Berry's connection, K\"ahler geometry and the Nahm construction of monopoles

TL;DR

This paper explores the connection between Kähler geometry, supersymmetric quantum mechanics, and monopole solutions, demonstrating how the Nahm equations emerge from supersymmetric deformations and generalizing monopole constructions to higher charges.

Contribution

It extends the Nahm construction of monopoles within supersymmetric quantum mechanics to cases with higher magnetic charge, using matrix-valued deformations related to Kähler geometry.

Findings

The twisted mass deformation generalizes to Nahm equation-based matrix functions.

The Berry connection satisfies BPS monopole equations under supersymmetry.

The analysis reduces to the standard Nahm construction for the Riemann sphere.

Abstract

We study supersymmetric deformations of N = 4 quantum mechanics with a Kahler target space admitting a holomorphic isometry. We show that the twisted mass deformation generalises to a deformation constructed from matrix-valued functions of the moment map, which obey the Nahm equations. We also explain how N = 4 supersymmetry implies that the Berry connection on the vacuum bundle for this theory satisfies the BPS monopole equations. In the case where the target space is a Riemann sphere, our analysis reduces to the standard Nahm construction of monopoles. This generalises an earlier result by Sonner and Tong to the case of monopoles of magnetic charge greater than one.