A note on sphere free energy of $p$-form gauge theory and Hodge duality

TL;DR

This paper calculates the free energy of free p-form gauge theories on a d-sphere, revealing how it depends on p, d, and R, and exploring the effects of Hodge duality and conformal invariance.

Contribution

It provides a general computation of the sphere free energy for p-form gauge theories and analyzes the effects of Hodge duality and conformal invariance across dimensions.

Findings

Free energy contains a log R term with coefficient proportional to (2p+2-d).

p-form and (d-p-2)-form theories agree for odd d, disagree for even d.

Results are consistent with known literature on conformal invariance and duality.

Abstract

We consider a free $p$-form gauge theory on a $d$-dimensional sphere of radius $R$ and calculate its free energy. We perform the calculation for generic values of $p$ and obtain the free energy as a function of $d, p$ and $R$. The result contains a $\log R$ term with coefficient proportional to $\left(2p+2-d \right)$, which is consistent with lack of conformal invariance for $p$ form theories in dimensions other than $2p+2$. We also compare the result for $p$-form and $(d-p-2)$-form theory which are classically Hodge dual to each other in $d$ dimensions and find that they agree for odd values of $d$. Instead, for even $d$, we find that the results disagree by an amount that is consistent with the reported values in the literature.