Quantum TBA for refined BPS indices

TL;DR

This paper reformulates the quantum Riemann-Hilbert problem associated with refined BPS indices as a non-commutative TBA-like equation, providing a formal solution and a generating function in the unrefined case.

Contribution

It introduces a non-commutative deformation of TBA equations linked to refined BPS indices and offers a formal solution expansion.

Findings

Derived a formal solution as an expansion in refined indices

Constructed a generating function for unrefined TBA solutions

Linked non-commutative deformation to quantum Riemann-Hilbert problems

Abstract

Refined BPS indices give rise to a quantum Riemann-Hilbert problem that is inherently related to a non-commutative deformation of moduli spaces arising in gauge and string theory compactifications. We reformulate this problem in terms of a non-commutative deformation of a TBA-like equation and obtain its formal solution as an expansion in refined indices. As an application of this construction, we derive a generating function of solutions of the TBA equation in the unrefined case.