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

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

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?

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?

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

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

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

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

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.

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)

What can we say about supergravity?

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.

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’’

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

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.

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

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

Polarity

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

Weak Jacobi Forms

Do such weak Jacobi forms exist? Extremal Elliptic Genus

Polar Region and Polar Polynomial

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

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

Counting Weak Jacobi Forms

Counting Polar Polynomials

Digression on Number Theory

Extremal Polar Polynomial

Search for the extremal elliptic genus

Polarity-Ordered Basis of V m

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

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

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

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….

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 - 

 -extremal theories

Number of equations = Number of variables.

Estimate  * (m)

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?

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.

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

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.

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

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

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

Supergravity Fock Space Supplies polar polynomial

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

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.

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.