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

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.