Communications in Number Theory and Physics

Volume 17 (2023)

Number 4

Numerical experiments on coefficients of instanton partition functions

Pages: 941 – 983

DOI: https://dx.doi.org/10.4310/CNTP.2023.v17.n4.a3

Authors

Aradhita Chattopadhyaya (School of Theoretical Physics, Dublin Institute for Advanced Studies, Dublin, Ireland; and Department of Theoretical Physics, National University of Ireland, Maynooth, Kildare, Ireland)

Jan Manschot (School of Mathematics, Trinity College, Dublin, Ireland; Hamilton Mathematical Institute, Trinity College, Dublin, Ireland; and School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey, U.S.A.)

Abstract

We analyze the coefficients of partition functions of Vafa–Witten (VW) theory on a four-manifold. These partition functions factorize into a product of a function enumerating pointlike instantons and a function enumerating smooth instantons. For gauge groups $SU(2)$ and $SU(3)$ and four-manifold the complex projective plane $\mathbb{CP}^2$, we experimentally study the latter functions, which are examples of mock modular forms of depth $1$, weight $3/2$, and depth $2$, weight $3$ 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 of these mock modular forms suggest that the coefficients of these functions grow as $O(n^{k-1})$ for the respective weights $k = 3/2$ and $3$. This growth is similar to that of a modular form of weight $k$. On the other hand the coefficients of the mock cusp form of weight $3$ appear to grow as $O(n^{3/2})$, 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.

Keywords

modularity, instanton partition functions, mock modular forms

2010 Mathematics Subject Classification

Primary 11F37. Secondary 14D21.

Full Text (PDF format)

The majority of this work was carried out while AC was a postdoctoral fellow in the School of Mathematics, Trinity College Dublin. During this time, AC and JM were supported by the Laureate Award 15175 “Modularity in Quantum Field Theory and Gravity” of the Irish Research Council. JM is also supported by the Ambrose Monell Foundation. AC is presently funded by a fellowship from DIAS. We thank Prof. Werner Nahm and Pranav Pandit for useful discussions. We also thank the participants of the conference “Modular forms in Number Theory and Beyond”, Bielefeld, Germany especially Olivia Beckwith, Michael Mertens, Caner Nazaroglu and Jan Vonk for useful insights. AC also thanks the International Centre for Theoretical Sciences (ICTS) Bangalore and Indian Institute of Science, Bangalore, India for hospitality during the course of the work.

Received 27 February 2023

Accepted 27 November 2023

Published 24 January 2024