Effective Lagrangian for 3d N=4 SYM theories for any gauge group and monopole moduli spaces

TL;DR

This paper constructs explicit low-energy effective Lagrangians for 3d N=4 supersymmetric Yang-Mills theories with any gauge group, relating them to monopole moduli spaces and hyper-Kahler reductions, and conjectures nonperturbative extensions.

Contribution

It provides a unified construction of effective Lagrangians for all gauge groups and details the hyper-Kahler reduction process, extending known results for unitary groups.

Findings

Explicit asymptotic metrics for various gauge groups

Relationship between effective Lagrangians and monopole moduli spaces

Conjecture on nonperturbative metrics via hyper-Kahler reduction

Abstract

We construct low energy effective Lagrangians for 3d N=4 supersymmetric Yang-Mills theory with any gauge group. They represent supersymmetric sigma models at hyper-Kahlerian manifolds of dimension 4r (r is the rang of the group). In the asymptotic region, perturbatively exact explicit expression for the metric are written. We establish the relationship of this metric with the TAUB-NUT metric describing the perturbatively exact effective Lagrangians for unitary groups and monopole moduli spaces: the former is obtained out of the latter by a proper hyper-Kahlerian reduction. We describe in details the reduction procedure for SO/Sp/G_2 gauge groups, where it can also be given a natural interpretation in D-brane language. We conjecture that the exact nonperturbative metrics can be obtained by a similar hyper-Kahlerian reduction from the corresponding multidimensional Atiyah-Hitchin metrics.