LLM geometries in M-theory and probe branes inside them

1. LLM geometries in M-theory and probe branes inside them Jun-Bao Wu IHEP, CAS Nov. 24, 2010, KITPC

2. Based on • B. Chen, E. O Colgain, JW, H. Yavartanoo, JHEP04(2010)078, 1001.0906. • E. O Colgain, JW, H. Yavartanoo, JHEP08(2010)114, 1005.4527. • E. O Colgain, JW, H. Yavartanoo, 1010.5982.

3. Outline • Vanishing of a particular flux in 11d LLM geometries • Probe branes in Maldacena-Nunez background • Conclusions and discussions

4. 11d LLM geometry • Lin, Lunin and Maldacena (2004) found a large class of half-BPS solutions with isometry SO(6)*SO(3)*R of 11d SUGRA. • The geometry is warped product of S5, S2 and M4. • This geometry plays an important role in AdS/CFT correspondence.

5. Holographic dual of N=2 theories • Gaiotto (2009) constructed a huge class of 4d N=2 gauge theories by wrapping N M5 branes on a (punctured) Riemann surface. • Gaiotto and Maldacena (2009) suggested the dual geometries fall into double-Wick-rotated LLM solutions (S5 becomes AdS5, and M4 becomes Euclidean).

6. Dual geometries • For cases without punctures, the dual geometries are solutions obtained by Maldacena and Nunez (2000), which are special cases of double-Wick-rotated LLM solutions. • For case with punctures, the full dual geometries haven’t been obtained.

7. Fluxes • From [Gaiotto, Maldacena]

8. No such a flux • We show that there are no solutions with such a flux. • Aside remark: LLM noticed that if there is such a flux, the geometry is singular. So in certain sense, this singularity is ruled out by the sixteen supercharges (and the isometry).

9. 11d supergravity • The bosonic sector of the 11d SUGRA includes the metric g and a 3-form potential C with field strength F(4)=dC. • The action for this sector is: • Killing spinor equation:

10. Ansatz • LLM looked for half-BPS solutions with isometry SO(6)*SO(3), so they began with the following ansatz

11. Decomposition • The decomposition of the gamma matrices: • We decompose the 11d Killing spinor using Killing spinors on S5 and S2:

12. Reduction of KSE • The 11d Killing spinor equations now reduce to:

13. The bispinors (scalars and vectors)

14. Algebraic relations among scalars

15. Algebraic relations among vectors

16. Vanishing of I • For general case, we have • If we assume I is nonzero, • By solving the above algebraic equations, we get

17. Gaiotto’s N=2 dualities • Gaiotto studied a huge class of N=2 theory obtained from wrapping M5 branes on (punctured) Riemann surface. • Only a small fraction of these theories have known descriptions in terms of UV Lagrangian. • Gaiotto found generalization of various known S-dualities. • Non-perturbative results can be obtained from M-theory.

18. Simplest example • SU(2) theory with 4 flavors is corresponding to a sphere with 4 punctures. (In the right figure, SO(4)*SO(4) subgroup of flavor group SO(8) is picked out.)

19. S-duality (I) • S-duality SL(2, Z) group acts on • SL(2, Z) acts through triality on SO(8) flavor group, and exchanges quarks, monopoles and dyons.

20. S-duality (II)

21. More complicated quiver

22. TN theory

23. The case without punctures

24. Maldacena-Nenuz background

25. A bit more on the geometry • S4 part of the six-dimetional internal space:

26. Non-local operators/probe branes • There are non-local operators (objects) with various dimensions in these N=2 field theories: Wilson-’t Hooft loops, surface operators, domain walls … • In certain conditions they should be dual to probe M2 or M5 branes. • The M2 branes dual to loop operators: [Drukker, Morrison, Okuda]

27. Killing spinors

28. M5 branes • We focus on M5-brane in this MN background. • There are self-dual 3-form h field in the worldvolume of M5-brane. • The equations of motion are quite complicated, so we do not give the details.

29. BPS condition • The supersymmetries preserved by the M5 brane are determined by the following condition

30. Half-BPS AdS3 probe • The brane is along AdS3 (inside AdS5) Σ2 and directions with θ=π/2 :

31. Field theory dual • Half of the supersymmetries are broken by this brane, while SU(2)*U(1) R-symmetry is preserved. • The brane should be dual to some two-dimensional operators in the field theory side. Maybe it is dual to half-BPS surface operator.

32. Back reaction • It is interesting to study the ¼-BPS solution of 11d SUGRA describing the back reaction of this BPS M5 brane. • It should be warped product of AdS3, S2 and a six-dimensional internal space including Σ2. • We tried to search such solution following the ideas of LLM.

33. Two known solutions • We began with the bispinors and using the tool of G-structures. • We re-obtained two known solutions: 1. SU(3)-structure: AdS3*S2*CY3[Maldacena, Strominger, Witten] 2. SU(2)-structure: the one studied by [Gauntlett, etal][Kim3] • The wanted solution is not in either class. • We are still searching for it …

34. Summary • We showed that there are no certain flux in LLM geometries (closed the previous loophole). • We studied the probe branes in a special LLM background.

35. Future directions • Continue to study the gravity dual for the case with punctures. Related works: [Donos, Simon] [Reid-Edwards et al] • Further studies on the correspondence between non-local operators and probe branes.

36. THE END Thank you very much!