Boundary Conditions and Partition Functions in Higher Spin AdS$_3$/CFT$_2$

TL;DR

This paper explores how different boundary conditions and definitions of the partition function in higher spin AdS$_3$/CFT$_2$ holography affect the correspondence, clarifying previous ambiguities in the literature.

Contribution

It provides a detailed analysis of boundary terms and conditions in Chern-Simons theory for higher spin holography, distinguishing between Hamiltonian and action deformations.

Findings

Identifies the correct boundary terms for various partition functions.

Clarifies the relationship between Hamiltonian and action deformations.

Resolves ambiguities in the holographic description of higher spin theories.

Abstract

We discuss alternative definitions of the semiclassical partition function in two-dimensional CFTs with higher spin symmetry, in the presence of sources for the higher spin currents. Theories of this type can often be described via Hamiltonian reduction of current algebras, and a holographic description in terms of three-dimensional Chern-Simons theory with generalized AdS boundary conditions becomes available. By studying the CFT Ward identities in the presence of non-trivial sources, we determine the appropriate choice of boundary terms and boundary conditions in Chern-Simons theory for the various types of partition functions considered. In particular, we compare the Chern-Simons description of deformations of the field theory Hamiltonian versus those encoding deformations of the CFT action. Our analysis clarifies various issues and confusions that have permeated the literature on this subject.