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

11 pages, 1 figure. v2: minor corrections, references added

References in corpus (3)

• Cardy Formulae for SUSY Theories in d=4 and d=6
• On Large N Expansion of the Sphere Free Energy
• Global Aspects of Abelian Duality in Dimension Three

Cited by in corpus (5)

• Asymptotic Symmetries in $p$-Form Theories
• Electromagnetic Duality and Entanglement Anomalies
• Dynamical Edge Modes in $p$-form Gauge Theories
• Odd dimensional analogue of the Euler characteristic
• Partition functions of $p$-forms from Harish-Chandra characters