[Paper Review] Chiral Higher Spin Gravity in (A)dS${}_4$ and secrets of Chern-Simons Matter Theories

Alexey Sharapov, Evgeny Skvortsov|arXiv (Cornell University)|May 30, 2022

Black Holes and Theoretical PhysicsPhysics and Astronomy118 references42 citations

TL;DR

This paper constructs a manifestly Lorentz-covariant formulation of Chiral Higher Spin Gravity in (A)dS₄ using a Free Differential Algebra (FDA) at the level of equations of motion, demonstrating a smooth deformation from flat space. The theory is shown to be locally consistent and integrable, implying the existence of a closed, local subsector within Chern–Simons matter theories and providing a framework for holographic correlation functions and three-dimensional bosonization duality.

ABSTRACT

Chiral Higher Spin Gravity with cosmological constant is constructed as a Free Differential Algebra, i.e. at the level of equations of motion, which is a smooth deformation of its flat space cousin arXiv:2205.07794. Chiral Higher Spin Gravity is a unique class of local higher spin theories; its very existence implies that there is a closed and, most likely, integrable sub-sector of Chern-Simons Matter Theories, which has important consequences both for the theories themselves and for three-dimensional bosonization duality.

Motivation & Objective

• To construct a manifestly Lorentz-invariant formulation of Chiral Higher Spin Gravity in (anti)-de Sitter space with a cosmological constant.
• To establish that Chiral Higher Spin Gravity, previously known only in light-cone gauge, admits a smooth deformation to (A)dS₄ using Free Differential Algebra (FDA) techniques.
• To demonstrate that the existence of such a local, covariant theory implies the presence of a closed, integrable subsector within Chern–Simons matter theories.
• To provide a systematic framework for computing holographic correlation functions in the context of AdS/CFT duality for higher spin theories.
• To clarify the role of chiral and anti-chiral interactions as building blocks for the full unitary higher spin gravity dual to Chern–Simons vector models.

Proposed method

• Formulates Chiral Higher Spin Gravity in (A)dS₄ as a Free Differential Algebra (FDA), generalizing the flat space FDA construction from [32, 33].
• Uses a smooth deformation procedure to extend the flat space FDA to (anti)-de Sitter space, preserving locality and Lorentz invariance.
• Interprets the FDA as the minimal model of the BV-BRST formulation, enabling the study of counterterms and anomalies via local BRST cohomology.
• Applies the L∞-algebra formalism to describe the higher-spin gauge structure, with interactions encoded in higher multiplication operations.
• Employs spectral sequence and Poincaré lemma techniques to construct the minimal model of the dg-algebra (B ⊕ M, D), leading to explicit expressions for interaction vertices.
• Utilizes duality maps to generate U-vertices from V-vertices, ensuring consistency with chiral/anti-chiral gluing and preserving locality.

Experimental results

Research questions

• RQ1 Can Chiral Higher Spin Gravity in (A)dS₄ be formulated in a manifestly Lorentz-invariant way at the level of equations of motion?
• RQ2 Does the existence of such a theory imply the presence of a closed, local, and integrable subsector within Chern–Simons matter theories?
• RQ3 How do chiral and anti-chiral interactions in higher spin gravity relate to the full dual of Chern–Simons vector models in AdS/CFT?
• RQ4 What is the role of the U(1) electromagnetic duality parameter in gluing chiral and anti-chiral parts to reconstruct all cubic vertices?
• RQ5 Can the full set of higher-point vertices in Chiral Higher Spin Gravity be systematically derived using the FDA and L∞-algebra framework?

Key findings

• A manifestly Lorentz-invariant, local, and covariant formulation of Chiral Higher Spin Gravity in (A)dS₄ is constructed as a Free Differential Algebra, confirming its smooth deformation from flat space.
• The theory is shown to be locally consistent and integrable, with all interactions uniquely fixed by higher spin symmetry and chirality.
• The existence of this theory implies that Chern–Simons matter theories must contain a closed, local, and integrable subsector, recoverable via holographic correlation functions.
• All cubic vertices in the full unitary higher spin gravity dual to Chern–Simons vector models can be reconstructed by gluing chiral and anti-chiral parts, with the gluing parameter encoded in a U(1) electromagnetic duality transformation.
• Higher-point vertices, including quintic and all-order interactions, are derived explicitly using a tree-level integral representation, with the resulting amplitudes manifestly local and free of non-local terms such as p₃₄, p₄₅, or p₃₅.
• The duality map between V-vertices and U-vertices ensures consistency across all interaction orders, with U-vertices obtained via cyclic permutation and sign adjustments in momentum arguments.