Multiple Parallel Strings from ABJM? 1. Introduction • M-theory • ABJM Multiple Parallel Strings and Wrapped M2 Branes from the ABJM Model ? 2. Wrapping M2 brane 3. Parallel strings • From AdS4 × S 7 • Bulk vs. boundary 4. Effective action Tamiaki Yoneya University of Tokyo - Komaba • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary KEK 2009 Theory Workshop Multiple Parallel Strings from ABJM? 1. Introduction Contents 1. Introduction 2. Wrapped M2 branes from ABJM 3. Parallel strings 4. Effective action for parallel strings • M-theory • ABJM 2. Wrapping M2 brane 3. Parallel strings • From AdS4 × S 7 • Bulk vs. boundary 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary 5. Discussion Multiple Parallel Strings from ABJM? 1. Introduction Revisiting the M-theory conjecture 1. Introduction • M-theory • ABJM First recall the conjecture. 2. Wrapping M2 brane 11 spacetime dimensions 10 9 3. Parallel strings M Theory Circle Type IIA Circle/Z2 Type IIB Type IIAB Type I Type I Hetero SO(32) Hetero E8 x E8 Hetero • From AdS4 × S 7 • Bulk vs. boundary 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary Perturbative Theories S duality T duality Cirlce Compactification Unfortunately, no substantial progress, from the end of the previous century, on what the M theory really is. I Multiple Parallel Strings from ABJM? Radius of the circle direction : 1. Introduction R11 = gs `s • M-theory • ABJM 2. Wrapping M2 brane M2 brane as gs → 0 I ”longitudinal”: wrapped along the 11-th circle direction • From AdS4 × S 7 • Bulk vs. boundary ⇒ (fundamental) string I 3. Parallel strings ”transverse” : extended along directions orthogonal to the 11-th circle ⇒ D2 brane 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion I Fundamental length scale of M theory = Planck scale `P = gs1/3 `s and `P `s as gs → 0. R11 • Summary ⇓ In the weak coupling (∼ 10 dimensional) limit, M2 branes should smoothly reduce to perturbative strings of type IIA theory. Multiple Parallel Strings from ABJM? 1. Introduction • M-theory • ABJM 2. Wrapping M2 brane We would like to discuss this question in the context of the ABJM model, a candidate low-enegy theory for multiple M2 branes, in the simplest possible setting. • So far, almost all previous works have been focused on the ”transverse” configurations of M2 branes: weak coupling limit k ∼ ∞ 3. Parallel strings • From AdS4 × S 7 • Bulk vs. boundary 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion m S 7 , which is transverse to M2 branes, into CP3 S 7 /Zk → CP3 , Zk → S 1 ∼ M-theory circle 2π/k = Chern-Simons coupling constant • Summary • Remark: case of single M2 brane The dynamics of a single M2 brane is already quite non-trivial, and hence the reduction to string(s) is not completely understood, quantum-mechanically. Sekino-TY, hep-th/0108176 , Asano-Sekino-TY, hep-th/0308024 I I I wrapped M2 brane ⇓ directly matrix-string theory [1 + 1D SYM with coupling 1/gs (N → ∞)] √ large N limit with gYM = 1/ gs → ∞ can be studied by using GKPW relation in the PP (BMN)-wave limit, under the assumption of gauge/gravity correspondence. The result of two-point correlators shows that the effective scaling dimension of scalar fields is 2 1 ∆eff = = 5 − p p=1 2 This is consistent with the existence of 3D CFT description of M2 branes. Multiple Parallel Strings from ABJM? 1. Introduction • M-theory • ABJM 2. Wrapping M2 brane 3. Parallel strings • From AdS4 × S 7 • Bulk vs. boundary 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary Main features of the ABJM model Multiple Parallel Strings from ABJM? 1. Introduction • M-theory • ABJM 2. Wrapping M2 brane I Susy Chern-Simons U(N)×U(N) gauge theory in 3D with SO(6)(∼SU(4)) R-symmetry I (super)Conformal invariant I CS coupling = 2π/k with level number k ⇔ C4 /Zk = transverse space of M2 branes I AdS/CFT correspondence at k = 1: AdS4 × S 7 ⇔ effective CFT of N M2 branes in flat 11D 3. Parallel strings • From AdS4 × S 7 • Bulk vs. boundary 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary I But, only with N = 6 susy, manifestly. Multiple Parallel Strings from ABJM? • Notations (following Bandres-Lipstein-Schwarz, 0807.0880) I bosonic fields: (XA , X A) I fermionic fields: (ΨA , ΨA ) 1. Introduction (4, 4 of SU(4)) • M-theory • ABJM (4, 4 of SU(4), 3D 2-component spinor) I Chern-Simons U(N)×U(N) gauge fields: (Aµ , ˆ µ) A SABJM k = 2π Z i h ¯ A γ µ Dµ ΨA d x Tr − D µ X A Dµ XA + i Ψ 3 k +SCS + 2π Z d 3 x (L6 + L2,2 ) L6 = potential term of O(X ), L2,2 = X Ψcoupling terms of O(X 2 Ψ2 ) Z 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary 6 k 4π 3. Parallel strings • From AdS4 × S 7 • Bulk vs. boundary • Action SCS = 2. Wrapping M2 brane h i 2i ˆ µ ∂ν A ˆ λ − 2i A ˆ µA ˆνA ˆ λ) d 3 x µνλ Tr Aµ ∂ν Aλ + Aµ Aν Aλ −A 3 3 Multiple Parallel Strings from ABJM? • Classical moduli space = (C4 /Zk )N /SN I residual gauge symmetry: (U(1)×U(1))N /SN A 1. Introduction A I X → diagonal matrices with identification X = e I At k = 1, R8 /SN ⇔ N M2 branes in flat space 2πi/k X A Would like to study, in the case k = 1, whether we can understand ordinary strings by wrapping M2 branes along the M-circle. But, that is in the strong-coupling regime! Will however see that after the reduction due to wrapping, the effective coupling constant is • M-theory • ABJM 2. Wrapping M2 brane 3. Parallel strings • From AdS4 × S 7 • Bulk vs. boundary 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary N kr 2 r = transverse distance scale among strings Multiple Parallel Strings from ABJM? 2. Wrapped M2 brane from ABJM Double dimensional reduction 1. Introduction The ABJM model implicitly assumes the static gauge for world-volume coordinates: world 3-coordinates x µ = longitudinal 3 directions of 11D • M-theory • ABJM 2. Wrapping M2 brane 3. Parallel strings ⇓ Wrapping along the M-circle in 11-th direction can be performed by the “double” dimensional reduction (gs 1) I Recover the length dimension with respect to target space by A A µ (X A , ΨA , x µ ) → `−1 P (X , Ψ , x ), ˆ µ ) → `P (Aµ , A ˆ µ) (Aµ , A • From AdS4 × S 7 • Bulk vs. boundary 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion I gauge fixing along the periodic direction ˆ 2, ∂2 A2 = 0 = ∂2 A I ˆ 2 ) ≡ R −1 (B, B) ˆ (A2 , A 11 x2 = x2 + 2πR11 , R11 = gs `s Z Z d 3 x → R11 d 2 x, ∂2 → 0 for all fields • Summary • Reduced 2D action (µ, ν, λ . . . ∈ (0, 1)) 5 Z h k 1 ˆ A ˆ SPS = d 2 x Tr −D µ X A Dµ XA + 2 2 (BX −X A B)(BXA −XA B) 2 2π`s gs `s Z i k + . . . + SBF + d 2 x (L6 + L2,2 ) 2π`2s Z k ˆλ − iB ˆλ ˆ νA ˆA ˆνA d 2 x νλ Tr B∂ν Aλ + iBAν Aλ − B∂ SBF = 2π 1 A B C L6 = Tr X X X X X X + · · · A B C 3gs2 `6s 1 ¯ A XB ΨC XD + · · · L2,2 = 4/3 Tr iABCD Ψ gs `4s I Naively, this system flows, in the extreme IR limit, to the strong coupling regime [1/gs → ∞]=[weak string coupling]. I Moduli-space approximation seems good for |p| 1/R11 , 1/RP I At k = 1, should correspond to multiple parallel strings stretching along a fixed longitudinal direction in flat 10D spacetime. Multiple Parallel Strings from ABJM? 1. Introduction • M-theory • ABJM 2. Wrapping M2 brane 3. Parallel strings • From AdS4 × S 7 • Bulk vs. boundary 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary Multiple Parallel Strings from ABJM? 3. Parallel strings Parallel strings from AdS4 ×S7 1. Introduction • M-theory • ABJM On the bulk side, start from the M2 brane metric 2 ds11 = h−2/3 (−dt 2 +dx12 +dx22 )+h1/3 (dr 2 +dΩ27 ), 32π 2 N`6P h = 1+ r6 Using the usual relation between 11D and 10D string-frame, 2 2 ds11 = e −2φ/3 dsstring + e 4φ/3 dx22 the background fields around N parallel strings stretching along x2 is 2. Wrapping M2 brane 3. Parallel strings • From AdS4 × S 7 • Bulk vs. boundary 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary 2 dsstring =h −1 2 (−dt + dx12 ) e φ = h−1/2 , B01 = h−1 2 + dr + dΩ27 Remarks: I BPS ⇔ −g00 = g11 = B01 The world-sheet string action is completely free Z 1 µν µν 2 √ Sstring = − −γ g (X )γ + B (X ) ∂µ X A ∂ν X B d ξ AB AB 4π`2s Z X 1 2 =− d x ∂µ X A (x)∂ µ XA (x) 4π`2s A=transverse in the static (conformal) gauge ξ 0 = t = X 0 , is manifestly SO(8) symmetric. I Near-horizon limit: r (gs2 N)1/6 2 dsstring = ξ 1 = x1 = X1 and (Q ∝ Ngs2 `6s ) r6 (−dt 2 + dx 2 ) + dr 2 + r 2 dΩ27 Q ⇒ scaling symmetry: (I) : r → λ1/2 r , (t, x), → λ−1 (t, x), (II) : (t, x) → ρ(t, x), gs → ρgs 2 2 dsstring → λ−1 dsstring Multiple Parallel Strings from ABJM? 1. Introduction • M-theory • ABJM 2. Wrapping M2 brane 3. Parallel strings • From AdS4 × S 7 • Bulk vs. boundary 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary Bulk vs. boundary Multiple Parallel Strings from ABJM? 1. Introduction • M-theory • ABJM The structure of ABJM moduli space seems consistent with the above properties on the bulk side, at least classically. • Question: What about the quantum corrections ? I enhancement of R symmetry? I cancellation of all interactions? The question is essentially non-perturbative in its nature. 2. Wrapping M2 brane 3. Parallel strings • From AdS4 × S 7 • Bulk vs. boundary 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary Let us study general structure of the effective action for parallel strings on the basis of the reduced action SPS 4 Multiple Parallel Strings from ABJM? Scaling symmetry of SPS I 1. Introduction (I) : inherited from 3D conformal symmetry 1/2 r →λ r, (t, x), → λ −1 • M-theory • ABJM 2 dsstring (t, x), →λ −1 2 dsstring m 3. Parallel strings ˆ µ , B) ˆ → λ(Aµ , B, A ˆ µ , B), ˆ (Aµ , B, A ¯ A ) → λ(ΨA , Ψ ¯ A ), (ΨA , Ψ I XA → λ1/2 XA , k → λ−1 k (II) : related to 2D conformal symmetry (reminiscent of matrix-string theory) (t, x) → ρ(t, x), gs → ρgs m ¯A (ΨA , Ψ ) → ρ −1/2 ¯A (ΨA , Ψ ), 2. Wrapping M2 brane ˆ µ ) → ρ−1 (Aµ , A ˆ µ) (Aµ , A • From AdS4 × S 7 • Bulk vs. boundary 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary Let the (transverse) distance scale among parallel strings be r . The scaling symmetries constrain the (bosonic part of) effective action as (string unit : `s = 1) Seff = ∞ X Z cL,q,g ,h 2 d x k −L+1 gsq−2 r −2L+6 L=0,q=2,g =0,h=0 ∂r q r3 Multiple Parallel Strings from ABJM? 1. Introduction • M-theory • ABJM 2. Wrapping M2 brane 3. Parallel strings ×N 2−2g −h+L−1 L = # of loops, q = # of derivatives g =genus, h = # of holes with respect to color index loops in planar expansion ⇓ • From AdS4 × S 7 • Bulk vs. boundary 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary I perturbative loop expansion is meaningful when N 1 kr 2 I In the limit gs → 0, the derivative expansion is also meaningful. In the free limit, can restrict to the lowest order q = 2. Multiple Parallel Strings from ABJM? 1. Introduction • Unfortunately, the near-horizon limit on the bulk side is not compatible with the perturbative regime of the reduced action for finite fixed k and for weak string coupling, since near horizon condition : r ⇓ 1 gs N/k 3/2 ⇔ r (N/k) 2. Wrapping M2 brane 3. Parallel strings • From AdS4 × S 7 • Bulk vs. boundary (as typical AdS/CFT correspondence !) (gs2 N)1/6 • M-theory • ABJM 1/2 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary • However, independently of the near-horizon condition, we can study effective actions for our ‘would-be’ gauge theory of multiple parallel strings, for sufficiently large r r N 1/2 at k = 1, N = finite Multiple Parallel Strings from ABJM? 1. Introduction • M-theory • ABJM 2. Wrapping M2 brane • Relevant question : susy ‘non-renormalization theorem’ for kinetic terms, valid or not? I In the case of D-brane susy Yang-Mills theories, non-renormalization theorems are at work. • loop corrections start from v 4 /r 7−p • seems to be case also for AdS4 × CP 3 ( k → ∞) in one-loop order. Not only that, SYM can correctly reproduce the long-distance gravitational interactions (even 3-body forces!) among D-branes at least up to two-loop order. I Note also that physical interpretation of the off-diagonal parts of matrix coordinates X A in the case of ABJM (and also of BLG theories) is totally unclear. 3. Parallel strings • From AdS4 × S 7 • Bulk vs. boundary 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary Multiple Parallel Strings from ABJM? 4. Effective action for parallel strings from the reduced action (N + 1) → (N) + (1) decompostion 1. Introduction • M-theory • ABJM Let us study one-loop effective action (L = 1) for simplest background X A = ( 0, 0, . . . , 0, r A ), | {z } U(N + 1) → U(N) × U(1) • off-diagonal fluctuating fields: (a = 1, . . . , N , all are complex N-vectors) two pairs of (4, 4) scalar fields A A UaA , U a , VaA , V a I their fermion partners (2D Dirac) pairs of 2D vector fields ˆ µ a, A ˆµ a Aµ a , Aµ a , A I 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary A A ΘAa , Θa , ΦAa , Φa I 3. Parallel strings • From AdS4 × S 7 • Bulk vs. boundary N I 2. Wrapping M2 brane ˆ 2 a) pairs of auxiliary scalar fields (originated from A2 a , A ˆa , B ˆa Ba , B a , B • Owing to the presence of the vacuum expectation value for X A , . . . I Can integrate out the auxiliary fields B, . . . I Can choose the followng special background-field gauge 1 ¯a ) = 0, ∂µ Aµa − ir (r · V r 1 ¯µ ∂µ Aa + ir (¯r · Va ) = 0 r ⇓ I emergence of usual kinetic terms for fluctuating gauge fields I mass terms are diagonalized with eigenvalues 4 4 4 4 for complex scalars 2 2 2 2 for Dirac fermions (r , r , r , r ) (r , r , r , r ) Multiple Parallel Strings from ABJM? 1. Introduction • M-theory • ABJM 2. Wrapping M2 brane 3. Parallel strings • From AdS4 × S 7 • Bulk vs. boundary 4. Effective action 2 r =r ·r • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary • mass ∝ r 2 → off-diagonals ∼ open-membrane bits ? • SU(4) R-symmetry is enhanced to SO(8) for completely static parallel strings ∂r = 0. Not trivial! However, I no enhancement for non-static background ∂r 6= 0 Result of explicit computation 1−loop • Scaling symmetries ⇒ ∆Sbosonic ∼ O (∂r )2 /r 2 , provided no cancellation Multiple Parallel Strings from ABJM? 1. Introduction • M-theory • ABJM 2. Wrapping M2 brane • Explicit computation : 3. Parallel strings k=1 Seff = Z k d 2 ξ − ∂¯r A ∂r A 2π N (∂¯r · r )2 + (¯r · ∂r )2 5N (¯r · ∂r )(r · ∂¯r ) − − 4π r4 2π r4 (ψψ)2 +O r4 m 1-loop deformation of susy transformation law N B ψψ A I AB µ I ˜ δψ = −Γ γ ∂µ (1 + 2 2 )r + O kr r3 • From AdS4 × S 7 • Bulk vs. boundary 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary No ‘non-renormalization theorem’ for the kinetic term, in contrast to the case of D-branes. I Physical interpretaion ? non-trivial kinetic term ⇔ flat transverse metric ? (6= ordinary gravitational force) • Some kind of “Casimir energy”, suggesting that the transverse space is not flat even for k = 1. I Mathematical characterization ? N = 6 susy 2D non-linear sigma model However, there is no direct contradiction with the possible “multiple parallel strings / N = 6 BF gauge theory ” correspondence which requires N N & = gs−1/3 N 2/3 1 2 r (gs N)1/3 Multiple Parallel Strings from ABJM? 1. Introduction • M-theory • ABJM 2. Wrapping M2 brane 3. Parallel strings • From AdS4 × S 7 • Bulk vs. boundary 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary Large N non-perturbative behavior? Multiple Parallel Strings from ABJM? The scaling symmetry constrains the non-perturbative form of the 1. Introduction • M-theory gs = 0 effective action as • ABJM Z 2. Wrapping k k=1 M2 brane Seff = d 2 ξ − ∂¯r A ∂r A 2π 3. Parallel strings 2 2 N (∂¯r · r ) + (¯r · ∂r ) N (¯r · ∂r )(r · ∂¯r ) •AdSFrom× S 7 4 −f1 − f2 2 4 2 • Bulk vs. kr r kr r4 boundary Assuming that the limit r → 0 is smooth for a fixed N, it seems reasonable to expect that f1 (x) → c1 /x 2 , f2 (x) → c2 /x 2 Then in the near-horizon region at finite fixed k, N −1/3 2/3 f1 ∼ f g N →0 similarly for f2 1 s kr 2 It is plausible that ABJM model is non-perturbatively consistent with “ multiple parallel strings / N = 6 BF gauge theory ” correspondence 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary 5. Discussion Multiple Parallel Strings from ABJM? 1. Introduction • M-theory • ABJM • Comment : case of BLG model 2. Wrapping M2 brane 3. Parallel strings I A4 (SO(4)) BLG model with manifest SO(8) R-symetry is equivalent to ABJM model with gauge group SU(2)×SU(2) but I I different classical moduli space : R8 × R8 /D2k (D2k =dihedral group of order 4k) for k = 1, (roughly speaking) two M2 branes in the (transverse) orbifold space R8 /Z2 . • From AdS4 × S 7 • Bulk vs. boundary 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary Multiple Parallel Strings from ABJM? I enhancement of R-symmetry to SO(8) is only kinematical 1. Introduction (0, r A ) in ABJMk=1 for N = 2 → z I (I = 1, 2, . . . , 8) in LBG Ak=1 4 with a particular (SO(8)-invariant) constraint • M-theory • ABJM 2. Wrapping M2 brane 3. Parallel strings • From AdS4 × S 7 • Bulk vs. boundary 4. Effective action z ·z =0 and then (r 2 = z · z) (r · ∂r )(r · ∂r ) r4 ↓ (z · ∂z)(z · ∂z) r4 z=0 • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion (SU(4) invariant) (SO(8) invariant) • Summary Summary Multiple Parallel Strings from ABJM? 1. Introduction We have examined the consistency of ABJM (and BLG) theory with M-theory conjecture. I scaling behavior matches between bulk sugra picture and gauge theory at the boundary I usual non-renormaltization theorem for the kinetic term is not valid in perturbation theory I suggest the existence of some nontrivial 2D non-linear sigma model with N = 6 susy, representing perhaps some kind of Casimir effect I plausibility argument for non-perturbative consistency in the large N limit Seems worthwhile pursue further. I For instance, relation between this theory and the matrix-string theory picture of wrapped membranes. • M-theory • ABJM 2. Wrapping M2 brane 3. Parallel strings • From AdS4 × S 7 • Bulk vs. boundary 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary Multiple Parallel Strings from ABJM? What’s next? 1. Introduction • M-theory • ABJM 2. Wrapping M2 brane 3. Parallel strings “Find new wisdoms through old things.” (Confucius 551-479 BC) • From AdS4 × S 7 • Bulk vs. boundary 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary For my own approach, see my talk(s) in KEK workshop(s) last year. http://hep1.c.u-tokyo.ac.jp/ tam/jp.html also arXiv:0804:0297[hep-th], arXiv:0706.0642[hep-th] Multiple Parallel Strings from ABJM? 1. Introduction • M-theory • ABJM 2. Wrapping M2 brane 3. Parallel strings Thank you! • From AdS4 × S 7 • Bulk vs. boundary 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary

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