Character Integral Representation of Zeta function in AdS${}_{d+1}$: 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 arXiv: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 $d=2,3,4,5,6$.

31 pages; v2: typos corrected; v3: discussion on the free energy difference induced by a change of boundary condition for a (partially) massless field in even dimensional AdS added