N=1 theories of class Sk

We construct classes of N = 1 superconformal theories elements of which are labeled by punctured Riemann surfaces. Degenerations of the surfaces correspond, in some cases, to weak coupling limits. Different classes are labeled by two integers (N, k). The k = 1 case coincides with A(N-1) N = 2 theories of class S and simple examples of theories with k > 1 are Z(k) orbifolds of some of the A(N-1) class S theories. For the space of N = 1 theories to be complete in an appropriate sense we find it necessary to conjecture existence of new N = 1 strongly coupled SCFTs. These SCFTs when coupled to additional matter can be related by dualities to gauge theories. We discuss in detail the A(1) case with k = 2 using the supersymmetric index as our analysis tool. The index of theories in classes with k > 1 can be constructed using eigenfunctions of elliptic quantum mechanical models generalizing the Ruijsenaars-Schneider integrable model. When the elliptic curve of the model degenerates these eigenfunctions become polynomials with coefficients being algebraic expressions in fugacities, generalizing the Macdonald polynomials with rational coefficients appearing when k = 1.