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

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.