Multiple Parallel Strings from ABJM? 1. Introduction • M-theory • ABJM 2. Wrapping Multiple Parallel Strings and M2 brane 3. Parallel strings Wrapped M2 Branes 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) E8 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) string 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 ∼ 1 • 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-string theory • Bulk vs. boundary [1 + 1D SYM with coupling 1=gs (N ! 1)] 4. Effective action p • (N + 1) ! I large N limit with gYM = 1= gs ! 1 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) gauge theory 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 ^ 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 −A^ @ A^ − A^ A^ A^ ) CS 4π µ ν λ 3 µ ν λ µ ν λ 3 µ ν λ Multiple Parallel Strings 4 N from ABJM? • Classical moduli space = (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 ^ ^ 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 ^ ^ −1 ^ @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 : : : 2 (0; 1)) from ABJM? Z k 2 h µ A 1 A A SPS = d x Tr −D X DµXA+ (BX^ −X B)(BXA−XAB^) 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 ^ ^ ^ ^ ^ 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 ! 1]=[weak string coupling]. I Moduli-space approximation seems good for jpj 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 p 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.