ABJM at finite $N$ via 4d supergravity

TL;DR

This paper uses 4d supergravity to derive exact holographic predictions for ABJM theory at finite N, including partition functions and indices, by fixing higher derivative and quantum corrections through localization results.

Contribution

It introduces a method to determine higher derivative and quantum corrections in supergravity for ABJM, leading to precise holographic predictions for various observables at finite N.

Findings

Partition function expressed as an Airy function with deformations

Topologically twisted and superconformal indices as products of Airy functions

Agreement with numerical calculations and general fixed-point formulas

Abstract

We apply the conjecture of arXiv:2111.06903 for gravitational building blocks to the effective supergravity description of M-theory on S$^7/\mathbb{Z}_k$. Utilizing known localization results for the holographically dual ABJM theory, we determine a complete tower of higher derivative corrections to the AdS$_4$ supergravity and a further set of quantum corrections. This uniquely fixes the gravitational block, leading to holographic predictions for a number of exact ABJM observables, excluding only constant and non-perturbative corrections in the gauge group rank $N$. The predicted S$^3$ partition function is an Airy function that reproduces previous results and generalizes them to include arbitrary squashing and mass deformations/R-charge assignments. The topologically twisted and superconformal indices are instead products of two different Airy functions, in agreement with direct numeric calculations in the unrefined limit of the former object. The general fixed-point formula for an arbitrary supersymmetric background is similarly given as a product of Airy functions.