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

2. 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

3. 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

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

5. 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)

6. 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.

7. 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

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

9. 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

10. 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.

11. 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

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

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

14. 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

15. ・ 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

16. 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 ○ ○ ○ ○ ○ ○

17. 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

18. ・ 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

19. 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

22. 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.

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

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

25. 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)

26. 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)

27. 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

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

30. 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)

31. Three Algebra Metric is positive definite Anti-symmetry Fundamental identity

33. 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)

34. 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!!

35. 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 ?

36. 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!!