Numerical experiments on coefficients of instanton partition functions

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.