Interactions in Higher-Spin Gravity: a Holographic Perspective

TL;DR

This work leverages the HS AdS/CFT duality between the type A minimal bosonic HS theory on AdS_{d+1} and the free scalar O(N) vector model to reconstruct interactions in metric-like HS gravity. By computing CFT correlators in the free vector model and matching them to tree-level AdS Witten diagrams, it fixes the complete cubic couplings for all HS fields and derives the quartic self-interaction of the AdS_4 scalar. The ambient space formalism and conformal partial wave (CPW) techniques enable efficient evaluation of spinning three-point and four-point amplitudes, including scalar exchange and HS exchanges, and their CPW decompositions. The resulting framework provides a concrete, holographically constrained set of interaction vertices, offers insights into locality properties in HS theories, and yields methodological tools that connect CFT data, Witten diagrams, and bulk Lagrangian structure, with broader relevance to string theory and quantum gravity. Key findings include a finite, local cubic vertex for fixed spins, a nonlocal yet controlled quartic scalar vertex in AdS_4, and a generating function for quartic coefficients that reproduces the expected double-trace CPW structure while canceling single-trace contributions in the appropriate channel. The work thus advances a concrete, gauge-consistent path to understanding HS interactions from a CFT vantage, informing broader questions about HS locality, holography, and the tensionless limit of string theory.

Abstract

This thesis is an elaboration of recent results on the holographic re-construction of metric-like interactions in higher-spin gauge theories on anti-de Sitter space (AdS), employing their conjectured holographic duality with free conformal field theories (CFTs). After reviewing the general approach and establishing the necessary intermediate results, we extract explicit expressions for the complete cubic action on AdS$_{d+1}$ and the quartic self-interaction of the scalar on AdS$_4$ for the type A minimal bosonic higher-spin theory from the three- and four- point correlation functions of single-trace operators in the free scalar $O\left(N\right)$ vector model. For this purpose tools were developed to evaluate tree-level three-point Witten diagrams involving fields of arbitrary integer spin and the conformal partial wave expansions of tree-level four-point Witten diagrams, which are underpinned by the ambient space formulation of AdS space and CFT. We also discuss the implications of the holographic duality on the locality properties of interactions in higher-spin gauge theories.

Figures (14)

• Figure 1: For a given dual pair \ref{['adscft0']}, we can think of the CFT$_{d}$ as living on the boundary (solid black boundary of disc) of the gravity theory in asymptotically AdS$_{d+1}$ (entire disc). This picture can be obtained by considering Euclidean AdS and compactifying the $\mathbb{R}^d$ boundary to $S^d$ by adding a point at infinity.
• Figure 2: Holographic interpretation of the three-point function of operators ${\cal J}_{s_i}$ in the free $O\left(N\right)$ model at large $N$. The operators ${\cal J}_{s_i}$ are inserted on the boundary of AdS$_{d+1}$, sourced by the boundary values of their dual fields ${\varphi_{s_i}}$ in AdS. The wavy lines are the propagators of the ${\varphi_{s_i}}$. With the knowledge of $\langle {\cal J}_{s_1}{\cal J}_{s_2}{\cal J}_{s_3} \rangle$ the on-shell form of the cubic interaction ${\cal V}_{s_1,s_2,s_3}$ can then in principle be determined.
• Figure 3: Contributions to the full four-point function \ref{['vec4pt2']} of the scalar single-trace operator ${\cal O}$. The first line constitutes the three disconnected terms, while the second line comprises the connected part of the correlator.
• Figure 4: Bulk interpretation of the $\langle {\cal O}{\cal O} \rangle$ CFT two-point function, in the large $N_{\text{dof}}$ limit. The propagator departs from one boundary insertion point to the other.
• Figure 5: Tree-level three-point Witten diagram generated by the bulk vertex $g\phi^3$. This gives the holographic computation of the dual scalar operator three-point function at leading order in $1/N_{\text{dof}}$.