Genus Two Partition Functions of Extremal Conformal Field Theories

TL;DR

This paper explicitly computes genus two partition functions for extremal conformal field theories at specific central charges, providing evidence for their existence and uniqueness based on modular invariance and degenerating limits.

Contribution

It determines the genus two partition functions for k=2 and k=3 ECFTs, supporting their potential existence and uniqueness for certain central charges.

Findings

Partition functions for k=2 and k=3 ECFTs explicitly computed.

Results support the conjecture of ECFT existence at these levels.

Genus two partition functions are uniquely fixed for k<11 if ECFTs exist.

Abstract

Recently Witten conjectured the existence of a family of "extremal" conformal field theories (ECFTs) of central charge c=24k, which are supposed to be dual to three-dimensional pure quantum gravity in AdS3. Assuming their existence, we determine explicitly the genus two partition functions of k=2 and k=3 ECFTs, using modular invariance and the behavior of the partition function in degenerating limits of the Riemann surface. The result passes highly nontrivial tests and in particular provides a piece of evidence for the existence of the k=3 ECFT. We also argue that the genus two partition function of ECFTs with k<11 are uniquely fixed (if they exist).