Whenever available, refined BPS indices provide considerably more information on the spectrum of BPS states than their unrefined version. Extending earlier work on the modularity of generalized Donaldson-Thomas invariants counting D4-D2-D0 brane bound states in type IIA strings on a Calabi-Yau threefold $\mathfrak{Y}$, we construct the modular completion of generating functions of refined BPS indices supported on a divisor class. Although for compact $\mathfrak{Y}$ the refined indices are not protected, switching on the refinement considerably simplifies the construction of the modular completion. Furthermore, it leads to a non-commutative analogue of the TBA equations, which suggests a quantization of the moduli space consistent with S-duality. In contrast, for a local CY threefold given by the total space of the canonical bundle over a complex surface $S$, refined BPS indices are well-defined, and equal to Vafa-Witten invariants of $S$. Our construction provides a modular completion of the generating function of these refined invariants for arbitrary rank. In cases where all reducible components of the divisor class are collinear (which occurs e.g. when $b_2(\mathfrak{Y})=1$, or in the local case), we show that the holomorphic anomaly equation satisfied by the completed generating function truncates at quadratic order. In the local case, it agrees with an earlier proposal by Minahan et al for unrefined invariants, and extends it to the refined level using the afore-mentioned non-commutative structure. Finally, we show that these general predictions reproduce known results for $U(2)$ and $U(3)$ Vafa-Witten theory on $\mathbb{P}^2$, and make them explicit for $U(4)$.

中文翻译：

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S-Duality 和 Refined BPS 指数

只要可用，精炼的 BPS 指数就比未精炼的版本提供更多关于 BPS 状态范围的信息。扩展早期关于在 Calabi-Yau 三重 $\mathfrak{Y}$ 上计算类型 IIA 字符串中 D4-D2-D0 膜束缚状态的广义唐纳森-托马斯不变量的模块化的工作，我们构建了精制 BPS 生成函数的模完成除数类支持的索引。尽管对于紧凑的 $\mathfrak{Y}$ 精炼索引不受保护，但打开精炼大大简化了模块化完成的构造。此外，它导致 TBA 方程的非交换模拟，这表明模空间的量化与 S 对偶一致。相比之下，对于由复杂表面 $S$ 上的规范丛的总空间给出的局部 CY 三倍，精制 BPS 指数是明确定义的，并且等于 $S$ 的 Vafa-Witten 不变量。我们的构造为任意等级的这些精制不变量的生成函数提供了模块化完成。在除数类的所有可约分量共线的情况下（例如，当 $b_2(\mathfrak{Y})=1$ 时，或在局部情况下），我们证明完整的生成函数满足全纯异常方程以二次阶截断。在局部情况下，它同意 Minahan 等人早先提出的未精化不变量的提议，并使用上述非交换结构将其扩展到精化级别。最后，