Sum Rules from an Extra Dimension

TL;DR

This work demonstrates that a broad holographic class of thermal field theories yields Green's function analyticity and sum rules that follow from gravity. By encoding bosonic correlators in a single second-order bulk equation and treating massless fermions via Dirac equations in black-brane backgrounds, the authors prove holomorphy in the upper half-plane, exclude upper-half-plane poles, and construct contraction maps to bound large-frequency behavior. They derive gravity-driven sum rules in various dimensions, recover established results such as the Romatschke–Son stress-tensor sum rule, and extend the framework to arbitrary $d$, including fermionic cases, with explicit large-$\omega$ asymptotics. The findings highlight a deeper gravitational origin for nonperturbative field-theory constraints and suggest avenues to explore dual interpretations of these sum rules and their generalizations to other backgrounds. Overall, the paper strengthens the bridge between causality- and unitarity-imposed Green's function properties and their holographic gravitational realization.

Abstract

Using the gravity side of the AdS/CFT correspondence, we investigate the analytic properties of thermal retarded Green's functions for scalars, conserved currents, the stress tensor, and massless fermions. We provide some results concerning their large and small frequency behavior and their pole structure. From these results, it is straightforward to prove the validity of various sum rules on the field theory side of the duality. We introduce a novel contraction mapping we use to study the large frequency behavior of the Green's functions.