Quartic AdS Interactions in Higher-Spin Gravity from Conformal Field Theory

TL;DR

The paper investigates locality in higher-spin gravity by holographically reconstructing the quartic self-interaction of the AdS scalar within the type A minimal bosonic HS theory on $AdS_4$, dual to the free $O(N)$ vector model in $d=3$. It fixes the bulk cubic couplings by matching to CFT three-point functions, computes tree-level four-point Witten diagrams (exchanges plus contact), and uses conformal-block and split-representation techniques to extract the bulk scalar quartic vertex as a derivative-expansion of the form $\mathcal{V}=\sum_{s} J_{s}(x,\partial_u)\,a_s(\Box)\,J_{s}(x,u)$. By comparing with the boundary four-point function and employing a channel-reduction strategy, the authors determine the generating function $a_s(\Box)$, showing that the resulting vertex, though unbounded in derivatives, yields a four-point amplitude that is analytic in Mellin space, i.e. is “weakly local.” The work thus provides evidence that higher-spin holography imposes strong locality-like constraints on bulk interactions via CFT data and crossing, and it outlines a framework applicable to more general higher-spin vertices and dimensions.

Abstract

Clarifying the locality properties of higher-spin gravity is a pressing task, but notoriously difficult due to the absence of a weakly-coupled flat regime. The simplest non-trivial case where this question can be addressed is the quartic self-interaction of the AdS scalar field present in the higher-spin multiplet. We investigate this issue in the context of the holographic duality between the minimal bosonic higher-spin theory on AdS$_4$ and the free $O\left(N\right)$ vector model in three dimensions. In particular, we determine the exact explicit form of the derivative expansion of the bulk scalar quartic vertex. The quartic vertex is obtained from the field theory four-point function of the operator dual to the bulk scalar, by making use of our previous results for the Witten diagrams of higher-spin exchanges. This is facilitated by establishing the conformal block expansions of both the boundary four-point function and the dual bulk Witten diagram amplitudes. We show that the vertex we find satisfies a generalised notion of locality.

Figures (8)

• Figure 1: Four-point Witten diagrams contributing to the holographic computation of the connected scalar singlet ${\cal O}$ four-point function. Diagrams (a), (b) and (c) are exchanges of massless spin-$s$ fields between two pairs of the real scalar $\varphi_0$, in the s-, t- and u-channel respectively. The contact diagram (d) is the amplitude associated to the quartic vertex ${\cal V}$ of $\varphi_0$, which we seek to establish by matching with the dual CFT result.
• Figure 2: The split representation of four-point bulk amplitudes: A given four-point exchange or contact Witten diagram can be decomposed into products of two three-point diagrams involving two of the original external fields and a spin-$k$ field of dual operator dimension $\tfrac{d}{2} \pm i\nu$, integrating over the common boundary point. Analogous to the conformal block expansion of a CFT correlator \ref{['introcb']}, the function $g_{k}\left(\nu\right)$ encodes the intermediate states.
• Figure 3: The split representation of exchange Witten diagrams: In expressing the bulk-to-bulk propagators in a basis of harmonic functions \ref{['split']}, the exchange amplitude decomposes into products of two three-point Witten diagrams. These involve two of the original external fields and a field of dual operator dimension $\tfrac{d}{2} \pm i\nu$, integrated over their common boundary point.
• Figure 4: Four-point contact diagram generated by a quartic vertex ${\cal V}_{m,s}$ of the basis \ref{['quartbasis']}. The vertical line through the interaction point serves to illustrate that, for the given labeling of external legs, the quartic interaction corresponds to gluing a bulk current $J_s$, associated the two external legs $y_{1,2}$, to another bulk current $J_s$, associated with the two remaining external legs $y_{3,4}$, with a power of the Laplacian $\Box^m$ in between them, c.f. \ref{['::contactsch']}. In accordance with the Feynman rules, the remaining diagrams contributing to the total contact amplitude for this vertex can be obtained from this amplitude by permuting the external legs.
• Figure 5: Contributions to the full scalar single-trace operator four-point function \ref{['full']} using Wick contractions. The first line constitutes the disconnected $O\left(N^0\right)$ terms, while the second line comprises the connected part of the correlator, which is $O\left(1/N\right)$.