No abstract

Making use of the exact solutions of the N = 2 supersymmetric gauge theories we construct new classes of superconformal field theories (SCFTs) by fine-tuning the moduli parameters and bringing the theories to critical points. SCFTs we have constructed represent universality classes of the 4-dimensional N = 2 SCFTs.

Recently Witten proposed to consider elliptic genus in N = 2 superconformal field theory to understand the relation between N = 2 minimal models and Landau-Ginzburg theories. In this paper we first discuss the basic properties satisfied by elliptic genera in N = 2 theories. These properties are confirmed by some fundamental class of examples.Then we introduce a generic procedure to compute the elliptic genera of a particular class of orbifold theories, i.e. the ones orbifoldized by e 2πiJ 0 in the Neveu-Schwarz sector.This enables us to calculate the elliptic genera for Landau-Ginzburg orbifolds. When the Landau-Ginzburg orbifolds allow an interpretation as target manifolds with SU (N ) holonomy we can compare the expressions with the ones obtained by orbifoldizing tensor products of N = 2 minimal models. We also give sigma model expressions of the elliptic genera for manifolds of SU (N ) holonomy.

We present simple observations on the topological nature of N=2 superconformal field theories. We point out that under a suitable redefinition of the energy-momentum tensor the central charge of N=2 theory vanishes and the theory is transformed into a topological quantum field theory. BRST invariant observables are given by chiral primary fields. We also discuss the relevant perturbations of N=2 and SU(2) coset models.

We discuss the topological CP 1 model which consists of the holomorphic maps from Riemann surfaces onto CP 1 . We construct a large-N matrix model which reproduces precisely the partition function of the CP 1 model at all genera of Riemann surfaces. The action of our matrix model has the form Tr V (M ) = −2Tr M (log M − 1) + 2 t n,P Tr M n (log M −c n )+ 1/n·t n−1,Q Tr M n (c n = n 1 1/j) where M is an N × N hermitian matrix and t n,P (t n,Q ), (n = 0, 1, 2 · · ·) are the coupling constants of the n-th descendant of the puncture (Kähler) operator.