Supersymmetric Casimir Energy and the Anomaly Polynomial

TL;DR

This paper proposes a universal relation between supersymmetric Casimir energy and the anomaly polynomial for superconformal field theories in even dimensions, supported by extensive computations across multiple dimensions.

Contribution

It introduces a conjecture linking supersymmetric Casimir energy to an equivariant integral of the anomaly polynomial, applicable to theories with or without Lagrangian descriptions.

Findings

Confirmed the conjecture in 2, 4, and 6 dimensions.

Computed Casimir energies for various superconformal theories.

Validated the relation for theories with and without known Lagrangians.

Abstract

We conjecture that for superconformal field theories in even dimensions, the supersymmetric Casimir energy on a space with topology $S^1\times S^{D-1}$ is equal to an equivariant integral of the anomaly polynomial. The equivariant integration is defined with respect to the Cartan subalgebra of the global symmetry algebra that commutes with a given supercharge. We test our proposal extensively by computing the supersymmetric Casimir energy for large classes of superconformal field theories, with and without known Lagrangian descriptions, in two, four and six dimensions.