The Holographic Dictionary for Beta Functions of Multi-trace Coupling Constants

TL;DR

This paper explores how multi-trace coupling constants in large N gauge theories with holographic duals exhibit non-trivial beta functions, and how this running is represented holographically through boundary conditions and bulk couplings, filling a gap in the holographic dictionary.

Contribution

It establishes a holographic dictionary entry linking the running of multi-trace couplings to specific bulk coupling constants and boundary conditions in AdS space.

Findings

Multi-trace couplings can have non-vanishing beta functions.

Boundary conditions in AdS depend on the cutoff scale, affecting conformal invariance.

Bulk coupling constants encode the running of multi-trace couplings.

Abstract

Field theories with weakly coupled holographic duals, such as large N gauge theories, have a natural separation of their operators into `single-trace operators' (dual to single-particle states) and `multi-trace operators' (dual to multi-particle states). There are examples of large N gauge theories where the beta functions of single-trace coupling constants all vanish, but marginal multi-trace coupling constants have non-vanishing beta functions that spoil conformal invariance (even when all multi-trace coupling constants vanish). The holographic dual of such theories should be a classical solution in anti-de Sitter space, in which the boundary conditions that correspond to the multi-trace coupling constants depend on the cutoff scale, in a way that spoils conformal invariance. We argue that this is realized through specific bulk coupling constants that lead to a running of the multi-trace coupling constants. This fills a missing entry in the holographic dictionary.