Character integral representation of zeta function in AdSd+1. Part II. Application to partially-massless higher-spin gravities

Abstract

Abstract:We compute the one-loop free energies of the type-A ℓ and type-B ℓ higherspin gravities in (d + 1)-dimensional anti-de Sitter (AdS d+1 ) spacetime. For large d and ℓ, these theories have a complicated field content, and hence it is difficult to compute their zeta functions using the usual methods. Applying the character integral representation of zeta function developed in the companion paper [arXiv:1805.05646] to these theories, we show how the computation of their zeta function can be shortened considerably. We find that the results previously obtained for the massless theories (ℓ = 1) generalize to their partially-massless counterparts (arbitrary ℓ) in arbitrary dimensions.

Character integral representation of zeta function in AdSd+1. Part I. Derivation of the general formula

“…, r. The contour of each α n is a circle enclosing the origin counter-clockwise with |α 0 | < · · · < |α r |. We would like to stress that the expressions (1.1) and (1.3) allow for a large room for various complex integral tricks: in the companion paper [28], we will compute the zeta function of partially-massless higher-spin gravities in arbitrary dimensions using complex integrals. The contour integrals in α i appearing in the above formulae (1.1) and (1.3) may be evaluated by the residue theorem, and hence reduce to a linear combination of derivatives in α i .…”

On the (A)dS decoupling limits of massive gravity

“…Higher spin AdS/CFT examples including partially massless states have been studied in[63][64][65][66][67][68][69][70].…”

Partition functions of p-forms from Harish-Chandra characters

“…According to the notation of [22], the 2-form coincides with p = 3, ∆ = 3, r = 2, x = 0 in the exceptional series. 5 It corresponds to a Young tableaux of a single column with 2 boxes or Y 3 = (1, 1). The contribution to the character comes from the second summation of equation ( 13) of [22].…”

“…One of the early instances was in [2] where the coincident heat kernel for arbitrary spin fields on thermal AdS 3 was written as a transform of the Harish-Chandra character of SL(2, C). In higher dimensional AdS character integral representations were found in [3][4][5] and [6,7], in the former series of works, the integral also involved angular integrations over the additional Cartan directions.…”