Supersymmetric Casimir Energy and $\mathrm{SL(3,\mathbb{Z})}$ Transformations

TL;DR

This paper introduces a method using $ ext{SL(3,Z)}$ transformations to compute supersymmetric Casimir energies directly from superconformal indices, unifying various known relations and extending to non-Lagrangian theories.

Contribution

It provides a novel framework connecting superconformal indices, partition functions, and Casimir energies via $ ext{SL(3,Z)}$ transformations, applicable to a broad class of supersymmetric theories.

Findings

Explicit calculation of Casimir energy for $ ext{N=1}$ SQCD and $ ext{N=4}$ SYM for all classical gauge groups.

Derived Casimir energy for non-Lagrangian $ ext{N=2}$ SCFT with $ ext{E}_6$ symmetry.

Predicted Casimir energies for $ ext{SP(2N)}$ and $ ext{E}_7$ symmetric theories within duality networks.

Abstract

We provide a recipe to extract the supersymmetric Casimir energy of theories defined on primary Hopf surfaces directly from the superconformal index. It involves an $\mathrm{SL(3,\mathbb{Z})}$ transformation acting on the complex structure moduli of the background geometry. In particular, the known relation between Casimir energy, index and partition function emerges naturally from this framework, allowing rewriting of the latter as a modified elliptic hypergeometric integral. We show this explicitly for $\mathcal{N}=1$ SQCD and $\mathcal{N}=4$ supersymmetric Yang-Mills theory for all classical gauge groups, and conjecture that it holds more generally. We also use our method to derive an expression for the Casimir energy of the nonlagrangian $\mathcal{N}=2$ SCFT with $\mathrm{E_6}$ flavour symmetry. Furthermore, we predict an expression for Casimir energy of the $\mathcal{N}=1$ $\mathrm{SP(2N)}$ theory with $\mathrm{SU(8)\times U(1)}$ flavour symmetry that is part of a multiple duality network, and for the doubled $\mathcal{N}=1$ theory with enhanced $\mathrm{E}_7$ flavour symmetry.