Character Integral Representation of Zeta function in AdS$_{d+1}$: I. Derivation of the general formula

TL;DR

This paper derives a general integral formula for the zeta function of fields in AdS$_{d+1}$, connecting it to $so(2,d)$ characters, extending previous results to arbitrary dimensions with explicit derivative forms.

Contribution

It generalizes the integral representation of the zeta function in AdS spaces to any dimension using $so(2,d)$ characters, providing explicit derivative formulas for multiple dimensions.

Findings

Derived a universal integral transform formula for the zeta function in AdS$_{d+1}$.

Extended previous results from specific dimensions to arbitrary $d$.

Provided explicit derivative expressions for dimensions $d=2$ to $6$.

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 $d=2,3,4,5,6$.