Large $N$ Partition Functions of the ABJM Theory

TL;DR

This paper derives explicit large N expressions for the ABJM theory's partition functions and topologically twisted index, revealing connections to Airy functions, string theory, and black hole physics.

Contribution

It provides a conjecture for the all-orders large N partition function on a squashed sphere and an explicit form for the topologically twisted index, advancing understanding of holography and AdS black holes.

Findings

Partition function expressed via Airy function for all orders in large N

Explicit compact formula for the topologically twisted index at fixed k

Results match string theory free energies and black hole entropy calculations

Abstract

We study the large $N$ limit of some supersymmetric partition functions of the $\mathrm{U}(N)_{k}\times \mathrm{U}(N)_{-k}$ ABJM theory computed by supersymmetric localization. We conjecture an explicit expression, valid to all orders in the large $N$ limit, for the partition function on the $\mathrm{U}(1)\times \mathrm{U}(1)$ invariant squashed sphere in the presence of real masses in terms of an Airy function. Several non-trivial tests of this conjecture are presented. In addition, we derive an explicit compact expression for the topologically twisted index of the ABJM theory valid at fixed $k$ to all orders in the $1/N$ expansion. We use these results to derive the topologically twisted index and the sphere partition function in the 't Hooft limit which correspond to genus $\tt g$ type IIA string theory free energies to all orders in the $\alpha'$ expansion. We discuss the implications of our results for holography and the physics of AdS$_4$ black holes.