The character of the supersymmetric Casimir energy

TL;DR

This paper analyzes the supersymmetric Casimir energy in $ ext{N}=1$ theories on $S^1 imes M_3$, linking it to twisted holomorphic modes and index-characters, and explores its relation to anomalies and generalized indices.

Contribution

It introduces a Hamiltonian approach to compute $E_{susy}$ on ambi-Hermitian backgrounds and connects it to index-characters and anomaly polynomials.

Findings

$E_{susy}$ arises from twisted holomorphic modes on $ ext{R} imes M_3$

$E_{susy}$ can be obtained as a limit of an index-character

Explicit computation of $E_{susy}$ on secondary Hopf surfaces

Abstract

We study the supersymmetric Casimir energy $E_\mathrm{susy}$ of $\mathcal{N}=1$ field theories with an R-symmetry, defined on rigid supersymmetric backgrounds $S^1\times M_3$, using a Hamiltonian formalism. These backgrounds admit an ambi-Hermitian geometry, and we show that the net contributions to $E_\mathrm{susy}$ arise from certain twisted holomorphic modes on $\mathbb{R}\times M_3$, with respect to both complex structures. The supersymmetric Casimir energy may then be identified as a limit of an index-character that counts these modes. In particular this explains a recent observation relating $E_\mathrm{susy}$ on $S^1\times S^3$ to the anomaly polynomial. As further applications we compute $E_\mathrm{susy}$ for certain secondary Hopf surfaces, and discuss how the index-character may also be used to compute generalized supersymmetric indices.