S-duality and refined BPS indices

TL;DR

This paper develops a modular completion framework for refined BPS indices in string theory, revealing new structures and equations that enhance understanding of BPS spectra and their dualities, especially in Calabi-Yau and local geometries.

Contribution

It constructs the modular completion of generating functions of refined BPS indices, introduces a non-commutative TBA equation, and extends holomorphic anomaly equations to the refined setting.

Findings

Modular completion of refined BPS index generating functions.

Introduction of a non-commutative TBA equation suggesting quantization.

Verification of predictions against known Vafa-Witten results for U(2), U(3), and explicit for U(4).

Abstract

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 $U(4)$.