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

Abstract

The zeta function of an arbitrary field in (d + 1)-dimensional anti-de Sitter (AdS) spacetime is expressed as an integral transform of the corresponding so(2, d) representation character, thereby extending the results of [1603.05387] for AdS 4 and AdS 5 to arbitrary dimensions. The integration in the variables associated with the so(d) part of the character can be recast into a more explicit form using derivatives. The explicit derivative expressions are presented for AdS d+1 with d = 2, 3, 4, 5, 6. A Character identities 26 B Generalized L'Hôpital's rule 28 B.1 Computing the denominator 28 B.2 Simplifying the numerator 29 -i -

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

“…Characters of the so(d) algebra can be found in [57] or the textbooks [104,105]. For more details on character of the conformal algebra, see e.g.…”

“…8) where the anti-clockwise w contour encloses the origin but excludes ±β. Using the result of [57], one can further express the first derivative of the zeta function in terms of contour integrals, namely…”

“…Our analysis provides a proof of this for arbitrary ℓ, d = 2r and m. As mentioned above, evaluation of the contour integral form of CIRZ is generically complicated because the β = 0 pole can contribute. For this reason, to evaluate the free energy contribution from cyc n for generic n, we fix d = 4 and use the derivative form of CIRZ obtained in [52], which reads (in the notations of [57]) In general, expressions for the vacuum energy Γ (1) cyc n are rather complicated functions of ℓ, for instance Γ (1) cyc 5 is already a polynomial of order 21 in ℓ. This can be traced back to the fact whenever the order n has divisors k such that n/k is odd, the pole in β = 0 of (6.13) contributes, and the order of this pole depends on n and k. As a consequence, the order of the resulting polynomial ℓ grows with these parameters.…”

Conformal higher-spin gravity: linearized spectrum = symmetry algebra

“…19 Taking a definite boundary condition 20 we find that the so(2, 3) character of this theory reads where we have written the character in terms of the temperature β and the variable α defined through q = e −β and x = e iα . Using the method introduced in [81,82], we can compute the one-loop free energy of this theory in Euclidean AdS 4 using the above character. To do so, one needs to evaluate the character (4.8) and its derivatives (with respect to the variable α) in α → 0, which is singular here (contrary to MHS theories and their partially-massless cousins).…”

“…Since the free energy with Neumann boundary condition is simply minus that with Dirichlet condition (the contribution of Killing tensor module simply vanishes), the a-anomaly of the d-dimensional CHS gravity is just minus two times the free energy of MHS gravity in Euclidean AdS d+1 . The cancellation of the latter can be shown using the method of character integral representation of zeta function [81][82][83].…”