N=4 Versus N=2 Phases, Hyperk\"Ahler Quotients and the 2D Topological Twist

TL;DR

This paper compares N=2 and N=4 supersymmetric gauge theories in two dimensions, focusing on their phases, topological twists, and applications to ALE-manifolds, highlighting the unique features of the N=4 case.

Contribution

It introduces a detailed analysis of N=4 supersymmetric models, their phases, topological twists, and applications to ALE-manifolds, expanding understanding beyond the N=2 case.

Findings

N=4 theories have a single phase, unlike N=2 theories with multiple phases.

The paper clarifies the topological twists and their relation to gravitational instantons.

Application to ALE-manifolds shows how to study their deformations using topologically twisted theories.

Abstract

We consider N=2 and N=4 supersymmetric gauge theories in two-dimensions, coupled to matter multiplets. In analogy with the N=2 case also in the N=4 case one can introduce Fayet-Iliopoulos terms.The associated three-parameters have the meaning of momentum-map levels in a HyperK\"ahler quotient construction. Differently from the N=2 case, however, the N=4 has a single phase corresponding to an effective $\sigma$-model. There is no Landau-Ginzburg phase. The main possible application of our N=4 model is to an effective Lagrangian construction of a $\sigma$-model on ALE-manifolds. We discuss the A and B topological twists of these models clarifying some issues not yet discussed in the literature, in particular the identification of the topological systems emerging from the twist. Applying our results to the case of ALE-manifolds we indicate how one can use the topologically twisted theories to study the K\"ahler class and complex structure deformations of these gravitational instantons.