N=1 theories of class S_k

TL;DR

This paper constructs new classes of ${ m N}=1$ superconformal theories labeled by punctured Riemann surfaces, explores their dualities, and introduces novel strongly coupled SCFTs, extending the class S framework to ${ m N}=1$ theories.

Contribution

It introduces a new framework for ${ m N}=1$ theories labeled by two integers, generalizes class S theories, and conjectures the existence of new strongly coupled SCFTs.

Findings

Constructed ${ m N}=1$ theories from punctured Riemann surfaces.

Connected degenerations to weak coupling limits and dualities.

Developed index calculations using elliptic quantum models.

Abstract

We construct classes of ${\cal N}=1$ 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} ${\cal N}=2$ 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 ${\cal N}=1$ theories to be complete in an appropriate sense we find it necessary to conjecture existence of new ${\cal N}=1$ 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.