# Abelian M5-brane on $S^6$

## TL;DR

This paper analyzes the abelian M5 brane on a six-sphere, deriving the heat kernel expansion, fixing the conformal anomaly coefficient, and resolving a zero mode counting mismatch, with implications for dimensional reduction.

## Contribution

It provides a detailed spectral analysis of the abelian M5 brane on $S^6$, clarifies the conformal anomaly normalization, and addresses zero mode counting issues.

## Key findings

- Derived the heat kernel expansion for the M5 brane on $S^6$
- Resolved the zero mode overcounting in anomaly calculations
- Found the conformal anomaly vanishes upon dimensional reduction to five dimensions

## Abstract

We study the abelian M5 brane on $S^6$. From the spectrum we extract a series expansion for the heat kernel. In particular we determine the normalization for the coefficient $a$ in the M5 brane conformal anomaly. When we compare our result with what one gets by computing the Hadamard-Minakshisundaram-DeWitt-Seeley coefficients from local curvature invariants on $S^6$, we first find a mismatch of one unit. This mismatch is due to an overcounting of one zero mode. After subtracting this contribution, we finally find agreement. We perform dimensional reduction along a singular circle fiber to five dimensions where we find the conformal anomaly vanishes.