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.

中文翻译：

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瞬子配分函数系数的数值实验

我们分析了四流形上 Vafa-Witten (VW) 理论的配分函数系数。这些配分函数分解为枚举点状瞬子的函数和枚举平滑瞬子的函数的乘积。对于规范组 $SU(2)$ 和 $SU(3)$ 以及四流形复射影平面 $\mathbb{CP}^2$，我们通过实验研究了后面的函数，它们是深度的模拟模形式的示例分别为$1$，重量$3/2$，深度$2$，重量$3$。我们还引入了“模拟尖点形式”的概念，并研究了与 $SU(3)$ 配分函数相关的权重 $3$ 的示例。对这些模拟模块化形式的前 200 个系数进行的数值实验表明，对于各自的权重 $k = 3/2$ 和 $3$，这些函数的系数增长为 $O(n^{k-1})$。这种增长类似于权重 $k$ 的模块化形式。另一方面，权重 $3$ 的模拟尖点形式的系数似乎增长为 $O(n^{3/2})$，这超过了权重 $3$ 的经典尖点形式的增长。我们使用鞍点分析提供界限，但这很大程度上超出了实验观察。