Numerical experiments on coefficients of instanton partition functions

TL;DR

This paper investigates the growth patterns of coefficients in instanton partition functions related to Vafa-Witten theory on P^2, introducing the concept of mock cusp forms and analyzing their asymptotic behavior through numerical and analytical methods.

Contribution

It introduces the notion of mock cusp forms and studies their growth, providing new insights into the asymptotics of coefficients of mock modular forms in gauge theory.

Findings

Coefficients of mock modular forms grow as n^{k-1} for weight k.

Coefficients of mock cusp forms grow approximately as n^{3/2}.

Bounds from saddle point analysis are much larger than observed growth.

Abstract

We analyze the coefficients of partition functions of Vafa-Witten theory for the complex projective plane $\mathbb{CP}^2$. We experimentally study the growth of the coefficients for gauge group $SU(2)$ and $SU(3)$, which are examples of mock modular forms of depth $1$ and 2 respectively. We also introduce the notion of ``mock cusp form'', and study an example of weight 3 related to the $SU(3)$ partition function. Numerical experiments on the first 200 coefficients suggest that the coefficients of a mock modular form of weight $k$ grow as the coefficients of a modular form of weight $k$, that is to say as $n^{k-1}$. On the other hand the coefficients of the mock cusp form appear to grow as $n^{3/2}$, which exceeds the growth of classical cusp forms of weight 3. We provide bounds using saddle point analysis, which however largely exceed the experimental observation.