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

TL;DR

This work clarifies how semiclassical partition functions for 2d CFTs with higher spin symmetry are defined and computed, distinguishing deformations of the CFT Hamiltonian from deformations of the CFT action and linking each to distinct Chern-Simons boundary conditions via Drinfeld-Sokolov reduction. It derives the corresponding Ward identities and shows how they map to flatness conditions of SL(N)×SL(N) (or hs[λ]×hs[λ]) CS connections, enabling universal expressions for free energy and entropy in both canonical and holomorphic formalisms. The authors expose the subtleties of chiral vs non-chiral deformations, the role of boundary terms, and the impact of modular transformations, and they explain how field redefinitions relate the different holographic pictures while preserving entropy. These results clarify longstanding confusions in the literature and provide a coherent holographic framework for higher-spin thermodynamics, with implications for modular properties and potential extensions to non-Chiral setups and different embeddings. The analysis also highlights practical connections to condensed-mense dynamical ensembles and the broader structure of W-algebras in holography.

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.