Correlation Functions in Holographic Renormalization Group Flows

TL;DR

The paper develops a framework to compute all one- and two-point functions in holographic RG flows for a bulk theory of scalars coupled to gravity, treating scalar and metric boundary data as independent sources. By linearizing around Poincaré-invariant backgrounds with active scalars and employing a gauge that factorizes the fluctuation equations, it derives the quadratic on-shell action and, after renormalization, the non-local correlators $ig\\langle \,\mathcal{O} \mathcal{O}\,\big\rangle$, $\big\langle T^i_j \mathcal{O} \big\rangle$, and $\big\langle T^i_j T^k_l \big\rangle$ for RG flows. The method distinguishes operator flows from vev flows via condensates and is validated on the GPPZ and Coulomb-branch flows, where it reproduces known results and reveals relations such as $T^i_i = 2\,\sqrt{3}\,\mathcal{O}$ in the GPPZ case and a massless-dilaton pole in the Coulomb-branch case. This provides a comprehensive, gauge-invariant approach to holographic correlation functions and offers insights into holographic beta functions and potential $c$-function connections.

Abstract

We consider the holographic duality for a generic bulk theory of scalars coupled to gravity. By studying the fluctuations around Poincare invariant backgrounds with non-vanishing scalars, with the scalar and metric boundary conditions considered as being independent, we obtain all one- and two-point functions in the dual renormalization group flows of the boundary field theory. Operator and vev flows are explicitly distinguished by means of the physical condensates. The method is applied to the GPPZ and Coulomb branch flows, and field theoretical expectations are confirmed.