Lectures on holographic renormalization

Abstract

We provide a pedagogical introduction to the method of holographic renormalization, in its Hamiltonian incarnation. We begin by reviewing the description of local observables, global symmetries, and ultraviolet divergences in local quantum field theories, in a language that does not require a weak coupling Lagrangian description. In particular, we review the formulation of the Renormalization Group as a Hamiltonian flow, which allows us to present the holographic dictionary in a precise and suggestive language. The method of holographic renormalization is then introduced by first computing the renormalized two-point function of a scalar operator in conformal field theory and comparing with the holographic computation. We then proceed with the general method, formulating the bulk theory in a radial Hamiltonian language and deriving the Hamilton–Jacobi equation. Two methods for solving recursively the Hamilton–Jacobi equation are then presented, based on covariant expansions in eigenfunctions of certain functional operators on the space of field theory couplings. These algorithms constitute the core of the method of holographic renormalization and allow us to obtain the holographic Ward identities and the asymptotic expansions of the bulk fields.