Spectral zeta function and non-perturbative effects in ABJM Fermi-gas

TL;DR

This paper explores how non-perturbative effects in the ABJM Fermi-gas model can be understood through spectral analysis, revealing hidden poles in the spectral zeta function that influence the partition function.

Contribution

It introduces a spectral approach using Mellin-Barnes integrals and zeta functions to identify non-perturbative corrections in the ABJM Fermi-gas system, linking spectral properties to quantum effects.

Findings

Spectral zeta function has infinite non-perturbative poles.

Non-perturbative corrections emerge after summing perturbative series.

Framework reproduces non-perturbative free energy in topological string theory.

Abstract

The exact partition function in ABJM theory on three-sphere can be regarded as a canonical partition function of a non-interacting Fermi-gas with an unconventional Hamiltonian. All the information on the partition function is encoded in the discrete spectrum of this Hamiltonian. We explain how (quantum mechanical) non-perturbative corrections in the Fermi-gas system appear from a spectral consideration. Basic tools in our analysis are a Mellin-Barnes type integral representation and a spectral zeta function. From a consistency with known results, we conjecture that the spectral zeta function in the ABJM Fermi-gas has an infinite number of "non-perturbative" poles, which are invisible in the semi-classical expansion of the Planck constant. We observe that these poles indeed appear after summing up perturbative corrections. As a consequence, the perturbative resummation of the spectral zeta function causes non-perturbative corrections to the grand canonical partition function. We also present another example associated with a spectral problem in topological string theory. A conjectured non-perturbative free energy on the resolved conifold is successfully reproduced in this framework.