Multiple Parallel Strings from ABJM?

1. Introduction • M-theory • ABJM 2. Wrapping Multiple Parallel Strings and M2 3. Parallel strings Wrapped M2 from • From 7 AdS4 × S • Bulk vs. the ABJM Model ? boundary 4. Effective action • (N + 1) → Tamiaki Yoneya (N) + (1) • Explicit computation • Large N University of Tokyo - Komaba 5. Discussion • Summary KEK 2009 Theory Workshop Multiple Parallel Strings from ABJM?

1. Introduction Contents • M-theory • ABJM 2. Wrapping 1. Introduction M2 brane 3. Parallel strings • From 7 AdS4 × S 2. Wrapped M2 branes from ABJM • Bulk vs. boundary 4. Effective 3. Parallel strings action • (N + 1) → (N) + (1) • Explicit computation 4. Effective action for parallel strings • Large N 5. Discussion • Summary 5. Discussion Multiple 1. Introduction Parallel Strings from ABJM? Revisiting the M-theory conjecture

1. Introduction • M-theory First recall the conjecture. • ABJM 2. Wrapping M2 brane

3. Parallel 11 M Theory strings Circle Circle/Z2 • From 7 spacetime AdS4 × S dimensions • Bulk vs. boundary 10 Type IIA Type IIB Type I Hetero Hetero SO(32) x E8 4. Effective action • (N + 1) → (N) + (1) 9 Type • Explicit Type I Hetero IIAB 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. Multiple Parallel Strings from ABJM? I Radius of the circle direction :

1. Introduction R11 = gs `s • M-theory • ABJM 2. Wrapping M2 brane as gs → 0 M2 brane 3. Parallel I ”longitudinal”: wrapped along the 11-th circle direction strings • From 7 ⇒ (fundamental) AdS4 × S • Bulk vs. boundary

I ”transverse” : extended along directions orthogonal to the 11-th 4. Effective action circle • (N + 1) → (N) + (1) • Explicit ⇒ D2 brane computation • Large N 5. Discussion I Fundamental length scale of M theory = Planck scale • Summary

1/3 `P = gs `s  R11 and `P  `s

as gs → 0. Multiple Parallel Strings ⇓ from ABJM?

In the weak coupling (∼ 10 dimensional) limit, M2 branes should 1. Introduction smoothly reduce to perturbative strings of type IIA theory. • M-theory • ABJM 2. Wrapping M2 brane

We would like to discuss this question in the context of the ABJM 3. Parallel strings model, a candidate low-enegy theory for multiple M2 branes, in the • From 7 AdS4 × S simplest possible setting. • Bulk vs. boundary 4. Effective • So far, almost all previous works have been focused on the action • (N + 1) → ”transverse” configurations of M2 branes: (N) + (1) • Explicit computation weak coupling limit k ∼ ∞ • Large N 5. Discussion • Summary m

7 S , which is transverse to M2 branes, into CP3

7 1 S /Zk → CP3, Zk → S ∼ M-theory circle

2π/k = Chern-Simons coupling constant Multiple • Remark: case of single M2 brane Parallel Strings from ABJM? The dynamics of a single M2 brane is already quite non-trivial, and hence the reduction to string(s) is not completely understood, 1. Introduction quantum-mechanically. • M-theory • ABJM Sekino-TY, hep-th/0108176 , Asano-Sekino-TY, hep-th/0308024 2. Wrapping M2 brane I wrapped M2 brane 3. Parallel ⇓ directly strings • From 7 AdS4 × S matrix- • Bulk vs. boundary [1 + 1D SYM with coupling 1/gs (N → ∞)] 4. Effective action √ • (N + 1) → I large N limit with gYM = 1/ gs → ∞ can be studied by using (N) + (1) • Explicit GKPW relation in the PP (BMN)-wave limit, under the computation assumption of gauge/gravity correspondence. • Large N 5. Discussion • Summary I 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 Main features of the ABJM model from ABJM?

1. Introduction • M-theory • ABJM 2. Wrapping M2 brane

3. Parallel strings I Susy Chern-Simons U(N)×U(N) in 3D with • From 7 SO(6)(∼SU(4)) R-symmetry AdS4 × S • Bulk vs. boundary I (super)Conformal invariant 4. Effective action I CS coupling = 2π/k with level number k • (N + 1) → 4 (N) + (1) ⇔ C /Zk = transverse space of M2 branes • Explicit computation I AdS/CFT correspondence at k = 1: • Large N 7 AdS4 × S ⇔ effective CFT of N M2 branes in flat 11D 5. Discussion • Summary I But, only with N = 6 susy, manifestly. Multiple • Notations Parallel Strings from ABJM? (following Bandres-Lipstein-Schwarz, 0807.0880)

A 1. Introduction I bosonic fields: (XA, X ) (4, 4 of SU(4)) • M-theory A • ABJM I fermionic fields: (Ψ , Ψ ) A 2. Wrapping (4, 4 of SU(4), 3D 2-component spinor) M2 brane ˆ 3. Parallel I Chern-Simons U(N)×U(N) gauge fields: (Aµ, Aµ) strings • From 7 AdS4 × S • Bulk vs. • Action boundary Z 4. Effective k 3 h µ A µ Ai action SABJM = d x Tr − D X DµXA + iΨ¯ Aγ DµΨ • (N + 1) → 2π (N) + (1) • Explicit computation Z • Large N k 3 +SCS + d x (L6 + L2,2) 5. Discussion 2π • Summary

6 L6 = potential term of O(X ), 2 2 L2,2 = X Ψcoupling terms of O(X Ψ )

k Z h 2i 2i i S = d 3x µνλTr A ∂ A + A A A −Aˆ ∂ Aˆ − Aˆ Aˆ Aˆ ) CS 4π µ ν λ 3 µ ν λ µ ν λ 3 µ ν λ Multiple Parallel Strings 4 N from ABJM? • Classical = (C /Zk ) /SN N I residual gauge symmetry: (U(1)×U(1)) /SN 1. Introduction • M-theory A A 2πi/k A • ABJM I X → diagonal matrices with identification X = e X 2. Wrapping 8 M2 brane I At k = 1, R /SN ⇔ N M2 branes in flat space 3. Parallel strings • From 7 AdS4 × S Would like to study, in the case k = 1, whether we can understand • Bulk vs. boundary ordinary strings by wrapping M2 branes along the M-circle. 4. Effective action But, that is in the strong-coupling regime! • (N + 1) → (N) + (1) • Explicit computation Will however see that after the reduction due to wrapping, • Large N the effective coupling constant is 5. Discussion • Summary N kr 2 r = transverse distance scale among strings Multiple 2. Wrapped M2 brane from ABJM Parallel Strings from ABJM? Double dimensional reduction

1. Introduction • M-theory The ABJM model implicitly assumes the static gauge for world-volume • ABJM µ 2. Wrapping coordinates: world 3-coordinates x = longitudinal 3 directions of 11D M2 brane

3. Parallel ⇓ strings • From 7 AdS4 × S • Bulk vs. Wrapping along the M-circle in 11-th direction can be performed by boundary the “double” dimensional reduction (gs  1) 4. Effective action • (N + 1) → I Recover the length dimension with respect to target space by (N) + (1) • Explicit A A µ −1 A A µ ˆ ˆ computation (X , Ψ , x ) → `P (X , Ψ , x ), (Aµ, Aµ) → `P (Aµ, Aµ) • Large N 5. Discussion • Summary I gauge fixing along the periodic direction ˆ ˆ −1 ˆ ∂2A2 = 0 = ∂2A2, (A2, A2) ≡ R11 (B, B)

I x2 = x2 + 2πR11, R11 = gs `s Z Z 3 2 d x → R11 d x, ∂2 → 0 for all fields Multiple 5 Parallel Strings • Reduced 2D action (µ, ν, λ . . . ∈ (0, 1)) from ABJM? Z k 2 h µ A 1 A A SPS = d x Tr −D X DµXA+ (BXˆ −X B)(BXA−XABˆ) 1. Introduction 2π`2 g 2`2 • M-theory s s s • ABJM Z 2. Wrapping i k 2 M2 brane + ... + SBF + d x (L6 + L2,2) 2π`2 3. Parallel s strings • From k Z   AdS × S7 2 νλ ˆ ˆ ˆ ˆ ˆ 4 SBF = d x  Tr B∂ν Aλ + iBAν Aλ − B∂ν Aλ − iBAν Aλ • Bulk vs. 2π boundary 4. Effective 1  A B C  action L6 = Tr X XAX XB X XC + ··· • (N + 1) → 3g 2`6 (N) + (1) s s • Explicit computation 1  ABCD  • Large N L2,2 = Tr i Ψ¯ AXB ΨC XD + ··· 4/3 4 5. Discussion gs `s • Summary

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 3. Parallel strings Parallel Strings from ABJM? 7 Parallel strings from AdS4×S

1. Introduction • M-theory On the bulk side, start from the M2 brane metric • ABJM 2. Wrapping M2 brane 32π2N`6 ds2 = h−2/3(−dt2 +dx 2 +dx 2)+h1/3(dr 2 +dΩ2), h = 1+ P 3. Parallel 11 1 2 7 r 6 strings • From 7 AdS4 × S • Bulk vs. Using the usual relation between 11D and 10D string-frame, boundary 4. Effective 2 −2φ/3 2 4φ/3 2 action ds11 = e dsstring + e dx2 • (N + 1) → (N) + (1) • Explicit computation the background fields around N parallel strings • Large N stretching along x2 is 5. Discussion • Summary 2 −1 2 2 2 2 dsstring = h (−dt + dx1 ) + dr + dΩ7

φ −1/2 −1 e = h , B01 = h Remarks: Multiple Parallel Strings from ABJM? I BPS ⇔ −g00 = g11 = B01

The world-sheet string action is completely free 1. Introduction • M-theory Z • ABJM 1 2 √  µν µν  A B 2. Wrapping Sstring = − 2 d ξ −γ gAB (X )γ + BAB (X ) ∂µX ∂ν X 4π`s M2 brane Z 3. Parallel 1 2 X A µ strings = − d x ∂µX (x)∂ XA(x) • From 2 7 4π`s AdS4 × S A=transverse • Bulk vs. boundary 0 0 1 4. Effective in the static (conformal) gauge ξ = t = X , ξ = x1 = X1 and action is manifestly SO(8) symmetric. • (N + 1) → (N) + (1) • Explicit Near-horizon limit: r  (g 2N)1/6 (Q ∝ Ng 2`6) computation I s s s • Large N 5. Discussion 6 r • Summary ds2 = (−dt2 + dx 2) + dr 2 + r 2dΩ2 string Q 7 ⇒ scaling symmetry:

1/2 −1 2 −1 2 (I): r → λ r, (t, x), → λ (t, x), dsstring → λ dsstring

(II):(t, x) → ρ(t, x), gs → ρgs Multiple Parallel Strings Bulk vs. boundary from ABJM?

1. Introduction • M-theory • ABJM 2. Wrapping The structure of ABJM moduli space seems consistent with the above M2 brane 3. Parallel properties on the bulk side, at least classically. strings • From 7 AdS4 × S • Bulk vs. • Question: What about the quantum corrections ? boundary 4. Effective I enhancement of R symmetry? action • (N + 1) → (N) + (1) I cancellation of all interactions? • Explicit computation • Large N The question is essentially non-perturbative in its nature. 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 1. Introduction I (I) : inherited from 3D • M-theory • ABJM 1/2 −1 2 −1 2 r → λ r, (t, x), → λ (t, x), ds → λ ds 2. Wrapping string string M2 brane m 3. Parallel strings • From ˆ ˆ ˆ ˆ 1/2 7 (Aµ, B, Aµ, B) → λ(Aµ, B, Aµ, B), XA → λ XA, AdS4 × S • Bulk vs. ¯ A ¯ A −1 boundary (ΨA, Ψ ) → λ(ΨA, Ψ ), k → λ k 4. Effective action • (N + 1) → (N) + (1) • Explicit (II) : related to 2D conformal symmetry computation I • Large N (reminiscent of matrix-string theory) 5. Discussion • Summary (t, x) → ρ(t, x), gs → ρgs m A −1/2 A −1 (ΨA, Ψ¯ ) → ρ (ΨA, Ψ¯ ), (Aµ, Aˆµ) → ρ (Aµ, Aˆµ) Multiple Let the (transverse) distance scale among parallel strings be r. The Parallel Strings scaling symmetries constrain the (bosonic part of) effective action as from ABJM? (string unit : ` = 1) s 1. Introduction • M-theory ∞ Z q • ABJM X 2 −L+1 q−2 −2L+6∂r  Seff = cL,q,g,h d x k g r 2. Wrapping s r 3 M2 brane L=0,q=2,g=0,h=0 3. Parallel strings • From 2−2g−h+L−1 7 ×N AdS4 × S • Bulk vs. boundary L = # of loops, q = # of derivatives 4. Effective g =genus, h = # of holes action • (N + 1) → with respect to color index loops in planar expansion (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 • M-theory • Unfortunately, the near-horizon limit on the bulk side is not • ABJM 2. Wrapping compatible with the perturbative regime of the reduced action for M2 brane finite fixed k and for weak string coupling, 3. Parallel strings • From 7 AdS4 × S (as typical AdS/CFT correspondence !) • Bulk vs. since boundary 4. Effective action 2 1/6 1/2 • (N + 1) → near horizon condition : r  (gs N) ⇔ r  (N/k) (N) + (1) • Explicit computation ⇓ • Large N 5. Discussion 3/2 1  gs N/k • Summary Multiple • However, independently of the near-horizon condition, we can study Parallel Strings effective actions for our‘ would-be’ gauge theory of multiple parallel from ABJM? strings, for sufficiently large r 1. Introduction • M-theory r  N1/2 at k = 1, N = finite • ABJM 2. Wrapping M2 brane

• Relevant question : 3. Parallel strings susy ‘non-renormalization theorem’ for kinetic terms, valid or not? • From 7 AdS4 × S • Bulk vs. boundary I In the case of D-brane susy Yang-Mills theories, 4. Effective non-renormalization theorems are at work. action • (N + 1) → • loop corrections start from v 4/r 7−p (N) + (1) • Explicit • seems to be case also for AdS4 × P3 ( k → ∞) in one-loop order. computation C • Large N Not only that, SYM can correctly reproduce the long-distance 5. Discussion gravitational interactions (even 3-body forces!) among D-branes • Summary 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. Multiple 4. Effective action for parallel strings from Parallel Strings the reduced action from ABJM?

(N + 1) → (N) + (1) decompostion 1. Introduction • M-theory • ABJM 2. Wrapping Let us study one-loop effective action (L = 1) for simplest background M2 brane

A A 3. Parallel X = ( 0, 0,..., 0, r ), U(N + 1) → U(N) × U(1) strings • From | {z } 7 N AdS4 × S • Bulk vs. boundary • off-diagonal fluctuating fields: 4. Effective action (a = 1,..., N , all are complex N-vectors) • (N + 1) → (N) + (1) I two pairs of (4, 4) scalar fields • Explicit A A computation A A • Large N Ua , Ua , Va , V a 5. Discussion I their fermion partners (2D Dirac) • Summary A A A A Θa , Θa , Φa , Φa I pairs of 2D vector fields

Aµ a, Aµ a, Aˆµ a, Aˆµ a

I pairs of auxiliary scalar fields (originated from A2 a, Aˆ2 a)

Ba, Ba, Bˆa, Bˆ a A Multiple • Owing to the presence of the vacuum expectation value for X ,... Parallel Strings from ABJM? I Can integrate out the auxiliary fields B,... Can choose the followng special background-field gauge 1. Introduction I • M-theory • ABJM 1 µ 1 µ 2. Wrapping ∂ A − ir(r · V¯ ) = 0, ∂ A¯ + ir(¯r · V ) = 0 M2 brane r µ a a r µ a a 3. Parallel strings • From ⇓ 7 AdS4 × S • Bulk vs. I emergence of usual kinetic terms for fluctuating gauge fields boundary 4. Effective I mass terms are diagonalized with eigenvalues action • (N + 1) → (N) + (1) 4 4 4 4 2 • Explicit (r , r , r , r ) for complex scalars r = r · r computation • Large N 2 2 2 2 (r , r , r , r ) for Dirac fermions 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 Multiple Parallel Strings Result of explicit computation from ABJM?

1−loop 2 2 • Scaling symmetries ⇒ ∆Sbosonic ∼ O (∂r) /r , 1. Introduction • M-theory provided no cancellation • ABJM 2. Wrapping • Explicit computation : M2 brane

3. Parallel strings • From Z  7 k=1 2 k A A AdS4 × S Seff = d ξ − ∂¯r ∂r • Bulk vs. 2π boundary 2 2 4. Effective N (∂¯r · r) + (¯r · ∂r) 5N (¯r · ∂r)(r · ∂¯r) action − − • (N + 1) → 4π r 4 2π r 4 (N) + (1) • Explicit  2  computation (ψψ) • Large N + O r 4 5. Discussion • Summary m

1-loop deformation of susy transformation law

N ψψ  δψA = −Γ˜I AB γµI ∂ (1 + 2 )r B + O µ kr 2 r 3 Multiple No ‘non-renormalization theorem’ for the kinetic term, Parallel Strings from ABJM? in contrast to the case of D-branes.

I Physical interpretaion ? 1. Introduction • M-theory non-trivial kinetic term ⇔ flat transverse metric ? • ABJM (6= ordinary gravitational force) 2. Wrapping M2 brane • Some kind of “Casimir energy”, suggesting that the transverse 3. Parallel strings space is not flat even for k = 1. • From 7 AdS4 × S • Bulk vs. I Mathematical characterization ? boundary N = 6 susy 2D non-linear sigma model 4. Effective action • (N + 1) → (N) + (1) • Explicit computation However, there is no direct contradiction with the possible • Large N 5. Discussion “multiple parallel strings / N = 6 BF gauge theory ” • Summary correspondence which requires N N = g −1/3N2/3  1 2 & 1/3 s r (gs N) Multiple Parallel Strings Large N non-perturbative behavior? from ABJM?

The scaling symmetry constrains the non-perturbative form of the 1. Introduction gs = 0 effective action as • M-theory • ABJM Z  k 2. Wrapping k=1 2 A A M2 brane Seff = d ξ − ∂¯r ∂r 2π 3. Parallel strings 2 2     • From N (∂¯r · r) + (¯r · ∂r) N (¯r · ∂r)(r · ∂¯r) 7 −f1 − f2 AdS4 × S 2 4 2 4 • Bulk vs. kr r kr r boundary 4. Effective Assuming that the limit r → 0 is smooth for a fixed N, it seems action • (N + 1) → reasonable to expect that (N) + (1) • Explicit 2 2 computation f1(x) → c1/x , f2(x) → c2/x • Large N 5. Discussion Then in the near-horizon region at finite fixed k, • Summary  N    f ∼ f g −1/3N2/3 → 0 similarly for f 1 kr 2 1 s 2 It is plausible that ABJM model is non-perturbatively consistent with “ multiple parallel strings / N = 6 BF gauge theory ” correspondence Multiple Parallel Strings 5. Discussion from ABJM?

1. Introduction • M-theory • ABJM 2. Wrapping • Comment : case of BLG model M2 brane 3. Parallel strings • From I A4 (SO(4)) BLG model with manifest SO(8) R-symetry is 7 AdS4 × S equivalent to ABJM model with gauge group SU(2)×SU(2) • Bulk vs. boundary 4. Effective action but • (N + 1) → (N) + (1) • Explicit computation 8 8 • Large N I different classical moduli space: R × R /D2k (D2k =dihedral 5. Discussion group of order 4k) • Summary

I for k = 1, (roughly speaking) two M2 branes in the (transverse) 8 space R /Z2. Multiple Parallel Strings from ABJM? I enhancement of R-symmetry to SO(8) is only kinematical 1. Introduction A k=1 I k=1 • M-theory (0, r ) in ABJM for N = 2 → z (I = 1, 2,..., 8) in LBG A4 • ABJM 2. Wrapping with a particular (SO(8)-invariant) constraint M2 brane 3. Parallel strings • From 7 AdS4 × S • Bulk vs. boundary 4. Effective action • (N + 1) → 2 (N) + (1) z=0 z · z = 0 (r = z · z) • Explicit computation • Large N and then (r · ∂r)(r · ∂r) 5. Discussion (SU(4) invariant) • Summary r 4 ↓ (z · ∂z)(z · ∂z) (SO(8) invariant) r 4 Multiple Parallel Strings Summary from ABJM?

1. Introduction We have examined the consistency of ABJM (and BLG) theory with • M-theory • ABJM M-theory conjecture. 2. Wrapping M2 brane

I scaling behavior matches between bulk sugra picture and gauge 3. Parallel theory at the boundary strings • From 7 AdS4 × S I usual non-renormaltization theorem for the kinetic term is not • Bulk vs. valid in perturbation theory boundary 4. Effective action I suggest the existence of some nontrivial 2D non-linear sigma • (N + 1) → (N) + (1) model with N = 6 susy, representing perhaps some kind of • Explicit computation Casimir effect • Large N 5. Discussion I plausibility argument for non-perturbative consistency in the large • Summary N limit

Seems worthwhile pursue further.

I For instance, relation between this theory and the matrix-string theory picture of wrapped membranes. Multiple Parallel Strings from ABJM?

What’s next? 1. Introduction • M-theory • ABJM 2. Wrapping M2 brane

3. Parallel strings • From 7 AdS4 × S • Bulk vs. boundary “Find new wisdoms through old things.” 4. Effective (Confucius 551-479 BC) 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 • From 7 AdS4 × S • Bulk vs. Thank you! boundary 4. Effective action • (N + 1) → (N) + (1) • Explicit computation • Large N 5. Discussion • Summary