N=4 VERSUS N=2 PHASES, HYPER-KAHLER QUOTIENTS AND THE 2D TOPOLOGICAL TWIST

We consider the rheonomic construction of N = 2 and N = 4 supersymmetr ic 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-Kahler quotient construction just as the single parameter of t he N = 2 Fayet-Iliopoulos term has the meaning of momentum map level i n a Kahler quotient construction. Differently from the N = 2 case, how ever, the N = 4 has a single phase corresponding to an effective sigma 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 supersy mmetry. The main application of our N = 4 model is to an effective Lag rangian construction of a sigma-model on ALE manifolds or other gravit ational instantons. We discuss in detail the topological twists of the se theories (A and B models) emphasizing the role of R-symmetries and clarifying some subtleties, not yet discussed in the literature, relat ed 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 sigma-mode l type) coupled to a topological gauge theory. In the B twist, instead , we show that the theory describes a topological matter system (of th e topological Landau-Ginzburg type) coupled to an ordinary (non-topolo gical) gauge-theory: in addition, one has a massive topological vector , which decouples from the other fields. Applying our results to the c ase Of ALE manifolds, we indicate how one can use the topologically tw isted theories to study the Kahler class and complex structure deforma tions of these gravitational instantons. Our results are also preparat ory for a study of matter-coupled topological 2D gravity as the twist of matter coupled N = 2, D = 2 supergravity.