Generalized Schur partition functions and RG flows

TL;DR

This paper introduces a generalized partition function for ${ m ext{SCFTs}}$ that extends the Schur index, demonstrating its invariance under certain deformations and relating different theories through parameter mappings.

Contribution

It defines a new generalized partition function for ${ m ext{SCFTs}}$ that captures invariance under mass and vev deformations, linking different theories via parameter transformations.

Findings

The generalized partition function reduces to the Schur index at a specific parameter value.

Different ${ m ext{SCFTs}}$ in the same Coulomb branch share the same partition function with parameter mappings.

The Schur index of all ${ m ext{SCFTs}}$ in the Deligne-Cvitanovi series is obtained from special values of the generalized partition function.

Abstract

We revisit a double-scaled limit of the superconformal index of ${\cal N}=2$ superconformal field theories (SCFTs) which generalizes the Schur index. The resulting partition function, $\hat {\cal Z}(q,\alpha)$, has a standard $q$-expansion with coefficients depending on a continuous parameter $\alpha$. The Schur index is a special case with $\alpha=1$. Through explicit computations we argue that this partition function is an invariant of certain mass deformations and vacuum expectation value (vev) deformations of the SCFT. In particular, two SCFTs residing in different corners of the same Coulomb branch, satisfying certain restrictive conditions, have the same partition function with a non-trivial map of the parameters, $\hat {\cal Z}_1(q,\alpha_1)=\hat {\cal Z}_2(q,\alpha_2(\alpha_1))$. For example, we show that the Schur index of all the SCFTs in the Deligne-Cvitanovi\'c series is given by special values of $\alpha$ of the partition function of the SU(2) $N_f=4$ ${\cal N}=2$ SQCD.