# Quantum Line Defects and Refined BPS Spectra

## TL;DR

This paper investigates refined BPS invariants linked to quantum line defects in class  theories, utilizing geometric and algebraic methods to analyze their spectral properties and spin content.

## Contribution

It introduces a novel approach to compute refined BPS invariants for quantum line defects using K-theoretic geometry and quiver representations.

## Key findings

- Refined BPS invariants are derived from moduli spaces of quiver representations.
- The spin content of BPS states is characterized via Euler characteristics of sheaf complexes.
- A geometric framework connects UV defect data to IR BPS spectra.

## Abstract

In this note we study refined BPS invariants associated with certain quantum line defects in quantum field theories of class $\mathcal{S}$. Such defects can be specified via geometric engineering in the UV by assigning a path on a certain curve. In the IR they are described by framed BPS quivers. We study the associated BPS spectral problem, including the spin content. The relevant BPS invariants arise from the K-theoretic enumerative geometry of the moduli spaces of quiver representations, adapting a construction by Nekrasov and Okounkov. In particular refined framed BPS states are described via Euler characteristics of certain complexes of sheaves.