Toward M5-branes from ABJM action

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

2. 1/18 §1Introduction From M theory to type IIA superstring theory compactify on S1 Type IIA Superstring theory Fundamental string D2 brane D4 brane NS5 brane M Theory M2 brane M5 brane Wrap on S1 Unwrap on S1 Wrap on S1 Unwrap on S1

3. 2/18 N D2-branes ・・・ World Volume Theory of N D2 branes || 3 dimensional U(N) Supersymmetric Gauge theory IIA S1 compactification N M2 branes World Volume Theory of N M2 branes ↓ ・・・ M ABJM model !! (Hopeful candidate) By Aharony, Bergman, Jaffris, Maldacena ArXiv: 0806.1218 [hep-th]

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

5. Approach to M5-brane 4/18 N M2-branes (N →∞) ABJM model N D2-branes (N →∞) 3 dim SYM S1 compactification We found a classical solution!! M5-brane (with non-zero flux) D4-brane (with non-zero flux ∝ 1/Θ) S1 compactification

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

7. 6/18 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

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

9. 8/18 ABJM theory is proposed to be the world volume theory of N M2-branes probing C4/Zk k:Chern-Simons level ABJM theory satisfies various property which are expected to the M2-branes probing C4/Zk

10. 9/18 World volume theory of D2 branes (3dim SYM) is obtained from ABJM model by S1 compactification. ABJM model U(N) ×U(N) 3dim SYM theory U(N) R7×S1 C4/Zk v : the distance between the M2 and singularity (v.e.v of a scalar field) Scaling limit v → ∞, k → ∞, v / k : fixed Mukhi et.al. , ABJM, Homma-Iso-Sumitomo-Zhang

11. 10/18 §3Classical solution of ABJM model N M2-branes (N →∞) ABJM model N D2-branes (N →∞) 3 dim SYM S1 compactification v → ∞, k → ∞, v / k : fixed We found a classical solution!! v → ∞ D4-brane (with non-zero flux ∝ 1/Θ) M5-brane (with non-zero flux) S1 compactification

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

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

14. 13/18 Claim We found a “M5-brane solution”, whose configuraiton is 0 1 2 3(r) 4(r’) 5(θ) 6 7 8 9 10 M2 ○ ○ ○ M5 ○ ○ ○ ○ 　○ 　　○ Compactified S1 direction Subtlety We cannot see the S1 direction manifestly. (Similar situation for M2 ⊥M5 bound state)

15. 14/18 §4Evidence for the claim ・Corresponding configuration of M5-brane with constant flux is a solution of the e.o.m from thesingleM5-brane action. ・We find the agreementbetween the tension of the M5-brane solution in the ABJM action and the one computed from single M5-brane action.

16. 15/18 1st Evidence: Satisfying the e.o.m Configuration of M5-brane with constant flux (Non-linear self-duality condition) satisfies the equations of motion from the M5-brane action!! 0 1 2 3(r) 4(r’) 5(θ) 6 7 8 9 10 M2 ○ ○ ○ M5 ○ ○ ○ ○ 　○ 　　○ By dimensional reduction of the M5-brane world volume theory (4+1)dim Non-commutative SYM (Seiberg - Witten)

17. 2nd Evidence: Matching of Tension 16/18 Tension of the M5-brane from ABJM action Tension Volume factor Tension of the M5-brane world volume action Tension

18. §5 Conclusion and discussion 17/18 Conclusion ・We gave a classical solution of the ABJM model, which reduce to D4-branesolution [X1,X2] = 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.

19. 18/18 Discussion ・Multiple M5 branes ・Fluctuation from the classical solution 　→ World volume theory of M5-branes ・ S1 direction which M5-brane is wrapping ・Contribution of monopole operators ・Relation to three algebra

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

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

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 S1 direction!! Similar problem in Terashima (‘08) Nastase, Papageorgakis, Ramgoolam (‘09) (another M2-M5 bound state) The M5-brane should extend not only to the direction of r and r’ but also to S1 direction Compactified S1 direction

26. Why we cannot see the extension of the M5-brane to the compactified S1 direction? Explanation 0 Because three algebra structure is not manifest in theABJM model . Ｃｆ Nambu-Poisson bracket (Ho,Matsuo ‘08) Structure like is needed ? Explanation 1 Because we calculate the solution perturbatively from the D4-brane solution. Wrapping on S1 is the non-perturbative effect

27. Explanation 2 Because we take large N limit. (Nastase et.al ‘09) N →∞ with λ=N/k: fixed k → ∞ Type IIA limit (S1 cannot be seen) Or Strong coupling limit (Classical solution is no more reliable) N →∞ with k: fixed λ → ∞ Although we cannot see the S1 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.

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 toM5-brane !! (Ho, Matsuo)

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

33. TypeⅠ Superstirng (S1 compactified) SO(32) Heterotic Superstirng (S1 compactified) TypeⅡA Superstirng Compactify on S1 ×S1/Z2 (cylinder) Compactify on S1 Compactify on S1/Z2 × S1 (cylinder) M Theory Compactify on S1×S1 （T2） Compactify on S1/Z2 Low energy limit (no compactification) Type ⅡB Superstring (S1 compactified) E8×E8 Heterotic Superstirng 11 dimensional Supergravity

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

35. What is the low energy effective theory on multiple M2-branes? ABJM model !! (Hopeful candidate) ?? By Aharony, Bergman, Jaffris, Maldacena ArXiv: 0806.1218 [hep-th] Quantization of open membrane? ?? ?? Comment If you believe the strongest version of AdS/CFT correspondence, world volume theory gives M theory in AdS4×S7 background. ・・・ N M2 branes How about multiple M5-branes ?

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