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

Abstract

We consider the one-parameter generalization $S_q^4$ 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 $S_q^4$. 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 $S_q^4$ 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.

29 pages. v2: minor changes

References in corpus (29)

• The Holographic Weyl anomaly
• Entanglement entropy, conformal invariance and extrinsic geometry
• Conformal Higher Spin Theory
• One Loop Tests of Higher Spin AdS/CFT
• On one-loop renormalization of black-hole entropy
• Conformal anomaly of (2,0) tensor multiplet in six dimensions and AdS/CFT correspondence
• Effective action in a higher-spin background
• Higher spins in AdS_5 at one loop: vacuum energy, boundary conformal anomalies and AdS/CFT
• On partition function and Weyl anomaly of conformal higher spin fields
• On limits of superstring in AdS_5 x S^5
• Renyi Entropies for Free Field Theories
• AdS Description of Induced Higher-Spin Gauge Theory
• Spectral Geometry and One-loop Divergences on Manifolds with Conical Singularities
• Partition function of free conformal higher spin theory
• A universal feature of CFT Renyi entropy
• Partition Sums and Entropy Bounds in Weakly Coupled CFT
• Cones, Spins and Heat Kernels
• Invariants for minimal conformal supergravity in six dimensions
• Ordinary-derivative formulation of conformal totally symmetric arbitrary spin bosonic fields
• On triviality of S-matrix in conformal higher spin theory
• On conformal higher spin wave operators
• Weyl anomaly of conformal higher spins on six-sphere
• Arbitrary spin conformal fields in (A)dS
• Heat-kernel Coefficients and Spectra of the Vector Laplacians on Spherical Domains with Conical Singularities
• The component structure of conformal supergravity invariants in six dimensions
• Conformal anomaly c-coefficients of superconformal 6d theories
• On induced action for conformal higher spins in curved background
• $C_T$ for higher derivative conformal fields and anomalies of (1,0) superconformal 6d theories
• The C_2 heat-kernel coefficient in the presence of boundary discontinuities

Cited by in corpus (14)

• Free energy and boundary anomalies on ${\mathbb{S}}^a\times {\mathbb{H}}^b$ spaces
• Hyperbolic cylinders and entanglement entropy: gravitons, higher spins, $p$-forms
• Conformal Higher-Spin Gravity: Linearized Spectrum = Symmetry Algebra
• Partition functions of higher derivative conformal fields on conformally related spaces
• One-Loop Holographic Weyl Anomaly in Six Dimensions
• On the Weyl anomaly of 4D Conformal Higher Spins: a holographic approach
• Partition functions and entanglement entropy: Weyl graviton and conformal higher spin fields
• Nonabelian M5-brane on $S_q^6$
• GJMS-like operators on symmetric 2-tensors and their gravitational duals
• De Sitter Horizon Edge Partition Functions
• A study about black hole solutions with nonconstant transversal curvature and its conserved charges in Lovelock gravity
• A calculation of the Weyl anomaly for 6D Conformal Higher Spins
• Abelian M5-brane on $S_q^6$
• One-loop divergences in 7D Einstein and 6D Conformal Gravities