Multiple M5-branes and ABJM Theory

1. Multiple M5-branes and ABJM Theory Seiji Terashima (YITP, Kyoto) based on the works (JHEP 0912 (2009) 059 and arXiv:1012:3961, to appear in JHEP) with Futoshi Yagi (IHES) 2011 February 18 at NTU

2. 1. Introduction

3. Recent exciting progress in string theory: Low energy actions of multiple Membranes in M-theory was found ! Why this is so exciting?

4. For string theory, perturbation theory is well understood and we can compute, for example, scattering amplitudes of gravitons But, for M-theory, we do NOT have well defined perturbative description. (because quantization of membrane have serious problems, for example, no coupling constant and presence of continuous spectrum.)

5. For non-perturbative aspects of string theory, D-branes have been very important objects to understand Why D-branes are so useful? Because D-brane is described by perturbativeopen strings although they are non-perturbative objects → Yang-Mills action as multiple D-brane action! AdS/CFT, Matrix Model, MQCD, etc On the other hand, until very recently, multiple M2-brane action had not been obtained.

6. Recently, Bagger and Lambert (BLG) proposed multiple membrane actions, then Aharony, Bergman, Jafferis and Maldacena (ABJM) found different multiple membrane actions. We will understand many aspects of M-theory (and string theory) !

7. Many possible applications, ex. AdS4/CFT3 (3+1)d gravity theory ↔ (2+1)d field theory relevant to condensed matter physics, because the membrane action are Chern-Simons theories

8. M5-branesare more mysterious and interesting • For example, • On M5-branes, there is self dual 3-form field strength. • M5-branes on torus give N=4 SYM and S-duality should be manifest. • Seiberg-Witten curve is obtained for M5 on curve Thus, it is very interesting to find low energy action of multiple M5-branes

9. Single M5 → effective action is known (ex. Pasti-Sorokin-Tonin) From BLG action, single M5-action was obtained by Ho-Matsuo-Imamura-Shiba N M5-branes → effective action is NOT known. N³ degree of freedom We will consider effective action for multiple M5-branes via ABJM action

10. Is ABJM action useful to understand M5-branes? Bound states of M2-branes and M5-branes should be constructed in the M2-brane actions. (M-theory lift of D2-D4 bound state in IIA) We will have M5-brane action by considering fluctuations around the background representing M2-M5 bound states!

11. Indeed, we found solutions of the BPS equations of ABJM which describe the M5-branes ST, GRRV M2-branes M5-branes Fuzzy 3-sphere appears

12. M2-branes M5-branes Fuzzy 3-sphere appears This is an M-theory lift of D2-D4 described by t’Hooft-Polyakov Monopole or Nahm equation D2-branes D4-branes Fuzzy 2-sphere appears

13. How about the M-theory lift of usual D2-D4 bound state? This bound state is described by D4-brane with magnetic flux or noncommutative R² which would be easier to be analyzed. D4-branes with nonzero magnetic field F

14. D4-brane with magnetic flux or noncommutative R² = D2-D4 bound state D4-branes with nonzero magnetic field F M-theory lift of this? We construct such M2-M5 bound state in ABJM action! Yagi-ST The bound state is M5-branes with nonzero 3-form flux

15. Strategy to construct it N D2-branes (N →∞) 3 dim SYM N M2-branes (N →∞) ABJM model S1 compactification ? M5-brane (with non-zero 3-form flux) D4-brane (with non-zero flux ∝ 1/Θ) S1 compactification

16. Strategy to construct it N D2-branes (N →∞) 3 dim SYM N M2-branes (N →∞) ABJM model S1 compactification We found a classical solution! M5-brane (with non-zero 3-form flux) D4-brane (with non-zero flux ∝ 1/Θ) S1 compactification

17. Interestingly, Our solution is closely related to the Lie 3-algebra, although this is in ABJM, not BLG. Lie 3-bracket = self-dual 3-form flux and Nambu bracket is hidden. →3-algebra may describe multiple M5-brane action. We also calculate fluctuations from M5-brane solution. D4-brane-like action but the gauge coupling constant depends on the spacetime coordinate obtained. → consistent with the properties of M5-brane action.

18. 2. M2-branes and ABJM action

19. Consider M2-branes in M-theory compactified on S¹ M-theory on S¹ = IIA string in 10d (Radius of S¹ ~ string coupling) Thus, M-theory is the strong coupling limit of IIA string, and M2 wrapping S¹ = fund. string in IIA M2 at a point in S¹ = D2 in IIA M5 wrapping S¹ = D4 in IIA M5 at a point in S¹ = NS5 in IIA M2-M5 ← D2-D4

20. D2-brane effective action is (2+1)d N=8 Yang-Mills theory which has 7 scalars = location of D2-brane 16 SUSY and SO(7) global symmetry Not Conformal (Yang-Mills coupling is not dimensionless) low energy limit = l_s → 0 with Yang-Mills coupling fixed (cut-off: 1/l_s , g_YM^2: g_s/l_s )

21. effective action of M2-brane on flat space should have 8 scalars = location of M2-brane 16 SUSY and SO(8) global symmetry Conformal symmetry (=not Yang-Mills theory) For (2+1)d Yang-Mills theory, Strong coupling limit = low energy limit M2-brane action = low energy limit of D2-brane action. Thus, we should solve the strong coupling dynamics. → very difficult. We want to find a conformal action for M2-brane

22. Fields in ABJM action: 4 complex scalars (A=1,2,3,4) bi-fundamental rep. of U(N) x U(N) 4 (2+1)d Dirac spinors bi-fundamental rep. of U(N) x U(N) (2+1)d U(N) x U(N) gauge fields , , ,

23. ABJM action:

24. ( (2+1)d N=6 ) SUSY transformation: Gaiotto-Giobi-Yin, Hosomichi et.el, Bagger-Lambert, ST, Bandres-Lipstein-Schwarz

25. ABJM action has 12 SUSY and SU(4)xU(1) global symmetry and Conformal symmetry • This action with U(N)xU(N) gauge group • describes N M2-branes on • (2) ABJM derived this action • as a limit of a D-brane configuration c.f. BLG is SU(2)xSU(2)

26. (3) Bagger and Lambert showed that ABJM action also has Lie 3-algebra structure defined by Structure constant: which satisfy (i) and (ii) (i) fundamental identities (ii) NOT total anti-symmetric However, meaning or importance of the 3-algebra had been unclear for ABJM action.

27. 3. ABJM to 3d YM and M2-M5 bound state

28. Orbifold to R^7 x S¹ M2-branes probing R^7 x S¹ = D2-branes probing R^7 M2-branes probing (2+1)d ABJM theory (Chern-Simon) (2+1)d SuperYM theory 2 π v / k θ= 2 π / k Mukhi et.al. ABJM Scaling limit v → ∞, k → ∞, v / k : fixed where v is the distance between the M2 and singularity

29. Bosonic part of ABJM where and is the 3-bracket Consider and take a linear combination then, This v.e.v gives mass to gauge field

30. is massive and can be integrated out. Then we have 3D YM from CS theory through Higgsing! M2 → D2 in the limit From the known D4-D2 bound state solution, we want to find a M-theory lift of this solution

31. Potential of the ABJM action Ansatz (i.e. forget gauge fields and only consider Hermite and constant part of Y¹ and Y²)

32. e.o.m. additional Ansatz (the solution becomes D2-D4 in the limit v →∞) where f→ 0 for v →∞

33. N M2-branes (N →∞) ABJM model N D2-branes (N →∞) 3 dim SYM S1 compactification M5-brane (with non-zero flux) D4-brane (with non-zero flux ∝ 1/Θ) S1 compactification

34. e.o.m. (infinite order nonlinear PDE)

35. Two equations for one function f(x,y). Are these really consistent?

36. Two equations for one function f(x,y). Are these really consistent? We can show a following identity, which guarantees the existence of the solutions ! Thus, there exist perturbative solutions for these equations. Anologue for the D2-brane is This is followed from Jacobi identity.

37. This identity is shown from the fundamental identity of Lie 3-algebra and following identities including both 2-bracket and 3-bracket:

38. a perturbative solution is We can show that the solutions have only one parameter, although there seem two parameters. Another remark: solution is real

39. We claim that the solution represents an M5-brane with 3-from flux wrapping following space 0 1 2 3(r) 4(r’) 5(θ) 6 7 8 9 10 M2 ○ ○ ○ M5 ○ ○ ○ ○ 　○ 　　○ Compactified S1 direction although we can not see the S1 direction manifestly. This will been seen by non-perturbative effects, like monopole operator (vortex) in ABJM. Instanton particle in D4 (?)

40. We can find Commutator and anti-commutator is simplified in this limit. Then, the e.o.m. is reduced to 3rd order non-linear PDE This is still difficult. Nevertheless, we found a solution!:

41. general expression of the solutions with Poisson bracket The e.o.m. is approximated in the limit as take a following ansatz: then the solution is

42. Relation to Nambu-Poisson bracket The M5-branes wrap the space with Poisson bracket for the KK reduced space is This is not consistent with our solution On the other hand, Nambu-Poisson bracket on the space is i.e. we can choose the normalization such that means Thus, we should define

43. The induced metric on the M5-brane is The potential is evaluated as In the star-product representation, Tr is given as then, we have where we inserted This indeed corresponds to the M5-brane volume factor, the cofficient is (a part of effective) tension of the M5-brane.

44. The M5-brane will have a constant flux which implies by the non-linear self duality. This is expected because non-commutative parameter of D4-brane is constant

45. Then, we can show that the metric with the constant flux is the solution of the single M5-brane action, which is essentially Nambu-Goto action. Furthermore, tension of the M5-brane computed from the M5-brane action match with the one from the ABJM action!

46. Lie 3-algebra and 3-form flux The potential can be written by the 3-bracket: Now, substituting our solution we have From the U(1) gauge transformation, we recover θdependence as In the real coordinates, we have This matches with the 3-form flux where

47. 4. Multiple M5-brane action from ABJM

48. We will consider fluctuations around Θ→ 0 solution First, decompose Y to Hermite and anti-Hermite parts Since 3-bracket is a combination of commutator and anti-commutator: Potential is also written by them. We will expand the potential by the number of commutators.

49. The result is

50. Now, we assume order of the fluctuations as follows: This was chosen such that all fluctuations are same order, thus remain in the Θ → 0. Then, we find leading order of the potential (assuming only p have v.e.v):