$C_T$ for conformal higher spin fields from partition function on conically deformed sphere

Abstract

We consider the one-parameter generalization $S^4_q$ of 4-sphere with a conical singularity due to identification $\tau=\tau + 2 \pi q$ in one isometric angle. We compute the value of the spectral zeta-function at zero $z(q) = \zeta(0, q)$ that controls the coefficient of the logarithmic UV divergence of the one-loop partition function on $S^4_q$. While the value of the conformal anomaly a-coefficient is proportional to $z(1)$, we argue that in general the second $c = C_T$ anomaly coefficient is related to a particular combination of the second and first derivatives of $z(q) $ at $q=1$. The universality of this relation for $C_T$ 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 $a_s$ and $c_s$ coefficients, the regularized sum of $z_s(q)$ over the whole tower of conformal higher spins $s=1,2, ...$ is found to vanish, implying UV finiteness on $S^4_q$ and thus also the vanishing of the associated Re'nyi 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.