Holographic Renormalization Group with Fermions and Form Fields

TL;DR

This work develops a Hamilton-Jacobi framework for holographic renormalization group analysis in gravity theories coupled to fermions and form fields, clarifying how bulk constraints map to boundary Ward identities. It derives descent equations and a Callan-Symanzik type equation with beta functions that govern holographic RG flows, and shows how the divergent local action is constrained by first-class bulk constraints. A key result is that diffeomorphism Ward identities can acquire anomalies unless boundary couplings satisfy specific alignment conditions or are canceled by counterterms, with explicit relations linking bulk and boundary data. The analysis lays groundwork for including broader bulk symmetries and higher fermion interactions in holographic duals and highlights the role of fermions and form fields in shaping Ward identities and potential boundary anomalies.

Abstract

We find the Holographic Renormalization Group equations for the holographic duals of generic gravity theories coupled to form fields and spin-1/2 fermions. Using Hamilton-Jacobi theory we discuss the structure of Ward identities, anomalies, and the recursive equations for determining the divergent terms of the generating functional. In particular, the Ward identity associated to diffeomorphism invariance contains an anomalous contribution that, however, can be solved either by a suitable counter term or by imposing a condition on the boundary fields. Consistency conditions for the existence of the dual arise, if one requires that a Callan-Symanzik type equation follows from the Hamiltonian constraint. Under mild assumptions we are able to find a class of solutions to the constraint equations. The structure of the fermionic phase space and constraints is treated extensively for any dimension and signature.