# Quivers, Line Defects and Framed BPS Invariants

## TL;DR

This paper develops a combinatorial formalism to compute framed BPS invariants in N=2 quantum field theories using quiver representation theory and equivariant localization, providing explicit examples and a new computational approach.

## Contribution

It introduces a novel combinatorial method based on equivariant localization to calculate framed BPS invariants via quiver representations.

## Key findings

- Explicit formulas for framed BPS invariants in selected examples.
- A new formalism simplifies the computation of BPS spectra with line defects.
- Demonstrates the effectiveness of the approach through detailed examples.

## Abstract

A large class of N=2 quantum field theories admits a BPS quiver description and the study of their BPS spectra is then reduced to a representation theory problem. In such theories the coupling to a line defect can be modelled by framed quivers. The associated spectral problem characterises the line defect completely. Framed BPS states can be thought of as BPS particles bound to the defect. We identify the framed BPS degeneracies with certain enumerative invariants associated with the moduli spaces of stable quiver representations. We develop a formalism based on equivariant localization to compute explicitly such BPS invariants, for a particular choice of stability condition. Our framework gives a purely combinatorial solution of this problem. We detail our formalism with several explicit examples.