Supersymmetric Casimir Energy and the Anomaly Polynomial

TL;DR

The paper proposes and tests a universal relation: in even dimensions, the supersymmetric Casimir energy $E$ on $S^1\times S^{D-1}$ equals the equivariant integral of the anomaly polynomial $A_{D+2}$, with equivariance taken over the Cartan subalgebra commuting with the preserved supercharge. By relating the energy to the background chemical potentials via the partition function $Z(\beta,\mu_j)=e^{-\beta E(\mu_j)}I(\beta_j)$ and evaluating the equivariant integrals of $A_{D+2}$ across 2d, 4d, and 6d theories (including non-Lagrangian cases like $\mathcal{T}_3$ and E-string theories), the authors provide extensive checks using localization, indices, and strong-coupling analyses. They derive explicit energy formulas in terms of conformal and ’t Hooft anomaly coefficients, reproduce known limits (e.g., Schur limit and chiral algebras in 4d), and discuss instanton contributions and holographic implications. The work suggests a deep, geometric link between supersymmetric partition functions and anomaly polynomials, with potential extensions to defects, other backgrounds, and higher-dimensional Cardy-type limits, offering a unifying framework for computing protected energies from anomalies. The results have implications for holography, modular properties, and the structure of SCFT data across dimensions.

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.

Figures (1)

• Figure 1: S-duality transformation relating $SU(3)$ superconformal SQCD to a $\mathcal{T}_3$ coupled to a fundamental $SU(2)$ hypermultiplet. Subscripts correspond to chemical potentials in main text.