On a hyperholomorphic line bundle over the Coulomb branch

TL;DR

This paper constructs a hyperholomorphic line bundle with a special connection over the hyperkahler moduli space arising from 4D N=2 theories reduced on S^1, linking geometric structures to physical BPS states.

Contribution

It introduces a canonical hyperholomorphic line bundle with a novel connection on the Coulomb branch, incorporating BPS degeneracies, and conjectures its physical significance in NUT space reductions.

Findings

Constructed a hyperholomorphic line bundle V with a special connection.

Connected the bundle's geometry to BPS particle degeneracies.

Proposed a physical interpretation involving NUT space.

Abstract

Given an N=2 supersymmetric field theory in four dimensions, its dimensional reduction on S^1 is a sigma model with hyperkahler target space M. We describe a canonical line bundle V on M, equipped with a hyperholomorphic connection. The construction of this connection is similar to the known construction of the metric on M itself: one begins with a simple "semiflat" connection and then improves it by including contributions weighed by the degeneracies of BPS particles going around S^1. We conjecture that V describes the physics of the theory dimensionally reduced on NUT space.