Conformal anomalies for higher derivative free critical p-forms on even spheres

TL;DR

This paper computes conformal anomalies for higher derivative free p-forms on even spheres, revealing their dependence on deformation parameters and establishing connections with entanglement entropy, hyperbolic space integrals, and Casimir energy.

Contribution

It introduces a method to compute conformal anomalies for higher derivative p-forms on spheres using q-deformation and hyperbolic space integrals, providing new insights into their structure and related physical quantities.

Findings

Anomaly extremum at round sphere for certain derivative orders.

Conformal anomaly expressed as an integral over hyperbolic space.

Connections established between anomaly, entanglement entropy, and Casimir energy.

Abstract

The conformal anomaly is computed on even $d$--spheres for a $p$--form propagating according to the Branson--Gover higher derivative, conformally covariant operators. The system is set up on a $q$--deformed sphere and the conformal anomaly is computed as a rational function of the derivative order, $2k$, and of $q$. The anomaly is shown to be an extremum at the round sphere ($q=1$) only for $k<d/2$. At these integer values, therefore, the entanglement entropy is minus the conformal anomaly, as usual. The unconstrained $p$--form conformal anomaly on the full sphere is shown to be given by an integral over the Plancherel measure for a coexact form on hyperbolic space in one dimension higher.A natural ghost sum is constructed and leads to quantities which, for critical forms, i.e. when $2k=d-2p$, are, remarkably, a simple combination of standard quantities, for usual second order, $k=1$, propagation, when these are available. Our values coincide with a recent hyperbolic computation of David and Mukherjee.Values are suggested for the Casimir energy on the Einstein cylinder from the behaviour of the conformal anomaly as $q\to0$ and compared with known results written as alternating sums over scalar values.