Modular Properties of 3D Higher Spin Theory

TL;DR

This work extends modular properties from spin-2 gravity to 3D higher-spin ($\,\mathfrak{sl}(N)$) Chern-Simons theory by showing the conical surplus and BTZ-like black holes are related through the boundary torus modulus via the S-transformation $\tau\to -1/\tau$. It constructs a full $\textrm{SL}(2,\mathbb{Z})$ family of smooth solutions, related by coordinate transformations that act on the boundary homology, and demonstrates that their thermodynamics and free energies transform covariantly under modular maps. A canonical variational framework is established for all family members, yielding modular-invariant expressions for the on-shell action, entropy, and free energy, and enabling a modular-invariant, tree-level partition function obtained by summing modular images over both flux vectors $\vec{n}$ and modular images. The results connect holographic thermodynamics with modular forms and provide a robust platform for exploring phase structure and modular completions in higher-spin holography. The distinction between canonical and holomorphic approaches is clarified, highlighting the modular behavior that emerges in the canonical formalism and its impact on integrability and partition functions.

Abstract

In the three-dimensional sl(N) Chern-Simons higher-spin theory, we prove that the conical surplus and the black hole solution are related by the S-transformation of the modulus of the boundary torus. Then applying the modular group on a given conical surplus solution, we generate a 'SL(2,Z)' family of smooth constant solutions. We then show how these solutions are mapped into one another by coordinate transformations that act non-trivially on the homology of the boundary torus. After deriving a thermodynamics that applies to all the solutions in the 'SL(2,Z)' family, we compute their entropies and free energies, and determine how the latter transform under the modular transformations. Summing over all the modular images of the conical surplus, we write down a (tree-level) modular invariant partition function.