C-T for conformal higher spin fields from partition function on conically deformed sphere

Abstract

We consider the one-parameter generalization Sq4 of 4-sphere with a conical singularity due to identification τ = τ +2πq in one isometric angle. We compute the value of the spectral zeta-function at zero ζˆ(q)=ζ(0;q)ζ^(q)=ζ(0;q) that controls the coefficient of the logarithmic UV divergence of the one-loop partition function on Sq4. While the value of the conformal anomaly a-coefficient is proportional to ζˆ(1)ζ^(1), we argue that in general the second c ∼ CT anomaly coefficient is related to a particular combination of the second and first derivatives of ζˆ(q)ζ^(q) at q = 1. The universality of this relation for CT is supported also by examples in 6 and 2 dimensions. We use it to compute the c-coefficient for conformal higher spins finding that it coincides with the “r = −1” value of the one-parameter Ansatz suggested in arXiv:1309.0785. Like the sums of as and cs coefficients, the regularized sum of ζˆs(q)ζ^s(q) over the whole tower of conformal higher spins s = 1, 2,… is found to vanish, implying UV finiteness on Sq4 and thus also the vanishing of the associated Rényi entropy. Similar conclusions are found to apply to the standard 2-derivative massless higher spin tower. We also present an independent computation of the full set of conformal anomaly coefficients of the 6d Weyl graviton theory defined by a particular combination of the three 6d Weyl invariants that has a (2, 0) supersymmetric extension.