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Toward M5-branes from ABJM action Based on going project with Seiji Terashima (YITP, Kyoto U. ） Futoshi Yagi (YITP, Kyoto U.)

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Type IIA Superstring theory Fundamental string D2 brane D4 brane NS5 brane compactify on S 1 Wrap on S 1 Unwrap on S 1 M Theory M2 brane M5 brane Wrap on S 1 Unwrap on S 1 From M theory to type IIA superstring theory §1 Introduction 1/18

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N D2-branes World Volume Theory of N D2 branes | 3 dimensional U(N) Supersymmetric Gauge theory ABJM model !! (Hopeful candidate) N M2 branes By Aharony, Bergman, Jaffris, Maldacena ArXiv: 0806.1218 [hep-th] World Volume Theory of N M2 branes ↓ ・・・ S 1 compactification IIA M 2/18

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Theme Find multiple M5-branes action 3/18 “Toward” M5-branes from ABJM action

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N M2-branes (N →∞) ABJM model N D2-branes (N →∞) 3 dim SYM M5-brane (with non-zero flux) S 1 compactification Approach to M5-brane We found a classical solution!! 4/18 D4-brane (with non-zero flux ∝ 1/Θ)

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M2-branes M5-branes M2-branes 0 1 2 3 4 5 6 7 8 9 10 M2 ○ ○ ○ M5 ○ ○ ○ ○ ○ ○ 0 1 2 3 4 5 6 7 8 9 10 M2 ○ ○ ○ M5 ○ ○ ○ ○ ○ ○ Our classical solution A classical solution already studied Terashima, Gomis, Rodriguez-Gomez, Van Raamsdonk, Verlinde Hanaki, Lin Nastase, Papageorgakisb, Ramgoolamc 5/18 *Non-BPS solution * M5 brane looks like D4 brane.

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Plan of this talk §1 Introduction §2 Brief review of ABJM model §3 Classical solution of the ABJM model §4 Evidence for the claim §5 Conclusion and discussion 6/18

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By Aharony, Bergman, Jaffris, Maldacena ArXiv: 0806.1218 [hep-th] §2 Brief review of ABJM model 7/18 Complex scalars Dirac Spinors Gauge fields

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ABJM theory is proposed to be the world volume theory of N M2-branes probing C 4 /Z k k: Chern-Simons level 8/18 ABJM theory satisfies various property which are expected to the M2-branes probing C 4 /Z k

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v : the distance between the M2 and singularity (v.e.v of a scalar field) Scaling limit v → ∞, k → ∞, v / k : fixed ABJM model U(N) ×U(N) 3dim SYM theory U(N) Mukhi et.al., ABJM, Homma-Iso-Sumitomo-Zhang 9/18 C 4 /Z k R 7 ×S 1 World volume theory of D2 branes (3dim SYM) is obtained from ABJM model by S 1 compactification.

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N M2-branes (N →∞) ABJM model N D2-branes (N →∞) 3 dim SYM M5-brane (with non-zero flux) D4-brane (with non-zero flux ∝ 1/Θ) S 1 compactification v → ∞, k → ∞, v / k : fixed S 1 compactification We found a classical solution!! v → ∞ §3 Classical solution of ABJM model 10/18

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Bosonic potential of the ABJM action Ansatz (the solution becomes D2-D4 in the limit v → ∞) 11/18

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e.o.m. Due to the special relation a perturbative solution exists. Solution in the limit Θ→0 12/18

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We found a “ M5-brane solution ”, whose configuraiton is Claim Subtlety We cannot see the S 1 direction manifestly. (Similar situation for M2 ⊥ M5 bound state) 0 1 2 3(r) 4(r ’ ) 5(θ) 6 7 8 9 10 M2 ○ ○ ○ M5 ○ ○ ○ ○ ○ ○ Compactified S 1 direction 13/18

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・ Corresponding configuration of M5- brane with constant flux is a solution of the e.o.m from the single M5-brane action. ・ We find the agreement between the tension of the M5-brane solution in the ABJM action and the one computed from single M5-brane action. §4 Evidence for the claim 14/18

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Configuration of M5-brane with constant flux satisfies the equations of motion from the M5-brane action!! 1 st Evidence: Satisfying the e.o.m By dimensional reduction of the M5-brane world volume theory (4+1)dim Non-commutative SYM (Seiberg - Witten) 15/18 (Non-linear self-duality condition) 0 1 2 3(r) 4(r ’ ) 5(θ) 6 7 8 9 10 M2 ○ ○ ○ M5 ○ ○ ○ ○ ○ ○

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2 nd Evidence: Matching of Tension Tension of the M5-brane from ABJM action 16/18 Volume factor Tension Tension of the M5-brane world volume action Tension

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・ We gave a classical solution of the ABJM model, which reduce to D4-brane solution [X 1,X 2 ] = iΘ in the scaling limit. We interpret this solution as a “ M5-brane solution ” from ABJM model. ・ We gave a several consistency checks that it indeed represents M5-brane. ・ Corresponding configuration with constant magnetic flux is a solution of the e.o.m of M5-brane world volume action. ・ We find the agreement between the tension of the M5-brane solution in the ABJM action and the one computed from M5-brane world volume action. §5 Conclusion and discussion Conclusion 17/18

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Discussion ・ Multiple M5 branes ・ Fluctuation from the classical solution → World volume theory of M5-branes ・ S 1 direction which M5-brane is wrapping ・ Contribution of monopole operators ・ Relation to three algebra 18/18

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Thus, there exist perturbative solutions for these equations. Perturbative solution is We interpret that this solution represents M5-brane * The product in this equation is Weyl ordered product.

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M5-brane action auxiliary field Gauge symmetry for simplicity Pasti, Sorokin, Tonin ( ‘ 97)

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Non-linear self duality condition = Equations of motion for a(x)

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Crucial problem of our work We cannot see the extension to the compactified S 1 direction!! Compactified S 1 direction The M5-brane should extend not only to the direction of r and r ’ but also to S 1 direction Similar problem in Terashima ( ‘ 08) Nastase, Papageorgakis, Ramgoolam ( ‘ 09) (another M2-M5 bound state)

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Why we cannot see the extension of 1 the M5-brane to the compactified S 1 direction? Explanation 0 Because three algebra structure is not manifest in theABJM model. Structure like is needed ? Explanation 1 Because we calculate the solution perturbatively from the D4-brane solution. Wrapping on S 1 is the non-perturbative effect Ｃｆ Nambu-Poisson bracket (Ho,Matsuo ‘ 08)

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Explanation 2 Because we take large N limit. N →∞ with λ=N/k: fixedk → ∞ Type IIA limit (S 1 cannot be seen) Although we cannot see the S 1 direction, we still claim that this classical solution correspond to M5-brane and that some aspects of M5 brane can be seen from this solution. (Nastase et.al ‘ 09) N →∞ with k: fixedλ → ∞ Strong coupling limit (Classical solution is no more reliable) Or

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e.o.m. (infinite dimensional nonlinear PDE)

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Comment on BLG model Bagger, Lambert ( ‘ 07) Gustavson ( ‘ 07) ・ Candidate of multiple M2-branes proposed before the ABJM model ・ Gauge symmetry is based on three algebra ・ Only one three algebra! → Two M2 branes case Thus, we need ABJM model (N M2 branes are describable )!! (Or we should release the constraint to three algebra) Not positive definite → Reduce to D2 brane after removing the ghost Not totally antisymmetric → Turn out to include the ABJM model (Bagger, Lambert) Infinite dimensional (Nambu-Poisson bracket) → Correspond to M5-brane !!(Ho, Matsuo)

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Three Algebra Metric is positive definite Anti-symmetry Fundamental identity

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M Theory Type Ⅱ A Superstirng Type Ⅰ Superstirng (S 1 compactified) Compactify on S 1 × S 1 /Z 2 (cylinder) SO(32) Heterotic Superstirng (S 1 compactified) Type Ⅱ B Superstring (S 1 compactified) E 8 ×E 8 Heterotic Superstirng Compactify on S 1 Compactify on S 1 /Z 2 Compactify on S 1 ×S 1 （ T 2 ） Low energy limit (no compactification) 11 dimensional Supergravity Compactify on S 1 /Z 2 × S 1 (cylinder)

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N Dp branes in 10 dimensional Minkowski Space ・・・ N Dp-branes (p+1 dim. object) Quantize the oscillation mode of the string and pick up massless mode World Volume Theory of N Dp branes || p+1 dimensional U(N) Supersymmetric Gauge theory Open string ： End points are on the Dp branes Non-perturbative aspects of superstring theory can be captured by studying this SYM theory!!

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ABJM model !! (Hopeful candidate) N M2 branes What is the low energy effective theory on multiple M2-branes? ・・・ Quantization of open membrane? ?? Comment If you believe the strongest version of AdS/CFT correspondence, world volume theory gives M theory in AdS 4 ×S 7 background. By Aharony, Bergman, Jaffris, Maldacena ArXiv: 0806.1218 [hep-th] How about multiple M5-branes ?

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But ！！ Formulation of M theory has not been established yet. Quantization of membranes are difficult. (partially because there are no free parameters.) It is expected that 5 types of superstring theories can be understood in a unified manner through ``M theory’’ To study “quantum M-theory” is important and challenging problem!!

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