Partition functions of non-Lagrangian theories from the holomorphic anomaly

TL;DR

This paper develops a method to compute partition functions of non-Lagrangian quantum field theories using the holomorphic anomaly, providing explicit formulas and applications to various phases and backgrounds.

Contribution

It introduces a novel approach to derive gravitational corrections for non-Lagrangian theories via the holomorphic anomaly equation, expanding computational tools in quantum field theory.

Findings

Derived a general formula for the partition function as a sum of hypergeometric functions.

Explicit results obtained for the round sphere and Nekrasov-Shatashvili phases.

Applications demonstrated in extremal correlators and anharmonic oscillator studies.

Abstract

The computation of the partition function in certain quantum field theories, such as those of the Argyres-Douglas or Minahan-Nemeschansky type, is problematic due to the lack of a Lagrangian description. In this paper, we use the holomorphic anomaly equation to derive the gravitational corrections to the prepotential of such theories at rank one by deforming them from the conformal point. In the conformal limit, we find a general formula for the partition function as a sum of hypergeometric functions. We show explicit results for the round sphere and the Nekrasov-Shatashvili phases of the $\Omega$ background. The first case is relevant for the derivation of extremal correlators in flat space, whereas the second one has interesting applications for the study of anharmonic oscillators.