N = 4 versus N = 2 phases, hyper-Kähler quotients and the 2D topological twist

Abstract

We consider the rheonomic construction of N=2 and N=4 supersymmetric gauge theories in two dimensions, coupled to matter multiplets. In full analogy with the N=2 case studied by Witten, we show that also in the N=4 case one can introduce Fayet--Iliopoulos terms for each of the Abelian factors of the gauge group. The three-parameters of the N=4 Fayet--Iliopoulos term have the meaning of momentum-map levels in a hyper-Kähler quotient construction just as the single parameter of the N=2 Fayet--Iliopoulos term has the meaning of momentum map level in a Kähler quotient construction. Differently from the N=2 case, however, the N=4 has a single phase corresponding to an effective model. The Landau--Ginzburg phase possible in the N=2 case seems to be deleted in those N=2 theories that have an enhanced N=4 supersymmetry. The main application of our N=4 model is to an effective Lagrangian construction of a -model on ALE manifolds or other gravitational instantons. We discuss in detail the topological twists of these theories (A and B models) emphasizing the role of R-symmetries and clarifying some subtleties, not yet discussed in the literature, related with the redefinition of the ghost number and the identification of the topological systems after twisting. In the A twist, we show that one obtains a topological matter system (of the topological -model type) coupled to a topological gauge theory. In the B twist, instead, we show that the theory describes a topological matter system (of the topological Landau--Ginzburg type) coupled to an ordinary (non-topological) gauge-theory: in addition, one has a massive topological vector, which decouples from the other fields. Applying our results to the case of ALE manifolds, we indicate how one can use the topologically twisted theories to study the Kähler class and complex structure deformations of these gravitational instantons. Our results are also preparatory for a study of matter-coupled topological 2D gravity as the twist of matter coupled N=2, D=2 supergravity.