1. Extremal N=2 2D CFT’s and Constraints of Modularity Work done with M. Gaberdiel, S. Gukov, C. Keller and H. Ooguri arXiv:0805.4216 TexPoint fonts used in EMF: AA A A A A A AA A A A A IAS, October 3, 2008 …

2. Outline 1. Two Motivations 2. Results 3. Extremal N=0 and N=1 theories 4. 2D N=2 Theories: Elliptic Genus + Polarity 5. Counting Polar Terms 6. Search for Extremal N=2 theories 7. Near Extremal N=2 Theories 8. Possible Applications to the Landscape 9. Summary & Concluding Remarks

3. Motivation 1 An outstanding question in theoretical physics is the existence of three-dimensional AdS ``pure quantum gravity.’’ Witten proposed that it should be defined by a holographically dual ``extremal CFT.’’ We do not know if such CFT’s exist for general central charge c=24 k, k> 1. In AdS 3 one can define OSp(p|2)xOSp(q|2) sugra dual to theories with (p,q) supersymmetry. What can we say about those?

4. Motivation 2 There are widely-accepted claims of the existence of a ``landscape’’ of d=4 N=1 AdS solutions of string theory with all moduli fixed. The same techniques should apply to M-theory compactifications on – say – CY 4-folds to AdS 3. Such backgrounds would be holographically dual to 2D CFT Does modularity of partition functions put any interesting constraints on the landscape?

5. Result 1 We give a natural definition of an ``extremal N=(2,2) CFT’’ And we then show that there are at most a finite number of ``exceptional’’ examples

6. Result -2 We present evidence for the following conjecture: Any N=(2,2) CFT must contain a state of the form:

7. Result - 3 The bound is nearly optimal: There are candidate partition functions (elliptic genera) where all states with are descendents of the vacuum.

8. Extremal Conformal Field Theory Definition: An extremal conformal field theory of level k is a CFT with c=24k with a (weight zero) modular partition function ``as close as possible’’ to the vacuum Virasoro character. Not modular

9. Reconstruction Theorem Define the polar polynomial of Z k to be the sum of terms with nonpositive powers of q. The weight zero modular function Z k can be uniquely reconstructed from its polar polynomial: This is the step which will fail (almost always) in the N=(2,2) case.

10. Witten’s proposal for pure 3D quantum gravity The holographic dual of pure AdS 3 quantum gravity is a left-right product of extremal conformal field theories. Justification: Chern-Simons form of action Polar terms: Chiral edge states (Brown-Henneaux) Nonpolar terms: Black holes (c.f. Fareytail)

11. What can we say about supergravity?

12. N=1 Theories Witten already pointed out that there is an analog of the j-function for the modular group preserving a spin structure on the torus. Therefore one can construct the analog of extremal N=1 partition functions for the NS and R sectors. We will return to N=1 later, but for now let us focus on N=2.

13. Pure N=(2,2) AdS 3 supergravity N=(2,2) AdS 3 supergravity can be written as a Chern-Simons theory for OSp(2|2) x OSp(2|2) A natural extension of Witten’s conjecture is that pure N=(2,2) sugra is dual to an ``extremal (2,2) SCFT’’

14. Extremal N=(2,2) SCFT Define an extremal N=(2,2) theory to be a theory whose partition function is ``as close as possible’’ to the vacuum character: This is neither spectral flow invariant, nor modular invariant. It is useful to parametrize c= 6 m

15. Extremal N=(2,2) SCFT – II Impose spectral flow by hand So, more precisely, an extremal N=(2,2) CFT is a CFT with a modular and spectral flow invariant partition function of the above form.

16. What do we mean by ``nonpolar terms’’ ?

17. Cosmic Censorship Bound Black holes with near horizon geometry Must satisfy the cosmic censorship bound Cvetic & Larsen

18. Polarity

19. Elliptic Genus Modular: Spectral flow invariant: (Assume: m integer, U(1) charges integral.)

20. Weak Jacobi Forms

21. Do such weak Jacobi forms exist? Extremal Elliptic Genus

22. Polar Region and Polar Polynomial

23. Reconstruction Theorem Dijkgraaf, Maldacena, Moore, Verlinde; Manschot & Moore

24. Obstructions However, some polar polynomials cannot be extended to a full weak Jacobi form! Does not converge. It must be regulated. Knopp; Niebur; Manschot & Moore The regularization can spoil modular invariance

25. Counting Weak Jacobi Forms

26. Counting Polar Polynomials

27. Digression on Number Theory

30. Extremal Polar Polynomial

31. Search for the extremal elliptic genus

32. Polarity-Ordered Basis of V m

33. Computer Search Recall P(m)>j(m) for m>4. Is there magic?

34. Not Much Magic We find five ``exceptional solutions’’

35. Finiteness Theorem Theorem: There is an M such that for m> M an extremal elliptic genus does not exist. Difficult proof. Compare the elliptic genus at The NS and R cusps to derive the constraint

36. Numerical analysis strongly indicates that our five exceptional solutions are indeed the only ones. All this suggests that there are at most a finite number of pure N=(2,2) supergravity theories. But….

37. Near-Extremal Theories Perhaps our definition of ``extremal’’ was too restrictive… Maybe there are quantum corrections to the cosmic censorship bound… Define a  -extremal N=2 CFT by only demanding agreement of the polar terms with the vacuum character for polarity less than or equal to - 

38.  -extremal theories

39. Number of equations = Number of variables.

40. Estimate  * (m)

41. suggests a loophole: Perhaps there are quantum corrections to the cosmic censorship bound. Perhaps pure N=2 sugra exists, and P(m)-j(m) polar quantum numbers in fact admit states which are semiclassically described as black holes (or other nondescendent geometries). Escape Hatch for Pure N=2 Sugra?

42. Bound on Conformal Weight There must be some polar-state which is NOT a descendent and satisfies Also implies an interesting bound on the conformal weight of the first N=2 primary.

43. Bound on Conformal Weight-2 Conclusion: The conjecture implies that for any unitary N=(2,2) theory there must exist a state of the form

44. Explicit Construction of Nearly Extremal Elliptic Genera We can explicitly construct elliptic genera so that only descendents of the vacuum contribute to the polar subregion: Gritsenko So our bound is close to optimal.

46. Flux Compactifications (Discussions with T. Banks, F. Denef M. Gaberdiel, C. Keller, J. Maldacena)

47. Existence of a Hauptmodul K: There are no restrictions on the polar polynomial However, given the polar piece, modularity does make predictions about the degeneracies at h=c/24 +a, a=1/2, 1,3/2,… Characterize the flux vacuum by c & J

48. Single Particle Spectrum Vacuum Kahler modulus Complex modulus Kaluza-Klein mode Planck scale Large c, moderate J: ``Near extremal’’

49. Supergravity Fock Space Supplies polar polynomial

50. Optimistically – one could estimate the degeneracies at h= c/24 + a, a=1/2, 1, 3/2,… Modularity: Descendents: J<<1: Contradiction with EFT J>>1: No contradiction. But large degeneracy might possibly invalidate EFT

51. Conclusion There are at most 9 extremal N=(2,2) 2D CFT’s. We have strong evidence for an upper bound on the weight h of the first N=2 primary. It is plausible that ``pure N=2 supergravity’’ does not exist. It would be good to investigate more precisely quantum corrections to the CC bound.

52. Conclusions – 2 It is possible that there can be applications to the landscape, but our original motivation was naïve: such applications will require further input than just modularity of the partition function.