The effect of resummation on retarded Green's function and greybody factor in $AdS$ black holes

TL;DR

This paper explores how resummation techniques influence the retarded Green's function and greybody factor in AdS black holes, linking complex differential equations to supersymmetric gauge theories to derive asymptotic behaviors and corrections.

Contribution

It introduces a novel resummation approach to analyze the Green's function and greybody factor in AdS black holes using connections to supersymmetric gauge theory.

Findings

Derived asymptotic expansions for Green's function and greybody factor

Computed corrections from resummation of instanton contributions

Linked differential equations to gauge theory free energy

Abstract

We investigate the retarded Green's function and the greybody factor in asymptotically AdS black holes. Using the connection coefficients of the Heun equation, expressed in terms of the Nekrasov-Shatashvili (NS) free energy of an $SU(2)$ supersymmetric gauge theory with four fundamental hypermultiplets, we derive asymptotic expansions for both the retarded Green's function and the greybody factor in the small horizon limit. Furthermore, we compute the corrections to these asymptotic expansions resulting from the resummation procedure of the instanton part of the NS function.