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 CP2, 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.