S-duality and refined BPS indices

Understand

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.