu x
£uzt A B C tu D vx~v}x wt xwu x vwtwwux
Ö×
0
2
4
ÜÚ%(@ # Î-Û!({&ÜÝ &Ý&Ü**ÜÝ ÜÚ %(AÜÚÝ(ÙÚ T
wxwæ
04/KTL21 XHIH321
z{v }tx {uzvtx z|t{{twu |vutx |tzuvwtx xwu {wtx }}tx |z¦
ztxwtwu uutx y tx x{uzvtx }x v{zuwutx t SU (2) × U (1) È
• ~tx|t }u IR È
4
g=
• S1 × S3
È
g=
1 2
dt + t σ1 2 + σ2 2 + σ3 2 .
t
1
t
dt2 + σ1 2 + σ2 2 + σ3 2 ,
v = − dt .
2λ
2
β(t) = cste
} x~vu t }~wvt {uzvt xv¦vwxutvw t|
§dψ ≡ dΨ vx ν → 0 wx §Î©© È
• S ×S
2

2
2ν
1
ds2
2
2
2
2
g=
+ 1 − s dΨ + σ1 + σ2
.
λ 1 − s2
•
t vuutz S §M
4
g=
2
= n2 = −
1
1+
λ
12
s2
2
3
4λ
tu t
2
3
s2
+
4 λ 4 1 + s2 λ 2
12
2
2
2
.
σ1 + σ2 + σ3
=−
s2
ds +
4
2
wx §É©© È
äw tu xxv }~utwvz tw ww}wu M wx §Í© v λ < 0 v} x~vu t } {uzvt
ww¦|{|ut wuv¦t vuutz |zztxwwut

§s ≡ q r
vx a → 0 wx §Æ©© È
• vwv¦ u xz P (C) =
SU (3)
U (2)
2
1/4
g=
1 + λ6 r2
(
2 1+
r2
)
4 dr2 + r2 σ3 2
+ r2 σ1 2 + σ2 2
.
1 + λ6 r2
v λ < 0 v} x~vu t xw zutwvzt ww |{|u xz
4
λ
6
SU (2,1)
U (2)

ÑMÀN L¿7WÒÓL
utx }tx {uzvtx tz¦îï}tz tw v{twxvw Í |wwtx xxtwu {vwx
aZ îv}}vw t }~w wutz K = ∂ tz tu vw}t wu {wuz wx ðÆ ~v} uvu
}zx xxv}t t |}xxtz |tx {uzvtx tw t ztx vxuvw|ux w ttu v} wt
Öð
t
#># Î-Û!({&ÜÝ Á Ü!,ÏÐ*Ü!
v|u{vt z }tx {uzvtx |wxvztx t x{uzvt {vwv{}t U (1) xvwu ~t}}tx
xxtwu ww w îv}}vw uzv¦}{zt ¥t}wx z v}}tzx t |tx {uzvtx
wu uutx w utwxtz t ¥v||v w}
wvwx
04/KTL21 5N2W 5L IHK31 L3 ÊKJJK3 4/KPXHJHIH/.X2
x |wxvzwx v|v }tx {uzvtx z }txt}}tx } zvt t vt tx xuz|uztx
|{}ttx J xt}w K txu w}}t È
i
L Ji = 0 ,
i ∈ J1, 3K.
K
tx {uzvtx wu u |tzutx }xvtzx vx z vztwux utzx wx
×Í
î ×ð vu×Ô {vx wx tx |zwwtx£ vztwutx |t v vu t }~w
w~ x x uu t xvut ~v} x~vxxvu tw vu tx { {tx
} x~vu tx àº¾¶¼»Äµ¶·Äá z }txt}}tx } vxuw|t tu uzx x~|zvzt xx
} z{t
ds2 =
1
(dt + Θ)2 + V h ,
V
Θ
~ X
~ ,
h = dX.d
< V tu }tx |{xwutx t } Å¦z{t ®wvtx z Θ = Θ dX xwu vwtwwutx
t t zzvu tz¦îï}tz |}t t } zt}uvw w{twu}t tx {}uv|twuztx
i
i
? dΘ = ±dV .
h
tuut zt}uvw twuz/wt } w}}vu }}|vtw utwuvt} V È
∆V =0.
¤vwxv y uut w|uvw V z{wvt wx IR v} txu xxv}t ~xx|vtz wt {uzvt
tz¦îï}tz §t|}vvtwwt© tw v{twxvw Í t w{ t Ê{}uv|twuztxÌ ww y |tx {¦
uzvtx vtwu vu t }~w |wxvzt wz}t{twu tx utwuvt}x wu z z{t
h
3
V =λ+
X
i
µi
~ −X
~i
X
.
äw tu vwxv vx}vxtz |tx {uzvtx z w twxt{}t t Ê{xxtxÌ µ vxtzxtx tw
|tzuvwx vwux X~ §}tx Ê|twuztxÌ© }t utwuvt} y }~vw®wv }wu z v}}tzx λ
} txu xxv}t t {wuztz t }t utwxtz t |zzt tu }t utwxtz t t}
xxtwu wt u¦}vu xt y |t}}t tx xuz|uztx |{}ttx ~txu ~v}}tzx }t
vu t }t utwxtz t |zzt xvu §wuv¦©u¦} v twuz/wt } w}}vu utwxtz
t ¥v||v
z{v }tx {}uv|twuztx tw tzx t }~tx|t }u §V = λ V = 1 © v} tw
~
t v xxtwu }~vx{uzvt {v{}t SU (2) × U (1) tu t }~w |X| wx }
xt|uvw z|twut È
ÖÔ
i
i
ÜÚ%(@ # Î-Û!({&ÜÝ &Ý&Ü**ÜÝ ÜÚ %(AÜÚÝ(ÙÚ T
•
¦Ç È
V =λ+
1
.
~
|X|
t îv}}vw xx|v y }~vx{uzvt
txu }zx uzv¦}{zt uwvx t }tx
zuuvwx xuv}tx v }vxxtwu VU (1)
vwzvwu wwtwu }vt y tx vx{uzvtx ¦
}{ztx

• |v¦ wxw È
V =
1
~
~ − ξ|
|X
1
+
~
~ + ξ|
|X
.
t îv}}vw xx|v y }~vx{uzvt U (1) txu {vwutwwu }{zt tu |zztxw
y } x{uzvt t zuuvw uz t ξ~ ¤}zx t } {uzvt |zztxwwut wt
xt{}t xxtz ~wt x{uzvt U (1) ×U (1) v} u {wuz wx ×ð
t tx x{uzvtx Ê||txÌ {twuvtwu }tx x{uzvtx uzv¦}{ztxx~y
w SU (2)
~txu zx wx z×Ô v {wuz t } {uzvt utwt z |v
tu wxw uvu vtw w {}uv|twuzt
w |{vwwu |tx t utwuvt}x v} txu xxv}t ~utwvz wt {uzvt ~vx{uzvtx
U (1) × U (1) } x~vu
Ç È
• }t ¦
T
H
T
T
H
V =λ+
1
~
~ − ξ|
|X
+
1
~
~ + ξ|
|X
.
tx uzvx {uzvtx xwu |{}utx ttwwu xv }~w xt }v{vut |x y t |twuztx
}t utwuvt} }t }x wz} y vx{uzvtx (1) ×U (1) xxt uzvx z{uztx vxt
}~w tu tw|zt wx }t }t ¦UÇ
wwtz tx {xxtx vztwutx y ||w tx
|twuztx äw }zx z }t utwuvt}
1
ρ
§Ö©
V =λ+
+
.
T
~
~ − ξ|
|X
wvwv
H
~
~ + ξ|
|X
04/KTL21 1531 ÊKJJK3 4/KPXHJHIH/.X2
zxt ux }tx îv}}vw xwu }{ztx |~txu y vzt t }~w z }tx xuz|¦
uztx |{}ttx
L J =J
tu
,
L J = 0,
L J = −J
w wt |wwvu x t}v|vut{twu uutx }tx {uzvtx y Í v{twxvwx ~vx{uzvt {vwv¦
{}t U (1) ttwwu tz tu vw}t w {wuz t |t}}tx¦|v vtwu uzx
x~|zvzt xx } z{t ðÆ È
K
3
K
1
K
2
2
1
H
ds2 =
1
V
2
dt2 + Θ + V dz 2 + eu (dx2 + dy 2 ) ,
×Õ
#># Î-Û!({&ÜÝ Á Ü!,ÏÐ*Ü!
t| }tx zt}uvwx
 2
 ∂x u + ∂y2 u + ∂z2 eu = 0 ,
tu ± dΘ = ∂ V dy ∧ dz + ∂ V dz ∧ dx + ∂ (V e ) dx ∧ dy .
t xt} tt{}t t}v|vut t }~w |wwvxxt txu } {uzvt t ¤uv¦ vu|vw v
xxt w SU (2) t îv}}vw }{ztx tx zt}uvwx twuzt }tx |zwwtx wwtx
z }tx utzx
tu }tx |zwwtx x, y, z wu u utwtx wx ä}vÔÅ z{t xt}}t
¤
ðð
x~|zvu
È
¤
• uv¦ vu|vw È

V = ∂z u
x
y
z
u
H
2
2
2
ds = A B C
2
dk
k (1 − k 2 ) K 2
2
+ A2 σ12 + B 2 σ22 + C 2 σ32 ,
< K(k) tu E(k) xwu ztxt|uvt{twu }tx vwuz}tx t}}vuvtx |{}utx t zt{vzt
tu xt|wt tx|t È
K(k) =
tu <
Z
0
π/2
dq
p
1 − k 2 sin2 q
et E(k) =
Z
π/2
dq
0
A B = −K [E
− K] ,
B C = −K E − (1 − k 2 ) K ,
A C = −K E .
×Å
q
1 − k 2 sin2 q ,
ÜÚ%(@ # Î-Û!({&ÜÝ &Ý&Ü**ÜÝ ÜÚ %(AÜÚÝ(ÙÚ T
×Æ
kpo± ¯
mqonkr
ÜÚ%(@ Ø# !Û('*ÜÝ
×Í
Ø#"# &ÚÛ&A ÝÛ!&'Û&!Ü ÙB +,%&*( Ü% AÙ%Ü*Ý (Û ÝÁAAÜÛ!Á .!Ü (ÚE
NN NN 8 WN L
¿¿ X Nuclear Physics B 591 (2000) 491–514
www.elsevier.nl/locate/npe
Quantum structure of T-dualized models with
symmetry breaking
Pierre-Yves Casteill ∗ , Galliano Valent
Laboratoire de Physique Théorique et des Hautes Energies, Unité associée au CNRS UMR 7859, Université
Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05, France
Received 26 June 2000; revised 29 August 2000; accepted 12 September 2000
Abstract
We study the principal σ -models defined on any group manifold GL × GR /GD with breaking of
GR , and their T-dual transforms. For arbitrary breaking we can express the torsion and Ricci tensor
of the dual model in terms of the frame geometry of the initial principal model. Using these results
we give necessary and sufficient conditions for the dual model to be torsionless and prove that the
one-loop renormalizability of a given principal model is inherited by its dual partner, who shares
the same β functions. These results are shown to hold also if the principal model is endowed with
torsion. As an application we compute the β functions for the full Bianchi family and show that for
some choices of the breaking parameters the dilaton anomaly is absent: for these choices the dual
torsion vanishes. For the dualized Bianchi V model (which is torsionless for any breaking), we take
advantage of its simpler structure, to study its two-loops renormalizability.  2000 Elsevier Science
B.V. All rights reserved.
PACS: 02.40.-k; 03.50.Kk; 03.70; 11.10.L; 11.10.Kk
Keywords: Sigma models; T-duality; Renormalization
1. Introduction
The subject of classical versus quantum equivalence of T-dualized σ -models has been
strongly studied in recent years, and extensive reviews covering abelian, non-abelian
dualities and their applications to string theory and statistical physics are available [2,5,
20]. More recent developments on the geometrical aspects of duality can be found in [1].
The interpretation of T-duality as a canonical transformation, for constant backgrounds,
was first given by [21]. Its more general formulation [4] was applied to the non-abelian
case in [25,27].
∗ Corresponding author.
E-mail addresses: [email protected] (P.-Y. Casteill), [email protected] (G. Valent).
0550-3213/00/$ – see front matter  2000 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 0 ) 0 0 5 6 2 - 9
×Î
ÜÚ%(@ Ø# !Û('*ÜÝ
492
P.-Y. Casteill, G. Valent / Nuclear Physics B 591 (2000) 491–514
After the settling of the classical equivalence, the most interesting problem was its study
at the quantum level. This was done mostly for dualizations of Lie groups, with emphasis
put on SU(2). For this model the one-loop equivalence was established in [16,18]. The
way towards the general case was cleaned up with the derivation of the classical structure
of the non-abelian dual for any group [2,3,18,22] and for non-inhomogeneous geometries
in [26]. However, the analysis of Bianchi V in [19] revealed that for some renormalizable
dual theories the divergences could not be absorbed by a re-definition of the dilaton field!
It was further realized that this phenomenon occurs for non semi-simple Lie groups with
traceful structure constants (fsis = 0), and that it can be interpreted as a mixed gravitationalgauge anomaly [3].
A further decisive progress was made by Tyurin [28], who generalized the one-loop
equivalence to an arbitrary Lie group and derived the general structure of the dilaton
anomaly. However, as pointed out in [7], his analysis considers only models with explicit
invariance under the left group action (whose existence is crucial for the dualization
process) leaving aside the right action and the possible symmetry breaking schemes for
it. The one-loop equivalence problem in this more general setting has been examined
recently [7,23] for the group manifold SU(2)L ×SU(2)R /SU(2)D , where SU(2)R is broken
down to a U (1). The renormalizability and dilatonic properties do survive despite the
lowering of the right isometries. It is the purpose of the present article to analyze the
geometry of the dualized model for a large class of models built on GL × GR /GD , with
arbitrary breaking of GR . While in [28] supersymmetry considerations à la Busher [8,9]
were convenient to derive the dualized geometry, we will show that a direct computation in
local coordinates is fairly efficient to extract the Ricci tensor in the presence of symmetry
breaking.
The content of this article is the following: after setting the notations, in Section 2 we
study the geometry of the group manifold (GL × GR )/GD . This is most conveniently
done using frames and, despite symmetry breaking, one obtains a manageable form for the
Ricci tensor. In Section 3 the dualized theory is examined and its torsion and Ricci tensor
are computed, exhibiting their dependence with respect to the geometrical quantities of the
principal model. The possibility of torsionless dualized models is discussed. In Section 4
we use the previous results to show that the one-loop renormalizability of the principal
model is inherited by its T-dual. In Section 5 we generalize the previous analyses to deal
with a principal model endowed with torsion. In Section 6 we examine the models in the
Bianchi class, compute their beta functions, and for the non semi-simple algebras discuss
the dilaton anomaly. For some breaking choices this anomaly may vanish and in these
cases the dual models are torsionless. Since any dualized Bianchi V model is torsionless,
we study in Section 7, for the simplest breaking, its two-loops renormalizability.
2. Geometry of the broken principal models
Since we have in view perturbative applications, our considerations will be of a local
nature. Let us consider a Lie algebra G = {Xi , i = 1, . . . , ν} with structure constants
×Ö
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493
[Xi , Xj ] = fijs Xs .
Denoting by zi the local coordinates in a neighbourhood of the origin, we exponentiate to
the group by g = exp(z · T ), and define
g −1 ∂µ g = Jµi Xi .
(1)
For further use we introduce the adjoint representation by
(Ti )jk ≡ (ad Xi )jk = −fijk ,
(2)
which allows to write the Jacobi identity
[Ti , Tj ] = fijs Ts ,
i, j, s = 1, . . . , ν = dim(G).
(3)
Then the action of the corresponding principal model can be written
Z
1
(4)
d 2 xBij ηµν Jµi Jνj ,
S=
2
where the matrix B is symmetric and invertible. For field theoretic applications one
should add the restriction that B is positive definite [6], while this does not seem to be
necessary for stringy applications. This restriction implies, in the semi-simple case that its
simple components have to be compact. Our analysis will not make use of this positivity
hypothesis.
Taking the curl of the first relation in (1) gives the Bianchi identity
i
i
(J ) ≡ ∂µ Jνi − ∂ν Jµi + fsti Jµs Jνt = 0 ⇐⇒ µν Mµν
(J ) = 0.
Mµν
(5)
2.1. Isometries
Let us proceed to a discussion of the isometries of the action (4). The groups GL × GR
and GD act on g according to
g −→ g 0 = GL g G−1
R ,
g −→ g 0 = GD g G−1
D .
(6)
As a consequence
g −1 ∂µ g −→ GR g −1 ∂µ g G−1
R ,
and specializing to infinitesimal transformations one gets
GR ≈ I + Ri Ti H⇒ δJµk = fij k Ri Jµj .
(7)
It follows that the action (4) is invariant under GL , while the matrix Bij will generally
break GR down to some subgroup H (possibly trivial). Denoting by {Ts , s = 1, . . . , h} the
generators of its Lie algebra H, these should satisfy
(Ts )ik Bkj + (Ts )jk Bik = 0,
∀ Ts ∈ H.
(8)
Let us emphasis that the metric B can be freely chosen (as far as it is symmetric and
invertible!), but, if G is simple, the most symmetric choice is given by the bi-invariant
metric
1
gij = Tr(Ti Tj ) = ρ̃ Tr(ti tj ),
(9)
Bij = gij ,
ρ
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where gij is the Killing metric and the ti the defining representation of the simple algebra
under consideration. In the simple compact case we have
ρ̃
so(n)
su(n)
sp(n)
(n − 2)
2n
2(n + 1)
and with the standard normalization of the generators Tr(ti tj ) = −2δij , we see that the
choice ρ = −2ρ̃ gives Bij = δij . In the simple non-compact case the same choice of ρ
gives Bij = ηij , which is diagonal, with ηii = +1 for a compact generator ti and ηii = −1
for a non-compact one.
The bi-invariant metric has for isometry group the full GL × GR because (Ts )ik gkl =
−fsil is fully skew-symmetric and therefore (8) is true for all the generators of GR .
For a semi-simple G the situation is not very different, since it can be split into a direct
sum of simple algebras
G = S1 ⊕ · · · ⊕ Sk ,
[Si , Sj ] = 0,
i 6= j.
2.2. Geometry of frames
In order to have a better insight into the geometry of the principal models with action
(4), it is convenient to use a vielbein formalism, through the identification
Bij ηµν Jµi Jνj ←→ Bij ei ej ,
and now the Bianchi identities appear as the Maurer–Cartan equations
1
dei + fst i es ∧ et = 0.
2
(10)
We follow the notations of [12] and define the spin-connection ωij by
dei + ωis ∧ es = 0,
ωij = ωij,s es .
The frame indices are lowered or raised using the metric Bij and its inverse B ij = Bij−1 . A
straightforward computation gives
2ωij,k = fij,k + fik,j − fj k,i ,
fij,k = fij s Bsk .
(11)
For further use let us point out two consequences
ωij,k − ωik,j = −fj ki ,
ωsi,s = −fis s .
(12)
The curvature and the Ricci tensor are defined by
1
R ij = dωij + ωis ∧ ωsj = R ij,st es ∧ et ,
2
It follows that
ricij = R si,sj .
R ij,st = −ωij,a fst a − ωia,t ωaj,s + ωia,s ωaj,t .
×ð
(13)
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495
In the Ricci tensor the first two terms are gathered using (12) and give
ricij = −ωsi,t ωtj,s + ωts,t ωsi,j .
(14)
The i ↔ j symmetry of the first term is obvious while for the second it follows from
(15)
ωts,t ωsi,j − ωsj,i = fstt fijs = 0,
where the last equality is obtained by taking the trace of the Jacobi identity (3).
One can give the following explicit form of the Ricci tensor
1
1
ricij = Bst As B −1 At ij − Bis Tr B −1 As B −1 At Btj
2
4
1
1
− Tr(Ti Tj ) + Tr(Ts ) f si,j + f sj,i , f si,j = B −1 st ft i,j ,
(16)
2
2
which exhibits that it is an homogeneous function of degree 0 in the breaking matrix B.
The scalar curvature R = (B −1 )ij ricij is a constant, as it should for homogeneous spaces.
A drastic simplification takes place for the bi-invariant metric (9), for which we have
ρ
(17)
ricij = − Bij .
4
The metric is therefore Einstein, and such a simple structure will have a counterpart in the
dualized theory.
2.3. Dualization
For the reader’s convenience we present a quick derivation [2,18] of the dualized model.
The essence of the dualization process is to switch from the coordinates on the group,
which parametrize g, to new coordinates ψi defined as the Lagrange multipliers of the
Bianchi identities. Concretely this transformation is carried out starting from the action
Z
1
i
(J ) .
d 2 x Bij ηµν Jµi Jνj − µν ψi Mµν
S=
4
Using light-cone coordinates, with the following conventions
x± =
x0 ± x1
,
√
2
µσ σ ν = δνµ ,
01 = 1,
J± =
J0 ± J1
√ ,
2
one has
S=
with
1
2
Z
j
d 2 x (B + A · ψ)ij J+i J− − ψi ∂+ J−i + ∂− J+i ,
(As )ij = (Ti )js = −fijs ,
(A · ψ)ij = (As )ij ψs .
The field equations obtained from the variations with respect to the currents J±i give
J−i = (B + A · ψ)is ∂− ψs ,
J+i = −∂+ ψs (B + A · ψ)si ,
(B + A · ψ)is (B + A · ψ)sj = δji .
×Ô
(18)
(19)
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Using minkowskian coordinates on the worldsheet one has
J µi = B ij µν ∂ν ψj − (A · ψ)j k Jνk , B is Bsk = δki .
Using this relation, the action (18) can be written, up to total derivatives
Z
Z
1
1
2
i
(20)
d x ∂+ ψi J− =
d 2 x ∂+ ψi (B + A · ψ)ij ∂− ψj .
S=
2
2
Comparing this action with the one given in relation (4.16) of [28] we see that in this
reference only the unbroken case Bij = δij has been considered.
Let us emphasize the following points:
1. Before dualization, all the field dependence on the coordinates chosen to parametrize
G must be hidden in expressions involving solely the currents Jµi . If this is not the case the
dualization process is not possible.
2. The dualized action is completely defined by the breaking matrix B and the field
matrix A · ψ ∈ so(ν). There are as many coordinates as generators in G.
3. In the process of dualization the isometries corresponding to GL (which leave the Jµi
invariant) are lost. This has for consequence that starting from an homogeneous metric, we
are led to a non-homogeneous one.
3. Geometry of the dualized theory
In (20) we come back to standard notations and change the coordinates ψi to ψ i . Let us
write the dual action
Z
1
(21)
d 2 x Gij ∂+ ψ i ∂− ψ j , Gij = (B + A · ψ)−1
S=
ij .
2
For further use we define the matrices
G± = (B ± A · ψ)−1 ,
G ≡ G+ ,
Γ ± = B ± A · ψ,
(A · ψ)ij = −fijs ψ s .
Writing the dual action (21) in minkowskian coordinates
Z
1
d 2 x gij ηµν ∂µ ψ i ∂ν ψ j + hij µν ∂µ ψ i ∂ν ψ j ,
S=
2
gives for metric and torsion potential
1
1
hij = (Gij − Gj i ),
gij = (Gij + Gj i ),
2
2
Using matrix notations we have
g = G+ BG− = G− BG+ ,
h = −g(A · ψ)B −1 ,
(22)
Gij = gij + hij .
(23)
and for the inverse metric:
g −1 = Γ + B −1 Γ − = Γ − B −1 Γ + .
(24)
The determinant of the metric is
2
det B
= det B · det G± .
det g =
±
2
(det Γ )
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497
3.1. Connection
We work with the standard conventions
Γjik = γjik + Tjik ,
Tjik = g is Tsj k ,
1
Tij k = (∂i hj k + ∂k hij + ∂j hki ),
2
∂
,
∂ψ i
∂i ≡
(25)
or using differential forms
H=
1
hij dψ i ∧ dψ j ,
2!
1
1
Tij k dψ i ∧ dψ j ∧ dψ k = dH.
3!
2
T=
The torsion potential is not uniquely defined since the following gauge transformation
leaves invariant the torsion:
A = Ai dψ i
H → H + dA,
⇐⇒
hij → hij + ∂[i Aj ] .
(26)
The connection is given by
Γjik =
1 −1 g is ∂j Gks + ∂k Gsj − ∂s Gkj .
2
(27)
Using the relation
∂i Gj k = fsti Gj s Gt k = − GAi G
one gets
jk
(28)
,
1 j
1
1
Γjik = − fst Γ + B −1 is Gkt − fstk B −1 Γ + t i Gsj + g −1 iu fstu Gsj Gkt . (29)
2
2
2
The next step is to simplify the last term in (29). To this end we combine Jacobi identity
and the definition (11) to prove the identity
(±)
(±)
fijs Γsk − fkjs Γsi
(∓)
= 2ωik,j − fiku Γuj .
(30)
Starting from relation (24) for the inverse metric we can write
+
,
g −1 iu fstu = Γ + B −1 iv fstu Γuv
and use (30) to interchange the indices s ↔ v. Several simplifications occur then in
relation (29) and one is left with the simple result
(31)
Γjik = fisk − ωts,u Γit+ Gku Gsj .
The same procedure, using the second writing of g −1 in relation (24), gives another
interesting form
j
(32)
Γjik = −fis + ωts,u Γit− Guj Gks .
ðÅ
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3.2. Torsion
To get a useful form for the torsion we use relation (31) to compute
γ
2Tjik Γj+r Γks+ = fir k Γks+ − ω r,β Γiγ+ (Γ − G)sβ − (r ↔ s).
The identity (30) and the easy relation Γ − G = 2BG − I, transform the previous relation
into
γ
−
= −ω r,β Γiγ+ (BG)sβ − (r ↔ s) − ωrs,i .
Tjik Γrj− Γsk
−1
−1 = (BG)−1 Γ − ,
It is natural to multiply both sides by (BG)−1
ur (BG)t s . Observing that g
we get
γ
T ij k = Γ + B −1 ks ω s,j Γiγ+ − (j ↔ k) + Γ + B −1 j s Γ + B −1 kt ωrs,i .
This result shows that this tensor is much simpler than Tij k since it is a polynomial in the
fields ψ. The coefficient of the linear term vanishes from Jacobi’s identity and we are left
with
1
αβ
T ij k = fij,k − (A · ψ)iα (A · ψ)jβ ω i + · · · ,
2
where the dots indicate circular permutations of the indices i, j, k. We expand the spin
connection according to (11) and use the identity (30) to end up with
2T ij k = fij,k − A · ψB −1 it A · ψB −1 j u ft u,k
− fijs A · ψB −1 A · ψ sk + · · · .
(33)
Now we can discuss a possibility not yet considered in the literature: the vanishing of the
torsion in the dual model. The terms which are independent of ψ require f[ij,k] = 0, a first
condition which mixes the structure constants and the breaking matrix. Using this relation
and the Jacobi identity one can check that the last two terms in (33) are equal. We conclude
that the torsion vanishes iff
v)
(34)
f[ij,k] = 0 and fαs(u B −1 st ft [k fij ]α = 0, ∀(u, v)[ij k].
Clearly for a simple algebra, the first constraint never holds, but for solvable algebras both
conditions may be satisfied, as will be seen in Section 5 for the Bianchi family.
Let us conclude with an example of Lie algebra, for which the torsion vanishes for any
choice of the breaking matrix. Let its generators be {Xi , i = 1, . . . , ν} and take
[X1 , Xi ] = Xi ,
i = 2, . . . , ν,
[Xi , Xj ] = 0,
i 6= j 6= 1.
3.3. Ricci tensor
The covariant derivatives are defined by
j
j
Di v j = ∂i v j + Γis v s = ∇i v j + Tis v s ,
Di vj = ∂i vj − Γijs vs = ∇i vj − Tijs vs , (35)
and the Riemann curvature by
[Dk , Dl ]v i = Ris,kl v s − 2Tkls Ds v i .
ðÆ
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499
Its explicit form is given by
Rij,kl = ∂k Γlji − ∂l Γkji + Γksi Γljs − Γlsi Γkjs .
The Ricci tensor follows from
Ricij = Rsi,sj = ∂s Γjsi − ∂j Γsis + Γsts Γjti − Γjst Γsit .
(36)
Using
p
Γsts = γsts = ∂t ln det g ,
we get for it a useful form
p
Ricij = ∂s Γjsi − Γjst Γsit − Dj Di ln det g .
(37)
+
−
−
+ ωsu,t Γas
= fats Γsu
− fuas Γst+ ,
ωst,u Γas
(38)
In order to compute the first two terms in this relation, we use (31) for the first two
connections and (32) for the third one. Apart from trivial cancellations one has to use
the identity
in order to obtain further strong cancellations of terms, with the final simple result
∂s Γjsi − Γjst Γsit = −Gis ricst Gtj + 2fsts ωtu,v Giu Gvj .
(39)
Using (32) and (35), one can check that the last term can be written
2fsts ωtu,v Giu Gvj = Dj Vi ,
Vi = −2 Git fsts .
Therefore, we end up with
Ricij = −Gis ricst Gtj + Dj vi ,
vi = Vi − ∂i ln
p
det g .
(40)
This relation, which displays the relation between the frame geometry of the principal
model and the geometry of its dual, will play an essential role in the next section.
Let us conclude with some remarks:
1. This result is different, although related to the ones by Tyurin [28] and Alvarez [1],
who expressed the frame geometry of the dual model in terms of the frame geometry
of the principal model. The first reference uses supersymmetry while the second uses
purely frames. Our approach, using mainly local coordinates computations is valid for
any breaking matrix B, while the previous authors have considered only the case B = I.
Note also that, in view of the complexity of the dualized vielbein it’s a long way from the
vielbein components of the Ricci to our relation (40).
2. If we consider a simple algebra G, equipped with its bi-invariant metric (9).
Relation (17) shows that the corresponding principal model is Einstein and we will prove
that the dual metric is quasi-Einstein. To this aim we insert relation (17) into (40), use
fsis = 0 to get for the dual theory
ρ
Ricij = (GBG)ij + Dj vi .
4
Using relation (32) one can check that
1
1
(41)
Dj λi = Gij + (GBG)ij , λi = B −1 is ψ s ,
2
2
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from which we deduce
p
ρ
ρ
(42)
Ricij = − Gij + Dj Vi , Vi = ∂i − ln det g + B −1 st ψ s ψ t ,
4
4
which establishes the desired result.
3. One further important point, with respect to string theory, is the dilatonic property of
the dualized geometry, i.e., whether the vector Vi is a gradient or not. For the semi-simple
groups the dilatonic property does hold since we have fsts = 0.
The failure of this property was first discovered for the dualized Bianchi V metric [19]
(see also [13]). In [22,28] it was shown to appear when the isometries are not semi-simple
and have traceful structure constants fsts 6= 0, and its interpretation as an anomaly was
worked out in [3].
4. One loop divergences of the dualized models
We are now in position to discuss the quantum properties of the dualized models at the
one loop level.
Let us first consider the broken principal models with classical action (4). Its one loop
counterterm, first computed by Friedan [17], is
Z
1
(43)
d 2 x ricij ηµν Jµi Jνj , d = 2 − ,
4π
where the Ricci components are computed in the vielbein basis.
Renormalizability in the strict field theoretic sense requires that these divergences have
to be absorbed by (field independent) deformations of the coupling constants ρ̂s hidden in
the matrix B and possibly a non-linear field renormalization. The renormalizability of the
classical theory is ensured by
∂
Bij .
(44)
∂ ρ̂s
The one loop renormalizability is clear for two extreme choices of metrics:
1. The bi-invariant metric, for which relation (17) shows that the principal model is
Einstein.
2. The maximally broken metric, for which the matrix B contains ν(ν + 1)/2
independent coupling constants ρ̂s . Since the Ricci is also a symmetric matrix, it can
always be absorbed by a deformation of the coupling constants.
For partial breakings of the group GR , relation (44) may fail to hold and is indeed a
constraint which mixes conditions involving the breaking matrix B and the algebra through
its structure constants.
In order to compare to the renormalization properties of the dualized theory, let us recall
that the most general conditions giving one loop renormalizability are

∂

gij + D(i uj ) ,
 Ric(ij ) = χ̂s
∂ ρ̂s
(45)

 Ric[ij ] = χ̂s ∂ hij + us T s + ∂[i Uj ] ,
ij
∂ ρ̂s
ricij = χ̂s (ρ)
ðÍ
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where the ρ̂s are the coupling constants in the principal model we started from, appearing
now in a non-trivial way in the dualized model. The only constraint on the functions χ̂s is
that they should be field independent.
These relations can be gathered into the single one
Ricij = χ̂s
∂
Gij + Dj ui + ∂[i (u + U )j ] .
∂ ρ̂s
(46)
We are now in position to prove that the one-loop renormalizability of the principal
model implies the one-loop renormalizability of its dual. For the reader’s convenience we
recall relation (40)
p
Ricij = −Gis ricst Gtj + Dj vi , vi = −2Git fsts − ∂i ln det g ,
in which we insert (44) to get
Ricij = −χ̂l Gis
∂
Bst Gtj + Dj vi .
∂ ρ̂l
The first term is reduced using the identity
∂
∂
Gij (B, ψ) = −Gis (B, ψ)
Bst Gtj (B, ψ),
∂ ρ̂l
∂ ρ̂l
(47)
to the final form
Ricij = χ̂s
∂
Gij + Dj vi .
∂ρs
(48)
Comparing with relation (46) we conclude to the one-loop renormalizability of the dual
model. Furthermore, the vectors ui and Ui , defined in relation (45), which could be
independent, are in fact related up to a gauge transformation by Ui = −ui + ∂i τ.
Our next task is to prove that the β functions are the same, so we need a precise definition
of the coupling constants. To do this let us switch from the couplings {ρ̂i , i = 1, . . . , c} to
new couplings (λ, ρi ) defined by
1
ρi
,
ρ̂i+1 = , i = 1, . . . , c − 1.
λ
λ
We scale similarly the breaking matrix
ρ̂1 =
(49)
1
Sij (ρ),
λ
where, for simplicity, the matrix S can be taken linear in the couplings ρs . Then
relation (44) becomes

 ric (B) = ric (S) = χ + X χ ∂ S (ρ),
ij
ij
λ
s
ij
(50)
∂ρs
s

χλ = χ̂1 , χi = χ̂i − ρi χ̂1 , i = 1, . . . , c − 1.
Bij (ρ̂) =
The full one loop action is, therefore,
Z
λχλ
λ X
∂
11
Sij (ρ) +
χs
Sij (ρ) Jµi Jνj ,
d 2x 1 +
λ2
2π
2π s
∂ρs
ðÎ
= 2 − d,
(51)
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from which we see that the divergences can be absorbed through coupling constant
renormalizations:
λχi
λχλ
(0)
,
ρi = ρi Zi ,
.
Zi = 1 +
Zλ = 1 −
λ0 = µ λZλ ,
2π
2π
It follows that the corresponding beta functions are
∂λ
∂ (1)
λ2
= λ2 Zλ = − χλ ,
∂µ
∂λ
2π
∂
∂ρi
λ
(1) =λ
χi .
βi = µ
ρi Zi =
∂µ
∂λ
2π
βλ = µ
(52)
For a principal model built with the bi-invariant metric given by (9) one has just the
single coupling λ, and
λ2 ρ
.
(53)
2π 4
In order to compute the divergences of the dualized theory in terms of the coupling
constants defined in (49) we start from the dual classical action
βλ =
Gij (B, ψ̃)∂+ ψ̃ i ∂− ψ̃ j ,
which we transform according to
1
Gij (S, ψ) ∂+ ψ i ∂− ψ j .
λ
The one-loop counterterms follow from the ricci. We start from relation (40) written
Ricij = λ2 −Gis (S, ψ) ricst Gtj (S, ψ) + Dj vi .
G(B, ψ̃) = λ G(S, ψ),
ψ i = λψ̃ i
−→
Using (50) we write the first term
−χλ Gis (S, ψ)Sst Gtj (S, ψ) −
X
χu Gis (S, ψ)
u
∂Sst
Gtj .
∂ρu
While the second term is reduced using the identity (47), the first term requires more work.
One has first to define the vectors
wi = gis ψ s ,
Wi = ψ s Gsi ,
(54)
then check the relation
1
Dj wi + ∂[i Wj ] = Gij + ψ s (∂j Gis + ∂i Gsj ) − Γjti gt s ψ s ,
2
which upon use of (27) becomes
1
Dj wi + ∂[i Wj ] = Gij + ψ s ∂s Gij .
2
Eventually relation (28) gives
1
1
Dj wi + ∂[i Wj ] = Gij + (GBG)ij .
2
2
ðÖ
(55)
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503
Scaling appropriately this identity, we have
X
∂
χλ Gij (S, ψ) +
χu
Gij (S, ψ) + Dj (vi − 2χλ wi ) − 2χλ ∂[i Wj ] ∂+ ψ i ∂− ψ j .
∂ρu
u
We end up with the renormalized dual theory
 Z
λ X
λχλ
∂
λ
1
2
+
Dj Vi − 2χλ ∂[i Wj ] ∂+ ψ i ∂− ψ j ,
χs
d x 1+
Gij +
λ
2π 2π s
∂ρs
2π

Vi = vi − 2χλ wi .
(56)
Comparing this relation with (51) we conclude that, up to the non-linear field redefinition described by the vector Vi and the gauge transformation described by Wi , the
coupling constants renormalization are exactly the same as in the principal model we
started from. We have thus established, at the one-loop level, that the principal σ -model
renormalizability implies the renormalizability, in the strict field theoretic sense, of its dual
and proved that their β functions do coincide.
Remarks
1. What is really new with respect to [28] is that, even working with renormalizability
in the strict field theoretic sense, the (possibly strong) breaking of the right isometries GR
does not jeopardize the one-loop renormalizability, and even in this extreme situation the β
functions of the principal model and its dual remain the same. This was not obvious since
the symmetry breaking is a “hard” breaking, by couplings of power counting dimension
two.
2. As already observed in Section 2.3, the isometries of GL are lost in the dualization
process. Hence for the maximal breaking of GR , no trace seems to remain of the original
isometries in the dualized theory. These dual theories constitute a nice example of nonhomogeneous metrics with torsion, with no isometries to account for their one-loop
renormalizability. Our computation, which puts forward an experimental fact (the oneloop renormalizability) needs some basic theoretical explanation since we know that
renormalizability is never accidental but the result of some underlying deeper symmetry.
3. As first observed in [7] for the dualized SU(2) model with symmetry breaking, there
appears in the final form of the divergences (56) a gauge transformation Wi . This term is
absent for models built on simple Lie groups with their bi-invariant metric Bij . Indeed in
this case we have the identities
1 −1 s t
B
ψ
ψ
≡ λi ,
ψ s Gsi = Gis ψ s = B −1 is ψ s H⇒ wi = Wi = ∂i
st
2
which implies ∂[i Wj ] = 0. Then the general identity (55) reduces, for this particular case,
to relation (41).
4. The situation at the two-loop level is still unclear since despite negative results in
several models [7,23] a more promising and new approach to the problem [24] seems to
yield a positive answer.
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5. It is well known that the unbroken principal models are integrable (for a review
see [29]). On the contrary the broken ones are not believed to be generically integrable,
a notable exception being SU(2), whose integrability was shown in [10] for the most
general breaking. If this belief is confirmed, our results show that the one-loop quantum
equivalence survives to symmetry breaking and therefore the root of this equivalence
cannot be integrability.
5. Extension to principal models with torsion
The previous results can be generalized to cover principal models with torsion, with
action
Z
1
d 2 x Bij ηµν + Cij µν Jµi Jνj , Cij = −Cj i ,
S=
2
where the matrix C has constant components. Taking into account the vielbein interpretation of the currents, we define the torsion tij k as usual by
1
1
tij k ei ∧ ej ∧ ek = dC,
3!
2
which gives
t=
1
C = Cij ei ∧ ej ,
2
1
fij s Csk + fj ks Csi + fki s Csj .
2
One should first observe that the parallelizing torsion [30] is not of this kind, and second
that we have to exclude the case where
tij k = −
Cij = fij s γs .
(57)
Indeed, if this relation holds the Bianchi identity (5) gives
Cij µν Jµi Jνj = γs fij s µν Jµi Jνj = −2γs µν ∂µ Jνs ,
which is a total divergence. Correspondingly the torsion vanishes as a consequence of the
Jacobi identity.
Even if (57) is valid for a semi-simple algebra G, it is not valid for any algebra. To see
this let us suppose that the center of G, is non-trivial, i.e., there is some generator Xα which
commutes with all the other generators. It follows that fαis γs ≡ 0 for all values of i, while
Cαi can be non-vanishing.
Let us describe briefly how our analysis can be generalized.
The spin connection Ω ij now verifies
dei + Ω ij ∧ ej = B ij tj ,
ti = tist es ∧ et .
Let us define
Ω
(±)
ij,k
= ωij,k ± tij k ,
Ω
(±) i
j,k
= B −1
is
Ω ±sj,k ,
i
es .
then the spin connection one-forms are Ω ij = Ω (−)j,s
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505
The Ricci tensor has now for components
ricij = Ω
(+) s
t,s
Ω
(−) t
i,j
−Ω
(−) s
(+) t
i,t Ω
j,s ,
Ω
(+) s
t,s
=Ω
(−) s
t,s
and is no longer symmetric.
Introducing the notations Γ ± = B ± (C + A · ψ), we have for the dualized metric G =
(Γ + )−1 . The connection in the dual theory becomes
j
t
t
Γit+ Gku Gsj = −fis + Ω (−)s,u
Γit− Guj Gks ,
Γjik = fisk − Ω (+)s,u
from which, after tedious computations, one gets for the Ricci tensor
p
Ricij = −Gis ricst Gtj + Dj vi , vi = −2Git fsts − ∂i ln det g ,
(58)
which is strikingly similar to (40).
Let us denote by ρsB the couplings present in the matrix B, by ρsC the couplings present
in the matrix C, and ρs the couplings present in both matrices. The renormalizability of
the principal model with torsion is ensured by

∂
∂

 ric(ij ) = χs
+ ηs
Bij ,

B
∂ρs
∂ρs
(59)
∂
∂


+ ξs C Cij .
 ric[ij ] = ηs
∂ρs
∂ρs
Inserting relation (59) into (58) one ends up with
∂
∂
∂
+ ξs C Gij + Dj vi .
Ricij = χs B + ηs
∂ρs
∂ρs
∂ρs
(60)
It follows, by the same arguments as in Section 5, that the dual model is also renormalizable
and has the same β functions as the initial principal model with torsion.
6. Dualized Bianchi metrics
Particular dualized models in the Bianchi family have been studied with emphasis either
put on the renormalizability properties of the dualized models with symmetry breaking
[7,23] or on the dilaton anomaly [13,19]. The aim of this section is to give some detailed
analysis of both aspects for the full family.
All the Lie algebras with 3 generators were classified by Bianchi (1897). In a modern
presentation [14,15] these algebras are described in terms of the parameter a and the vector
nE = (n1 , n2 , n3 ) according to
[X1 , X2 ] = aX2 + n3 X3 ,
[X3 , X1 ] = n2 X2 − aX3 ,
[X2 , X3 ] = n1 X1 ,
fsts = −2aδt 1.
The Jacobi identity requires a · n1 = 0.
The algebras of interest appear in the following table
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Class A: a = 0
Class B: n1 = 0, a > 0
Type
n1
n2
n3
Type
a
n2
n3
I
II
VI0
VII0
VIII
IX
0
1
0
0
1
1
0
0
1
1
1
1
0
0
−1
1
−1
1
V
IV
III
VIa
VIIa
1
1
1
a 6= 1
0
0
1
1
1
0
1
−1
−1
1
su(1, 1)
su(2)
The adjoint representation is given by




0 a n3
0 0 0
T2 =  0 0 0  ,
T1 =  0 −a −n3  ,
0 n2 −a
−n1 0 0


0 −n2 a

T3 = n1 0 0  .
0 0 0
(61)
The Killing metric gij = Tr(Ti Tj ) is diagonal with
g11 = 2(a 2 − n2 n3 ),
g22 = −2n3 n1 ,
g33 = −2n1 n2 .
It follows that B VIII and B IX are semi-simple (in fact, simple). Among the remaining
non semi-simple algebras, only those in class B have traceful structure constants.
To simplify matters, still keeping the main peculiarities of symmetry breaking, we take
the diagonal metric Bij = ri δij . The dual metric tensor is then



r2 r3 + x 2 r3 z − xy −r2 y − zx
 x = n1 ψ 1 ,
1
2
 −r3 z − xy r3 r1 + y −r1 x + yz  ,
G=
y = n2 ψ 2 − aψ 3 , (62)

∆+
2
r1 x + yz r1 r2 + z
r2 y − zx
z = aψ 2 + n3 ψ 3
with
∆± = r1 r2 r3 + r1 x 2 ± r2 y 2 + r3 z2 .
From (25) we get the torsion
Tij k = tij k , t = N/∆2+ ,
N = ν∆− + 2r2 r3 n3 y 2 + n2 z2 − n1 r12 ,
ν = r1 n1 + r2 n2 + r3 n3 .
(63)
This result shows that for Bianchi V the dualized metric is torsion-free!
Relation (16) gives for the non-vanishing vielbein components of the initial Ricci tensor

n2 r 2 − (n2 r2 − n3 r3 )2

 ric11 = −2a 2 + 1 1
,


2r2 r3



(n2 r2 − n3 r3 )
r2 n2 r 2 − (n3 r3 − n1 r1 )2
, ric23 = ric32 = a
, (64)
ric22 = −2a 2 + 2 2

r
2r
r
r1

1
3 1


2 2
2


 ric33 = −2a 2 r3 + n3 r3 − (n1 r1 − n2 r2 ) .
r1
2r1 r2
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507
6.1. Class A dual models and their β functions
We see at a glance from the Ricci that the class A principal models, with 3 independent
diagonal couplings are renormalizable at one-loop. From the the previous section this
ensures the renormalizability of the dualized model, with the same β functions.
For Bianchi I and II we define
1
g
g0
,
r2 = ,
r3 = .
λ
λ
λ
Then one gets for the β functions

 Bianchi I: βλ = βg = βg 0 = 0,
λ2 1
λ 1
, βg = −
,
 Bianchi II: βλ = −
4π gg 0
8π g 0
r1 =
βg 0 = −
λ 1
.
8π g
The result for Bianchi I is obvious, since its metric is flat.
For Bianchi IX (with σ = +1) and Bianchi VIII (with σ = −1), we parametrize the
couplings according to 1
σ
σ (1 + g)
1 + g0
,
r2 =
,
r3 =
.
λ
λ
λ
With three independent couplings, the SU(2)R isometries are fully broken. If one takes
g = g 0 the corresponding model has a residual U (1)R isometry and has been studied in [7],
where the quantum equivalence was proved at the one-loop order.
Using (64) and (52) it is a simple matter to compute

λ2 (1 + g − g 0 )(1 − g + g 0 )


,
 βλ = −
4π
(1 + g)(1 + g 0 )
(65)
0
λ g 0 (1 − g + g 0 )
λ g(1 + g − g )


 βg =
, βg 0 =
.
2π (1 + g 0 )
2π
(1 + g)
r1 =
For g = 0 and σ = 1 these results agree with [7].
For the remaining models we parametrize the couplings according to
1
1+g
1 + g0
r2 = σ
,
r3 =
,
r1 = σ ,
λ
λ
λ
where σ = +1 (respectively, σ = −1) correspond to Bianchi VII0 (respectively, Bianchi
VI0 ). We get for the β functions
λ2
(g − g 0 )2
,
4π (1 + g)(1 + g 0 )
λ (1 + g)
(g − g 0 ),
βg =
2π (1 + g 0 )
βλ =
βg 0 = −
λ (1 + g 0 )
(g − g 0 ).
2π (1 + g)
(66)
Let us observe that for g 0 = g the metric of the principal model is flat, which explains the
vanishing of all the β functions.
1 In the g = g 0 = 0 limit we recover the bi-invariant metrics.
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6.2. Class B dual models and their β functions
Let us begin with Bianchi V, which has diagonal ricci, and is therefore renormalizable
with three independent couplings
r1 =
1
,
λ
r2 =
1+g
,
λ
r3 =
1 + g0
.
λ
One gets
βλ =
λ2
,
π
βg = βg 0 = 0.
(67)
For the remaining models in this class the ricci is not diagonal, therefore we conclude to
the non-renormalizability of the remaining models with three independent couplings.
However, if we restrict ourselves to two couplings, tuned in such a way to have ric23 = 0,
most of the class B models become renormalizable:

1+g
1+g
1


, r3 = −
,
r1 = , r2 =
 Bianchi III:
λ
λ
λ

1+g
1+g
1

 Bianchi VIa (σ = −1), VIIa (σ = +1) : r1 = , r2 =
, r3 = σ
.
λ
λ
λ
Their beta functions are
βλ =
λ2 2
a ,
π
βg = 0.
(68)
For Bianchi IV no choice of diagonal breaking matrix leads to renormalizability.
6.3. Class B dual models and dilaton anomaly
Let us first get a convenient characterization of the absence of the dilaton anomaly. Using
relation (28) one has the equivalence
j
v
fsti Gj t − fst Git = 0.
Vi = −2Git fsts = ∂i 8 ⇐⇒ ∂i Vj − ∂j Vi = 0 ⇐⇒ Gsu fvu
Upon multiplication by Γai Γbj and use of (15), (38) one gets
t
t
v
Γat − fsa
Γbt = 0 −→ ωab,s Gst futu = 0.
fsb
Gsu fvu
It follows that the equivalence becomes
Vi = −2Git fsts = ∂i 8 ⇐⇒ ωab,s Vs = 0,
∀a, b.
(69)
Despite the convenient form of the final relation (69), it is fairly difficult to discuss in
general. Let us simply observe that the matrices ωa , with matrix elements defined by
(ωa )bs = ωab,s are singular. So the analysis of (69) depends strongly on the size of the
kernel of the ωa and therefore of the algebra and of the breaking matrix considered.
To discuss this point for the class B of the Bianchi family, we will consider the most
general breaking matrix B and we denote its off-diagonal terms by
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B12 = s3 ,
det B
B23 = s1 ,
= r1 r2 r3 − r1 s12
B31 = s2 ,
− r2 s22 − r3 s32 + 2s1 s2 s3 6= 0.
Let us notice that this last condition forbids the simultaneous vanishing of s1 , r2 and r3 .
The matrices ωa are given generally by
X
X
ν
ns Bsk sij + ai Bj k − aj Bik , ai = aδi1 , ν =
ns Bss .
(ωi )j k = ωij,k = − ij k +
2
s
s
For class B we have ν = n2 r2 + n3 r3 . Taking into account the relations
(r2 r3 − s12 )
(r3 s3 − s1 s2 + s1 y + r3 z)
,
G21 = −
,
det Γ
det Γ
(s3 s1 − r2 s2 + r2 y + s1 z)
G31 =
,
det Γ
G11 =
det Γ = det B + r2 y 2 + r3 z2 + 2s1 yz,
it is a purely algebraic matter, using (69), to prove that the dilaton anomaly is absent iff
ν ≡ n2 r2 + n3 r3 = 0 and µ ≡ s12 − r2 r3 = 0.
(70)
These constraints show that Bianchi VIIa is always anomalous, but also that an appropriate
choice of the couplings can get rid of the anomaly in the other models!
One can summarize the constraints (70) for the class B models and their possibly nonvanishing ricci component:
Model
Bianchi III
Bianchi IV
Bianchi V
Bianchi VIa (a 6= 1)
Bianchi VIIa
Constraint ( = ±1)
ric11
det B 6= 0
r3 = r2 , s1 = r2
r3 = 0, s1 = 0
√
s1 = r2 r3
r3 = r2 , s1 = r2
2( − 1)
0
0
2(a − 1)
r2 (s2 − s3 ) 6= 0
r2 · s2√6= 0
√
|r2 | s2 − |r3 | s3 6= 0
r2 (s2 − s3 ) 6= 0
impossible
It follows that the models B IV, B V and B III with = +1 are flat.
We want to show that the restrictions (70) are equivalent to the vanishing of the torsion.
To see this we use the constraints (34), which give, when specialized to class B:
3f[ij,k] = ν,
3 A · ψ B −1 A · ψ
f s =µ
s[k ij ]
(a 2 + n2 n3 )(n2 (ψ 2 )2 + n3 (ψ 3 )2 )
.
det B
In this case it is interesting to compare the vectors Vi = −2Git fsts and gi =
√
Di ln( det g ). One can check that the difference Vi − 2gi is then covariantly constant,
giving for final geometry
p
Ricij = −Gis ricst Gtj + Dj Di ln det g .
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7. Dualized Bianchi V model at two loops
As observed in the previous sections, dualized models may be torsionless: it is therefore
important to ascertain which models lead to this phenomenon. To this end we use the
constraints (34). Algebraic computations lead to the following conclusions:
1. For the class A models, no choice of the non-singular matrix B leads to vanishing
torsion.
2. For the class B models, except Bianchi V, the necessary and sufficient conditions for
vanishing torsion are given by the relations (70).
3. Among all the class B models only Bianchi V has a vanishing torsion for an arbitrary
breaking matrix B. In this case the torsion potential is an exact 2-form with

s1 s2 − r3 s3
s3 s1 − r2 s2
 H = 1 dA,
ν2 = 2
,
ν3 = 2
,
2
s1 − r2 r3
s1 − r2 r3
p

γ = ln det g .
A = γ dψ 1 − ν3 dψ 2 − ν2 dψ 3 ,
The case where s12 = r2 r3 6= 0 is special, with
x +α
1
· ψ 2 dψ 2 ,
A = γ dψ 1 +
s3 ln x 2 − α 2 − ln
r2
x −α
x = r2 ψ 3 − s1 ψ 2 ,
α = s1 s3 − r2 s2 .
It follows that the dual model, at least perturbatively, can be analyzed as if it had no
WZW coupling! This situation is fairly original: the principal Bianchi V model, which
is homogeneous and torsionless, is mapped by T-duality to an inhomogeneous but still
torsionless σ -model. It is therefore attractive to check the two-loop equivalence of the
models using the firmly established counterterms given by Friedan [17]. Let us consider
the simplest Bianchi V dual model, with Bij = rδij . Its dualized metric, taken from (62),
reads:
2
2
2
2 1 2
r
dψ 1 +
r dψ 2 + r 2 dψ 3 + ψ 3 dψ 2 − ψ 2 dψ 3 ,
∆
r∆
2 2
3 2
+ ψ
+ r 2.
∆= ψ
g=
(71)
Following [13] we take for new coordinates
ψ 1 = z,
ψ 2 + iψ 3 = ρeiφ
H⇒
g=
ρ2
ρ2
r
dz2 + dρ 2 + (dφ)2 ,
2
+r
r
(72)
which bring the metric to a simple diagonal form, with the obvious vielbein
g=
3
X
a=1
ea2 ,
e1 = p
√
r
ρ2
+ r2
dz,
√
r
e2 = p
dρ,
2
ρ + r2
ρ
e3 = √ dφ.
r
(73)
One can prove that this metric has two isometries, described by the vector fields ∂/∂z and
∂/∂φ.
The geometrical quantities of interest are
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
p

ρ
1
ρ2 + r 2
1


ω23 = − √
e3 ,
ω12 = − √ p
e1 ,



ρ
r
r ρ2 + r 2




1
1
σ

R31 = e3 ∧ e1 ,
R12 = − e1 ∧ e2 ,
R23 = − e2 ∧ e3 ,
r
r
r


1−σ
1+σ



,
Ric22 = −
,
Ric33 = 0,
Ric11 =


r
r


σ

 R = Ricss = −2 .
r
σ=
ρ2 − r 2
,
ρ2 + r 2
(74)
The one-loop renormalizability relations
Ricij = χ (1)
∂
gij + ∇(i vj ) ,
∂r
become, using vielbein components

 Ric = χ (1) e−1 j ∂ e + e−1 j ∂ e
+ D(a vb) ,
ab
b ∂r aj
a ∂r bj
j

∂ˆa = e−1 a ∂i .
Da vb = ∂ˆa vb + ωbs,a vs ,
(75)
Relation (75) works with
χ (1) = −2,
ρ
2
v ≡ va ea = − √ p
dρ.
r ρ2 + r 2
(76)
Let us remark that while χ (1) is uniquely defined, the vector vi is not unique and we took
its simplest form. As it should, the renormalization of the coupling constant r is the same
as in the principal model as can be seen from relation (64).
The two-loops counterterms, first computed by Friedan [17], are
Z
1
d 2 x Ris,t u Rj s,t uηµν Jµi Jνj ,
16π 2 where the Ris,t u are the vielbein components of the Riemann tensor.
For three dimensional geometries, this counterterm is most easily obtained from the
identity
R2
1
(77)
δab ,
(RR)ab ≡ Ras,t uRbs,t u = R Ricab − Ric2 ab + Tr Ric2 −
2
2
which gives
(RR)11 = (RR)22 =
1
(1 + σ 2 ),
r2
(RR)33 =
2
.
r2
In order to prove renormalizability we have to solve for χ (2) and wa such that
j ∂
j ∂
(RR)ab = χ (2) e−1 b eaj + e−1 a ebj + D(a wb) .
∂r
∂r
Explicitly, these equations give the differential system
ÔÎ
(78)
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
σ
1


1 + σ 2 − χ (2) = ∂ˆ1 w1 + ω12,1 w2 ,
0 = ∂ˆ1 w2 + ∂ˆ2 w1 − ω12,1 w1 ,

2

r
r
σ
1
(79)
0 = ∂ˆ3 w1 + ∂ˆ1 w3 ,
1 + σ 2 − χ (2) = ∂ˆ2 w2 ,
2

r
r


1
2


0 = ∂ˆ3 w2 + ∂ˆ2 w3 + ω23,3 w3 .
+ χ (2) = ∂ˆ3 w3 − ω23,3 w2 ,
r
r2
Integrating some relations with respect to the variable ρ we obtain

q
q
d 2
W1 (z, φ)
(0)
(0)

2

p
w1 = −w1 (ρ)∂z W2 (z, φ) +
,
ρ + r w1 (ρ) = ρ 2 + r 2 ,


2 + r2
dρ

ρ


√

d (0)
r
1 + σ2
(0)
(2) σ
(80)
w (ρ) = p
−χ
,
w2 = w2 (ρ) + W2 (z, φ),
2 + r2

dρ 2
r2
r
ρ


p


2
2


 w3 = − ρ + r ∂φ W2 (z, φ) + ρW3 (z, φ).
r
Inserting these relations into the last left relation of (79) one has
χ (2) √
2
− r ∂φ W3 = M,
W2 − ∂φ2 W2 = N,
+
r2
r
√
ρ
(0)
M,
w2 (ρ) + N = r p
ρ2 + r 2
(81)
where M and N are coordinate independent. Differentiating this last relation with respect
to ρ yields a constraint which does not hold, irrespectively of the values taken for M and
χ (2) .
The failure of relations (78) means that the two-loops quantum extension chosen for the
dual model does not lift the classical equivalence to the quantum level.
In fact we should consider 2 the whole family of metrics
gij −→ gij + γij ,
where γij is a one-loop deformation of the classical metric gij which describes different
possible quantum extensions of the same classical dual model. For this modified theory we
get an extra contribution at the two-loops level which is
Z
1
d 2 x Ricij (g + γ ) − Ricij (g) ηµν Jµi Jνj .
4π
Let us examine whether the two-loops renormalizability can be implemented or not. As
is well known, one has
1
Ricij (g + γ ) − Ricij (g) = − ∆L γij + ∇(i αj ) ,
2
where ∆L is Lichnerowicz’s laplacian
1
αi = ∇ s γsi − ∇i γ ss ,
2
∆L γij = ∇ s ∇s γij + 2Ris,j t γ st − Ricis γ sj − Ricj s γ si .
The connection ∇, the Riemann tensor and the raising or lowering of indices are related to
the unperturbed metric g.
2 We thank G. Bonneau for suggesting to us this idea.
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P.-Y. Casteill, G. Valent / Nuclear Physics B 591 (2000) 491–514
Up to a scaling of γ , the two-loops renormalizability constraints become
(RR)ij = ∆L γij + χ (2)
∂
gij + ∇(i wj ) .
∂r
(82)
We will exhibit a solution of these equations for any choice of χ (2) . For this we consider
the vielbein components of the deformation
1
γ = γ1 e12 + γ2 e22 + γ3 e32 ,
r
and we use the notations
2
χ (2) = (1 − χ),
r
3+χ
8(x) =
2(1 − x)2
x=
Zx
ρ2
ρ2
∈ [0, 1[,
+ r2
ln(1 − u)
du.
u
0
One should notice that for the principal Bianchi V model at two-loops we have χ = 0.
Let us define

1+x
4(4 + 3χ)x − (5χ − 1)x 2


 γ1 (x) = −
γ
(0)
−
2


1−x
8(1 − x)2



3
+
χ
−
x


ln(1 − x) + 8(x),
−



2(1 − x)
4(11 + 4χ)x + (11 + 5χ)x 2 − 4x 3

γ2 (x) = γ2 (0) +


8(1 − x)2



8
+
3χ
+
x


ln(1 − x) − (1 + 2x)8(x),
+



2(1 − x)

 γ (x) = −γ (x).
3
1
The vector vielbein components are
w1 = w3 = 0,
√
d
r 3/2 x w2 = 4(1 − χ)x − 2x 2 − (2 + χ) ln(1 − x) − 2x(1 − x) γ2 (x).
dx
The reader can check that the deformation and the vector given above are indeed solution
of (82) for any value of χ .
Two main points need to be checked. The first one is the analyticity of the γi (x) in a
neighbourhood of x = 0. This follows from the analyticity of 8(x) and is explicit on the
other terms. The second point is that we are using polar coordinates; in order to secure
an analytic dependence with respect to the cartesian coordinates ψ 1 , ψ 2 ≈ 0 we have
imposed γ3 (0) = γ2 (0). The free parameters in this solution are γ2 (0) and χ.
As a side remark, let us observe that the deformation obtained above, cannot be written
in the form
∂
γij = A gij + D(i Wj ) ,
∂r
which means that it cannot be interpreted as a finite renormalization of the initial metric
gij .
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P.-Y. Casteill, G. Valent / Nuclear Physics B 591 (2000) 491–514
So we can conclude that it is always possible to have a quantum extension of the dualized
Bianchi V which does preserve the two-loops renormalizability. Unfortunately nothing, in
this process, enforces χ to have the same value as in the principal model we started from.
This shows that further constraints are needed to define uniquely the two-loops quantum
dual theory.
Acknowledgements
We are indebted to O. Alvarez, G. Bonneau, F. Delduc and E. Ivanov for enlightening
discussions.
References
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[20] A. Giveon, M. Porrati, E. Rabinovici, Phys. Rep. 244 (1994) 77, hep-th/9401139.
[21] A. Giveon, E. Rabinovici, G. Veneziano, Nucl. Phys. B 322 (1989) 167, cern-th.5106/88.
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[24] Z. Horváth, R.L. Karp, L. Palla, hep-th/0001021.
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Physics Letters B 543 (2002) 241–248
www.elsevier.com/locate/npe
Renormalisability of non-homogeneous T-dualised sigma-models
Pierre-Yves Casteill
Laboratoire de Physique Théorique et des Hautes Energies, Unité associée au CNRS UMR 7589, Université Paris 7, 2 Place Jussieu,
75251 Paris Cedex 05, France
Received 4 March 2002; accepted 30 July 2002
Editor: L. Alvarez-Gaumé
Abstract
The quantum equivalence between σ -models and their non-Abelian T-dualised partners is examined for a large class of four
dimensional non-homogeneous and quasi-Einstein metrics with an isometry group SU(2) × U (1). We prove that the one-loop
renormalisability of the initial torsionless σ -models is equivalent to the one-loop renormalisability of the T-dualised torsionful
model. For a subclass of Kähler original metrics, the dual partners are still Kähler (with torsion).
 2002 Elsevier Science B.V. All rights reserved.
PACS: 0240; 11.10.Gh; 11.10.Kk; 11.10.Lm
Keywords: Non-homogeneous sigma models; T-duality; Renormalisation
1. Introduction
The subject of target space duality, or T-duality,
in String Theory and in Conformal Field Theory has
generated much interest in recent years and extensive reviews covering Abelian, non-Abelian dualities
and their applications to string theory and statistical
physics are available in the literature [1–3]. The geometrical aspects of this duality can be found in [4].
T-duality provides a method for relating inequivalent string theories. First discovered for the case of
σ -models with some Abelian isometry, the concept
of T-duality has been recently enlarged to theories
with non-Abelian isometries [5–7]. A very important
and interesting property of T-duality applied on nonAbelian isometry is that it can map a geometry with
E-mail address: [email protected] (P.-Y. Casteill).
such isometries to another which has none. Therefore, non-Abelian T-duality cannot be inverted as in
the Abelian case.
By showing that T-duality is a canonical transformation [5,8,9], it was proved that theories in such way
related where classically equivalent. Furthermore, this
equivalence was still remaining at the one-loop level,
in a strict renormalisability sense, in all the many example that have been tested up to now to this duality, with an emphasis put on SU (2) [1,7,10–13]. For
example, this one-loop equivalence still remains for
principal σ -models whatever strongly broken the right
isometries may be [14]. The non-Abelian dualisation
of non-homogeneous metrics such as the Schwarzschild black hole or Taub-NUT was performed in
[7,12] and in [15]. We propose here the dualisation of
the general SU(2) × U (1) metrics.
Problems arise when one addresses the question of
the renormalisability of dualised theories beyond the
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PII: S 0 3 7 0 - 2 6 9 3 ( 0 2 ) 0 2 4 2 2 - X
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242
P.-Y. Casteill / Physics Letters B 543 (2002) 241–248
Therefore σ is a SU(2)L singlet and a SU(2)R triplet.
If β(t) = γ (t), the SU(2)R isometries will be broken
down to a U (1) and the total isometry group of the
metric will then be SU(2)L × U (1). Indeed, in order
to keep the metric invariant, one then must have R =
{0, 0, µ}. If ε = −1, σ is changed under infinitesimal
transformations of su(2)L ⊕ su(2)R as
one-loop order. It had been proved that even for the
simplest (SU(2) × SU(2))/SU(2) principal σ -model,
the dualised theory is not two-loop renormalisable, in
the minimal dimensional scheme [16,17]. However, as
shown in [18], a finite deformation at the h̄ order of
the dualised metric is sufficient for recovering a twoloop renormalisability for this particular model. As it
will be shown, the SU(2) × U (1) σ -models are not in
general two-loop renormalisable, even though the oneloop renormalisability remains for their dual partners!
The content of this Letter is the following: in Section 2, we recall the general expression of the SU(2) ×
U (1) metrics and set the notations. In Section 3, we
make a review of such metrics which give rise to
one-loop renormalisable σ -models, as for example the
celebrated Taub-NUT and Eguchi–Hanson metrics. In
Section 4, we show that only the particular metrics
where homogeneity is recovered by some enhancement of the isometries are two-loop renormalisable. In
Section 5, we dualise the original theory and show in
Section 6 that the one-loop renormalisability survives
during the dualisation process. When the original metric is Kähler, we investigate in Section 7 if such a property is still present for the dual partner. Some concluding remarks are offered in Section 8.
δ σ = L ∧ σ ,
and therefore the isometry group of the metric will
be SU(2)R × U (1). The choice of ε switches also the
autodual components of the Weyl tensor (W+ ↔ W− ).
In all cases, when β(t) = γ (t), the metric has for
isometry group SU(2)L × SU(2)R and is conformally
flat.
It is then possible to define the σ -model corresponding to these metrics
1
dx 2 ηµν gij ∂µ φ i ∂ν φ j ,
S=
(1)
T
with {φ 0 = t, φ 1 = θ, φ 2 = ϕ, φ 3 = ψ}, and address
the question of its one-loop and two-loop renormalisability.
In order to derive the Ricci tensor, we define the
vierbein {ea | a ∈ {0, 1, 2, 3}} as
e2 = β(t) σ2 ,
e0 = α(t) dt,
e1 = β(t) σ1 ,
e3 = γ (t) σ3 .
2. The SU(2) × U (1) metric
We consider the four dimensions metrics with
cohomogeneity one under a SU(2) × U (1) isometry.
In the more general way, these can write
g = α(t) dt 2 + β(t) σ1 2 + σ2 2 + γ (t)σ3 2 ,
In the absence of torsion, the condition for giving
one-loop renormalisability is the quasi-Einstein property of the metric
Ricab = λgab + D(a vb) ,
where the σi are 1-forms such that
1
dσi = ε ij k σj ∧ σj ,
2
(2)
where the Einstein constant λ will renormalise the
coupling while the vector v will renormalise the field.
ε = ±1.
One can always writes σ1 2 + σ2 2 and σ3 under the
well-known specific shape
3. One-loop renormalisation
We will only consider metrics satisfying condition (2) so that the corresponding σ -models are oneloop renormalisable. Of course, as we want to keep
the SU(2) symmetry while renormalising, we will
only consider here vectors v that depends only on
the t coordinate: v = v(t). As the expression of the
SU(2) × U (1) metric (3) we chose does not mix dt,
σ1 2 + σ2 2 = dθ 2 + sin2 θ dϕ 2 ,
σ3 = dψ + cos θ dϕ.
If ε = +1, the triplet of 1-forms σ is changed under
infinitesimal transformations of su(2)L ⊕ su(2)R as
δ σ = R ∧ σ .
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P.-Y. Casteill / Physics Letters B 543 (2002) 241–248
σ1 , σ2 and σ3 , both the metric g and the Ricci tensor Ric will be diagonal in the {dt, σ1 , σ2 , σ3 } basis
and this will hold in the vierbein. As a consequence,
D(a vb) must be also diagonal; this √
is true only for vectors of the form v = √
v0 (t)e0 + ρ γ (t) e3 . The constant ρ is arbitrary as γ (t)e3 is in fact the form dual
to the Killing ∂ψ . We will take ρ = 0.
In order to simplify matters, from now on, we will
choose the coordinate t so that β(t) = t. The metric
now writes
g = α(t) dt 2 + t σ1 2 + σ2 2 + γ (t)σ3 2 .
(3)
All this being settled, the quasi-Einstein character
of the metric (2) can now be expressed as a set of three
non-linear differential equations which are

1 γ (t ) α (t ) γ (t )2
γ (t )
1


2 + t + 2γ (t ) α(t ) +
2 − γ (t )

t
2γ (t )


√


 = 2λα(t) + 2 α(t) v0 (t),




 22 − γ (t ) α(t) + α (t ) − γ (t )
t
α(t )
γ (t )
√

 = 4λtα(t) + 2 α(t) v0 (t),



 2
2 γ (t )2


−
+
α(t) + αα(t(t)) + γγ (t(t)) − 2γγ (t(t))

t
t 2 γ (t )



 = 4λ γ (t ) α(t) + 2√α(t) v (t).
0
γ (t )
(4)
This system is difficult to solve, even though it can
still be done for some limited cases as the Einstein
one (v0 = 0) and the quasi-Einstein Kähler one. It is
possible to eliminate α(t) and v0 (t) in the system (4),
leading to a single, deeply non-linear, differential
equation of the fourth order in γ (t). The general
SU(2) × U (1) quasi-Einstein metric should therefore
depend on four parameters.
In order to convince the reader of the large class
of models that will be dualised, we will now give a
short review of the SU(2) × U (1) Einstein and quasiEinstein Kähler metrics.
3.1. Einstein metrics
The metric g will be Einstein if Ric = λg. It is possible to integrate the differential system (4) imposing
v0 = 0 and one gets
α(t) =
1
1
,
1 + At γ (t)
√
4λt 2 3 + 1 + At
4t
−
γ (t) =
√
2
3 (1 + √1 + At)3
(1 + 1 + At)
B√
1 + At,
+
(5)
t
A and B being the integration constants. This family contains many metrics of interest which we recall
briefly.
If A = 0, we recover the Kähler–Einstein extension
of Eguchi–Hanson [20]. If A = 0 then g identifies
with the large class of Einstein metrics derived by
Carter [21]. By making the change of coordinates
1
and B = −8(M − n)n3 ,
n2
one can have for g a more simple expression

2
2
t 2 −n2
2
2
2

 g = f (t ) dt + t − n σ1 + σ2

2
(6)
+ t 24n
f (t)σ3 2 ,
−n2



f (t) = t 2 − 2Mt + n2 − λ3 (t − n)3 (t + 3n).
t → t 2 − n2 ,
with A =
Notice that as A and B are real constants, M and n can
be both reals or pure imaginaries. Defining 2n dψ =
dΨ and taking the limit n → 0 gives the Schwarzschild metric with cosmological constant
g=
1
1−
2M
t
+ 1−
−
λ 2
3t
dt 2 + t 2 dθ 2 + sin2 θ dϕ 2
2M λ 2
− t dΨ 2 .
t
3
(7)
Other limits of (6) lead to the Page metric on
P2 (C)#P2 (C) and to the Taub-NUT metric.
3.2. Quasi-Einstein Kähler metrics
These are the only SU(2) × U (1) quasi-Einstein
metrics known up to now [22]. We suppose here
that there is a choice of holomorphic coordinates on
which the isometries SU(2) × U (1) act linearly. It
happens that this hypothesis implies the integrability
of the complex structure. A necessary condition of the
Kähler property is the closing of the Kähler form
d(e0 ∧ e3 + εe1 ∧ e2 )
= d α(t)γ (t) dt ∧ σ3 + β(t) dσ3 = 0.
It is clear that this relation will hold iff β (t)2 =
α(t)γ (t), i.e., α(t) = γ 1(t ) . It is then possible to solve
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P.-Y. Casteill / Physics Letters B 543 (2002) 241–248
system (4) and one gets for the metric and for the
vector v:
1
g=
dt 2 + t σ1 2 + σ2 2 + γ (t)σ3 2 ,
γ (t)
v = −C γ (t) e0 = −C dt,
(8)
4.1. Einstein metrics
In the vierbein basis, one can compute the two-loop
divergences for the metric given in (6) and find
1
Ram,np Rbm,np
2
with
DeCt
+t
t
1
2λ
2
eCt − 1 − Ct − C 2 t 2 ,
+ 2 1−
C
2
C t
where C and D are the integration constants.
In the limit C → 0, we have v = 0 and thus
we are back to the Kähler–Einstein metrics, i.e.,
the Kähler–Einstein extension of Eguchi–Hanson (the
correspondence between the parameters is then D =
B).1
γ (t) =
=3
(M − n)2
(n − t)6
+
M +n+
8n3 λ
3
6
(n + t)
2
+
λ2
δab .
9
Quite surprisingly, the two-loop divergences are conformal to the original metric.
Relation (9) in the vierbein basis becomes
1
1
Ram,np Rbm,np = λ̃δab + Ea j (χM ∂M + χn ∂n )Ebj
2
2
1
+ Da ṽb + (a ↔ b),
2
where Eai is defined by ea = Eai dφ i . As for the oneloop renormalisation conditions (4), this last relation
gives us three equations. These can easily be reduced
to two by eliminating ṽ. The remaining equations will
only depend on the variable t and on the constants
λ̃, χn and χM . As these must vanish irrespectively of
the values taken by t, one can show that they will be
verified in only two particular cases where M and n
are fixed such that
4. Two-loop renormalisation
The two-loop divergences, first computed by Friedan [19], are
h̄2 T
Ris,t u Rj s,t u,
d = 2 − .
8π 2 In order to reabsorb these divergences, the counterterms may come from the renormalisation of the
but also from the
coupling T and the fields φ,
renormalisation of the parameters that were let in the
metric at one-loop. For example, if one starts with
the Einstein metric (6), one should allow for counterterms renormalising the parameters M, n. In general,
if we define such parameters as ρc , the theory will
be renormalisable at two loops iff one can find some
vector ṽ = ṽ(t) and some constants λ̃ and χc such that
Divij2 = −
M 2 = n2 = −
3
4λ
or M = n = 0.
2
In both cases, (9) will be satisfied with λ̃ = λ3 and
χM = χn = ṽ = 0, but it is not surprising as these
choice for M and n are the one which enlarge the
SU(2) × U (1) isometries to SO(5), making the metric
homogeneous (de Sitter metric).
1
Ris,t uRj s,t u = λ̃gij + χc ∂ρc gij + D(i ṽj ) .
(9)
2
We will show that, except for the few particular cases
where the metric is homogeneous,2 the SU(2) × U (1)
Einstein and Kähler metrics do not give in a direct way
two-loop renormalisable σ -models.
4.2. Kähler metrics
Proceeding as for the Einstein metrics, one can
compute the two-loop divergence using the metric (8).
Once again, the parameters C and D must have special
values for the action to be two-loop renormalisable.
Indeed, one must have (C = 2λ, D = 0) or (C → 0,
D = 0). In the first case, we recover flat space. In
the second case, we get the Fubiny–Study metric on
P2 (C) and its non-compact partner which are also
two-loop renormalisable with λ̃ = 23 λ2 and ṽ = 0.
1 This shows that the four parameters of the general solution
of (4) cannot be A, B, C and D as these are not independent.
2 It was proven in [23] that homogeneous metrics are always
renormalisable to all loop order.
ÅÕÉ
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P.-Y. Casteill / Physics Letters B 543 (2002) 241–248
The Einstein and Kähler metrics with no more
isometries than SU(2) × U (1) are therefore not renormalisable in the minimal scheme at two loops. This
could of course be cured by adding some infinite deformation of the metric itself as in D. Friedan’s approach to σ models quantisation, but it is the author
belief that a finite deformation keeping the isometries,
as explained in [18], would be sufficient.3
5. The dual metric
We dualise the initial metric (3) over the SU(2)
isometries, keeping aside the U (1). Practically, it
consists in dualising the three-dimensional metric [15]
g3 = t σ1 2 + σ2 2 + γ (t)σ3 2 ,
leaving the term α(t) dt 2 unchanged. If we define the
new fields of the dual metric λi , i ∈ {1, 2, 3}, the dual
theory of g3 will writes, in light-cone coordinates
1
dx 2 G3ij ∂+ λi ∂− λj ,
S3 =
T
where
G3ij =
t
−λ3
λ2
λ3
t
−λ1
−λ2
λ1
γ (t)
−1
.
+
where
(r 2 + t 2 )γ (t)
dr ∧ dz
∆
ryγ (t)
dy ∧ dz.
(11)
∆
We define ĝij as the tensor associated to the metric
(10) and ĥij as the torsion potential. Let Gij =
be the new Ricci tensor which is
ĝij + ĥij and Ric
not symmetric anymore because of the presence of
torsion in the dualised model. Eventually, the dualised
action of our SU(2) × U (1) theory is, in light-cone
coordinates
1
S=
(12)
dx 2 Gij ∂+ φ̂ i ∂− φ̂ j ,
T
+
where the coordinates are {φ̂ 0 = t, φ̂ 1 = r, φ̂ 2 =
y, φ̂ 3 = z}. It could be useful to notice that
t 2y2
α(t)γ (t).
∆2
It was proved in [12] that the dualised Eguchi–Hanson
model is conformally flat. We have checked that, in the
class studied here, this is the only case where the Weyl
tensor vanishes.
det ĝ =
5.1. The SO(3) dual of Schwarzschild
λ2 = y cos(z),
λ3 = r,
one has for the total dual metric ĝ = α(t) dt 2 +
G3(ij ) dλi dλj
ĝ = α(t) dt 2 +
H = d(z dr) +
ij
After the following change in coordinates
λ1 = y sin(z),
The torsion is defined by T = 21 dH where H =
∧ dλj is the torsion potential 2-form
1
i
2 G3[ij ] dλ
ry
r2 + t2
dr + 2
dy
∆
r + t2
ty 2 γ (t) 2
t
2
dz ,
dy
+
r2 + t2
∆
2
(10)
∆ = y 2 t + r 2 + t 2 γ (t).
3 Here, one should start with the general metric, solution of (4),
Among all the SU(2) × U (1) metrics, the Schwarzschild one has an interesting peculiarity as its dual can
be obtained in two ways. Indeed, in the original metric (7), due to the split of σ3 2 , the SU(2) isometries
appear only in the (σ1 2 + σ2 3 ) term. One can therefore
first dualise the “sub-metric” corresponding to this last
term and then add the dt 2 and dΨ 2 terms in order to
obtain the dualised Schwarzschild metric. Doing this,
only two Lagrange multipliers λi will appear during
the dualisation procedure [15]. But it is still possible to
obtain it by first dualising the metric (6) and then taking the appropriate limit (n → 0). As γ (t) → 0, one
has first to make the change of coordinates dz = dψ
2n
before taking the limit. Doing this, one gets for ĝ:
ĝ =
if no new parameters is a required condition for the renormalisation
process.
1
1−
+
ÅÕÍ
2M
t
t2
r2
− λ3 t 2
+ t4
dt 2 +
ry
r2 + t4
dr + 2
dy
t 2y2
r + t4
dy 2 + 1 −
2M λ 2
− t dΨ 2 .
t
3
2
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246
P.-Y. Casteill / Physics Letters B 543 (2002) 241–248
Finally,
by making the coordinate change y =
√
s 2 − r 2 , we get
ĝ =
1
1−
+
2M
t
−
λ 2
3t
dt 2 + 1 −
We have checked that (14) is verified taking


 λ̂ = λ,
v̂i = −2λĝij Xj + Di log ∆ + vi ,


ŵj = −2λXj Gj i ,
2M λ 2
− t dΨ 2
t
3
4 2
1
t dr + s 2 ds 2 .
t 2 (s 2 − r 2 )
where X is defined by X = r∂r + y∂y .
Conversely, let us now suppose that λ̂, v̂ and ŵ
are defined by (16) where λ and v are supposed to
be arbitrary. It is possible to show that if (14) holds,
then the original metric is quasi-Einstein with Ricij =
λgij + D(i vj ) . In order to demonstrate this, we first
define the three functions fA (t), fB (t) and fC (t) such
that
 
 α (t) = A t, α(t), γ (t), v0 (t), v0 (t) + fA (t),
γ (t) = B t, α(t), γ (t), v0 (t), v0 (t) + fB (t), (17)

 γ (t) = C t, α(t), γ (t), v0 (t), v0 (t) + fC (t).
(13)
In the special case λ = 0, we recover the SO(3) dual
of Schwarzschild which was one of the first examples
for non-Abelian duality [7]. While making n → 0, the
torsion potential 2-form H (11) writes as d( Ψ2ndr ) +
O(n), and therefore, as H is only defined up to a total
derivative, the torsion vanishes, which is consistent
with the result found in [7].
We will now address the question of the one loop
renormalisability of the dual theory S.
Assuming that (14) holds, and after having replaced
each occurrence of α (t), γ (t) and γ (t) by its
value in (17), we get some equation system where
the unknowns are the functions fX (t). As this last
system must hold irrespectively of the values taken
by r and y which are free variables, one can then prove
that fA (t) = fB (t) = fC (t) = 0. This shows that (15)
holds and therefore the quasi-Einstein property of the
original metric.
We have proven, for arbitrary functions α(t) and
γ (t), the equivalence
6. One-loop renormalisation of the dual metric
We want to prove that the one-loop renormalisation
property does survive to the dualisation process. In
other words, if the torsionless action (1) is quasiEinstein, then so is the action (12). In the presence
of torsion, this now means that one can find some
constant λ̂ and some vectors v̂ and ŵ such that
ij = λ̂Gij + Dj v̂i + ∂[i ŵj ] .
Ric
(16)
(14)
Ricij = λgij + D(i vj ) ⇐⇒
ij = λ̂Gij + Dj v̂i + ∂[i ŵj ] ,
Ric
This equality gives a system of equations much more
complicated than (4), but what is important is that
now α(t) and γ (t) are not considered as unknown
functions. Furthermore, as we suppose the original
metric to be quasi-Einstein, the system (4) is assumed
to be verified and one can easily derive from it, in an
algebraic way, the three functions A, B and C such
that
 
 α (t) = A t, α(t), γ (t), v0 (t), v0 (t) ,
(15)
γ (t) = B t, α(t), γ (t), v0 (t), v0 (t) ,

 γ (t) = C t, α(t), γ (t), v0 (t), v0 (t) .
(18)
where λ, λ̂, v and v̂ are related by (16).
6.1. Remarks
• The cosmological constant does not change
through the dualisation process as it was already
proved for T-dualised homogeneous metrics [14].
That means that the coupling will renormalise in
exactly the same way that in the initial theory: the
one-loop Callan–Symanzik β function is the same
for the initial and dualised SU(2) × U (1) theories.
• As one could expect, the coordinate t which
was a spectator coordinate during the dualisation
process plays a special role: ŵt = 0 and, up to the
Dt log ∆ term, v̂t and vt are equal.
The procedure is the following: we choose some
ansatz for λ̂, v̂ and ŵ and express relation (14). Then,
in this last expression, we replace each occurrence of
α (t), γ (t) and γ (t) by its expression in (15) and
check if (14) holds.
ÅÕÎ
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P.-Y. Casteill / Physics Letters B 543 (2002) 241–248
• The SU(2) symmetries where lost during the
dualisation process, so at the end, there is just
a U (1) symmetry left and therefore the Killing
∂z is unique. Indeed, v̂ and ŵ are defined up to
this Killing vector, which dual 1-form is K =
y 2 t γ (t )
dz. One then has D(i Kj ) = 0 and D[j Ki] +
∆
s
∂[i K Gsj ] = 0.
• One can address the question of the unicity of λ̂,
v̂ and ŵ which satisfy (14). There will be multiple
solutions if one can find some D, V and W such
that
DGij + Dj Vi + ∂[i Wj ] = 0.
On the one hand, ŵ alone is obviously defined
up to a gradient while v̂ and ŵ together are
defined up to the Killing vector K; on the other
hand, equivalence (18) shows that if multiple
solutions exist for λ̂ and v̂ in the dualised metric,
then such ambiguity will appear for the original
metric. We have checked that, in our case of
SU(2) × U (1) metrics, only flat metric leads
to such possibilities.4 Therefore, except for this
trivial original metric and up to the already noticed
freedom in v̂ and ŵ, (16) is the unique solution
of (14).
• The SO(3) dual of the Schwarzschild metric (13)
gives us a nice example of a torsionless quasiEinstein metric with a U (1) as minimal isometry.
7. Conservation of the Kähler property
Bakas and Sfetsos described, for SUSY applications, how the complex structures were changed when
hyper-Kähler metrics were T-dualised [24]. We propose here to show that when one starts with the original metric (8), the dual partner is still Kähler.
If we define
σ̂i = −Gsi d φ̂ s ,
it is possible to write the dual metric of (8) under the
specific shape
1
dt 2 + t σ̂12 + σ̂22 + γ (t)σ̂32 .
ĝ =
γ (t)
4 For flat space (β(t) = γ (t) = 1/α(t) = t), we have λg +
ij
D(i vj ) = 0 with v = −2λ dt, ∀λ ∈ R.
247
One can then check that the 2-form
1
ρ̂ = dt ∧ σ̂3 + t σ̂1 ∧ σ̂2 = Jˆij d φ̂ i ∧ d φ̂ j
2
is a Kähler form with torsion for the dual metric.
Indeed, for the almost complex structure Jˆ , we have

j
ˆ ˆ sj

 Jis J = −δi ,
Jˆ(ij ) = 0,


Di Jˆj k = 0,
where D is the covariant derivative with torsion. One
should notice here that, in the presence of torsion,
the closing condition on the Kähler form is replaced
by
d ρ̂ = (FdH ) ∧ ρ̂.
The torsion potential 2-form H is given by the Eq. (11).
8. Concluding remarks
We have considered all of the four-dimensional
non-homogeneous metrics with an isometry group
SU(2) × U (1). We have shown that the dual partners
are quasi-Einstein (with torsion) iff the original metrics are quasi-Einstein (without torsion). Let us emphasize that this was possible despite the fact that
the explicit form of these metrics are not all known
yet.
In [17], it was proven that, in the minimaldimensional scheme, the dualised SU(2) principal σ model is not two-loop renormalisable although this
property holds for its original model. Here, the oneloop renormalisability remains although the starting
models are not in general two-loop renormalisable.
This is another suggestion that the renormalisability
beyond one loop for the original and dualised models
are not linked. Indeed, it is our ansatz that for the dualised models investigated here, one could still define a
proper theory up to two loops. This could be achieved
by adding some finite deformation to the dualised metric, as it was done in [18] for the SU(2) principal σ model, irrespectively of the two-loop renormalisability
of the original theory.
ÅÕÖ
Ø#># ´ÜÚÙ!A*(Ý.(*(ÛÁ ÙB ÚÙÚ,ÙAÙEÜÚÜÙ&Ý +,%&*(ÝÜ% Ý(EA,AÙ%Ü*Ý
248
P.-Y. Casteill / Physics Letters B 543 (2002) 241–248
Acknowledgements
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(1989) 167, cern-th.5106/88.
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(1994) 155, hep-th/9403155.
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[14] P.-Y. Casteill, G. Valent, Nucl. Phys. B 591 (2000) 491, hepth/0006186.
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Suppl.) 49 (1996) 16, hep-th/9601091.
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I am indebted to G. Valent for suggesting this work
and to G. Bonneau for enlightening discussions and
remarks.
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ÅÕ×
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ÅÕð
Ø#³# ~&*(ÝÜ% σ,AÙ%Ü*Ý Û ÛÜ Û Ù,*ÙÙ Ù!%Ü!
K N W L W7 σ
Nuclear Physics B 607 (2001) 293–304
www.elsevier.com/locate/npe
Dualised σ -models at the two-loop order
Guy Bonneau, Pierre-Yves Casteill
Laboratoire de Physique Théorique et des Hautes Energies, Unité associée au CNRS UMR 7589,
Université Paris 7, 2 Place Jussieu, 75251 Paris Cedex 05, France
Received 2 April 2001; accepted 3 May 2001
Abstract
We adress ourselves the question of the quantum equivalence of non-abelian dualised σ -models
on the simple example of the T-dualised SU(2) σ -model. This theory is classically canonically
equivalent to the standard chiral SU(2) σ -model. It is known that the equivalence also holds at the
first order in perturbations with the same β functions. However, this model has been claimed to be
non-renormalisable at the two-loop order. The aim of the present work is the proof that it is — at
least up to this order — still possible to define a correct quantum theory. Its target space metric being
only modified in a finite manner, all divergences are reabsorbed into coupling and fields (infinite)
renormalisations.  2001 Elsevier Science B.V. All rights reserved.
PACS: 11.10.Gh; 11.10.Kk; 11.10.Lm
Keywords: Sigma models; T-duality; Renormalisation
1. Introduction
The subject of classical versus quantum equivalence of T-dualised σ -models has been
strongly studied in recent years, and extensive reviews covering abelian, non-abelian
dualities and their applications to string theory and statistical physics are available [1–3].
More recent developments on the geometrical aspects of duality can be found in [4].
The interpretation of T-duality as a canonical transformation, for constant backgrounds,
was first given by [5,6]. Its more general formulation [7] was applied to the non-abelian
case in [8,9].
After the settling of the classical equivalence, the most interesting problem was its study
at the quantum level. This was done mostly for dualisations of Lie groups, with emphasis
put on SU(2). For this model the one-loop equivalence was established in [10,11]. This
one-loop quantum equivalence was recently settled for the general class of models built on
GL × GR /GD , with an arbitrary breaking of GR [12]. An interesting intermediary result
E-mail addresses: [email protected] (G. Bonneau), [email protected] (P.-Y. Casteill).
0550-3213/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 1 ) 0 0 2 1 6 - 4
ÅÕÔ
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294
G. Bonneau, P.-Y. Casteill / Nuclear Physics B 607 (2001) 293–304
is an expression for the Ricci tensor of the dualised geometry (with torsion) exhibiting its
dependence with respect to the geometrical quantities of the original model. In the same
work, the two-loop renormalisability problem was tackled and the need for extra (nonminimal) one-loop order finite counter-terms was emphasized. Some years ago, it was
noted that in the minimal dimensional scheme, two-loop renormalisability does not hold
for the SU(2) T-dualised model [13].
The aim of the present work is a more precise analysis of this two-loop (in)equivalence
for the non-abelian T-duality, still on the simple example of the original SU(2) T-dualised
model.
The main remark is that, part of the isometries being somehow lost, the T-dualised
models are not — as they should be if one wants to give an all-order analysis — defined
by a sufficient system of Ward identities. For example, in our simple case there is, a priori
only a linear SU(2) [or O(3)] invariance, and any O(3) invariant action is allowed (let us
remind the reader that in higher-loop corrections to a classical action, all the terms which
are not prohibited by some reason such as power counting, isometries or conservation
laws . . . , would appear). To our present knowledge, the extra constraints coming from the
origin of the model (dualisation of an (SU(2)L × SU(2)R )/SU(2)D chiral model) are not
understood. 1 As it is highly probable that they are linked with the space–time dimension,
it is not surprising that a minimal dimensional renormalisation scheme fails: as is well
known, when the regularization method does not respect all the properties that define the
theory, extra finite counter-terms are needed [15].
The content of this article is the following: in Section 2 we recall the expression of
the classical action of the dualised theory and set the notations. In Section 3, we start
from the corresponding a priori quantum bare action and obtain through h̄ expansion the
possible counter-terms that may be added to the classical action in order to reabsorb the
divergences. Then in Section 4 we give the 2-loop divergences and in Section 5 we discuss
how they match with the candidates in Section 3. Our result is that coupling constant and
field renormalisations (infinite and finite ones) are not sufficient to ensure the two-loop
existence of the T-dualised theory but the metric itself has to be deformed (in a finite way).
Some concluding remarks are offered in Section 6.
2. The classical action
At the classical level and in light-cone co-ordinates, the dual action can be writen
[10,12]:
1
S=
Gij ∂+ φ i ∂− φ j ,
λ
1 In [14] the quantisation of a U(1)-invariant non-linear σ model, the so-called Complex sine-Gordon model,
was performed by imposing as extra constraints its classical property of factorisation and non-production; there it
was shown that definite extra finite one-loop counter-terms are needed to enforce this property to one-loop order
and then they also restore the two-loop renormalisability.
ÅÅÕ
Ø#³# ~&*(ÝÜ% σ,AÙ%Ü*Ý Û ÛÜ Û Ù,*ÙÙ Ù!%Ü!
G. Bonneau, P.-Y. Casteill / Nuclear Physics B 607 (2001) 293–304
295
where gij = G(ij ) is the target space metric and hij = G[ij ] is the torsion potential. The
torsion Tij k is defined by Tij k = 23 ∂[i hj k] . The connections with torsion Γjik and without
torsion γjik respectively write:
1
Γjik = g is (∂j Gks + ∂k Gsj − ∂s Gkj ) = γjik + Tjik ,
2
1
γjik = g is (∂j gsk + ∂k gsj − ∂s gj k ),
2
and the corresponding covariant derivatives are:
Di kj = ∂i kj − Γijs ks = ∇i kj − Tijs vs ,
j
j
Di k j = ∂i k j + Γis k s = ∇i k j + Tis v s .
The Riemann tensor without torsion will be noted Rij,kl whereas we will denote the one
with torsion as R̄ij,kl .
The expression of the dualised target space metric Gij as a function of the original
one is well known and in [12] the various geometrical quantities (Ricci tensor, . . .) were
also related. In the special case considered here, where the original model is the SU(2) ×
SU(2)/SU(2) non-linear σ model, the metric writes:
Gij [φ] =
1
1 + φ2
δij + φ i φ j + ij k φ k ,
(1)
where φ is a SU(2) (real) vector representation and the φ i , i = 1, 2, 3, are the co-ordinates
on the dualised manifold. Then φ 2 is a SO(3) invariant and the symmetry is linearly
realised. Torsion breaks parity, but the model is invariant under the simultaneous change
φ → −φ and ij k → −ij k . Let us emphasize that no other local symmetry exists for that
model.
3. The two-loop order bare action
In order to analyse the two-loop renormalisability of the dualised SU(2) σ -model, we
first examine all the possible ways to reabsorb the divergences through local counter-terms.
As usual, we allow for finite and infinite renormalisations of both fields and coupling. But,
as we shall see later on, this appears as insufficient to reabsorb the various divergences.
Thus, we also allow for a finite deformation of the classical metric and torsion potential
gij + hij = Gij to describe its quantum extension: of course, this à la Friedan [16]
extension of the notion of renormalisability involves a priori an infinite number of new
parameters. Let us emphasize that we shall consider only finite deformations.
Even if by doing so we obviously introduce too many parameters, we first let them all
independent in order to show the announced need for such intrinsic metric deformation.
Let us first write the bare action:
1
S o = o Goij ∂+ φ oi ∂− φ oj ,
(2)
λ
ÅÅÅ
ÜÚ%(@ Ø# !Û('*ÜÝ
296
G. Bonneau, P.-Y. Casteill / Nuclear Physics B 607 (2001) 293–304
where:

h̄λ 2 c
1
h̄λ Λ1
1
Λ2



+
b
+
+ d + ··· ,
=
+
1
+

o
2

λ
λ
2π ε
2π
ε
ε



h̄λ 2 v2 (φ) w2 (φ)
h̄λ v1 (φ)
φo = φ +
+ w1 (φ) +
+ x(φ) + · · · , (3)
+

2π
ε
2π
ε2
ε



2

h̄λ

h̄λ

 Goij = Gij +
Gij + · · · .
Gij + 2π
2π
To express (2) we shall need the Lie derivative L and a “second order” Lie derivative
k
L(2) . Indeed, for any tensor Sij defined on a manifold with co-ordinates φ j , in a change of
k
co-ordinates:
Sij0 (φ 0 )∂+ φ oi ∂− φ oj = Sij (φ)∂+ φ i ∂− φ j ,
and if φ 0 = φ + ηk (note that k is not a vector field on the manifold):
1
Sij (φ) = Sij0 (φ) − η L Sij0 (φ) + η2 L(2) Sij0 (φ) + O η3 .
2
k
k
(4)
We remind the reader that
s
s
s
L(Sij ) = k ∇s Sij + Ssj ∇i k + Sis ∇j k .
(5)
k
One can show that
L(2) (Sij ) = L L(Sij ) − L (Sij ).
k
k k
(6)
k s ∂s k
With ∇i gj k = 0, we rewrite Eqs. (5), (6) for Sij ≡ Gij as:

(Gij ) = 2Dj ki + ∂[i ζj ] ,
ζi = 2k l Gli ,

L


 k (2)
L (Gij ) = 2k s k u Rsi,j u + 2Di k s Dj ks − 4Tius k u Dj k s
k




+ L (Gij ) + ∂[i ζ̂j ] .

(7)
(k s k u γsu )
ζ̂i is some quantity whose computation is useless as, in the same manner as ζi , it gives a
vanishing contribution to the action or, the torsion potential being always defined up to a
gauge transformation, such term can always be put into hij (moreover, in our particular
situation, the O(3) symmetry implies that such ∂[i ζj ] terms vanish). Then, we shall not
write them anymore.
Then, expending (2) with the help of (3), (4), one gets the possible counter-terms at
lowest orders:
h̄λ
:
• 0 order in
2π
1
Gij ∂+ φ i ∂− φ j ;
λ
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G. Bonneau, P.-Y. Casteill / Nuclear Physics B 607 (2001) 293–304
• first order in
1
λ
297
h̄λ
:
2π
Λ1
+ b Gij +
ε
• at second order in
i
j
L (Gij ) + Gij ∂+ φ ∂− φ ;
(8)
v1
ε +w1
h̄λ
:
2π
1 1
Λ1
Λ1
(· · ·) +
Gij + L (Gij ) +
L (Gij ) + b L (Gij )
λ ε2
ε
v1
ε w1
v1
ε
ε
Λ2
1 (2)
+
Gij + L (Gij ) + L (Gij ) + (· · ·) ∂+ φ i ∂− φ j ,
1
ε
w2
2 v1 +w1
ε
ε
ε
(9)
where Q| 1 means that we only take the term in
ε
1
ε
in the expression Q.
As we don’t consider the 3-loop order, in expression (9) we only need the coefficient of
1
h̄2
ε (the double poles ε 2 are not new quantities as they are directly related to first order
simple poles and it has already been proved that the dualised SU(2) σ -model is one-loop
renormalisable [10]).
Using the following identity between Lie derivatives:
LL−LL = L
XY
Y X
with
Z
Z i = Xj ∂j Y i − Y j ∂j Xi ,
the term with the “second order” Lie derivative may be re-expressed:
(2)
ε L (Gij ) = L L (Gij ) + L L (Gij ) −
(Gij )
L
v1
1
k ∂ w +w k ∂ v )
v1 w1
w1 v1
(v
+
w
k
k
1
1
1
1
1
ε
ε
= 2 L L (Gij ) − L (Gij ) .
v1 w1
v1k ∂k w1
So, the O(h̄) term (8) may be rewritten as:
1 1
Λ1 Gij + L(Gij ) + L (Gij ) + bGij + Gij ∂+ φ i ∂− φ j ,
λ ε
v1
w1
and the O(h̄)2 term (9) as:
1 Λ1 L (Gij ) + bGij + Gij + (Λ2 − bΛ1)Gij
λε
w1
+ L L (Gij ) + bGij + Gij +
(Gij ) .
L
v1 w1
(w2 −v1k ∂k w1 )
As a consequence, as expected, any term L (Gij ) + bGij may be reabsorbed into the finite
w1
deformation Gij (and vice-versa) to the expense of a change in the O(h̄)2 parameters:
Gij + L (Gij ) + bGij → Ḡij ⇒ Λ2 → Λ2 − bΛ1 , w2 → w2 − v1k ∂k w1 .
(10)
w1
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G. Bonneau, P.-Y. Casteill / Nuclear Physics B 607 (2001) 293–304
h̄λ 2
Finally, for the term in 1ε ( 2π
) in the bare action, one has the following expression:
1
Λ1 Gij + L(Gij ) + Λ2 Gij + L (Gij ) + Hij (v1 , w1 ) ,
λ
v1
W2
where
W2 = w2 + b v1 + Λ1 w1 + v1s w1u γsu ,
Hij (v1 , w1 ) = v1s w1u Ris,uj + Di v1s Dj w1s − 2Tius v1u Dj w1s + (v1 ↔ w1 ).
(11)
(12)
4. The two-loop order divergences
We use the expression of the covariant divergences given by Hull and Townsend [17], 2
in the background field method and in the minimal dimensional scheme, up to the two-loop
order:

 Div1 = − h̄ Ric ,
ij
ij
2πε (13)
2

h̄ λ
1
klm R
k
lmn R
Div2ij = − 8π
kij l .
klmj − 2 Rlmkj + 2Tmn T
2 ε Ri
In order to ensure the renormalizability of the theory, these divergences should match
with the candidate counter-terms given by (8) and (11):

h̄ 

Λ1 Gij + L(Gij ) ,
 CTij1 =

2πε
v1
(14)
2

2 = h̄ λ Λ G + (G ) + Λ G +

CT
(G
)
+
H
(v
,
w
)
.

L ij
2 ij
ij 1
1
 ij 4π 2 ε 1 ij vL ij
1
W2
It has been previously proven [12] that the dualised metric is quasi-Einstein as soon as the
original metric is Einstein. In our special case, we get:
Ric ij = ΛGij + 2Dj vi ,
1
Λ = Λ1 = ,
2
v = v1 =
1 1 − φ2
φ.
2 1 + φ2
(15)
The addition to the effective action of a h̄ finite deformation of the metric and of some
finite renormalisations for the coupling and fields (non-minimal scheme) modifies the h̄2
divergences. The additional term is easily obtained as
h̄
h̄λ −
Ricij Gkl +
L (Gkl ) + bGkl + Gkl − Ricij (Gkl )
2πε
2π w1
≡−
h̄2 λ
2ij + O h̄3 .
4π 2 ε
Here also, only the combination Gij + L (Gij ) + bGij appears. Then, we could decide
w1
to reabsorb bGij and L (Gij ) into Gij , but, as announced at the beginning of Section 3, in
w1
order to see if they would be sufficient by themselves, we keep them apart in a first step.
2 We checked for our example that the two other calculations in [18,19] give the same result.
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G. Bonneau, P.-Y. Casteill / Nuclear Physics B 607 (2001) 293–304
Finally, the dualised SU2 σ -model will be renormalisable at two loops if and only if we
can find {Gij [φ], b, w1 [φ]; Λ2 , W2 [φ]} such that:
h̄2 λ
2ij
4π 2 ε
h̄2 λ
+
Λ1 Gij + L(Gij ) + Λ2 Gij + L (Gij ) + Hij (v1 , w1 ) = 0.
4π 2 ε
v1
W2
Div2ij −
(16)
5. Results
According to the linearly realised symmetry of the T-dualised SU(2) σ -model, the finite
deformation of the metric Gij and the vectors w1 (φ) and W2 (φ) respectively write:
Gij = α(τ )δij + β(τ )φ i φ j + ij k γ (τ )φ k ,
w1 = w1 (τ )φ,
W2 = W2 (τ )φ,
where τ = φ 2 . Moreover, the symmetry also implies that terms of the form ∂[i kj ] or of the
t are equal to zero. It is then possible to re-express (16) as a set of three linear
form k s K u γsu
differential equations :
W2 (τ ) +
=
(1 + τ )Λ2 45 + 68τ − 18τ 2 − 12τ 3 − 3τ 4
1−τ
+
w1 (τ )
−
3
2
(1 + τ )2
16(1 + τ )
3 + 10τ + 5τ 2 + 2τ 3
4 + 5τ + 6τ 2 + τ 3
α(τ ) +
β(τ )
4(1 + τ )
4(1 + τ )
3(1 + τ )(3 + τ )
4 + 11τ + 5τ 2 − τ 3 τ
γ (τ ) −
α (τ ) + β (τ )
2
2
2
2 − τ (1 + τ )(3 + τ )γ (τ ) − τ (1 + τ ) α (τ ),
−
3Λ2 −
3(−5 + 60τ
+ 10τ 2
+ 12τ 3
(17)
+ 3τ 4 )
8(1 + τ )4
(7 + 10τ )
(12 + 5τ )
=
α(τ ) +
β(τ ) − 3(11 + 5τ )γ (τ )
2
2
+ (−17 − 22τ + 9τ 2 )α (τ ) + (5 + 4τ + τ 2 )β (τ ) − 2(5 + 2τ )(3 + 5τ )γ (τ )
+ 2(−5 − 19τ − 12τ 2 + τ 3 )α (τ ) + 2τβ (τ ) − 4τ (1 + τ )(3 + τ )γ (τ )
− 4τ (1 + τ )2 α (3) (τ ),
Λ2 +
3(1 − τ )(13 + 6τ
(18)
+ τ 2)
8(1 + τ )3
(6 + τ )
(5 + 2τ )
α(τ ) +
β(τ ) − (17 + 3τ )γ (τ )
=
2
2
+ (−7 + τ )(3 + τ )α (τ ) − 2(−5 + 6τ + τ 2 )γ (τ )
− 2τ (3 + τ )α (τ ) + 4τ γ (τ ).
(19)
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G. Bonneau, P.-Y. Casteill / Nuclear Physics B 607 (2001) 293–304
The need for a true deformation Gij immediately appears: setting both α(τ ), β(τ ) and
γ (τ ) to zero, Eqs. (18) and (19) cannot be satisfied, even if 3 we allowed for some finite
renormalisations of the coupling (b) and field (w1 (φ)), both hidden into the vector W2 (φ)
(see Eq. (12)). Then, as first proven in [13], we have checked that:
In a purely dimensional scheme (even with non minimal subtractions), the dualised
SU(2) σ model is not renormalisable at the two-loop order.
So, from the discussion in the previous sections, and without restricting the generality
of our analysis, one can take b and w1 (φ) as vanishing quantities.
Remarks.
• As Λ2 is not a function, but a constant, differentiating Eqs. (18) and (19) will relate
α(τ ), β(τ ) and γ (τ ). Then, as soon as Gij , the finite one loop renormalisation, has
been definitely set, Eq. (17) will give the infinite two-loop renormalisations W2 (φ)
and Λ2 .
• From the previous discussions, we know that Gij will be fixed up to some b̌ Gij +
L (Gij ); it is then natural to use this freedom, for example to reabsorb α(τ ), and to
W̌ (φ)
redefine Gij such that:
Gij = b̌ Gij + L (φ)(Gij ) + Ǧij ,
W̌
W̌ (φ) = W̌ (τ )φ
with
W̌ (τ ) =
b̌(1 + τ )
(1 + τ )2
α(τ ) −
⇒ Ǧij = β̌(τ )φ i φ j + ij k γ̌ (τ )φ k
2
2
with
(3 + τ )W̌
b̌
−
.
1+τ
(1 + τ )2
(20)
We know that, when expressed as functions of β̌(τ ) and γ̌ (τ ), Eqs. (17), (18), (19)
remain unchanged, up to the substitutions discussed in Section 3 (Eq. (10)):
β̌ = β −
b̌
2(2 + τ )W̌
−
− 4W̌ 1+τ
(1 + τ )2
and γ̌ = γ −
b̌
Λ2 → Λ̌2 = Λ2 + ,
2
1
1−τ
+ W̌ (τ )
2(1 + τ ) 2
1−τ +
W̌ (τ ) + 2τ W̌ (τ ) .
2(1 + τ )
W2 (φ) → W̌2 (τ ) = W2 (τ ) + b̌
(21)
3 One notices also that the parameters b and w do not appear in (18) and (19). So, the existence of some
1
solution to this set of differential equations is independent of the finite renormalisations of both coupling and
fields, as is usual in perturbation theory. This freedom corresponds to a change of renormalisation scheme. This
absence is only true if we take the very vector v1 (15) that reabsorbs the divergences at the one-loop order:
otherwise, w1 (φ) would appear in (18) and (19). This is a check of a correct renormalisation at the one-loop
order.
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G. Bonneau, P.-Y. Casteill / Nuclear Physics B 607 (2001) 293–304
301
Eqs. (18) and (19) give β̌(τ ) as a function of γ̌ (τ ) which itself satisfies a nonhomogeneous linear fourth order differential equation:
b̌ + 2Λ2 3(1 − τ )(13 + 6τ + τ 2 ) 2(17 + 3τ )
=
γ̌ (τ )
+
6+τ
6+τ
4(6 + τ )(1 + τ )3
4(5 − 6τ − τ 2 ) 8τ (22a)
−
γ̌ (τ ) −
γ̌ (τ ),
6+τ
6+τ
(6 − τ )(7 + τ ) (3)
1260 − 276τ − 91τ 2 + 3τ 3 + τ 4 γ̌ (4) (τ ) +
γ̌ (τ ) +
γ̌ (τ )
τ (6 + τ )
4τ 2 (6 + τ )2
−120 + 254τ + 57τ 2 + 3τ 3 138 + 25τ + τ 2
b̌ + 2Λ2
+
γ̌
(τ
)
−
γ̌ (τ ) −
2
8τ 2 (6 + τ )2
8τ 2 (6 + τ )2
β̌(τ ) −
=−
3(6402 − 8681τ − 5856τ 2 − 22τ 3 + 390τ 4 + 39τ 5 )
.
64τ 2 (1 + τ )5 (6 + τ )2
Note that under the change
b̌ + 2Λ2
,
Γ (τ ) = γ̌ (τ ) −
2
B(τ ) = β̌(τ ) − 3(b̌ + 2Λ2 ) ,
(22b)
(23)
the parameter b̌ and the constant Λ2 disappear from the set (22). Then, Λ2 being an
unknown constant, the general solution of the differential equation (22b) will be
γ̌ (τ ) = Γ (τ ) + c,
where c is an arbitrary constant,
and the two-loop coupling constant renormalisation Λ2 will be:
b̌
Λ2 = c − .
2
The model will be renormalisable up to two loops iff equation (22b), where γ̌ (τ ) has
been replaced by Γ (τ ) according to (23), has a solution which is analytic near τ =
0. In order to reach such a conclusion, we use the method of Frobenius for linear
differential equations [20]. τ = 0 is a regular singularity (notice that we are only interested
in τ 0). The indicial equation of the linear differential equation (22b) around the
singular point 0 has four different solutions: ν = − 32 , − 12 , 0, 1. For each one, we can
n
find convergent series τ ν ∞
n=0 cn τ that are independent solutions of the homogeneous
equation associated to (22b). We give here the first terms of such series (it happens that for
ν = − 32 we have an exact solution):
√
35 2
τ
1
1
1
11
γ̌− 3 (τ ) = 3 + √ −
,
γ̌− 1 (τ ) = √ 1 − τ +
τ + ··· ,
2
2
20
6
108
20
τ
τ
τ2
1 2
23 2
1
γ̌0 (τ ) = 1 +
(24)
τ + ···,
τ + ··· .
γ̌1 (τ ) = τ 1 + τ −
840
42
324
Then, we use the method of variation of parameters to find λ− 3 (τ ), λ− 1 (τ ), λ0 (τ ) and
2
2
λ1 (τ ) such that
Γ (τ ) = λ− 3 (τ )γ̌− 3 (τ ) + λ− 1 (τ )γ̌− 1 (τ ) + λ0 (τ )γ̌0 (τ ) + λ1 (τ )γ̌1 (τ )
2
2
2
2
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G. Bonneau, P.-Y. Casteill / Nuclear Physics B 607 (2001) 293–304
is the general solution of the inhomogeneous equation (22b) where γ̌ (τ ) has been replaced
by Γ (τ ) according to (23).
The first terms in the expansion of these functions are:
1067 13691
−
τ + ··· ,
1680
3780
5
1067 2543509
λ− 1 (τ ) = λo− 1 + τ 2 −
+
τ + ··· ,
2
240
100800
2
1067 2 9805 3
λ0 (τ ) = λo0 +
τ −
τ + ···,
192
288
27887 2
1067
τ+
τ + ···.
λ1 (τ ) = λo1 −
480
5760
7
λ− 3 (τ ) = λo− 3 + τ 2
2
2
(25)
The analyticity requirement near τ = 0 enforces the choice λo
− 32
= λo
− 12
= 0; γ̌ (τ ) is
then expressed as a convergent series in τ , and the same will be true for β̌(τ ). The final
expression for the deformation Gij depends on 3 constants [c, λo0 and λo1 ] and an arbitrary
function [W̌ (τ )] and is given by the three functions:
α(τ ) =
b̌
2W̌
+
,
1+τ
(1 + τ )2
β(τ ) = 6c +
2(2 + τ )W̌
b̌
+
+ 4W̌ 1+τ
(1 + τ )2
4(5 − 6τ − τ 2 ) 3(1 − τ )(13 + 6τ + τ 2 ) 2(17 + 3τ )
Γ
(τ
)
−
Γ (τ )
+
4(6 + τ )(1 + τ )3
6+τ
6+τ
8τ
Γ (τ ),
−
6+τ
+
γ (τ ) = c +
(3 + τ )W̌
b̌
+
+ Γ (τ ).
1+τ
(1 + τ )2
(26)
We now use the up to now free parameter b̌ to reabsorb the parameter c. Let us define
b̄ = b̌ − 2c,
W (τ ) = W̌ (τ ) + c(1 + τ ),
we get
Gij = Gij + b̄Gij + L Gij
W
with W = W (τ )φ and
Gij = Gij Eq. (26) for c=b̌=W̌ (τ )≡0
.
The dualised SU(2) σ -model is therefore renormalisable at the two-loop order if and
only if we add a finite h̄ deformation of the classical metric, depending on two new
parameters λo0 and λo1 .
ÅÅð
Ø#³# ~&*(ÝÜ% σ,AÙ%Ü*Ý Û ÛÜ Û Ù,*ÙÙ Ù!%Ü!
G. Bonneau, P.-Y. Casteill / Nuclear Physics B 607 (2001) 293–304
303
6. Concluding remarks
We have been able to exhibit some set of counter-terms that ensures the two-loop
renormalisability of the T-dualised chiral non-linear σ model. The one-loop effective
metric is defined up to two constants (λo0 and λo1 ), and some finite arbitrary field
and coupling renormalisations. As is well known (e.g., in [21]), the two-loop Callan–
Symanzik β function (related to Λ2 ) 4 depends on these finite counterterms.
We emphasize that, contrarily to D. Friedan’s approach to σ models quantisation,
where the classical metric receives infinite perturbative deformations, our candidate for the
deformation of the classical metric is a finite one, depending on only two parameters (plus
the usual infinite, and finite, renormalisations of the fields and of the coupling constant):
our ansatz is that a proper understanding of the dualisation process will precisely offer
the extra constraints that uniquely define the quantum extension of the classical theory,
order by order in perturbation theory, in the same spirit as Ward identities determine what
otherwise would appear as new parameters (see also footnote 1).
Note added in proof
For completeness, let us mention that for abelian T-duality similar works were achieved
in Refs. [22,23].
Acknowledgements
It is a pleasure to thank Galliano Valent whose interest in that subject was really
stimulating for us.
References
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4 The two loops quantities Λ and W are fixed as:
2
2
Λ2 =
b̄
,
2
W2 obtained through (21).
Notice that the normalisation condition b̄ = 0 (no h̄ extra finite coupling constant renormalisation) enforces
Λ2 = 0.
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ÜÚ%(@ Ø# !Û('*ÜÝ
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ÅÆÕ
Ø#³# ~&*(ÝÜ% σ,AÙ%Ü*Ý Û ÛÜ Û Ù,*ÙÙ Ù!%Ü!
ÅÆÅ
V
ÜÚ%(@ Ø# !Û('*ÜÝ
N 6 8 L NX NXWå5
31 May 2001
Physics Letters B 508 (2001) 354–364
www.elsevier.nl/locate/npe
Quaternionic extension of the double Taub-NUT metric
Pierre-Yves Casteill a , Evgeny Ivanov a,b , Galliano Valent a
a Laboratoire de Physique Théorique et des Hautes Energies, Unité associée au CNRS URA 280, Université Paris 7,
2 Place Jussieu, 75251 Paris cedex 05, France
b Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, 141 980 Moscow region, Russia
Received 9 April 2001; accepted 29 April 2001
Editor: L. Alvarez-Gaumé
Abstract
Starting from the generic harmonic superspace action of the quaternion-Kähler sigma models and using the quotient approach
we present, in an explicit form, a quaternion-Kähler extension of the double Taub-NUT metric. It possesses U (1) × U (1)
isometry and supplies a new example of non-homogeneous Einstein metric with self-dual Weyl tensor.  2001 Published by
Elsevier Science B.V.
1. Introduction
In view of the distinguished role of hyper-Kähler (HK) and quaternion-Kähler (QK) manifolds in string theory
(see, e.g., [1–3]), it is important to know the explicit form of the corresponding metrics. One of the approaches to
this problem proceeds from the generic actions of bosonic nonlinear sigma models with the HK or QK targets.
A generic action for the bosonic QK sigma models was constructed in [4], based upon the well-known oneto-one correspondence [5] between the QK manifolds and local N = 2, d = 4 supersymmetry. This relationship
was made manifest in [6,7], where the most general off-shell action for the hypermultiplet N = 2 sigma models
coupled to N = 2 supergravity was constructed in the framework of N = 2 harmonic superspace (HSS) [8]. The
generic QK sigma model bosonic action was derived in [4] by discarding the fermionic fields and part of the
bosonic ones in the general HSS sigma model action. The action of physical bosons parametrizing the target QK
manifold arises, like in the HK case [9], after elimination of infinite sets of auxiliary fields present in the off-shell
hypermultiplet superfields. This amounts to solving some differential equations on the internal sphere S 2 of the
SU(2) harmonic variables. It is a difficult problem in general to solve such equations. As was shown in [4], in the
case of metrics with isometries the computations can be greatly simplified by using the HSS version of the QK
quotient construction [10,11]. An attractive feature of the HSS quotient is that the isometries of the corresponding
metric come out as manifest internal symmetries of the HSS sigma model action.
In [4], using these techniques, we explicitly constructed QK extensions of the Taub-NUT and Eguchi–Hanson
(EH) HK metrics [12]. In this note we apply the HSS quotient approach to construct a QK extension of the
E-mail addresses: [email protected] (P.-Y. Casteill), [email protected], [email protected] (E. Ivanov),
[email protected] (G. Valent).
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0370-2693/01/$ – see front matter  2001 Published by Elsevier Science B.V.
PII: S 0 3 7 0 - 2 6 9 3 ( 0 1 ) 0 0 5 3 7 - 8
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355
4-dimensional “double Taub-NUT” HK metric. The latter was derived from the HSS approach in [13] by directly
solving the corresponding harmonic differential equations. It turns out that the HSS quotient allows one to
reproduce the same answer much easier, and it remarkably works in the QK case as well. We gauge two commuting
U (1) symmetries of the system of three “free” hypermultiplets and, after solving two algebraic constraints and
fully fixing gauges, are left with a 4-dimensional QK metric having two U (1) isometries and going onto the double
Taub-NUT in the HK limit. It is a new explicit example of non-homogeneous QK metrics. Based on the results of
Przanowski [14], Tod [15] and Flaherty [16], this metric gives also a new explicit solution of the coupled Einstein–
Maxwell system with self-dual Weyl tensor.
2. The gauged HSS action of the QK double Taub-NUT
Details of the general construction can be found in [4]. Here we apply the HSS quotient approach to explicitly
construct a sigma model giving rise to a QK generalization of the “double Taub-NUT” HK metric. The latter
belongs to the class of two-center ALF metrics with the U (1) × U (1) isometry (one U (1) is triholomorphic) and
was treated in the HSS approach in [13].
We start with the action of three hypermultiplet superfields,
Q+a
A (ζ ),
g +r (ζ ),
a = 1, 2,
r = 1, 2,
A = 1, 2,
(1)
possessing no any self-interaction. So, by reasoning of [4,17], this action corresponds to the “flat” QK manifold
HH 3 ∼ Sp(1, 3)/Sp(1)×Sp(3). In (1), the indices a and r are the doublet indices of two Pauli–Gürsey-type SU(2) s
+r
realized on Q+a
A and g , the index A is an extra SO(2) index. These superfields are given on the harmonic analytic
N = 2 superspace
(ζ ) = x m , θ +µ , θ̄ +µ̇ , u+i , u−k ,
(2)
the coordinates u+i , u−k , u+i u−
i = 1, i, k = 1, 2, being the SU(2)/U (1) harmonic variables, and they satisfy the
pseudo-reality conditions
(a)
+ ab +
Q+a
A ≡ QaA = Qb A ,
(b)
+
g +r ≡ gr = rs gs+ ,
(3)
where ab bc = δca , 12 = −1. The generalized conjugation ∼ is the product of the ordinary complex conjugation
and a Weyl reflection of the sphere S 2 ∼ SU(2)/U (1) parametrized by u±i . In the QK sigma model action below
we shall need only the bosonic components in the θ -expansion of the above superfields:
+ m + −a
2 2 (−3a)
+a
Bm A (x, u) + θ + θ̄ + GA
Q+a
(x, u),
A (ζ ) = FA (x, u) + i θ σ θ̄
+ m + −r
+ 2 + 2 (−3r)
+r
+r
θ̄
g
(x, u)
g (ζ ) = g0 (x, u) + i θ σ θ̄ gm (x, u) + θ
(4)
(possible terms ∼ (θ + )2 and ∼ (θ̄ + )2 can be shown not to contribute to the final action). The component fields still
have a general harmonic expansion off shell. The physical bosonic components FAai (x), g ri (x) are defined as the
first components in the harmonic expansions of FA+a (x, u) and g0+r (x, u)
FA+a (x, u) = FAai (x)u+
i + ···,
(FAai (x)) = ab ik FAbk (x),
g0+r (x, u) = g ri (x)u+
i + ···,
(g ri (x)) = rs ik g sk (x).
(5)
The selection of two commuting U (1) symmetries to be gauged and the form of the final gauge-invariant
HSS action are uniquely determined by the natural requirement that the resulting action have two different
limits corresponding to the earlier considered HSS quotient actions of the QK extensions of Taub-NUT and EH
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P.-Y. Casteill et al. / Physics Letters B 508 (2001) 354–364
metrics [4]. The full action SdTN has the following form
1
1
−
d 4 x D(x) + V m(ij ) (x)Vm(ij ) (x) ,
SdTN =
(6)
dζ (−4)L+4
dTN
2
2
2κ
+ ++ +a
L+4
q
dTN = − qa D
2 +
2 −
++ +a
2 (ij ) + +
+ κ u · q + Q+
QA + gr+ D++ g +r + W ++ Q+a
gi gj + c(ij ) vi+ vj+
aA D
A QaB AB − κ c
+ (7)
+ V ++ 2 v + · g + − a (ab)Q+
aA QbA .
Here, dζ (−4) = d 4 x d 2 θ + d 2 θ̄ + du is the measure of integration over (2), (a · b) ≡ ai bi , the covariant harmonic
derivative D++ is defined by
2 2 −
D++ = D ++ + θ + θ̄ + D(x) ∂ −− + 6V m (ij ) (x)u−
(8)
i uj ∂m ,
with D ++ = ∂ ++ − 2iθ + σ m θ̄ + ∂m , ∂ ±± = u±i /∂u∓i , the non-propagating fields D, V m(ij ) are inherited from the
N = 2 supergravity Weyl multiplet, κ 2 is the Einstein constant (or, from the geometric point of view, the parameter
of contraction to the HK case) and the new harmonic v +i is defined by
v +a =
(u+ · q + )
q +a
= u+a − − + u−a .
+
·q )
(u · q )
(u−
The superfield q +a = f +a (x, u) + · · · = f ai (x)u+
i + · · · is an extra compensating hypermultiplet, with the θ
expansion and reality properties entirely analogous to (3), (4). Like in [4], we fully fix the local SU(2)c symmetry
of (6) (which is present in any QK sigma model action) by the gauge condition
fai (x) = δai ω(x).
(9)
The objects defined so far are necessary ingredients of the generic QK sigma model action. The specificity of
the given case is revealed in the particular form of L+4 in (7). It includes two analytic gauge abelian superfields
V ++ (ζ ) and W ++ (ζ ) and two sets of SU(2) breaking parameters c(ij ) and a (ab) satisfying the pseudo-reality
condition
c(ij ) = ik j l c(kl)
(10)
(and the same for a (ab) ). The Lagrangian (7) can be checked to be invariant under the following two commuting
gauge U (1) transformations, with the parameters ε(ζ ) and ϕ(ζ ):
+a
2 +− +a
δg +r = εκ 2 c(rn) gn+ − c+− g +r ,
δQ+a
QA ,
A = ε AB QB − κ c
δq +a = εκ 2 c(ab)qb+ ,
(11)
δW ++ = D++ ε c+− ≡ c(ik) vi+ u−
k ,
+a +r
(ab) +
+a
+r
2 −
+ +r
2 −
+
δQA = ϕ a Qb A − κ u · g QA ,
δg = ϕ v − κ u · g g ,
+a
2 −
+ +a
++
++
δq = ϕκ u · q g ,
(12)
δV
= D ϕ.
This gauge freedom will be fully fixed at the end. The only surviving global symmetries are two commuting
+a
U (1). One of them comes from the Pauli–Gürsey SU(2) acting on QA
and broken by the constant triplet a (bc) .
Another U (1) is the result of breaking of the SU(2) which uniformly rotates the doublet indices of harmonics
and those of q +a and g +r . It does not commute with supersymmetry and forms the diagonal subgroup in the
product of three independent SU(2)s realized on these quantities in the “free” case; this product gets broken
down to the diagonal SU(2) and further to U (1) due to the presence of explicit harmonics and constants c(ik)
in the interaction terms in (7). These two U (1) symmetries will be isometries of the final QK metric, the first one
becoming triholomorphic in the HK limit. The fields D(x) and Vm(ik) (x) are inert under any isometry (modulo some
rotations in the indices i, j ), and so are D++ and the D, V part of (6).
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357
It can be shown that the action (6), (7) is a generalization of both the HSS quotient actions describing the QK
extensions of the EH and Taub-NUT sigma models: putting g +r = a (ab) = 0 yields the EH action as it was given in
+a
(ik) = 0 yields the Taub-NUT action [4]. Also, fixing the gauge with respect to the
[4,17], putting Q+a
A=2 (QA=1 ) = c
λ transformations by the condition (u− · g + ) = 0, varying with respect to the non-propagating superfield V ++ and
eliminating altogether (v + · g + ) by the resulting algebraic constraint, we arrive at the form of the action which in
the HK limit κ 2 → 0 exactly coincides with the HSS action describing the “double Taub-NUT” manifold [13,18].
Thus (6), (7) is the natural QK generalization of the action of [13,18] and therefore the relevant metric is expected
to be a QK generalization of the double Taub-NUT HK metric.
3. Towards the target metric
We are going to profit from the opportunity to choose a WZ gauge for W ++ and V ++ , in which harmonic
differential equations for f +a (x, u), FA+b (x, u) and g +r (x, u) are drastically simplified.
In this gauge W ++ and V ++ have the following short expansion
2 2
−
W ++ = iθ + σ m θ̄ + Wm (x) + θ + θ̄ + P (ik) (x)u−
i uk ,
2
2
−
V ++ = iθ + σ m θ̄ + Vm (x) + θ + θ̄ + T (ik) (x)u−
(13)
i uk
(once again, possible terms proportional to (θ + )2 and (θ̄ + )2 can be omitted). The hypermultiplet superfields have
the same expansions as in (4). At the intermediate step it is convenient to redefine these superfields as follows
+a +r +a +r , ĝ .
QA , g
(14)
= κ(u− · q + ) Q
A
Due to the structure of the WZ-gauge superfields (13), the highest components in the θ expansions of the
redefined HM superfields appear only in the kinetic part of (7). This results in the linear harmonic equations
+b (x, u), ĝ +r (x, u):
for f +a (x, u), F
A
∂ ++ f +a = 0 ⇒ f +a = u+a ω(x),
∂ ++ ĝ +r = 0 ⇒ ĝ +r = ĝ ri (x)u+
i ,
+a = F
Aai (x)u+ ,
a+a = 0 ⇒ F
∂ ++ F
A
i
(15)
where we have simultaneously fixed the gauge (9).
Next steps are technical and quite similar to those explained in detail in [4] on the examples of the QK extensions
of the Taub-NUT and EH metrics. One substitutes the solution (15) back into the action (with the θ and u integrals
performed), varies with respect to the rest of non-propagating fields and also substitutes the resulting relations back
into the action. At the final stages it proves appropriate to redefine the basic fields once again
2 ri
Aai = 1 FAai ,
g
ĝ ri =
F
(16)
κω
κω
and to fully fix the residual gauge freedom of the WZ gauge for the ϕ transformations (with the singlet gauge
parameter ϕ(x)), so as to gauge away the singlet part of g ri (x):
g ri (x) = g (ri) (x)
(17)
(the residual SO(2) gauge freedom, with the parameter ε(x), will be kept for the moment). In particular, in terms of
(ij )
the thus defined fields we have the following expressions for the fields ω and Vm which are obtained by varying
(ij )
the full action (6) with respect to D and Vm :
κω =
1
1 − λ2 g 2 − 2λF 2
,
(ij )
Vm
1
j)
j)
= −16λ2 ω2 FAa(i ∂m FaA + g r(i ∂m gr ,
4
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(18)
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P.-Y. Casteill et al. / Physics Letters B 508 (2001) 354–364
where
F 2 ≡ FAai FaiA ,
g 2 ≡ g ri gri ,
λ≡
κ2
.
4
(19)
The final form of the sigma model Lagrangian in terms of the fields FAai (x) and g (rk) (x) is as follows (we
replaced altogether ‘∂m ’ by ‘d’, thus passing to the distance in the target QK space instead of its x-space pullback)
Y
1
2Y
+λ g
D X+Z+
+ 2T
(20)
4
8
D2
with
1
1
λ 2
g − 2λF 2 ,
X = dFai A dFAai ,
Y = dgij dg ij ,
2
2
2
1
Z=
γ (J · K) − α(J · J ) − β(K · K) ,
4α β − γ 2
1
aj
T = Fai B dFB Fai A dFjaA + gir dg rj .
2
D=1−
Here
1
J = a ab Fai A dFbi A ,
2
(21)
1
λ
K = − AB FAai dFai B − cij g is dg sj ,
2
2
(22)
and
1 F2
λ2 2 2
1
â 2 2 λ 2
2
− λ ĉ +
ĉ g ,
F − g ,
α=
β=
1+
2 4
2
4
4
2
1 ab i
γ = a Fa A Fbi B AB − λ(c · g),
4
where
ĉ2 ≡ cik cik ,
(23)
â 2 = a ab aab .
(24)
On top of this, there are two algebraic constraints on the involved fields
j)
FAa(i Fa B AB − λ g (li) g (rj ) c(lr) + c(ij ) = 0,
(25)
j
g ij = a ab Fai B Fb B ,
(26)
P (ik) (x)
T (ik) (x)
and
in the WZ
which come out by varying the action with respect to the auxiliary fields
gauge (13). Keeping in mind these 6 constraints and one residual gauge (SO(2)) invariance, we are left with just
four independent bosonic target coordinates as compared with 11 such coordinates in (20). The problem is now
to explicitly solve (25), (26). But before turning to this issue, let us notice that the sought metric includes three
parameters. These are the Einstein constant, related to λ, and two breaking parameters: the triplet c(ij ) , which
breaks the SU(2)SUSY to U (1), and the triplet a (ab), which breaks the Pauli–Gürsey SU(2) to U (1). The final
isometry group is therefore U (1) × U (1). For convenience we choose the following frame with respect to the
broken SU(2) groups
c12 = ic,
c11 = c22 = 0,
ĉ2 = 2c2 ,
â 2 = 2a 2.
a 12 = ia,
a 11 = a 22 = 0,
with real parameters a and c, and we shift λ → λ/a 2 . Hereafter we shall use this frame, in which, in particular, the
squares (24) become
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359
4. Solving the constraints
We need to find true coordinates to compute the metric. This step is non-trivial, due to the fact that (25)
becomes quartic after substitution of (26). Instead of solving this quartic equation, it proves more fruitful to take
as independent coordinates just the components of the triplet g (ri)
g 12 = g 21 ≡ iah,
h = h,
g 11 ≡ g,
g 22 = g,
and one angular variable from FAai . Then, relabelling the components of the latter fields as follows

1
1
a=1 i=2
a=1 i=1

 FA=1
FA=1
= (F + K),
= (P + V),


2
2

1
1
a=1 i=2
a=1 i=1
FA=2
FA=2
= (F − K),
= (P − V),


2i
2i


 a=2 i=1
FA
FAa=2 i=2 = FAa=1 i=1 ,
= −FAa=1 i=2 ,
we substitute this into (25), (26), and find the following general solution (it amounts to solving a quadratic equation
and we choose the solution which is regular in the limit g = ḡ = h = 0)
P = −iM ei(φ+α/ρ− +µρ+ ) ,
K = iS ei(φ−α/ρ− −µρ+ ) ,
F = R ei(φ+µρ− ) ,
V = L ei(φ−µρ− ) ,
ρ± = 1 ± 4
λc
a2
(27)
and
g = at ei(α/ρ− +8λc/a
2 µ)
(28)
.
The various functions involved are
1 1 L=
R=
∆− + B− ,
∆+ + B+ ,
2
2
1
1 M=
S=
∆+ − B+ ,
∆− − B− ,
2
2
with
A± = 1 ± 2λc h,
B± = c 1 + λr 2 ± h A∓ ,
2 + t 2 A2 ,
∆± = B±
∓
r 2 = h2 + t 2 ,
g ḡ = a 2 t 2 .
The true coordinates are (φ, α, h, t). An extra angle µ parametrizes the local SO(2) transformations (they act as
shifts of µ by the parameter ε(x)). In view of the gauge invariance of (20), the final form of the metric should not
depend on µ and we can choose the latter at will. For instance, we can change the precise dependence of phases
in (27), (28) on φ and α. In what follows we shall stick just to the above parametrization.
5. The resulting metric
To get the full metric is fairly involved and Mathematica was intensively used! The final result is
2
1 P
t2 A
Q
2
2
2 2
2
1 + λr dα .
dα + 2 dh + dt +
g=
dφ +
4D2 A
4P
D
P
It depends on 4 functions
D,
A,
P,
Q,
ÅÆ×
(29)
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P.-Y. Casteill et al. / Physics Letters B 508 (2001) 354–364
given by
λc2 4λt 2 − (1 + λ r 2 )2
1
a2 1 1
1
1
− λch √
+ 2
+√
−√
,
+ 1 − 4λc2 1 − λr 2 √
√ √
4
8
a
∆+
∆−
∆+
∆−
∆+ ∆−
2λc h + c(1 − λr 2 ) h − c(1 − λr 2 ) 2
2 2
P = 1 + λr
−
√
√
1− 2
a
∆+
∆−
2
2
2
2
2
4λ c t 1 − λr − 4λch 1 − λr + 4λch 2
+
,
−
√
√
a4
∆+
∆−
2 h + c(1 − λr 2 ) h − c(1 − λr 2 )
Q = − 1 + λr 2
+
√
√
∆+
∆−
2 + 4λch 2 − 4λch
1
−
λr
1
−
λr
.
+ 4λct 2
−
√
√
∆+
∆−
The overall conformal factor is
λ D = 1 − λr 2 − 2 2
∆+ + ∆− .
a
To simplify matter we first rescale c → c/2. The relations
A=
∆± = (1 + λc2 )t 2 + (h ± c/2(1 − λr 2 ))2
suggest the following change of coordinates
2h
2t
,
H=
,
ρ = T 2 + H 2,
2
2
1 − λr
1 − λr
which has the virtue of reducing the quartic non-linearities according to
T=
(30)
(1 − λr 2 )2
δ± , δ± = 1 + λc2 T 2 + (H ± c)2 .
4
Further, to get rid of the square roots we use spheroidal coordinates (s, x) defined by
1 + λc2 T = (s 2 − c2 )(1 − x 2 ),
H = sx, s c, x ∈ [−1, +1].
∆± =
For convenience reasons we scale the angles φ and α according to
φ
⇒ φ,
√
1 + λc2
α
⇒ α,
√
1 + λc2
and to have a smooth limit for a → 0 we come back to the original λ, λ → λa 2 .
Putting these changes together, we get the final form of the metric
2
2
P
Q
A 2
dφ +
dα +
s − c2 1 − x 2 1 + λa 2 c2 x 2 (dα)2
4d g = 1 + λa 2 s 2
A
4P
P
2
ds
dx 2
+A
+
,
(s 2 − c2 )(1 + λa 2 s 2 ) (1 − x 2 )(1 + λa 2 c2 x 2 )
with

2
2 2
2

 d = 1 − 2λs, Q = −2 1 + λa c s − c x,
4A = 2 + a 2 s s − 2λc2 − a 2 c2 d 2 x 2 ,


P = c2 1 − x 2 1 + λa 2 c2 x 2 d 2 + s 2 − c2 1 + λa 2 c2 x 2 − 4λ2 c2 1 − x 2 .
The isometry group U (1) × U (1) acts as translations of φ and α.
ÅÆð
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6. Geometric structure of the metric
We know that this metric is QK by construction, but in view of the many steps involved, it is a good selfconsistency check to verify that it is Einstein with self-dual Weyl tensor. The details will be presented in [19], let
us describe the main result. We take for the vierbein
Q
e0 = a(s, x) dφ +
dα ,
e3 = b(s, x) dα,
e1 = µ ds,
e2 = ν dx,
4P
with

1 √
1
P
A


2
 a(s, x) =
,
µ=
,
1 + λs
2d
A
2d A

1 2
1 A
A

 b(s, x) =
(s − c2 )B
,
ν=
,
2d
P
2d B
The spin connection being defined as usual by
dea + ωab ∧ eb = 0,
A = s 2 − c2 1 + λs 2 ,
B = 1 − x 2 1 + λc2 x 2 .
a, b = 0, 1, 2, 3,
one has to compute the anti-self-dual spin connection and curvature
1
ij k ωj k ,
2
One gets the crucial relation
Ri− ≡ R0i −
ωi− = ω0i −
Ri− = −16λ e0 ∧ ei −
1
ij k Rj k = dωi− + ij k ωj− ∧ ωk− ,
2
1
ij k ej ∧ ek ,
2
i, j, k = 1, 2, 3.
(32)
which shows at the same time that the metric is Einstein, with
Ric = Λg,
Λ
= −16λ = −4κ 2,
3
and that the Weyl tensor is self-dual, i.e. Wi− = 0.
Let us now consider a few limiting cases.
The quaternionic Taub-NUT limit
Let us show that in the limit c → 0 we get the quaternionic Taub-NUT. We first write the metric (29) in the form
2
h
1 (1 + λ r 2 )2
dψ
+
dα
+
A
γ
g(c → 0) =
0 0 ,
4D2
A0
r
with


λr
1
− λr,
D = 1 − λr 2 − 4 2 ,
r
a

r 2 = h2 + t 2 .
γ0 = dh2 + dt 2 + t 2 dα 2 ,
ψ = −2φ,
A0 = a 2 +
Switching to the spherical coordinates r, θ, α for which
t = r sin θ,
h = r cos θ
allows one to get the final form
2
(1 + λr 2 )2 1 2
A0
2
2 2
g(c → 0) =
,
σ +
dr + r σ1 + σ2
A0 3 (1 + λr 2 )2
4D2
ÅÆÔ
(33)
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P.-Y. Casteill et al. / Physics Letters B 508 (2001) 354–364
with
σ3 = dψ + cos θ dα,
σ12 + σ22 = dθ 2 + sin2 θ dα 2 .
The derivation of the quaternionic Taub-NUT metric from harmonic superspace was given in [20]. It contains 2
parameters λ̃, R, and in the limit R → 0 it reduces to Taub-NUT. One can see that, upon the identifications
a 2 = 4λ̃2 ,
s = r,
λ = −R λ̃2 ,
the metric 2g(c → 0) is nothing but the quaternionic Taub-NUT.
The quaternionic Eguchi–Hanson limit
This metric was derived using harmonic superspace in [4], and can be written as
(s̃ 2 − c̃2 ) 2
d s̃ 2
2
2
2
4C g =
σ̃3 + s̃B 2
+ σ̃1 + σ̃2 ,
s̃B
s̃ − c̃2
where
s̃B = s̃ − κ 2 c̃2 ,
C = 1 − κ 2 s̃,
σ̃3 = dφ + cos θ dψ,
(34)
σ̃12 + σ̃22 = dθ 2 + sin2 θ dψ 2 .
The writing (34) is adapted to the Killing ∂φ ; if we switch to the Killing ∂ψ we can write the metric as
2
D s̃B
s̃ 2 − c̃2
d s̃ 2
2
2
2
dψ
+
B
+
+
dθ
+
θ
dφ
,
sin
D
4s̃BC 2
4C 2 s̃ 2 − c̃2
with
s̃ 2 − c̃2
D = s̃ 2 − c̃2 cos2 θ + (s̃B)2 sin2 θ,
cos θ dφ.
B=
D
If we now take, in the metric (31), the limit a → 0 it becomes proportional to the metric (34) upon the following
identifications
s = 2s̃,
c = 2c̃,
λ=
κ2
,
4
φ→
ψ
,
2
α → −φ,
x → cos θ.
The hyper-Kähler limit
Relation (32) makes it clear that in the limit λ → 0 we recover a Riemann self-dual geometry, which is therefore
hyper-Kähler. At the level of the metric, it is most convenient to discuss it using the coordinates (30). Indeed, we
obtain the multicentre structure [21–23]
1 1
(dφ + A)2 + V γ0 ,
4 V
with the flat 3-metric
γ0 = dH 2 + dT 2 + T 2 dα 2 .
The potential V and the connection A are, respectively,
1 2
1
1
1 H +c H −c
V=
√
+ √
,
A=−
dα,
a +√ +√
4
4
δ+
δ−
δ+
δ−
δ± = T 2 + (H ± c)2 .
(35)
The potential shows two centres and V (∞) = a 2 /4. An easy computation gives
dV = − G dA,
γ0
which is the fundamental relation of the multicentre metrics. For a = 0 we have the double Taub-NUT metric,
while for a = 0 we are back to the EH metric.
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P.-Y. Casteill et al. / Physics Letters B 508 (2001) 354–364
363
Comparison with other known QK metrics
The QK metric considered here is Einstein with self-dual Weyl tensor. From a general result due to
Przanowski [14] and Tod [15], this class of metrics is conformally related to a subclass of Kähler scalar-flat
ones. From a result of Flaherty [16], any Kähler scalar-flat metric is a solution of the coupled Einstein–Maxwell
equations, with the restriction that the Weyl tensor be self-dual. The explicit solutions of the coupled Einstein–
Maxwell equations known so far fall in two classes: the Perjés–Israel–Wilson metrics [24,25] and the Plebanski–
Demianski [26] metrics. In general they are not Weyl-self-dual.
For the first class we have checked (details will be given in [19]), that the Weyl-self-dual metrics are conformal
to the multicentre metrics. For the metrics in the second class, imposing Weyl self-duality indeed gives rise to a
QK metric. In the HK limit, with the same coordinates as in (35), we have found its potential to be
1
m
V=√ +√ .
δ+
δ−
For m = 0 we recover flat space while for m = 1 it describes a deformation of Eguchi–Hanson with two unequal
masses. Thus our metric is also outside the Plebanski–Demianski ansatz, since their HK limits are different. We
conclude that it supplies a novel explicit example of the Einstein metrics with the self-dual Weyl tensor and,
simultaneously, of the solution of the coupled Einstein–Maxwell system.
Acknowledgements
G.V. thanks Gary Gibbons for useful discussions and for bringing his attention to the references [25,26]. Work of
E.I. was partially supported by grants RFBR-CNRS 98-02-22034, RFBR-DFG-99-02-04022, RFBR 99-02-18417
and NATO grant PST. CLG 974874. He thanks the Directorate of LPTHE for the hospitality extended to him within
the Project PAST-RI 99/01.
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Ù
À
U (1)×U (1) N 8 L N77
Nuclear Physics B 627 [PM] (2002) 403–444
www.elsevier.com/locate/npe
U (1) × U (1) quaternionic metrics from harmonic
superspace
Pierre-Yves Casteill a , Evgeny Ivanov b,a , Galliano Valent a
a Laboratoire de Physique Théorique et des Hautes Energies, Unité associée au CNRS URA 280,
Université Paris 7 2 Place Jussieu, 75251 Paris Cedex 05, France
b Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, 141 980 Moscow region, Russia
Received 9 November 2001; accepted 9 January 2002
Abstract
We construct, using harmonic superspace and the quaternionic quotient approach, a quaternionicKähler extension of the most general two centres hyper-Kähler metric. It possesses U (1) × U (1)
isometry, contains as special cases the quaternionic-Kähler extensions of the Taub-NUT and Eguchi–
Hanson metrics and exhibits an extra one-parameter freedom which disappears in the hyper-Kähler
limit. Some emphasis is put on the relation between this class of quaternionic-Kähler metrics and selfdual Weyl solutions of the coupled Einstein–Maxwell equations. The relation between our explicit
results and the recent general ansatz of Calderbank and Pedersen for quaternionic-Kähler metrics
with U (1) × U (1) isometries is traced in detail.  2002 Elsevier Science B.V. All rights reserved.
PACS: 11.30.Pb
1. Introduction
Recently, there was a surge of interest in the explicit construction of metrics for various
classes of the hyper-Kähler (HK) and quaternionic-Kähler (QK) manifolds, caused by the
important role these manifolds play in string theory (see, e.g., [1–5]). At present, there exist
a few approaches to tackling this difficult problem [6–25]. One of them proceeds from the
generic actions of bosonic non-linear sigma models with the HK and QK target manifolds
[8–13,19–23].
E-mail addresses: [email protected] (P.-Y. Casteill), [email protected],
[email protected] (E. Ivanov), [email protected] (G. Valent).
0550-3213/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved.
PII: S 0 5 5 0 - 3 2 1 3 ( 0 2 ) 0 0 0 1 3 - 5
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Such generic actions, respectively for the HK and QK sigma models, were constructed
in [8,12,13] and [19–21,23] within the harmonic superspace (HSS) method [26,27],
based on the renowned one-to-one correspondence [28,29] between the HK and QK
manifolds on the one hand, and global and local N = 2, d = 4 supersymmetries on
the other. It was proved in [28,29] that the most general self-coupling of N = 2 matter
supermultiplets (hypermultiplets) in the rigid or local N = 2 supersymmetry, necessarily
implies, respectively, the HK or QK target geometry for the hypermultiplet physical
bosonic fields. Conversely, any HK or QK bosonic sigma model can be lifted to a rigidly or
locally N = 2 supersymmetric non-linear sigma model. Most general off-shell actions for
such N = 2 sigma models were constructed in [13,19] in the framework of N = 2 harmonic
superspace (HSS) [26] as the only one to offer such an opportunity. As was proved in [13,
21] starting from the general definition of HK or QK geometries as the properly constrained
Riemannian ones, the corresponding analytic superfield Lagrangians of interaction have
a nice geometric interpretation as the HK or QK potentials. These are the fundamental
objects of the HK and QK geometries (like the Kähler potential in Kähler geometry).
They encode the entire information about the local properties of the relevant bosonic
metric, in particular, about its isometries. Then, based on the one-to-one correspondence
mentioned above, the generic HK and QK sigma model bosonic actions can be obtained
simply by discarding the fermionic fields in the general harmonic superspace sigma model
actions. For the QK case such a generic bosonic action was constructed in [23]. The
actions of physical bosons containing the explicit HK or QK metric associated with the
given harmonic potential appear in general as the result of elimination of infinite sets of
auxiliary fields contained in the off-shell hypermultiplet harmonic analytic superfields.
This procedure amounts to solving some differential equations on the internal sphere
S 2 parametrized by the SU(2) harmonic variables. It is a difficult problem in general to
solve such equations. However, as was shown in [9,23], in the cases with isometries the
computations can be radically simplified by using the harmonic superspace version of the
HK [6,7] or QK [16–18] quotient constructions. One of the attractive features of the HSS
quotient is that it allows one, at all steps of computation, to keep manifest the corresponding
isometries of the metric which come out as internal symmetries of the HSS sigma model
Lagrangian with a transparent origin. It is especially interesting and tempting to apply this
method for the explicit calculation of new inhomogeneous QK metrics. Indeed, whereas
a lot of the HK metrics of this sort was explicitly constructed (both in 4- and higherdimensional cases, see, e.g., [14,30–33]), not too many analogous QK metrics are known
to date.
In [23], using the HSS quotient techniques, we constructed QK extensions of the wellknown [32] Taub-NUT and Eguchi–Hanson 4-dimensional HK metrics and discussed some
their distinguished geometric features. In one or another (though rather implicit) form
these QK metrics already appeared in the literature (see, e.g., [17,22,34]) and our detailed
treatment of them was a preparatory step to reveal capacities of the HSS approach for
working out more interesting and less known examples.
In [11], the double Taub-NUT HK metric was derived from the HSS approach by
directly solving the corresponding harmonic differential equations. It turns out that the
HSS quotient approach allows one to reproduce the same answer much easier, and it nicely
works as well in the QK case, where solving similar harmonic equations would bear a much
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P.-Y. Casteill et al. / Nuclear Physics B 627 [PM] (2002) 403–444
405
more involved problem. In [35] we constructed a QK extension of the double Taub-NUT
metric using the HSS quotient approach.
The present paper is intended, on the one hand, to give the detailed proof of some
statements made in the letter [35] and to perform a further comparison with the available
ansatzes for QK metrics. On the other hand, we demonstrate here that the HSS quotient
approach suggests a further extension of the class of explicit QK metrics presented in [35].
All of them possess U (1) × U (1) isometry and are characterized by two additional free
parameters. In the HK limit they go over into a generalization of the standard double TaubNUT metric with two unequal “masses”, one of the new parameters being just the ratio
of these “masses”. Another parameter does not show up in the HK limit, but it proves
essential at the non-vanishing contraction parameter (Einstein constant). Thus we observe
the existence of a one-parameter class of non-equivalent QK metrics having the same HK
limit.
In Section 2 we remind the basic facts about the HSS action of generic QK sigma
model, as it was derived in [23]. In Section 3 we construct the HSS quotient for the
considered case of the QK double-Taub-NUT sigma model: proceed from a sum of the HSS
“free” actions of three Q+ hypermultiplets (having the hyperbolic HH 3 manifold as the
target space) and then gauge two common commuting one-parameter symmetries of these
actions by two non-propagating N = 2 vector multiplets. The freedom in embedding these
two symmetries in the variety of symmetries of the “free” action is characterized by two
arbitrary constants which specify the most general QK extension of the double Taub-NUT
metric.1 The intermediate steps leading to the final 4-dimensional metrics are described
in Section 4. The metric is read off after fixing the appropriate gauges and solving two
sets of algebraic constraints appearing as the equations of motion for the auxiliary fields
of the gauge multiplets. In Section 5 we bring the metrics into the final form. Using the
Przanowski–Tod ansatz [34,36], we make an independent check that the metrics are indeed
self-dual Einstein. Several limiting cases are also discussed. In Section 6 we examine our
metrics in the context of the literature related to self-dual Einstein geometries [37–43],
including Flaherty’s equivalence to the (self-dual Weyl) solutions of the coupled Einstein–
Maxwell equations [40].
Just after publication of our letter [35] reporting the construction of a QK extension of
the double Taub-NUT metric in the HSS approach, Calderbank and Pedersen [43] have
obtained the exact linearization of any four-dimensional QK metric with two commuting
Killing vectors. After a short review of their results in Section 6.5, we give the precise
relation between their coordinates and ours.
2. The generic HSS action of QK sigma models
In [23] the generic action of QK sigma models with 4n-dimensional target manifold of
physical bosons was obtained as a pure bosonic part of the general off-shell HSS action
1 The QK metric presented in [35] corresponds to the minimal case, when both extra parameters are equal to
zero.
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of n self-interacting matter hypermultiplets coupled to the so-called principal version of
N = 2 Einstein supergravity [19]. The gauge multiplet of the latter, in the language of
N = 2 conformal SG, consists of the N = 2 Weyl multiplet (24 + 24 off-shell components),
the compensating vector multiplet (8 + 8 off-shell components) and the compensating
hypermultiplet (∞ + ∞ off-shell components). It is the only version which admits the most
general hypermultiplet matter self-couplings and thus, in accord with the theorem of [29],
the most general QK metric in the sector of physical bosons. The matter and compensating
+
hypermultiplets are described by the superfields Q+
r (ζ ) and qa (ζ ), r = 1, . . . , 2n, a = 1, 2,
given on the harmonic analytic N = 2 superspace
(ζ ) = x m , θ +µ , θ̄ +µ̇ , u+i , u−k ,
(2.1)
where the coordinates u+i , u−k , u+i u−
i = 1, i, k = 1, 2, are the SU(2)/U (1) harmonic
variables. These superfields obey the pseudo-reality conditions
+
+
(a) Q+r ≡ (2.2)
Qr = Ω rs Q+
(b) q +a ≡ qa = ab qb+ ,
s ,
where Ω rs and ab ( 12 = −12 = −1) are the skew-symmetric constant Sp(n) and
Sp(1) ∼ SU(2) tensors. The generalized conjugation is the product of the ordinary complex
conjugation and a Weyl reflection of the sphere S 2 ∼ SU(2)/U (1) parametrized by u±i .
The superspace (2.1) is real with respect to this generalized conjugation which acts in the
following way on the superspace coordinates:
m = x m,
x
+µ = θ̄ +µ̇ ,
θ
+µ̇ = −θ +µ ,
θ̄
±
±i
u
i =u ,
±i = −u± .
u
i
In the QH sigma model action to be given below we shall need to know only the bosonic
components in the θ -expansion of the above superfields:
+ 2 + 2 −3a
θ̄
q +a (ζ ) = f +a (x, u) + i θ + σ m θ̄ + A−a
g (x, u),
m (x, u) + θ
2
2
−r
Q+r (ζ ) = F +r (x, u) + i θ + σ m θ̄ + Bm
(2.3)
(x, u) + θ + θ̄ + G−3r (x, u)
(possible terms ∼ (θ + )2 or ∼ (θ̄ + )2 can be shown to fully drop out from the final action
and so can be discarded from the very beginning). The component fields still have general
harmonic expansions off shell. The physical bosonic components F ri (x), f ai (x) are
defined as the lowest components in the harmonic expansions of F +r (x, u), f +a (x, u)
f +a (x, u) = f ai (x)u+
F +r (x, u) = F ri (x)u+
i + ···,
i + ···,
sk
bk
F ri (x) = Ωrs ik F (x),
f ai (x) = ab ik f (x).
Further details can be found in [23] and [26].
The bosonic QK sigma model action derived in [23] consists of the two parts
1
(−4)
SQK =
−qa+ D++ q +a
dζ
2
+ ++ +r
κ2 +i 2
Qr D Q + L+4 Q+ , v + , u−
+ 2 u−
i q
γ
1
− 2 d 4 x D(x) + V m ij (x)Vm ij (x) ≡ Sq,Q + SSG .
2κ
ÅÉ×
(2.4)
(2.5)
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407
Here, dζ (−4) = d 4 x d 2 θ + d 2 θ̄ + du is the measure of integration over (2.1), the covariant
harmonic derivative D++ is defined by
2 2
−
D++ = D ++ + θ + θ̄ + D(x)∂ −− + 6V m(ij ) (x)u−
(2.6)
i uj ∂m ,
with D ++ = ∂ ++ − 2iθ + σ m θ̄ + ∂m , ∂ ±± = u±i /∂u∓i , the non-propagating fields D,
ij
ji
Vm = Vm are inherited from the N = 2 Weyl multiplet, κ 2 ([κ] = −1) is the Einstein
constant (or, from the geometric standpoint, the parameter of contraction to the HK case),
γ ([γ ] = −1) is the sigma model constant (chosen equal to 1 from now on), and the “target”
harmonic variable v +a is defined by
v +a =
+i
u+
q +a
i q
+a
−a
=
u
−
− +i u ,
+i
u−
q
u
q
i
i
v +a u−
a = 1.
(2.7)
The function L+4 (Q+ , v + , u− ) is the analytic QK potential, the object which encodes the
full information about the relevant QK metric.
The action (2.5) possesses a local SU(2) invariance, the remnant of the N = 2
ij
supergravity gauge group, with Vm (x) as the gauge field. The precise form of the
SU(2)loc transformations leaving the Sq,Q part of (2.5) invariant can be inferred from
the realization of the group of N = 2 conformal SG as restricted diffeomorphisms of the
analytic superspace (2.1) [44]. This can be achieved by fixing a WZ gauge for the Weyl
ik (x), e a (x) → δ a and all
multiplet and neglecting all its field components besides D(x), Vm
m
m
the residual gauge invariance parameters besides the SU(2)loc one λik (x) = λki (x). These
transformations read2
++ −
δu+
ui ,
i =Λ
δu−
i = 0,
Λ++ = λ++ + 2iθ + σ m θ̄ + ∂m λ+−
2 2 − θ + θ̄ + 2λ−− + 4V −−m ∂m λ−− − 2V +−m ∂m λ−− − λ−− D ,
µ
2 δθ +µ = λ+− θ +µ − i θ + σ m θ̄ + ∂m λ−− ≡ λ+µ (ζ ),
+µ ) = λ̄+µ̇ (ζ ),
δ θ̄ +µ̇ = (δθ
δx m = −2iθ + σ m θ̄ + λ−− + 6(θ + )2 (θ̄ + )2 V −− m λ−− ≡ λm (ζ ),
δD++ = −Λ++ D 0 ,
∂
∂
∂
∂
D 0 = u+i +i − u−i −i + θ +µ +µ + θ̄ +µ̇ +µ̇ ,
∂u
∂u
∂θ
∂ θ̄
1
δq +a (ζ ) q +a (ζ ) − q +a (ζ ) = − Λ(ζ )q +a (ζ ),
2
δQ+r (ζ ) Q+r (ζ ) − Q+r (ζ ) = 0,
m
Λ(ζ ) = ∂m λ + ∂
−−
++
Λ
+µ
− ∂+µ λ
+µ̇
− ∂+µ̇ λ̄
.
2 They were not explicitly given in [23] and earlier papers on the subject.
ÅÉð
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
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Here
±
λ±± = λik (x)u±
i uk ,
− −
−−
ik
Vm = Vm (x)ui uk ,
−
λ+− = λik (x)u+
i uk ,
+ −
+−
ik
Vm = Vm (x)ui uk .
To these transformations one should add the transformation laws of the fields D(x) and
ik (x)
Vm
m
δ ∗ D(x) = 2∂m λik (x)Vik
(x),
(i
k)j
ik
δ ∗ Vm
(x) = −∂m λik (x) + 2λj (x)Vm (x), (2.13)
which uniquely follow from the transformation law (2.9).3 It is easy to see that the SSG
part of (2.5) is invariant under (2.13), implying the SU(2)loc invariance of the full action
(2.5). Note that the QK potential L+4 (Q+ , v + , u− ) in (2.5) is SU(2)loc invariant because
its arguments Q+r , v +a and u−i behave as scalars under the above transformations. The
transformations (2.8)–(2.12) entail the following simple SU(2)loc transformation rules for
the lowest components f +a (x, u), F +r (x, u) in the θ -expansion (2.3)
δ ∗ f +a = λ+− f +a − λ++ ∂ −− f +a ,
δ ∗ F +r = −λ++ ∂ −− F +r .
(2.14)
The procedure of obtaining the QK metric from the action (2.5) goes through a few
steps. First one integrates over θ s in Sq,Q , then varies with respect to the non-propagating
ij
−r
fields g −3a (x, u), G−3r (x, u), A−a
m (x, u), Bm (x, u), D(x) and Vm (x), solve the resulting
non-dynamical equations and substitute the solution back into (2.5), thus expressing
everything in terms of the physical components f ai (x) and F ri (x). Varying with respect
ik (x) yields the important constraint relating f +a and F +r :
to D(x) and Vm
2
1
du f +a ∂ −− fa+ − κ 2 u− f + F +r ∂ −− Fr+ = 2
(2.15)
κ
ik in terms of the hypermultiplet fields
and the general expression for Vm
2
ik
Vm
(x) = 3κ 2 du u−i u−k f +a ∂m fa+ − κ 2 u− f + F +r ∂m Fr+ .
(2.16)
As the next step, one fixes a gauge with respect to the SU(2)loc transformations defined
above. Most convenient is the gauge leaving only the singlet part in f ai (x)
fai (x) = δai ω(x)
(2.17)
(in what follows, we shall permanently use just this gauge). Finally, using the constraint
(2.15), one expresses ω in terms of F ri (x), substitutes this expression into the action and
reads off the QK metric on the 4n-dimensional target space parametrized by F ri (x).
An essential assumption is that ω is a constant in the flat (hyper-Kähler) limit which is
achieved by putting altogether
|κ|ω = 1,
(2.18)
3 Though looking rather involved, the transformations (2.8)–(2.13) can be straightforwardly checked to be
il k
il k
closed, with the Lie bracket parameter λik
br = λ2 λ1l − λ1 λ2l .
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409
and then setting
κ = 0.
(2.19)
Note that in order to approach the HK limit in (2.5) in the unambiguous way, one should
ik (x) by its algebraic equation of motion and
firstly eliminate the non-propagating field Vm
also perform varying with respect to the auxiliary field D(x). Taking into account that
ik (x) ∼ O(κ 2 ) and q +a → u+a |κ|−1 in the HK limit, one observes
the composite field Vm
ik disappears in this limit, and (2.5) goes into
that any dependence on q +a , D and Vm
the HSS action of generic HK sigma model of n hypermultiplets Q+r (r = 1, . . . , 2n)
[12,13]. The constraint (2.15) becomes just the identity 1 = 1. Another possibility is to
ik (x) from (2.5) by equating them to zero. In this case one
remove the fields D(x), Vm
reproduces the HSS action of the most general conformally-invariant HK sigma model
with n + 1 hypermultiplets [23,27,45] (the former compensator q +a (ζ ) enters it on equal
footing with other hypermultiplets). One can reverse the argument, i.e., start from such
HK sigma model action and reproduce the QK sigma model one (2.5) by coupling the HK
ik (x) in order to restore the local SU(2)
action to the non-propagating fields D(x) and Vm
symmetry and to be able to remove the remaining (non-gauge) bosonic degree of freedom
in f +a by the constraint (2.15). This is the content of the so-called “N = 2 superconformal
quotient” approach to the construction of 4n-dimensional QK manifolds from the 4(n + 1)dimensional HK ones [15,24,25,46,47]. In what follows we shall not need to resort to such
an interpretation and shall proceed from the general QK sigma model action (2.5).
3. QK extensions of the “double Taub-NUT” sigma model from HSS quotient
As already mentioned, on the road to the explicit QK metrics one needs to solve the
differential equations on S 2 for f +a (x, u), F +r (x, u) which follow by varying the QK
sigma model action with respect to the non-propagating fields g −3a (x, u) and G−3r (x, u).
No regular methods of solving such non-linear equation are known so far, and this can
(and does) bear some troubles in general. However, in a number of interesting examples
there is a way around this difficulty, the HSS quotient method (it should not be confused
with the “superconformal quotient” mentioned in the end of the previous section). It
can be applied both in the HK [9] and QK [20,23] cases. In it, one proceeds from a
system of several “free” hypermultiplets (with L+4 = 0 in (2.5), which corresponds to a
HH n ∼ Sp(1, n)/Sp(1) × Sp(n) sigma model) and gauges some symmetries of this system
in the analytic superspace by non-propagating N = 2 vector multiplets represented by the
gauge superfields V ++ (ζ ) (once again, only bosonic components of these superfields
are of relevance). In one of possible gauges these superfields can be fully integrated
out, producing a non-trivial QK (or HK) potential L+4 with the necessity to solve nonlinear harmonic equations. But in another gauge (Wess–Zumino gauge) the harmonic
equations remarkably become linear and can be easily solved. All the non-linearity in this
gauge proves to be concentrated in non-linear algebraic constraints on the hypermultiplet
physical fields. These constraints are enforced by the auxiliary fields of vector multiplets
as Lagrange multipliers. They are much easier to solve as compared to the differential
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equations on S 2 . This allows one to get the explicit form of the QK (or HK) metric at cost
of a comparatively little effort.
In [23], we exemplified the HSS quotient approach by QK extensions of the Taub-NUT
and Eguchi–Hanson (EH) metrics. Here we elaborate on a more interesting and non-trivial
case of the QK non-linear sigma model generalizing the HK model with the “double TaubNUT” target manifold. The HSS action of the latter model was proposed in [9], and the
relevant HK metric was directly computed in [11] (it belongs to the class of two-center
ALF metrics, with the triholomorphic U (1) × U (1) isometry). Here we construct, using
the HSS QK quotient method, the QK sigma model action going into that of [9,11] in the
HK limit. We find an interesting degeneracy suggested by the QK quotient: there is a oneparameter family of the QK metrics, all having U (1) × U (1) isometry and reproducing
the double Taub-NUT metric in the HK limit. More general QK action contains one more
parameter which survives in the HK limit and corresponds to a generalization of the double
Taub-NUT metric by non-equal “masses” in its two-centre potential.
3.1. Minimal QK double-Taub-NUT HSS action
The actions we wish to construct have as their “parent action” the QK action including
three hypermultiplet superfields of the type Q+r with the vanishing L+4 . So it corresponds
to the “flat” QK manifold HH 3 ∼ Sp(1, 3)/Sp(1) × Sp(3). For our specific purposes we
relabel this superfield triade as
Q+a
A ,
g +r ,
a = 1, 2; r = 1, 2; A = 1, 2.
(3.1)
The indices a and r are the doublet indices of two (initially independent) Pauli–Gürsey
type SU(2) groups realized on Q+ and g + , the index A is an extra SO(2) index. Each of
these three superfields satisfies the pseudo-reality condition (2.2).
We wish to end up with a 4-dimensional quaternionic metric. So, following the general
strategy of the quotient method, we need to gauge two commuting one-parameter (U (1))
symmetries of this action. In this case the total number of algebraic constraints and residual
gauge invariances in the WZ gauge is expected to be just 8, which is needed for reducing
the original 12-dimensional physical bosons target space to the 4-dimensional one. These
U (1) symmetries should be commuting, otherwise their gauging would entail gauging
the symmetries appearing in their commutator. This would result in further constraints
trivializing the theory.
The selection of two commuting symmetries to be gauged and the form of the final
gauge-invariant HSS action are to a great extent specified by the natural requirement that
the resulting action has two different limits corresponding to the earlier considered HSS
quotient actions of the QK extensions of Taub-NUT and Eguchi–Hanson metrics [23]. The
simplest gauged action SdTN which meets this demand is
1
1
d 4 x D(x) + V m ij (x)Vm ij (x) ,
−
dζ (−4) L+4
SdTN =
(3.2)
dTN
2
2
2κ
where
+ ++ +a
L+4
q
dTN = −qa D
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411
2 ++ +r
QA + gr+ D++ g +r
+ κ 2 u− · q + Q+
rA D
+
2 (ij ) + +
+ W ++ Q+a
gi gj + c(ij ) vi+ vj+
A QaB AB − κ c
+ + V ++ 2 v + · g + − a (rf ) Q+
(3.3)
rA Qf A
and the second term (SSG ) is common for all QK sigma model actions. In (3.3), V ++ (ζ )
and W ++ (ζ ) are two analytic gauge abelian superfields, c(ij ) and a (rm) are two sets of
independent SU(2) breaking parameters satisfying the pseudo-reality conditions
(c(ij ) ) = ik j l c(kl),
(a (rm)) = rn ms a (ms).
(3.4)
The Lagrangian can be checked to be invariant under the following two commuting gauge
U (1) transformations, with the parameters ε(ζ ) and ϕ(ζ ):4
+r
2 ij + − +r
δε Q+r
A = ε AB QB − κ c vi uj QA ,
+r
δε g +r = εκ 2 c(rn) gn+ − cij vi+ u−
,
j g
δε q +a = εκ 2 c(ab)qb+ ,
δε W ++ = D++ ε,
(rb) +
δϕ Q+r
Qb A − ϕκ 2 u− · g + Q+r
A = ϕa
A ,
δϕ g +r = ϕ v +r − κ 2 (u− · g + )g +r ,
δϕ q +a = ϕκ 2 u− · q + g +a ,
δϕ V ++ = D++ ϕ.
(3.5)
(3.6)
This gauge freedom will be fully fixed at the end. The only surviving global symmetries
of the action will be two commuting U (1). One of them comes from the Pauli–Gürsey
(bc) . Another U (1) is the result
SU(2) acting on Q+a
A and broken by the constant triplet a
of breaking of the SU(2) which uniformly rotates the doublet indices of harmonics and
those of q +a and g +r . It does not commute with supersymmetry (in the full N = 2
supersymmetric version of (3.3)) and forms the diagonal subgroup in the product of three
independent SU(2)s realized on these quantities in the “free” case; this product gets broken
down to the diagonal SU(2), and further to U (1), due to the presence of explicit harmonics
and constants c(ik) in the interaction terms in (3.3). These two U (1) symmetries are going
to be isometries of the final QK metric, the first one becoming triholomorphic in the HK
(ik)
limit. The fields D(x) and Vm
(x) are inert under any isometry (modulo some rotations
in the indices i, j after fixing the gauge (2.17)), and so are D++ and the SSG part of
(3.2). The harmonics v +a , as follows from their definition (2.7), undergo some appropriate
transformations induced by those of q +a in (3.5) and (3.6). Note that the presence of the
g-field term in the supercurrent (Killing potential) to which W ++ couples in (3.3), in
4 To avoid a possible confusion, let us recall that the original general QK sigma model action (2.5) contains
a dimensionful sigma model constant γ , [γ ] = −1, which we have put equal to 1 for convenience. Actually, it
is present in an implicit form in the appropriate places of Eq. (3.3) and subsequent formulae, thus removing an
apparent discrepancy in the dimensions of various involved quantities. From now on, we assign the following
dimensions to the basic involved objects and the gauge transformation parameters (in mass units): [q] = [Q] = 1,
[W ++ ] = 0, [V ++ ] = 1, [c] = 2, [a] = −1, [ε] = 0, [ϕ] = 1. With this choice, γ nowhere reappears on its own
right.
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parallel with the vi+ term (becoming the Fayet–Iliopoulos term in the HK limit), is required
for ensuring the invariance of this supercurrent under the ϕ gauge transformations. This
in turn implies the non-trivial transformation property of g +r under the ε gauge group
in (3.5). In the HK limit the g-field term drops out and g +r becomes inert under the ε
transformations.
By fixing the appropriate broken SU(2) symmetries in (3.3), we can leave only one
real component in each of the SU(2) breaking vectors a rf and cik . Thus the relevant QK
metric is characterized by three real parameters: two SU(2) breaking ones and the Einstein
constant κ 2 . The SU(2) breaking parameters survive in the HK limit.
The QK EH and Taub-NUT sigma model limits
It is easy to see that the action (3.2), (3.3) is indeed a generalization of the HSS quotient
actions describing QK extensions of the EH and Taub-NUT sigma models.
Putting g +r = a (rm) = 0 yields the QK EH action as it was given in [20,23]:
L+4
dTN
⇒
+ ++ +a
L+4
q
EH = −qa D
2 ++ +r
QA
+ κ 2 u− · q + Q+
rA D
+a +
++
QA QaB AB + c(ij ) vi+ vj+ . (3.7)
+W
+a
(ik) = 0 yields the QK Taub-NUT action [23]
Putting Q+a
2 (Q1 ) = c
L+4
dTN
⇒
+ ++ +a
L+4
q
TN = −qa D
2 + κ 2 u− · q + gr+ D++ g +r
+ + V ++ 2(v + · g + ) − a (rf ) Q+
r1 Qf 1 . (3.8)
The HSS action with g +r eliminated
Representing g +r as
g +r = u− · g + v +r − v + · g + u−r ,
fixing the gauge with respect to the ϕ transformations by the condition
− +
u · g = 0,
varying with respect to the non-propagating superfield V ++ and eliminating altogether
(v + · g + ) by the resulting algebraic equation,
+ +
1
+
v · g ≡ L++ = a rf Q+
rA Qf A ,
2
we arrive at the following equivalent form of (3.3), with only two matter hypermultiplets
Q+a
A being involved
+ ++ +a
L+4
q
dTN = −qa D
2 +
++ +r
QA + L++ L++ + W ++ Q+a
+ κ 2 u− · q + Q+
rA D
A QaB AB
− ++ ++
− κ 2 c(ij ) u−
L + c(ij ) vi+ vj+ .
i uj L
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(3.9)
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In the HK limit κ 2 → 0 (q +a → |κ|−1 u+a , |κ|(u− · q + ) → 1) the corresponding action
goes into the HSS action describing the double Taub-NUT manifold [9,11].5 Thus (3.2),
(3.3) is the natural QK generalization of the action of [9,11] and therefore the relevant
metric is expected to be a QK generalization of the double Taub-NUT HK metric. We
shall calculated it and its some generalizations in the next sections by choosing another,
Wess–Zumino gauge in the relevant gauged QK sigma model actions.
3.2. Generalizations
In order to better understand the symmetry structure of the action (3.3) and to construct
its generalizations, let us make the field redefinition
+r
− + +a
Q+a
(3.10)
ĝ +r = |κ| u− · q + gA
.
A = |κ| u · q QA ,
In terms of the redefined superfields, Eqs. (3.3), (3.5) and (3.6) are simplified to
+ ++ +a
++ +r
L+4
q + Q+
QA + ĝr+ D++ ĝ +r
rA D
dTN = −qa D
+
2 (ij ) + +
ĝi ĝj − qi+ qj+
+ W ++ Q+a
A QaB AB − κ c
+ + V ++ 2|κ| q + · ĝ + − a (rf ) Q+
rA Qf A ,
+r
δε Q+r
A = ε AB QB ,
δϕ Q+r
A
= ϕa
(rb)
Q+
b A,
δε ĝ +r = ε κ 2 c(rn) ĝn+ ,
δϕ ĝ
+r
= ϕ|κ|q
+r
,
(3.11)
δε q +a = εκ 2 c(ab)qb+ ,
δϕ q
+a
= ϕ |κ|ĝ
+a
(3.12)
(3.13)
(the gauge superfields W ++ , V ++ have the same transformation laws as before).
This form of gauge transformations clearly shows that the corresponding rigid
transformations are linear combinations of four independent mutually commuting oneparameter symmetries which are enjoyed by the free part of the Lagrangian (3.11):
(a) SO(2) symmetry realized on the capital index of Q+r
A ; (b) a diagonal U (1) subgroup in
the product of two commuting SU(2)PG groups realized on q +a and ĝ +r , with cik as the
rs
U (1) generator; (c) U (1) subgroup of the SU(2)PG group acting on Q+r
A , with a as the
U (1) generator; (d) a hyperbolic rotation of q +a and ĝ +r ,
δ ĝ +r = ϕ|κ|q +r ,
δq +a = ϕ |κ|ĝ +a .
(3.14)
Note that the bilinear form invariant under (3.14) is just c(ij ) (ĝi+ ĝj+ − qi+ qj+ ). This explains
the presence of this expression in the ε-Killing potential (first square brackets in (3.11)): the
q + term which is needed for making one of two basic constraints of the theory meaningful
and solvable (see below) should be accompanied by the proper ĝ + term in order to comply
with the symmetry (3.14). One is led to ε-gauge the diagonal U (1) subgroup in the product
of two independent SU(2)PG groups realized on q +a and ĝ +r just in order to gain this
expression in the relevant Killing potential. In the HK limit |κ|q +a → u+a , κ → 0 the
symmetry (3.14) becomes gauging of the familiar shift symmetry of the free hypermultiplet
5 For the precise correspondence one should choose a 12 = ia, a 11 = a 22 = 0 by appropriately fixing the
frame with respect to the broken Pauli–Gürsey SU(2) symmetry of Q+r .
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action:
δ ĝ +r = ϕu+r ,
δu+a = 0.
(3.15)
Thus we come to the conclusion that our original Lagrangian (3.3) is the simplest
and natural choice yielding the double Taub-NUT HK action in the κ → 0 limit, but it
is by no means the unique one. Indeed, one could gauge two most general independent
combinations of the four commuting U (1) symmetries just mentioned. The corresponding
generalization of (3.11) which still has a smooth κ → 0 limit is as follows
+ ++ +a
++ +r
L+4
q + Q+
QA + ĝr+ D++ ĝ +r
rA D
dTN = −qa D
+ +
2 (ij ) + +
ĝi ĝj − qi+ qj+ − β0 a (rf ) Q+
+ W ++ Q+a
rA Qf A
A QaB AB − κ c
+
2 (ij ) + +
+ V ++ 2|κ| q + · ĝ + − a (rf ) Q+
ĝi ĝj − qi+ qj+ ,
rA Qf A − α0 κ c
(3.16)
with α0 and β0 ([α0 ] = −1, [β0 ] = 1) being two new real independent parameters. It is
straightforward to find the precise modification of the gauge transformation rules (3.12),
(3.13):
+r
(rb) +
δ̃ε Q+r
Qb A ,
A = δε QA + ε β0 a
δ̃ϕ q +a = δϕ q +a + ϕ α0 κ 2 c(ab)qb+
δ̃ϕ ĝ +r = δϕ ĝ +r + ϕ α0 κ 2 c(rn) ĝn+ ,
(3.17)
(the rest of transformations remains unchanged).
Limits and truncations
In the HK limit the generalized Lagrangian is reduced to
+
++ +r
L+4
QA + ĝr+ D ++ ĝ +r
dTN (κ → 0) = QrA D
+
(ij ) + +
(rf ) +
ui uj
+ W ++ Q+a
QrA Q+
fA + c
A QaB AB − β0 a
+
(ij ) + +
+ V ++ 2 u+ · ĝ + − a (rf ) Q+
ui uj .
rA Qf A + α0 c
(3.18)
It is easy to see that the α0 term in the second bracket in (3.18) can be removed by the
redefinition
1
ĝ +r ⇒ ĝ +r − α0 cri u+
(3.19)
i ,
2
which does not affect the kinetic term of ĝ +r . At the same time, no such a redefinition
is possible in the QK Lagrangian (3.16), so α0 is the essentially new parameter of the
corresponding QK metric. This α0 -freedom disappears in the HK limit.
Thus the associated class of QK metrics includes two extra free parameters α0 and
β0 besides the SU(2) breaking parameters and Einstein constant which characterize the
minimal case treated before. But only one of them, β0 , is retained in the HK limit. Here we
encounter a new (to the best of our knowledge) phenomenon of violation of the one-to-one
correspondence between the HK manifolds and their QK counterparts.
It remains to understand the meaning of the parameter β0 . At β0 = 0, we have the α0 modified QK double Taub-NUT action. To see what happens at non-zero β0 , it is instructive
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415
to take a modified EH limit in (3.18). Let us redefine
a ik =
1 ik
ã
β0
and then put ĝ +r = 0, β0 → ∞ with keeping ã ik finite and non-vanishing. Then (3.18)
goes into
+
++ +r
L+4
QA
EH (κ → 0) = QrA D
+a +
+
++
(ij ) + +
QA QaB AB − ã (rf ) Q+
+W
ui uj .
rA Qf A + c
(3.20)
It is shown in Section 5.5 that this HSS action produces a generalization of the standard
two-centre Eguchi–Hanson metric by bringing in two unequal “masses” 1 − a and 1 + a
in the numerators of poles in the relevant two-centre potential, with a = 12 ã ik ãik (cik
specifies the centres like in the standard EH case [9]). Then it is clear that the action (3.18)
describes a similar non-equal masses modification of the double Taub-NUT metric as a
non-trivial “hybrid” of the Taub-NUT and unequal masses EH metrics, with β0 measuring
the ratio of the masses.
The general Lagrangian (3.16) has still two commuting rigid U (1) symmetries which
constitute the U (1) × U (1) isometry of the related QK metric. As distinct from the
QK Taub-NUT and EH truncations (3.8) and (3.7) of (3.2), in which the isometries are
enhanced to U (2) [20,23], the same truncations made in the Lagrangian (3.16) lead to
generalized QK Taub-NUT and EH metrics having only U (1) × U (1) isometries. In the
ik
ik
QK Taub-NUT truncation which is performed by putting Q+a
2 = β0 = 0, c = 0, α0 c ≡
ik
c̃ = 0 in (3.16),
L+4
dTN
⇒
+ ++ +a
++ +r
L+4
q + Q+
Q1 + ĝr+ D++ ĝ +r
TN = −qa D
r1 D
+
+ V ++ 2|κ| q + · ĝ + − a (rf ) Q+
r1 Qf 1
− κ 2 c̃(ij ) ĝi+ ĝj+ − qi+ qj+ ,
(3.21)
this isometry is again enhanced to U (2) after taking the HK limit, because any dependence
on the breaking parameter c̃ik disappears in this limit (after the redefinition like (3.19)). At
the same time, in the QK EH truncation (ĝ +r = α0 = a rf = 0, β0 a rf ≡ ã rf = 0 in (3.16))
the U (1) × U (1) isometry is retained in the HK limit, as clearly seen from the form of the
limiting HK Lagrangian (3.20) (parameters ã ik break SU(2)PG and cik break the SU(2)
which rotates harmonics).
Alternative HSS quotient
Finally, we wish to point out that the QK sigma model actions we considered up to now
give rise to the QK metrics which are one or another generalization of the HK double TaubNUT metric. This is closely related to the property that one of the symmetries of the free
+r
QK action of (q +a , Q+a
A , ĝ ) which we gauge always includes as a part the hyperbolic
+r
+a
ĝ , q rotation (3.14) becoming a pure shift (3.15) of g +r in the HK limit. This ensures
the existence of the QK Taub-NUT truncation for the considered class of QK metrics. An
essentially different class of QK metrics can be constructed by gauging two independent
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combinations of those mutually commuting U (1) symmetries of the free action which
are realized as the homogeneous phase transformations of the involved superfields. The
most general gauged QK sigma model of this kind is specified by the following superfield
Lagrangian
+ ++ +a
++ +r
QA + ĝr+ D++ ĝ +r
q + Q+
L+4
dEH = −qa D
rA D
+
+
(ik) + +
(ik) 2 + +
+ W ++ Q+a
ĝi ĝk + β0 a (rf ) Q+
κ qi qj
A QaB AB + γ0 d
rA Qf A + c
+
2 (ij ) + +
+ V ++ d (ik) ĝi+ ĝk+ − a (rf ) Q+
(3.22)
qi qj ,
rA Qf A + α0 κ c
where the involved constants are different from those in (3.16), despite being denoted by
the same letters. To see to which kind of the 4-dimensional HK sigma model the QK
Lagrangian (3.22) corresponds, let us examine its HK limit
+
++ +r
L+4
QA + ĝr+ D ++ ĝ +r
dEH (κ → 0) = QrA D
+
+
(ik) + +
+ W ++ Q+a
ĝi ĝk + βa (rf ) Q+
A QaB AB + γ d
rA Qf A
+
+ c(ik) u+
i uj
+
(ij ) + +
ui uj .
+ V ++ d (ik) ĝi+ ĝk+ − a (rf ) Q+
rA Qf A + αc
(3.23)
Under the truncation ĝ +r = 0, α0 = 0, β0 a rf ≡ ã rf = 0, a rf = 0 it goes into the
Lagrangian (3.20) which corresponds to the EH model with unequal masses, while under
+a
+a
ik
ik
ik
the truncation Q+a
2 = 0, Q1 ≡ Q , γ0 = β0 = 0, α0 c ≡ c̃ = 0, c = 0 it is reduced
to the following expression
+ ++ +r
L+4
Q + ĝr+ D ++ ĝ +r
EH = Qr D
+
(ij ) + +
+ V ++ d (ik) ĝi+ ĝk+ − a (rf ) Q+
ui uj .
rA Qf A + c̃
(3.24)
This HSS Lagrangian can be shown to yield again a EH sigma model with unequal masses.
The parameters of this model are different from those pertinent to the first truncation. Thus
(3.23) defines a “hybrid” of two different EH sigma models, and the associated QK sigma
model (3.22) could be called the “QK double EH sigma model”.6
As the final remark, we note that in the system of three hypermultiplets in the HK case
one can define mutually-commuting independent shifting symmetries of the form (3.15)
separately for each hypermultiplet. Accordingly, one can use them to define different
HSS quotient actions (actually, all such actions, with at least two independent shifting
symmetries (3.15) gauged, yield the Taub-NUT sigma model, while those where all three
such symmetries are gauged yield a trivial free 4-dimensional HK sigma model). No such
an option exists in the QK case: any other hyperbolic rotation like (3.14) (in the planes
+a
+i
+i
(Q+a
1 , q ) or (Q2 , q )) does not commute with (3.14) and the third one of the same
kind. For this reason, we are allowed to use only one such hyperbolic symmetry in the
gauged combinations of independent U (1) symmetries in the course of constructing the
relevant HSS quotient actions. Of course, this is related to the fact that the full symmetry
6 We expect that the related QK metrics fall into the class of QK metrics described by the Plebanski–
Demianski ansatz [39]; this is not the case for the QK double Taub-NUT metrics, see Section 6.4.
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+a +r
of the “flat” QK action of (q +a , QA
, ĝ ) is Sp(1, 3), while the analogous symmetry of
the relevant limiting HK action is a contraction of Sp(1, 3), with a bigger number of the
mutually commuting abelian subgroups.
4. From the HSS actions to QK metrics
4.1. Preparatory steps
As already mentioned, the basic advantage of the HSS quotient as compared to the
approach based on solving non-linear harmonic equations is the opportunity to choose the
WZ gauge for W ++ and V ++ by using the ε and ϕ gauge freedom (see (3.5), (3.6)). In this
gauge the harmonic differential equations for the lowest components f +a (x, u), FA+r (x, u)
+r
and ĝ +r (x, u) of the superfields q +a (ζ ), Q+r
A (ζ ) and ĝ (ζ ) become linear and can be
straightforwardly solved.
In the WZ gauge the gauge superfields has the following short expansion
−
W ++ (ζ ) = iθ + σ m θ̄ + Wm (x) + (θ + )2 (θ̄ + )2 P (ik) (x)u−
i uk ,
−
V ++ (ζ ) = iθ + σ m θ̄ + Vm (x) + (θ + )2 (θ̄ + )2 T (ik) (x)u−
i uk
(4.1)
(like in (2.3), we omitted possible terms proportional to the monomials
and (θ̄ + )2
because the equations of motion for the corresponding fields are irrelevant to our problem
of computing the final target QK metrics). At the intermediate steps it is convenient to deal
+r related to the original superfields by (3.10).
with the hypermultiplet superfields Q+a
A , ĝ
They have the same θ expansions (2.3), with “hat” above all the component fields. Due
to the structure of the WZ-gauge (4.1), the highest components in the θ expansions of the
+r and q +a (G−3a (x, u), ĝ −3r (x, u) and f −3a (x, u)) appear only in the
superfields Q+a
A , ĝ
A
kinetic part of (3.3). This results in the linear harmonic equations for f +a (x, u), FA+r (x, u)
and ĝ +r (x, u):
∂ ++ f +a = 0
⇒
f +a = f ai (x)u+
i ,
∂ ++ ĝ +r = 0
⇒
ĝ +r = ĝ ri (x)u+
i .
∂ ++ F +r = 0
⇒
(θ + )2
FA+b = FAbi (x)u+
i ,
(4.2)
It is easy to check that these equations are covariant under the SU(2)loc transformations
(2.14) which act on f +a , FA+r = |κ|(u− · f + )FA+r and ĝ +r = |κ|(u− · f + )g +r as follows:
δ ∗ f +a = λ+− f +a − λ++ ∂ −− f +a ,
δ ∗ ĝ +r = λ+− ĝ +r − λ++ ∂ −− ĝ +r
δ ∗ FA+r = λ+− FA+r − λ++ ∂ −− FA+r ,
(4.3)
(in checking this, one must use the properties ∂ ++ λ++ = 0, ∂ ++ λ+− = λ++ ,
[∂ ++ , ∂ −− ] = ∂ 0 ≡ u+i ∂/∂u+i −u−i ∂/∂u−i and ∂ 0 (f +a , F +r , ĝ +r ) = (f +a , F +r , ĝ +r )).
These transformations entail the following ones for the bosonic fields of physical dimension
δf ai (x) = λik (x)f ak (x),
δ ĝ ri (x) = λik (x)ĝ rk (x).
δ FAri (x) = λik (x)FArk (x),
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(4.4)
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This step is common for all QK sigma model actions considered in the previous section.
ik (x) in order to
The next common step is to vary with respect to the SG fields D(x) and Vm
obtain the appropriate particular forms of the constraint (2.15) and the expression (2.16).
Bearing in mind the harmonic “shortness” (4.2), we find
κ2 2
κ2 2
F + ĝ 2 ,
f =1+
2
2
a(i
j)
j)
j)
r(i
ik
2
Vm = κ f ∂m fa − FA ∂m FrA − ĝ r(i ∂m ĝr ,
(4.5)
(4.6)
where
f 2 = f ai fai ,
F 2 = FAri Fri A ,
ĝ ri ĝri .
Taking into account the constraint (4.5), it is easy to check that the SU(2)loc transformation
laws (4.3) imply just the transformation law (2.13) for the composite gauge field (4.6) .
One more common step is enforcing the gauge (2.17)
fai (x) = δai ω(x)
⇒
f ai fai
2ω2 .
⇒
(4.7)
For what follows it will be useful to give how the residual gauge symmetries of the WZ
gauge (4.1) with the parameters ε(x) = ε(ζ )| and ϕ(x) = ϕ(ζ )| are realized in the gauge
(4.7) (in the general case of gauge symmetries (3.17), (3.12), (3.13))
r
δ̃ε FAri = εAB FBri + ε β0 a rs FsiA − λik
ε F k A,
r
δ̃ε ĝ ri = ε κ 2 crn ĝni − λik
ε ĝ k ,
(4.8)
r
δ̃ϕ FAri = ϕ a rs FsiA − λik
ϕ F k A,
r
δ̃ϕ ĝ ri = ϕ |κ| ri ω + ϕ α0 κ 2 crs ĝsi − λik
ϕ ĝ k ,
1
δ̃ϕ ω = ϕ |κ| ia ĝ ai ,
2
(4.9)
ik
where λik
ε , λϕ are the parameters of two different induced SU(2)loc transformations needed
to preserve the gauge (4.7)
|κ| (ri)
2 ri
2 ik
ri
.
ĝ
+
α
κ
c
=
−ε
κ
c
,
λ
=
−ϕ
λik
(4.10)
0
ε
ϕ
ω
From now on, we fully fix the residual ϕ(x) gauge symmetry by gauging away the singlet
part of g ri (x):
ir g ri (x) = 0
⇒
g ri (x) = g (ri) (x).
(4.11)
The residual SO(2) gauge freedom, with the parameter ε(x), will be kept for the moment.
We shall explain further steps on the example of the simplest QK double Taub-NUT
action (3.2), (3.3) and then indicate the modifications which should be made in the resulting
physical bosons action in order to encompass the general case (3.16).
These steps are technical (though sometimes amounting to rather lengthy computations)
and quite similar to those expounded in [23] on the examples of the QK extensions of the
Taub-NUT and EH metrics. So here we shall describe them rather schematically.
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419
Firstly one substitutes the solution (4.2) back into the action (3.2), (3.3) (with the
θ -integration performed) and varies with respect to the remaining non-propagating (vector)
−a (x, u), B −a (x, u) and b̂ −r (x, u) in the
fields of the hypermultiplet superfields (Am
m
Am
+r , respectively). Then one substitutes the resulting
and
ĝ
θ -expansions of q +a , Q+a
A
ik (x),
expressions for these fields into the action (together with those for ω(x) and Vm
Eqs. (4.5), (2.16)) and performs the u-integration. At this stage it is convenient to redefine
the remaining fields as follows7
2 ri
1 ai
ĝ .
g ri =
(4.12)
FA ,
κω
κω
In terms of the redefined fields and with taking account of the gauges (4.7), (4.11), the
ij
composite fields ω and Vm are given by the following expressions:
1
1 r(i
j)
(ij )
j)
a(i
2 2
κω=
,
Vm = −16λ ω FA ∂m FaA + g ∂m gr ,
4
1 − λ2 g 2 − 2λF 2
(4.13)
where
κ2
F 2 ≡ FAai FaiA ,
(4.14)
g 2 ≡ g ri gri ,
λ≡ .
4
After substituting everything back into the action we get the following intermediate
expression for the x-space Lagrangian density LdTN (x):
FAai =
LdTN (x) = L0 (x) + Lvec (x),
where
L0 (x) =
with
D=1−
1
Y
2 Y
+
2T
·
D
X
+
+
λ
g
D2
4
8
aj
Lvec (x) =
with
1 m
∂ Fai A ∂m FAai ,
2
1
r
a
Fai A ∂m F j A + gir ∂m g j ,
2
λ 2
g − 2λ F 2 ,
2
T = Fai B ∂ m FB
and
(4.15)
X=
(4.16)
Y=
1 m
∂ gij ∂m g ij ,
2
1 m
α W Wm + β V m Vm + γ W m Vm + W m Km + V m Jm ,
D
(4.17)
(4.18)
1 ab i
1
λ
a Fa A ∂m Fbi A ,
Km = − AB FAai ∂m Fai B − cij g is ∂m g sj ,
2
2
2
2
2
1 F2
λ
λ
1
â
2
2 2
2
2
α=
− λ ĉ +
ĉ g ,
F − g ,
β=
1+
2 4
2
4
4
2
1
γ = a ab Fai A Fbi B AB − λ(c · g),
(4.19)
4
Jm =
7 This relation was misprinted in [35].
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ĉ2 ≡ cik cik ,
â 2 = a ab aab .
(4.20)
After integrating out the non-propagating gauge fields
acquires the typical non-linear sigma model form
Lvec
⇒
1
Z,
D
Z=
W m (x)
and
V m (x),
the part Lvec
1
γ (J · K) − α (J · J ) − β (K · K) .
4αβ −γ2
(4.21)
The resulting sigma model action should be supplemented by two algebraic constraints
on the involved fields
a(i
j)
FA Fa B AB − λ g (li) g (rj ) c(lr) + c(ij ) = 0,
(4.22)
j
g ij − a ab Fai B Fb B = 0,
(4.23)
which follow from varying the action with respect to the auxiliary fields P (ik) (x) and
T (ik) (x) in the WZ gauge (4.1). Keeping in mind these 6 constraints and one residual
gauge (SO(2)) invariance, one is left just with four independent bosonic target coordinates
as compared with eleven such coordinates explicitly present in (4.16), (4.21). The problem
now is to solve Eqs. (4.22), (4.23), and thus to obtain the final sigma model action with
4-dimensional QK target manifold. This will be the subject of our further presentation.
Here, as the convenient starting point for the geometrical treatment in Section 5, it is
worth to give how the full distance looks before solving the constraints (4.22), (4.23)
Y
Y
1
+ 2T .
g = 2 D X + Z +
(4.24)
+ λ g2 ·
D
4
8
The quantities with “prime” are obtained from those defined above by replacing altogether
“∂m ” by “d”, thus passing to the distance in the target space. For instance,
1
1
dFai A dFAai ,
Y = dgij dg ij .
(4.25)
2
2
Note that this metric includes three free parameters. These are the Einstein constant related
to λ (λ ≡ κ 2 /4), and two SU(2) breaking parameters: the triplet c(ij ) , which breaks the
SU(2)SUSY to U (1), and the triplet a (ab) , which breaks the Pauli–Gürsey SU(2) to U (1).
The final isometry group is therefore U (1)×U (1). Constraints (4.22), (4.23) are manifestly
covariant under these isometries. For convenience, from now on we choose the following
frame with respect to the broken SU(2) groups
X =
c12 = ic,
c11 = c22 = 0,
a 12 = ia,
a 11 = a 22 = 0,
(4.26)
with real parameters a and c. In this frame, the squares (4.20) become
ĉ2 = 2c2,
â 2 = 2a 2.
Let us now discuss which modifications the distance (4.24) undergoes if one starts from
the general QK double Taub-NUT action corresponding to the Lagrangian (3.16). Since
the difference between (3.2) and (3.16) is solely in the structure of supercurrents (Killing
potentials) to which gauge superfields W ++ and V ++ couple, the only modifications
entailed by passing to (3.16) are the appropriate changes in the Z -part of (4.24) and in
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421
the constraints (4.22), (4.23). Namely, one should make the following replacements in Z :
α
⇒
β
⇒
γ
⇒
Km
⇒
Jm
⇒
1 2 2 2 1
i
Ff iB AB ,
β â F + β0 a rf FrA
16 0
4
λ 2
1 2 2
β̂ = β − λα0 (g · c) − α0 ĉ 1 − g ,
2
2
1
λ
γ̂ = γ + β0 â 2 F 2 − λα0 ĉ2 1 − g 2 ,
8
2
1
i
Km = Km + β0 a rf FrA
dFf iA ,
2
1
Jm = Jm − λα0 clr glk dgrk
2
α̂ = α +
(4.27)
and pass to the following modification of the constraints (4.22), (4.23):
j)
j
i
FAa(i FaB AB − λg (li) g (rj ) c(lr) − β0 a ab FaB
FbB + c(ij ) = 0,
j
i
g ij − a ab FaB
FbB + α0 cij − λg (li) g (rj ) c(lr) = 0.
(4.28)
(4.29)
4.2. Solving the constraints
In order to find the final form of the QK target metric corresponding to the HSS
Lagrangian (3.3) or its generalization (3.16), we should solve the constraints (4.22), (4.23)
or their generalization (4.28), (4.29). It is a non-trivial step to find the true coordinates to
solve these constraints. Indeed, a direct substitution of g ij from (4.23) into (4.22) gives a
quartic constraint for FAai which is very difficult to solve as compared to the HK case [9,11]
where the analogous constraint is merely quadratic. In the general case (4.28), (4.29) the
situation is even worse.
In view of these difficulties, it proves more fruitful to take as independent coordinates
just the components of the triplet g (ri) ,
g 12 = g 21 ≡ iah,
h = h,
g 11 ≡ g,
g 22 = g,
and one angular variable from FAai . Then the above 6 constraints and one residual gauge
invariance (the ε(x) one) allow us to eliminate the remaining 7 components of FAai in terms
of 4 independent coordinates thus defined. Following the same strategy as in the previous
subsection, we shall first explain how to solve Eqs. (4.22) and (4.23) in this way and then
indicate the modifications giving rise to the solution of the general two-parameter set of
constraints (4.28), (4.29). We relabel the components of FAai as follows
1
a=1 i=2
= (F + K),
FA=1
2
1
a=1 i=2
FA=2
= (F − K),
2i
1
a=1 i=1
FA=1
= (P + V),
2
1
a=1 i=1
FA=2
= (P − V),
2i
FAa=2 i=1 = −FAa=1 i=2 ,
FAa=2 i=2 = FAa=1 i=1 ,
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and substitute this into (4.22), (4.23). After some simple algebra, the constraints can be
equivalently rewritten in the following form
i
A− ,
2a
= − i A+ ,
(b) V K
2a
= −
(a) P F
(and c.c.);
(4.30)
(and c.c.);
− PP
= B+ ;
(c) F F
− KK
= B− .
(d) V V
(4.31)
Here
A± = 1 ± 2λa 2 ch,
r 2 = h2 + t 2 ,
B± = c 1 + λa 2 r 2 ± hA∓ ,
g ḡ = a 2 t 2 .
and K
from (4.30) and substitutes them into (4.31), which gives two
Next, one expresses P
≡ X and V V
≡ Y,
quadratic equations for F F
1
1
X2 − XB+ − t 2 A2− = 0,
(4.32)
Y 2 − Y B− − t 2 A2+ = 0.
4
4
Solving these equations, selecting the solution which is regular in the limit g = ḡ =
h = 0 and properly fixing the phases of F , P, V and K in terms of the phase of g with
taking account of the residual ε(x) gauge freedom, we find the general solution of (4.22),
(4.23) in the following concise form
P = −iMei(φ+α/ρ− −µρ+ ) ,
K = iSei(φ−α/ρ− +µρ+ ) ,
ρ± = 1 ± 4λc
F = Rei(φ−µρ− ) ,
V = Lei(φ+µρ− ) ,
(4.33)
and
g = atei(α/ρ− −8µλc) .
(4.34)
The various functions involved are
1 1 ( ∆− + B− ),
( ∆+ + B+ ),
L=
R=
2
2
1 1 M=
( ∆+ − B+ ),
( ∆− − B− ),
S=
2
2
where
2
∆± = B±
+ t 2 A2∓ .
The true coordinates are (φ, α, h, t). An extra angle µ parametrizes the residual local
SO(2) transformations which act as shifts of µ by the parameter ε(x), µ → µ + ε. To see
this, one must rewrite the ε-transformation law of FAri following from that of FAri , Eq. (4.8)
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(at β0 = α0 = 0),
r
δε FAri = εAB FBri + εκ 2 cik FkA
,
in terms of the newly defined variables and in the SU(2) frame (4.26)
δε F = −iερ− F ,
δε h = 0,
δε V = iερ− V,
δε g = −8iελcg.
δε P = −iερ+ P,
δε K = iερ+ K,
(4.35)
As a consequence of gauge invariance of (4.16), the final form of the metric should not
depend on µ and we can choose the latter at will. For instance, we can change the precise
dependence of phases in (4.33), (4.34) on φ and α. In what follows we shall stick just to
the above parametrization. Explicitly keeping µ at the intermediate steps of calculations is
a good self-consistency check: this gauge parameter should fully drop out from the correct
final expression for the metric.
Finally, let us indicate the modifications which should be made in the above solution
to adapt it to the general set of constraints (4.28), (4.29). It is convenient to represent the
latter in the following equivalent form
a(i j )
FA FaB AB − β0 g (ij ) + (1 − α0 β0 ) c(ij ) − λg (li) g (rj ) c(lr) = 0,
j
i
g ij − a ab FaB
FbB + α0 cij − λg (li) g (rj ) c(lr) = 0.
(4.36)
(4.37)
Then, following the same line as in the case of β0 = α0 = 0, one gets the general solution
in the form
i(φ−µρ− ) e−iµβ0 a ,
i(φ+α/ρ− −µρ+ ) e−iµβ0 a , F = Re
P = −i Me
K = i
Sei(φ−α/ρ− +µρ+ ) e−iµβ0 a ,
i(φ+µρ− ) e−iµβ0 a ,
V = Le
(4.38)
where the functions with “tilde” are related to those defined earlier by the following
replacements
A±
⇒
B±
⇒
± = (1 ± aβ0 )(1 − 2α0 λach) ± 2λa 2 ch,
A
α0
α0
B± = 1 ± (1 ∓ aβ0 ) B± − a β0 + 2 (1 ∓ aβ0 ) h.
a
a
(4.39)
The appearance of an additional phase factor in (4.34) is due to the fact that in the general
case the ε transformations (4.35) acquire the common extra piece proportional to β0 :
δε F
⇒
δε F − iεβ0aF ,
etc. The QK Taub-NUT and QK EH truncations of the general solution correspond to
imposing the following conditions:
QK Taub-NUT: β0 = 0,
QK EH: α0 = 0,
c = 0,
α0 c ≡ α̃0 = 0,
β0 a ≡ β̃0 = 0,
a ⇒ 0.
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(4.41)
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Respectively, in these two limits we have
QK Taub-NUT:
± = (1 − 2λa α̃0 h) ≡ A,
A
2 + t 2 A
2
+ = ∆
− = B
∆
α̃0
2 2
B± = ± h + (1 + λa r ) ≡ ±B,
a
= M,
R
= ⇒ L
S,
QK EH:
± = (1 ± β̃0 ),
± = c ± (1 ∓ β˜0 )h,
A
B
2
± = c ± (1 ∓ β˜0 ) + (1 ∓ β̃0 )2 t 2 .
∆
(4.42)
(4.43)
Note that in the Taub-NUT case we can obviously choose, up to a gauge freedom,
P = V,
F =K
⇒
e2iµ = −ieiα ,
which, according to the above definition of P, V, F , V, corresponds just to the truncation
Q+a
A=2 = 0 at the level of the general HSS Lagrangian (3.16). Also note that for taking the
QK EH limit in the original form of constraints (4.36), (4.37) in the unambiguous way,
one should firstly rescale g ik → ag (ik) . The corresponding limiting QK metrics can be
obtained by taking the limits (4.40), (4.41) in the QK metric associated with the general
choice of α0 = 0, β0 = 0.
5. The structure of general metric
5.1. First set of coordinates
To obtain the metric, we substitute the explicit form (4.38) of the coordinates into
the distance (4.24) and compute it. The algebraic manipulations to be done in order to
cast the resulting expression in a readable form are rather involved, and Mathematica
was intensively used while doing this job. To simplify matters, we make the change of
coordinates
2t
2h
T=
(5.1)
T 2 + H2
,
H
=
,
ρ
=
1 − a 2 λr 2
1 − a 2 λr 2
and use the notations
1
aβ0
α0
,
c± =
β=
±
c,
1 − 4cλ
1 ∓ aβ0
a
2 2
4∆±
2 2
δ± = 2 = 1 + 4a λc± T + H ± 2c± .
1 − a 2 λr 2
The final result for the metric g can be presented in terms of 4 functions D, A, P , Q
2
Q
P
1 + a 2 λρ 2 2 2
4D 2 g =
(5.2)
dφ +
dα + A g0 +
T dα
A
4P
P
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425
where
g0 =
dH 2 + dT 2 + a 2 λ(T dH − H dT )2
1 + a 2 λρ 2
(5.3)
is the metric on the two-sphere (a 2λ < 0), on flat space (a 2 λ = 0), or on the hyperbolic
plane (a 2 λ > 0).
The various involved functions are as follows
D
= 1 − λ (1 + aβ0 ) δ− + (1 − aβ0 ) δ+ ,
D=
2
2
1 − a λr
2 2
1 − 4a 2λc+
1 − 4a 2λc−
a2 1
A=
+
+ (1 − aβ0r)
√
√
(1 + aβ0 )
4
4
δ−
δ+
2
(1 + aβ0 )c− (1 − aβ0 )c+
4c λ 1 + a 2 λH 2
+ a 2 λH
−
,
√
√
√ √
−
δ−
δ+
1 − a 2 β0 2 δ − δ +
H + 2c+
H − 2c− 2
P = 1 + a 2λρ 2 1 − 2cλ √
+ 2cλ √
δ+
δ−
2
2
1 − 2a 2 λc+ H
1 + 2a λc− H
2 2 2
+
,
+ 4c λ T −aα0 −
√
√
δ−
δ+
H − 2c−
Q = − 1 + a 2 λρ 2 2β(1 + 2cλ) + 1 + β(1 + 4cλ) √
δ−
H + 2c+
(H − 2c− )(H + 2c+ )
+ 1 − β(1 + 4cλ) √
√ √
− 4cβλ
δ+
δ− δ+
− 2caλT 2 a 2 α0 − 2 1 − 2cα0 2 βaλ
1 + 2c− a 2 λH
+ a + α0 + α0 β(1 + 4cλ)
√
δ−
1 − 2c+ a 2 λH
− a − α0 + α0 β(1 + 4cλ)
√
δ+
2
2
2β (1 + 2c− a λH )(1 − 2c+ a λH )
−
√ √
.
a
δ− δ+
The isometry group U (1) × U (1) acts by translations on φ and α.
(5.4)
5.2. Second set of coordinates
In order to verify that g is self-dual Einstein (see Section 5.4), it is more convenient to
use coordinates s and x defined by
T = s 1 − x 2,
(5.5)
H = sx.
We then get for the metric the expression
2
Q
P
dφ +
dα
4D 2 g =
A
4P
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s 2 dx 2 s 2 (1 + a 2 λs 2 )(1 − x 2 ) 2
ds 2
dα .
+
+
+A
P
1 + a 2 λs 2 1 − x 2
(5.6)
The functions A, P , Q and D are still the same as in (5.4), up to the substitution (5.5), and
the functions δ± can be written as
δ± =
2 1 2 2
2
2 2
1
+
a
−
1
∓
2a
.
1
+
4a
λs
λc
sx
λc
±
±
a 2λ
5.3. Third set of coordinates (α0 = β0 = 0 case)
In the limit α0 → 0 and β0 → 0, the metric g reduces to the quaternionic extension of
the double Taub-NUT metric given in [35]. For this particular case one can get rid of the
square roots by switching to the spheroidal coordinates (u, θ ),
√
u2 − 4c2
sin θ,
H = u cos θ.
T=√
(5.7)
1 + 4a 2λc2
In these coordinates:
δ± = u ± 2c cos θ.
It is convenient to scale the angles φ and α according to
φ
α
,
α̂ =
.
1 + 4a 2 λc2
1 + 4a 2λc2
Then the metric at α0 = β0 = 0 becomes
2
Q
2
2 2 P
4D g = 1 + a λu
d φ̂ +
d α̂
4P
A
(u2 − 4c2)(1 + 4a 2λc2 cos2 θ ) 2
+ A g0 +
sin θ d α̂ 2 ,
P
where
φ̂ =
and
g0 = u2 − 4c2 cos2 θ g0 =
(5.8)
dθ 2
du2
+
(u2 − 4c2 )(1 + a 2 λu2 ) 1 + 4a 2λc2 cos2 θ
4A = 4 u2 − 4c2 cos2 θ A = 2 + a 2 u u − 8c2 λ − 4a 2c2 D 2 cos2 θ,
(1 + 4a 2λc2 )(u2 − 4c2 cos2 θ )
P
1 + a 2 λu2
= 4c2 sin2 θ 1 + 4a 2 λc2 cos2 θ D 2
+ u2 − 4c2 1 + 4a 2λc2 cos2 θ − 16λ2 c2 sin2 θ ,
P=
(1 + 4a 2λc2 )(u2 − 4c2 cos2 θ )
Q = −2 u2 − 4c2 1 + 4a 2λc2 cos θ,
2
2
1 + a λu
D = 1 − 2λu.
Q=
ÅÎ×
(5.9)
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427
5.4. Einstein and self-dual Weyl properties of the metric
A four-dimensional QK metric is nothing but an Einstein metric with self-dual Weyl
tensor. This property should be inherent to the metric g given by (5.2), since we started
from the generic HSS action for QK sigma models. However, checking these properties
explicitly is a good test of the correctness of our computations.
We first consider the particular case α0 = β0 = 0 because the use of the spheroidallike coordinates (5.7) greatly simplifies the metric as can be seen from relations (5.8) and
(5.9). Despite these simplifications, intensive use of Mathematica was needed to compute
the spin connection, the anti-self-dual curvature Ri− and to check the crucial relation (see
Appendix A for the notation):
Ri− = −16λ Ξi− .
It simultaneously establishes that the metric is indeed self-dual Einstein, with
Λ
= −16λ,
Wi− = 0.
3
For non-vanishing α0 or β0 , such a check is no longer feasible because of the strong
increase in complexity of various functions appearing in the metric. Moreover, in this case
we failed to find any proper generalization of
coordinates (5.7) which
√
√the spheroidal-like
would allow us to get rid of the square roots δ+ and δ− .
In order to by-pass these difficulties we have used an approach due to Przanowski [36]
and Tod [34], which reduces the verification of the self-dual Einstein property to simpler
checks. We shall begin with a description of their construction.
One starts from an Einstein metric g (more precisely, Ric(g) = Λg). Furthermore it will
be supposed that this metric has (at least) one Killing vector with the associated 1-form
K = Kµ dx µ . Differentiating K gives
Ric(g) = Λ g,
1
Ξi± = e0 ∧ ei ± ij k ej ∧ ek ,
2
for some vierbein of the metric g. We can extract, from dK, an integrable complex
structure I and a coordinate w according to
dK = dKi+ Ξi+ + dKi− Ξi− ,
I=
dKi−
Ξi− ,
− 2
i (dKi )
w=−
Using these elements one can formulate
Λ
3
i
(dKi− )2
.
(5.10)
Proposition 1 ([34,36]). There exist real coordinates w, ν and µ such that any Einstein
metric g with self-dual Weyl tensor and a Killing vector ∂φ can be written as
1 1
(dφ + Θ)2 + W ev dν 2 + dµ2 + dw2 .
g= 2
(5.11)
w W
This metric will be self-dual Einstein iff
(a)
−2
Λ
W = 2 − w∂w v,
3
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(b)
(c)
2
∂ν + ∂µ2 v + ∂w2 ev = 0,
−dΘ = ∂ν Wdµ ∧ dw + ∂µ Wdw ∧ dν + ∂w Wev dν ∧ dµ.
(5.12)
The following remarks are in order:
1. The relation (5.12b) is the celebrated continuous Toda equation.
2. Except for this Toda equation, the checks of the self-dual Einstein property are reduced
to solving first-order partial differential equations.
3. Relation (5.11) shows that any self-dual Einstein metric with at least one Killing is
conformal to a subclass of Kähler scalar-flat metrics (see Section 6.1 for the proof).
Let us now use this approach to analyze our metric (5.6) in the (s, x) coordinates and to
check whether it obeys the conditions (5.12).
We take for vierbein
√
ds
A
1 dφ + Θ ,
e1 =
e0 = √
√
,
2D 1 + a 2 λs 2
W
√
√
A s dx
W 2 λs 2 1 − x 2 dα,
e2 =
,
e3 =
s
√
1
+
a
2D 1 − x 2
4D 2
2
4D A
,
W=
P
and consider the Killing ∂φ , with the 1-form
1
1
(dφ + Θ) = √ e0 .
W
W
− 2
The computation of i (dKi ) eventually leads to the identification
K=
w=−
where
Λ
D
,
3 4λ δ(ĉ)
(5.13)
2 1 1 + 4a 2λĉ2 1 + a 2 λs 2 − 1 − 2a 2 λĉsx ,
a 2λ
α0
2ĉ = 1 − aβ0 c+ − 1 + aβ0 c− = 2c .
(5.14)
a
Then, comparing the metric g in the form (5.6) with (5.11), we express the quantities W, µ
and ev entering (5.11) in terms of ours
s 2 1 + a 2λs 2 1 − x 2 w4
W
v
W = 2,
(5.15)
µ = α,
e =
.
w
16D 4
Simultaneously, we obtain the expressions for the partial derivatives of ν
δ(ĉ) =
∂x ν = −
4D 2 ∂s w
,
(1 − x 2 )w2
∂s ν =
4D 2 ∂x w
.
2
s (1 + a 2 λs 2 )w2
ÅÎÔ
(5.16)
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429
Two expressions for the mixed derivative ∂s ∂x ν coincide as a consequence of the relation:
s 2 1 + a 2 λs 2 ∂s2 D + 1 − x 2 ∂x2 D = 0.
(5.17)
Checking the relation (5.12a) suggests the identification
Λ
= −16λ.
(5.18)
3
Then the remaining equations (b) and (c) in (5.12) have been explicitly checked using
Mathematica, and shown to be valid. This proves that our general metric (5.6) is self-dual
Einstein.
5.5. Limiting cases
The hyper-Kähler limit Using the coordinates H and T (defined in (5.1)), in the limit
λ → 0, the metric (5.2) can be written as the multicentre structure
1
(dΦ + A)2 + V g0 (λ → 0),
V
with the flat 3-metric and the angle Φ defined by
4g(λ → 0) =
g0 (λ → 0) = dH 2 + dT 2 + T 2 dα 2 ,
Φ =φ−
aβ0
α.
2
The potential V and the connection A are, respectively,
1 2 1 + aβ0 1 − aβ0
a + √
,
V=
+ √
4
δ−
δ+
1
H + 2c+
H − 2c−
A = − (1 + aβ0 ) √
+ (1 − aβ0 ) √
dα,
4
δ−
δ+
(5.19)
(5.20)
with
δ± = (H ± 2c± )2 + T 2 ,
c± =
c
.
1 ∓ aβ0
Since α0 is an irrelevant parameter in the limit λ → 0 (it can be removed from the metric
by a shift of H ), we put it equal to zero from the very beginning.
The potential shows two centres at T = 0, H = ∓2c± with different masses 1 ∓ aβ0 /4
and V (∞) = a 2 /4. An easy computation gives the fundamental multicentre relation
dV = −
N
g0 (λ→0)
dA.
For a = 0, β0 = 0, we have the double Taub-NUT metric; for a = 0, c = 0 and a = 0, aβ0 =
±1, (c± finite), we have the Taub-NUT metric; for a = 0, we have the Eguchi–Hanson
metric.
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430
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The quaternionic Taub-NUT limit
In order to show that in the limit c → 0 we recover the quaternionic Taub-NUT metric,
we switch to new coordinates (ŝ, θ̂ ) defined by
H=
2 ŝ cos θ̂
,
a 1 − λŝ 2
T=
2 ŝ sin θ̂
.
a 1 − λŝ 2
1
The metric g coincides, up to a constant factor 2a
, with the metric given by relation (5.4)
in [22]:
ŝ A2 2
1 B
ŝ B 2
σ1 + σ22 +
2ag(c → 0) =
d ŝ 2 +
σ3 ,
2 ŝ C 2
C2
BC2
where
and
A = 1 − R λ̂2 ŝ 2 ,
B = 1 + λ̂2 ŝ(4 + R ŝ),
C = 1 + R ŝ + R λ̂2 ŝ 2 ,
σ12 + σ22 = d θ̂ 2 + sin2 θ̂ dα 2 ,
σ3 = (−2dφ + aβ0 dα) + cos θ̂ dα,
λ
a
R = −4 ,
(5.21)
λ̂2 = .
a
4
Various limits of the quaternionic Taub-NUT metric can be found in [22]. Let us just
remark here that in the limit R → 0 we once again recover the standard Taub-NUT metric.
The quaternionic Eguchi–Hanson limit
In the limit a → 0 with aβ0 = β̃0 = 0, it is more convenient to study the metric in
coordinates in which the square roots disappear. Thus, we define the coordinates s̃ and θ̃
by
2
2 2
s̃ − c2 sin θ̃ ,
H=
s̃ cos θ̃ + c− − c+ ,
T=
2
1 − β̃0
1 − β̃02
so that
δ± =
2
1 − β̃02
(s̃ ± c cos θ̃ ).
The metric can now be expressed as
with
2 g(a → 0, β̃0 ) =
4(1 − β̃02 )C
=1−
C
κ2
1 − β̃02
2
s̃ 2 − c2 2
2
2
2
d s̃
+
d
θ̃
+
sin
G + s̃ B
,
θ̃
H
s̃ 2 − c2
s̃ B
(s̃ − cβ̃0 cos θ̃ ),
= s̃ − κ 2 c2 + cβ̃0 cos θ̃,
s̃ B
G = −(1 + β cos θ̃ ) dα + 2 cos θ̃ dφ,
H=
1 −(s̃ − c)β dα + 2 s̃ − κ 2 c2 dφ ,
s̃ B
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431
where
β̃0
.
1 − 4cλ
One can see that in the limit a → 0, the parameter α0 fully drops out from the metric. If we
now take the limit β̃0 = aβ0 → 0, we reproduce the quaternionic Eguchi–Hanson metric
derived in [23] (see Eq. (4.7) of this reference):
2
2
2
2
2
2 g(a → 0) = s̃ − c σ̃32 + s̃ B
d s̃
4C
+
σ̃
+
σ̃
1
2 ,
s̃ 2 − c2
s̃ B
κ 2 = 4λ,
β=
with
= 1 − κ 2 s̃,
C
= s̃ − κ 2 c2 ,
s̃ B
σ̃3 = (−dα) + cos θ̃(2 dφ),
σ̃12 + σ̃22 = d θ̃ + sin2 θ̃(2 dφ)2 .
In conclusion, let us point out that, whereas the parameters a, c and β0 have a counterpart
in the HK limit, this is not the case for the parameter α0 . This distinguished parameter is
specific just for the QK metrics.
6. Connection with the literature
Metrics with self-dual Weyl tensor may appear as:
1. Kähler scalar-flat metrics;
2. self-dual Einstein metrics (considered in this work);
3. metrics in the system of coupled Einstein–Maxwell fields.
In order to exhibit the relationships between these classes and to find out how our metrics
correlate with them, let us begin with the description, due to LeBrun, of the Kähler scalarflat metrics with one Killing vector.
6.1. Kähler scalar-flat metrics in LeBrun setting
These metrics, with self-dual Weyl tensor, have received attention in [41]. There, it was
proved that any such metric, with at least one Killing vector K = ∂t , can be written as
3
g=
2
1
2 + W dw2 + ev dν 2 + dµ2 =
(dt + Θ)
eA ,
W
(6.1)
A=0
where the functions v and W must be solutions of the following equations
2
2
∂ν + ∂µ2 v + ∂w2 ev = 0,
∂ν + ∂µ2 W + ∂w2 Wev = 0.
is then obtained from
The connection one-form Θ
= ∂ν (W) dµ ∧ dw + ∂µ (W) dw ∧ dν + ∂w (Wev ) dν ∧ dµ.
dΘ
ÅÖÆ
(6.2)
(6.3)
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The vierbein, defined in relation (6.1), is taken to be
√
√
dt + Θ
e0 = √
e2 = Wev dµ,
,
e1 = Wev dν,
W
e3 =
√
W dw.
In terms of the self-dual two-forms Ξi± = e0 ∧ ei ± 12 ij k ej ∧ ek the Kähler form is antiself-dual,
+ Wev dν ∧ dµ = −Ξ − ,
Ω = dw ∧ (dt + Θ)
3
(6.4)
while the Ricci form is self-dual,
1
∂w v
∂w v
∂w v
1
+
+
2ρ̂ = √ ∂ν
(6.5)
· Ξ1 + √ ∂µ
· Ξ2 + ∂w
· Ξ3+ .
W
W
W
ev
ev
Now, comparing (5.11) and (6.1), we observe that any self-dual Einstein metric with at least
one Killing, in particular the metric (5.2), is conformally related to a subclass of Kähler
scalar-flat metrics, with the identifications:
= −Θ,
Θ
dt = −dφ,
q=
ev = q 2 ,
dµ = dα,
g = w2 g.
In [41], a large class of explicit solutions of (6.2) was obtained. Taking
√
2w,
dν 2 + dµ2 + dq 2
,
q2
where γ is the hyperbolic 3-space, these metrics have the form
1
q2
(6.6)
(dt + Θ)2 + V γ ,
V
where V is some real harmonic function on γ .
LeBrun obtained the potential V as a sum of monopoles in this hyperbolic space. In the
limit where the hyperbolic space becomes flat, one recovers the multicentre metrics. However, the possibility that these metrics could be conformally Einstein has been ruled out by
Pedersen and Tod in [42]. Therefore the metrics (6.6) bear no relation to our metric (5.2).
V = q 2 W,
γ=
6.2. Flaherty’s equivalence
Let us now examine Flaherty’s equivalence relating Kähler scalar-flat metrics and selfdual metrics solving the coupled Einstein–Maxwell field equations.
In [40] Flaherty has proved:
Proposition 2. The following two classes of metrics are equivalent:
1. Any Kähler scalar-flat metric.
2. Any metric which is a solution of the coupled Einstein–Maxwell equations
1
1
ρσ
ρσ
Ricµν =
,
Fµρ g Fνσ − gµν Fρσ F
2
4
dF − = 0,
dF + = 0,
with self-dual Weyl tensor
(W −
= 0).
ÅÖÉ
(6.7)
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433
In this equivalence the self-dual parts of the Maxwell field strength are given by
F − ∝ Ω,
F + ∝ ρ̂,
where Ω denotes the Kähler form and ρ̂ the Ricci form of the Kähler metric.
In the euclidean case, this equivalence can be easily checked for metrics with at least one
Killing vector, using the LeBrun framework. One can check Eqs. (6.7) and find the selfdual parts of the field strength two-forms:
2
m
(6.8)
Ω,
F + = ρ̂,
2
m
where m is an arbitrary real parameter.
This equivalence and the property that any self-dual Einstein metric with one Killing is
conformal to some Kähler scalar-flat metric suggest that the Weyl-self-dual metrics which
solve the Einstein–Maxwell system may hide, up to some conformal factor, a self-dual
Einstein metric. Let us now examine two known classes of the metrics giving solution
of the Einstein–Maxwell system (in general, they are not Weyl-self-dual) in order to see
whether the metric (5.2) is conformally related to any of them. We shall find that the answer
is negative in both cases. This means that (5.2) determines a new explicit solution of the
Einstein–Maxwell system, with the conformal factor w given in (5.13).
F− = −
6.3. The metrics of Perjès–Israel–Wilson
These metrics are solutions of the Einstein–Maxwell field equations. They were derived
independently, for the minkowskian signature, by Perjès [37] and Israel and Wilson [38].
Their continuation to the euclidean signature was given by Yuille [48] and Whitt [49]
who discussed their global properties and their possible applications in the path integral
approach to quantum gravity.
These metrics have at least one Killing vector ∂t . Their local form is given by
1
,
(dt + A)2 + V γ0 ,
V = UU
V
must be harmonic
The real functions U and U
γ0 = d x · d x.
g=
(6.9)
= 0,
∆U = ∆U
(6.10)
dU − U d U
.
N dA = U
(6.11)
and the connection one-form A is constrained by
γ0
The star and laplacian are taken with respect to the three-dimensional flat space with
are constant we come back to the multicentre
cartesian coordinates x . Clearly, when U or U
metrics.
In order to check the previous assertions, let us define the vierbein eA by
1
e0 = √ (dt + A),
V
ei =
√
V dxi ,
i = 1, 2, 3.
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It is an easy task to compute the matrices A, B and C giving the curvature (see
Appendix A for the definitions and notation). One finds, upon using the relations (6.10),
(6.11), the simple expressions
2
∂i U ∂j U
(∂l U )2
1 ∂ij U
Aij =
−3
+ δij
,
V
U
U2
U2
1
,
Bij = − 2 ∂i U ∂j U
V
2
∂j U
)2 ∂i U
(∂l U
1 ∂ij U
+ δij
Cij =
(6.12)
,
−3
2
2
V
U
U
U
where the derivatives are taken with respect to the cartesian coordinates x . The scalar
curvature R = 4(Tr A) vanishes as it should.
The first equation in (6.7) gives for the field strength
−1 F ≡ F − + F + = ∂i U −1 · Ξi− − ∂i U
· Ξi+ .
Using (6.10), (6.11) one can check that these field strengths indeed obey the Maxwell
equations:
dF + = dF − = 0.
Let us prove the following:
Proposition 3. The Perjès–Israel–Wilson metrics are self-dual Weyl only in the two cases:
is a constant: they are homothetic to the multicentre metrics.
1. When U
2. When U = m/|
x − x0 |: they are conformal to the multicentre metrics.
Proof. Let us impose, for instance, the condition that the Weyl tensor is self-dual (i.e.,
W − = 0). Using (6.12) and (6.10), the corresponding constraints can be written as
1
1
1
− δij ∆
= 0.
∂i ∂j
(6.13)
2
3
U
U2
Acting on the left-hand side by ∂i gives
1
1
=
0
⇒
∆
= const ≡ 6B.
∂j ∆
2
2
U
U
Then one can integrate relation (6.13) to
1
r 2 = x · x ,
= A + f · x + Br 2 ,
2
U
is harmonic (Eq. (6.10))
where A and f are integration constants. The requirement that U
amounts to the relation
f · f = 4AB.
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435
is evidently reduced to a constant which can be
If B vanishes, the harmonic function U
scaled to 1. Then the relations (6.10), (6.11) imply that the metric is homothetic to some
multicentre one.
= m/|
If B does not vanish, we can write U
x − x0 | which can be simplified to 1/r by
rescaling and translation of x . The metric (6.9) becomes
g = r 2 ĝ,
with
1
1
2
γ̂ = 4 γ̂0 ,
ĝ =
(dτ + A) + V γ̂ ,
r
V
Using spherical coordinates we have
γ̂0 = dr 2 + r 2 dΩ 2
⇒
V = rU.
γ̂ = dρ 2 + ρ 2 dΩ 2 ,
ρ = 1/r,
thus establishing that γ̂ is flat. Then relation (6.11) becomes
− N dA = d V ,
γ̂
showing that ĝ is some multicentre. This completes the proof. 2
Proposition 3 tells us that the metrics of Perjès–Israel–Wilson, when they have self-dual
Weyl tensor, are never conformal to Einstein metrics (with non-vanishing cosmological
constant). This implies that our metric (5.2) can never be transformed to the Perjès–Israel–
Wilson form.
6.4. The Plebanski–Demianski metric
In [39] Plebanski and Demianski have derived a minkowskian solution of the coupled
Einstein–Maxwell field equations. Its euclidean version, obtained by complex changes of
coordinates and parameters, can be written in the form
gPD =
3
2
eA
,
A=0
with the vierbein
1
e0 =
1 + pq
1
e2 =
1 + pq
where
p2 − q 2
dp,
X(p)
1
e1 =
1 + pq
X(p) dτ + q 2 dσ ,
2
2
p −q
1
e3 =
1 + pq
p2 − q 2
dq,
Y (q)
Y (q) dτ + p2 dσ ,
2
2
p −q
λ
λ 4
X(p) = g02 − γ +
− 2lp + p2 − 2mp3 − e2 + γ +
p ,
6
6
λ
λ 4
Y (q) = e2 + γ −
− 2mq − q 2 − 2lq 3 − g02 − γ −
q .
6
6
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It displays 6 real parameters besides the cosmological constant λ and possesses U (1) ×
U (1) isometry realized by shifts of τ and σ.
The meaning of the parameters e, g0 , l and m follows from:
Proposition 4. The Plebanski–Demianski metrics are
• Einstein for e = g0 = 0.
• Einstein with self-dual Weyl tensor (W − = 0) for e = g0 = 0 and l = m.
• Einstein with anti-self-dual Weyl tensor (W + = 0) for e = g0 = 0 and l = −m.
Proof. The proposition follows from the computation of the curvature matrices A, B and
C defined in Appendix A.
We are going to show that our metric (5.2) lies outside the above ansatz. To this end,
we shall work with an anti-self-dual Weyl tensor (W − = 0) and analyze the λ → 0 limit
of gPD . We switch to the triholomorphic Killing vector ∂φ by making the change of
coordinates
dφ = dτ,
dα = dσ + dτ.
It leads to the limiting metric
gPD (λ → 0) =
1
(dφ + A)2 + V γ0 ,
V
(6.14)
with the potential
(1 + pq)2 (p2 − q 2 )
,
D
the gauge field one-form
V=
2
2
D = 1 − q 2 X(p) + 1 − p2 Y (q),
q 2 (1 − q 2 )X(p) + p2 (1 − p2 )Y (q)
dα,
D
and the three-dimensional metric
D
dp2
X(p)Y (q) 2
dq 2
γ0 =
dα .
+
+
4
(1 + pq) X(p) Y (q)
(1 + pq)4
A=
(6.15)
(6.16)
(6.17)
One can explicitly check the relation
N dA = ±dV .
γ0
To prove that (6.14) is indeed multicentre, we define cartesian coordinates x by
x = A sin m2 + γ (2γ + )α ,
y = A cos m2 + γ (2γ + )α ,
z = B,
ÅÖ×
(6.18)
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P.-Y. Casteill et al. / Nuclear Physics B 627 [PM] (2002) 403–444
with
1
A=
(1 + pq)2
B =−
m2
X(p)Y (q)
,
+ γ (2γ + )
m(p − q)(1 − pq) + γ (p2 + q 2 ) + pq
.
m2 + γ (2γ + )(1 + pq)2
One can check that these coordinates make manifest the flatness of the metric (6.17)
γ0 = (dx)2 + (dy)2 + (dz)2 .
For comparing (6.14) with the HK limit of our metric we need to express the potential
V in terms of the coordinates (6.18).
For m = 0, as observed in the original paper [39], the metric (6.14) is flat: this is a
special case which needs a separate analysis. We have
1
1
V= √
,
2 γ ( − 2γ ) x 2 + y 2 + Z 2
Z=z+ √
,
2 γ (2γ + )
provided that the expressions within square roots are positive. This potential corresponds
to a mass at the origin, and is known to yield a flat four-dimensional metric [32].
For m = 0, we define new parameters by
( − 2γ )2 − 16m2
− 2γ
, φ 0,
c= ,
cosh φ =
4m
4 m2 + γ (2γ + )
and
2γ + Z=z+ ,
2
4 m + γ (2γ + )
d± = x 2 + y 2 + (Z ± c)2 .
In this notation, the potential (6.15) becomes
η
1/η
V =µ √ + √
,
d−
d+
with
2
e−φ
(6.19)
√
c
.
µ=
4m(sinh φ)3/2
η =√
,
c2 + 1 + c
The HK limit of the Plebanski–Demianski metric therefore gives an ALE generalization of
the Eguchi–Hanson metric (recovered for η = 1) with two different masses. It is reduced
to the flat metric, up to rescaling, in the limits η → 0 and η → ∞.
The potential (6.19) is a particular case a = 0, aβ0 = 0 of our limiting HK potential
(5.19):
1 2 1 + aβ0 1 − aβ0
+ √
a + √
.
V=
4
δ−
δ+
The conclusion is that our general metric (5.2) cannot be embedded into the Plebanski–
Demianski class of self-dual Einstein metrics because their HK limits are different. 2
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Summarizing the discussion in Sections 6.3 and 6.4, we observe that our metric
(5.2) cannot be reduced to either known class of metrics solving the Einstein–Maxwell
equations. Hence, by Flaherty’s equivalence, it provides (up to conformal factor (5.13)) a
new family of explicit solutions of this system. For the minimal case α0 = β0 = 0 this fact
was pointed out in [35].
6.5. The linearization by Calderbank and Pedersen
Quite recently, while we were typing this article, Calderbank and Pedersen [43] have
exhibited a class of self-dual Einstein metrics with two commuting (and hypersurface
generating) Killing vectors. To describe their metrics, two main ingredients are needed:
1. A function F (ρ, η) which is a solution of the linear differential equation
3
ρ 2 (Fρρ + Fηη ) = F.
4
(6.20)
It is an eigenfunction of the laplacian in the hyperbolic plane H2 with metric
g0 (H2 ) =
dρ 2 + dη2
,
ρ2
(6.21)
ρ > 0.
2. The set of one-forms
α=
√
ρ dα,
β=
dφ + η dα
.
√
ρ
In terms of these, the full metric is
g=
F 2 − 4ρ 2 (Fρ2 + Fη2 )
4F 2
+
g0 H2
[(F − 2ρFρ )α − 2ρFη β)2 + (2ρFη α − (F + 2ρFρ )β]2
.
F 2 [F 2 − 4ρ 2 (Fρ2 + Fη2 )]
(6.22)
The main result of [43] is a theorem which states that these metrics with two commuting
Killings are self-dual Einstein with non-vanishing scalar curvature and that any such metric
has locally the structure given by the expression (6.22).
In order to get a deeper insight into the construction of [43], it is convenient to pass to a
√
function G according to F = G/ ρ. The metric g becomes
GGη
1
2
2
(dφ + Θ) + Wγ ,
Θ=
− η dα,
G g=
(6.23)
W
G2ρ + G2η
with
W=
1 GGρ
− 1,
ρ G2ρ + G2η
γ = ρ 2 dα 2 + G2ρ + G2η dρ 2 + dη2 .
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Following Tod, we can now compute the anti-self-dual part of dK, where K is the
1-form associated with the Killing ∂φ . After some algebra, using (6.20), we obtain
K=
1
G2 W
(dφ + Θ),
dK − = −
1
G G2ρ + G2η
Gρ Ξ1− + Gη Ξ2− ,
from which we conclude that in fact G is proportional to Tod’s coordinate w, defined in
(5.10). Taking G = w, relation (6.20) becomes
1
(6.24)
wρ .
ρ
Using relations (6.4) and (6.5), and switching from Tod’s coordinates (w, ν) to the
(ρ, η) coordinates, we can obtain the Kähler form Ω and the Ricci form ρ̂ in this setting:
Ω = −dw ∧ dφ + ηwρ − ρwη dρ ∧ dα + (ρwρ + ηwη − w) dη ∧ dα,
1
1
ρ̂ = −d
(6.25)
(dφ + Θ) + (dφ − η dα) .
wW
w
wρρ + wηη =
The Kähler form Ω is closed as a consequence of (6.24). One can check that Ω and ρ̂
possess opposite self-dualities. In view of Flaherty’s equivalence, the metrics described by
the Calderbank–Pedersen ansatz are conformally related to a subclass of metrics solving
the coupled Einstein–Maxwell equations. Then the two-forms (6.25) specify the field
strengths of the corresponding Maxwell field (6.8).
We are now in a position to establish the precise connection between our coordinates
s and x and the coordinates ρ and η in the hyperbolic plane H2 . To this end, we have to
identify the pieces which are independent of the choice of basis for the Killing vectors, i.e.,
the pieces involving γ . One gets the correspondence:
4s 1 − x 2 1 + a 2 s 2 λ,
ρ=
δ(ĉ)
2 s(s + 2ĉx)
η=
(6.26)
−1 ,
ĉ
δ(ĉ)
where δ(ĉ) was defined in (5.14) and ĉ = α0 c/a. Let us notice that the coordinate η is
defined up to an additive constant that can always be reabsorbed through a redefinition of
the Killing ∂φ . The check of Eq. (5.12a) gives Λ = 3 ⇔ λ = −1/16. One can then invert
relations (6.26):
2
2
2a
2a
2 − |2 + cα |
η
+
+
ρ
+ ρ2
|2 − cα0 | η − 2−cα
0
2+cα
0
0
,
x=
2 (cα0 η + 2a)2 + c2 α02 ρ 2
s=8
cα0
2
cα0 2
ρ2
2
2a
|2 − cα0 | η − 2−cα
+ ρ 2 + |2 + cα0 | η +
0
a
η+2
+
a
2
2a
2+cα0
.
+ ρ2
As discussed in Section 5.5, in the analysis of the QK Eguchi–Hanson limit, for
a → 0 the parameter α0 becomes irrelevant since it disappears from the metric. The above
coordinate s is well defined in the limit a → 0 only if we first put α0 = 0.
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Having the explicit expressions for s, x, it is then possible to compute w(ρ, η) which
was given in (5.13) as a function of s, x:
2
1
2a
w = |2 − cα0 |
+ ρ2
η−
4
2 − cα0
2
1
2a
+ |2 + cα0 |
+ ρ2
η+
4
2 + cα0
2
2
2
2
|c|
|c|
η − (1 − aβ0 ) + ρ 2 +
η + (1 + aβ0 ) + ρ 2 .
+
8
c
8
c
It is easy to check that w(ρ, η) satisfies Eq. (6.24).
Let us finally give w(ρ, η) in the two interesting cases a → 0 (QK-EH) and c → 0
(QK-TN):
2
|c|
|c|
2
2 2
2
2
2
wQK-EH (ρ, η) = η + ρ +
+ρ +
+ ρ2,
η−
η+
8
c
8
c
2
2
1 1 1 wQK-TN (ρ, η) = +
η − a + ρ2 +
η + a + ρ2.
2 2
2
Using these relations we can, e.g., compute the forms Ω and ρ̂ (6.25) for our metrics
and, via the correspondence (6.8), to find the relevant Maxwell field strengths.
7. Conclusions
In this paper, proceeding from the general HSS formulation of QK sigma models, we
have constructed a wide class of U (1) × U (1) 4-dimensional QK metrics extending most
general two-centre HK metrics. These QK metrics supply, via Flaherty’s equivalence [40],
a new family of explicit solutions of the coupled Einstein–Maxwell equations. We have
given the precise embedding of our metrics in the framework of general U (1) × U (1)
ansatz of Calderbank and Pedersen [43].
The HSS approach gives QK metrics in the form which admits a transparent
interpretation of the involved parameters as the symmetry breaking ones and possesses
a clear hyper-Kähler limit, with the Einstein constant as a contraction parameter. However,
despite these attractive features, it does not immediately provide the natural coordinates
best suited to display the final linearization of the self-dual Einstein equations along the
line of Ref. [43]. It would be interesting to explore what the choice of such coordinates
means in the language of the original hypermultiplet superfields parametrizing the general
HSS action of QK sigma models. One more obvious direction of further study could consist
in applying our HSS methods for explicit construction of higher-dimensional QK metrics
generalizing, e.g., the HK metrics constructed in [14].
One of possible physical applications of the QK metrics presented here is to utilize them
in the context of gauged five-dimensional supergravities. The latter seemingly provide an
appropriate framework for supersymmetric extensions of the famous Randall–Sundrum
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441
scenario (for a recent review, see [50]). The presence of matter hypermultiplets seems
necessary for the existence of such (smooth) extensions (see, e.g., [51]). To analyse various
possibilities, it is important to know the structure of the scalar potential which is obtained
by gauging isometries of the QK manifold parametrized by the hypermultiplets. Until now,
in the actual computations (e.g., in [51,52]) there was mainly used the so-called universal
hypermultiplet [1] corresponding to the homogeneous QK manifold SU(2, 1)/U (2). It
would be tempting to study models with non-homogeneous QK manifolds possessing
isometries and, in particular, with those considered here. It is straightforward to gauge
the U (1) × U (1) isometries of our HSS actions following the general recipe of Ref. [20]
(in order to generate scalar potentials, the relevant gauge supermultiplets should be
propagating, in contrast to the non-propagating ones employed in the HSS quotient). The
SU(2, 1)/U (2) QK manifold is a special case [23] of the QK Eguchi–Hanson limit of our
U (1) × U (1) class of QK manifolds, so the scalar potentials associated with our metrics
and inheriting all free parameters of the latter may offer new possibilities as compared to
the case of universal hypermultiplet.
Acknowledgements
E.I. thanks Directorate of Laboratoire de Physique Théorique et des Hautes Energies,
Université Paris VII, for the hospitality extended to him during the course of this work
under the Project PAST-RI 99/01. His work was partially supported by the grants RFBR 9902-18417, RFBR-CNRS 98-02-22034, INTAS-00-0254, NATO Grant PST.CLG 974874
and PICS Project No. 593.
Appendix A. Definitions and notation
For a given metric g, the vierbein ea , a = 0, 1, 2, 3, is such that
g=
ea2 .
a
The spin connection ωab is defined by
dea + ωab ∧ eb = 0,
ωab = −ωba ,
with self-dual components
1
ωi± = ω0i ± ij k ωj k ,
2
and similarly for the curvature
1
Rab = dωab + ωas ∧ ωsb = Rab,st es ∧ et
2
We take for the Ricci tensor and scalar curvature
Ricab = Ras,bs ,
→
R = Ricss .
Å×Æ
1
Ri± = R0i ± ij k Rj k .
2
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It is useful to define the two-forms of definite self-duality by
1
Ξi± = e0 ∧ ei ± ij k ej ∧ ek .
2
Using this basis, the curvature and Ricci tensor are encoded in the three matrices A, B and
C such that
Ri+ = Aij Ξj+ + Bij Ξj− ,
Ri− = Bijt Ξj+ + Cij Ξj− ,
where the matrices A and C are symmetric.
The Ricci components in the vierbein basis are
1
Ric0i = − ij k Bj k − Bjt k ,
2
Ricij = Tr(A − B)δij + Bij + Bijt ,
Ric00 = Tr(A + B),
and the scalar curvature is
R = 4(Tr A) = 4(Tr C).
The Einstein condition Ricab = Λδab is seen to be equivalent to the vanishing of the matrix
B and we have Tr C = Tr A = Λ.
One further defines the Weyl tensor
Wab,cd = Rab,cd +
R
(δac δbd − δad δbc )
6
1
− (δac Ricbd − δad Ricbc + δbd Ricac − δbc Ricad ).
2
The corresponding two-forms
1
Wab = Wab,cd ec ∧ ed ,
2
and their self-dual parts are given by
1
Wi+ ≡ W0i + ij k Wj k = Wij+ Ξj+ ,
2
1
Wi− ≡ W0i − ij k Wj k = Wij− Ξj− ,
2
1
Wij+ = Aij − (Tr A)δij ,
3
1
Wij− = Cij − (Tr C)δij .
3
We conclude that for an Einstein space with self-dual Weyl tensor (i.e., Wi− = 0) we should
have
Λ
Λ
Cij = δij ⇐⇒ Ri− = Ξi− .
3
3
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[48]
[49]
[50]
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