Exact results in N=8 Chern-Simons-matter theories and quantum geometry

TL;DR

This paper derives exact non-perturbative results for N=8 supersymmetric ABJ(M) theories on the three-sphere, revealing simplified structures and explicit formulas involving spectral curves and theta functions, with implications for quantum geometry.

Contribution

It provides closed-form expressions for the grand potential and partition functions of N=8 ABJ(M) theories, connecting them to spectral curves and theta functions, and explores their analytic continuation.

Findings

Explicit formulas for the grand potential and partition functions.

Partition function as an entire function on the complex plane.

Exact quantization conditions derived from theta functions.

Abstract

We show that, in ABJ(M) theories with N=8 supersymmetry, the non-perturbative sector of the partition function on the three-sphere simplifies drastically. Due to this simplification, we are able to write closed form expressions for the grand potential of these theories, which determines the full large N asymptotics. Moreover, we find explicit formulae for the generating functionals of their partition functions, for all values of the rank N of the gauge group: they involve Jacobi theta functions on the spectral curve associated to the planar limit. Exact quantization conditions for the spectral problem of the Fermi gas are then obtained from the vanishing of the theta function. We also show that the partition function, as a function of N, can be extended in a natural way to an entire function on the full complex plane, and we explore some possible consequences of this fact for the quantum geometry of M-theory and for putative de Sitter extensions.