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 S 5, S 2 and M 4. This geometry plays an important role in AdS/CFT correspondence.

5. Holographic dual of N=2 theories (punctured) 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 (S 5 becomes AdS 5, and M 4 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: singular 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 S 5 and S 2 :

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. T N theory

23. The case without punctures

24. Maldacena-Nenuz background

25. A bit more on the geometry S 4 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 AdS 3 probe The brane is along AdS 3 (inside AdS 5 ) Σ 2 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 AdS 3, S 2 and a six-dimensional internal space including Σ 2. 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: AdS 3 *S 2 *CY 3 [Maldacena, Strominger, Witten] 2. SU(2)-structure: the one studied by [Gauntlett, etal][Kim 3 ] 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!