On higher spin partition functions

Abstract

We observe that the partition function of the set of all free massless higher spins s=0,1,2,3,... in flat space is equal to one: the ghost determinants cancel against the "physical" ones or, equivalently, the (regularized) total number of degrees of freedom vanishes. This reflects large underlying gauge symmetry and suggests analogy with supersymmetric or topological theory. The Z=1 property extends also to the AdS background, i.e. the 1-loop vacuum partition function of Vasiliev theory is equal to 1 (assuming a particular regularization of the sum over spins); this was noticed earlier as a consistency requirement for the vectorial AdS/CFT duality. We find that Z=1 is also true in the conformal higher spin theory (with higher-derivative d^{2s} kinetic terms) expanded near flat or conformally flat S^4 background. We also consider the partition function of free conformal theory of symmetric traceless rank s tensor field which has 2-derivative kinetic term but only scalar gauge invariance in flat 4d space. This non-unitary theory has a Weyl-invariant action in curved background and corresponds to "partially massless" field in AdS_5. We discuss in detail the special case of s=2 (or "conformal graviton"), compute the corresponding conformal anomaly coefficients and compare them with previously found expressions for generic representations of conformal group in 4 dimensions.

35 pages. v3: minor clarifications, comments, and references added

References in corpus (27)

• A Quartic Conformally Covariant Differential Operator for Arbitrary Pseudo-Riemannian Manifolds (Summary)
• How higher-spin gravity surpasses the spin two barrier: no-go theorems versus yes-go examples
• A universal result on central charges in the presence of double-trace deformations
• Higher Spin Gauge Theories in Various Dimensions
• One Loop Tests of Higher Spin AdS/CFT
• Effective action in a higher-spin background
• Higher spins in AdS_5 at one loop: vacuum energy, boundary conformal anomalies and AdS/CFT
• Higher order singletons, partially massless fields and their boundary values in the ambient approach
• On partition function and Weyl anomaly of conformal higher spin fields
• Bosonic conformal higher--spin fields of any symmetry
• Higher Spin AdS${}_{d+1}$/CFT${}_d$ at One Loop
• Partition Functions and Casimir Energies in Higher Spin AdS_{d+1}/CFT_d
• Partition function of free conformal higher spin theory
• On higher spin interactions with matter
• Conformal Invariance of Partially Massless Higher Spins
• Ordinary-derivative formulation of conformal totally symmetric arbitrary spin bosonic fields
• Partial Masslessness and Conformal Gravity
• Gravitational- and Self- Coupling of Partially Massless Spin 2
• Weyl anomaly of conformal higher spins on six-sphere
• Partition Functions for Higher-Spin theories in AdS
• Polyakov formulas for GJMS operators from AdS/CFT
• Quantum Effective Action in Spacetimes with Branes and Boundaries
• BRST invariant effective action of shadow fields, conformal fields, and AdS/CFT
• Conformal Powers of the Laplacian via Stereographic Projection
• Conformally invariant wave equation for a symmetric second rank tensor ("spin-2") in d-dimensional curved background
• Laplacian operators and Q-curvature on conformally Einstein manifolds
• Effective action for conformal spins on spheres with the multiplicative and conformal anomalies

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• Light-cone AdS/CFT-adapted approach to AdS fields/currents, shadows, and conformal fields
• BRST-BV approach to massless fields adapted to AdS/CFT correspondence