A Note On The Semiclassical Formulation Of BPS States In Four-Dimensional N=2 Theories

TL;DR

This paper extends the semiclassical framework for BPS states in 4D N=2 theories to include matter hypermultiplets and various line defects, providing a more comprehensive understanding of their vector spaces and applications.

Contribution

It introduces modifications to the semiclassical formulation of BPS states to incorporate matter hypermultiplets and arbitrary line defects in 4D N=2 theories.

Findings

Extended the semiclassical description to include matter hypermultiplets.

Provided modifications for arbitrary 't Hooft-Wilson line defects.

Illustrated applications of the extended framework.

Abstract

Vector spaces of (framed) BPS states of Lagrangian four-dimensional N=2 field theories can be defined in semiclassical chambers in terms of the $L^2$-cohomology of Dirac-like operators on monopole moduli spaces. This was spelled out previously for theories with only vectormultiplets, taking into account only a subset of the possible half-supersymmetric 't Hooft-Wilson line defects. This note completes the discussion by describing the modifications needed when including matter hypermultiplets together with arbitrary 't Hooft-Wilson line defects. Two applications of this extended discussion are given.