Supersymmetric Casimir energy and mathrm{S}mathrm{L}left(3,mathrm{mathbb{Z}}right) transformations

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{S}mathrm{L}left(3,mathrm{mathbb{Z}}right) 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 E6 flavour symmetry. Furthermore, we predict an expression for Casimir energy of the mathcal{N}=1 SP(2N) theory with SU(8) × U(1) flavour symmetry that is part of a multiple duality network, and for the doubled mathcal{N}=1 theory with enhanced E7 flavour symmetry.