Recently, a universal formula for a non-holomorphic modular completion of the generating functions of refined BPS indices in various theories with $N=2$ supersymmetry has been suggested. It expresses the completion through the holomorphic generating functions of lower ranks. Here we show that for $U(N)$ Vafa-Witten theory on Hirzebruch and del Pezzo surfaces this formula can be used to extract the holomorphic functions themselves, thereby providing the Betti numbers of instanton moduli spaces on such surfaces. As a result, we derive a closed formula for the generating functions and their completions for all $N$. Besides, our construction reveals in a simple way instances of fiber-base duality, which can be used to derive new non-trivial identities for generalized Appell functions. It also suggests the existence of new invariants, whose meaning however remains obscure.

中文翻译：

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来自模异常的 Vafa-Witten 不变量

最近，提出了一个通用公式，用于在各种理论中使用 $N=2$ 超对称性对精制 BPS 指数的生成函数进行非全纯模补全。它通过较低秩的全纯生成函数表示完成。在这里，我们表明对于 Hirzebruch 和 del Pezzo 曲面上的 $U(N)$ Vafa-Witten 理论，该公式可用于提取全纯函数本身，从而提供此类曲面上瞬子模空间的 Betti 数。因此，我们为所有 $N$ 的生成函数及其完成推导出了一个封闭的公式。此外，我们的构造以一种简单的方式揭示了基于纤维的对偶性实例，可用于为广义 Appell 函数推导出新的非平凡恒等式。它还表明存在新的不变量，