Witten indices of abelian M5 brane on $\mathbb{R}\times S^5$

TL;DR

This paper calculates Witten indices and partition functions for abelian 6d tensor and hypermultiplets on a squashed five-sphere, revealing their relation to gravitational anomalies and extending understanding of supersymmetric indices in curved backgrounds.

Contribution

It provides explicit computations of Witten indices for abelian 6d theories on squashed spheres, linking these to anomaly polynomials and exploring their behavior under Wick rotation.

Findings

Witten indices match gravitational anomaly polynomials.

Witten index on   is zero, consistent with anomaly cancellation.

Results extend to general squashing and include hypermultiplet mass effects.

Abstract

Witten indices and partition functions are computed for abelian 6d tensor and hypermultiplets on $\mathbb{R}\times S^5$ in Lorentzian signature in an R gauge field background which preserves some supersymmetry. We consider a generic supersymmetric squashing that also admits squashing of the Hopf fiber. Wick rotation to Euclidean M5 brane amounts to Wick rotation of squashing parameters and the hypermultiplet mass parameter. We compute Casimir energies for tensor and hypermultiplets separately for general squashing, and match these with the corresponding gravitational anomaly polynomials. We extract Witten indices on $\mathbb{R}\times \mathbb{CP}^2$ and find that this is zero, again matching with the vanishing anomaly polynomial on an odd dimensional space.