Exact low-energy effective actions for hypermultiplets in four dimensions

TL;DR

This paper derives exact low-energy effective actions for hypermultiplets in four-dimensional N=2 supersymmetric gauge theories, utilizing harmonic superspace and projective superspace techniques to classify hyper-Kähler metrics and monopole moduli spaces.

Contribution

It provides explicit forms of the hypermultiplet low-energy effective action and introduces a new classification of multi-monopole moduli space metrics using quartic polynomials.

Findings

Explicit LEEA forms for hypermultiplets with magnetic charge

New classification scheme for multi-monopole metrics

Encoding of n=2 case in terms of elliptic curves

Abstract

We consider the general hypermultiplet Low-Energy Effective Action (LEEA) that may appear in quantized, four-dimensional, N=2 supersymmetric, gauge theories, e.g. in the Coulomb and Higgs branches. Our main purpose is a description of the exact LEEA of n magnetically charged hypermultiplets. The hypermultiplet LEEA is given by the N=2 supersymmetric Non-Linear Sigma-Model (NLSM) with a 4n-dimensional hyper-K"ahler metric, subject to non-anomalous symmetries. Harmonic Superspace (HSS) and the NLSM isometries are very useful to constrain the hyper-K"ahler geometry of the LEEA. We use N=2 supersymmetric projections of HSS superfields to N=2 linear (tensor) O(2) and O(4) multiplets in N=2 Projective Superspace (PSS) to deduce the explicit form of the LEEA in some particular cases. As the by-product, a simple new classification of all multi-monopole moduli space metrics having su(2)_R symmetry is proposed in terms of real quartic polynomials of 2n variables, modulo Sp(n) transformations. The 4d hypermultiplet LEEA for n=2 can be encoded in terms of an elliptic curve.