BPS Indices, Modularity and Perturbations in Quantum K-theory

TL;DR

This paper explores the connections between 3d gauge theories, quantum K-theory, and modular forms, providing new formulas and insights into BPS indices, hypergeometric series, and mock modular behavior.

Contribution

It introduces a 3d Intriligator-Vafa formula for quantum K-theory of Grassmannians and links BPS indices to hypergeometric series and modular q-characters.

Findings

Derived a 3d Intriligator-Vafa formula for Grassmannians

Connected BPS indices to bilateral hypergeometric q-series

Identified mock modular behavior under Wilson line perturbations

Abstract

We study a perturbation family of N=2 3d gauge theories and its relation to quantum K-theory. A 3d version of the Intriligator-Vafa formula is given for the quantum K-theory ring of Grassmannians. The 3d BPS half-index of the gauge theory is connected to the theory of bilateral hypergeometric q-series, and to modular q-characters of a class of conformal field theories in a certain massless limit. Turning on 3d Wilson lines at torsion points leads to mock modular behavior. Perturbed correlators in the IR regime are computed by determining the UV-IR map in the presence of deformations.