Higher Spin Gauge Theory and Holography: The Three-Point Functions

TL;DR

This work provides substantial evidence for the Klebanov–Polyakov duality by computing tree-level three-point functions in Vasiliev’s higher-spin theory in AdS$_4$ and showing exact matches with the free and critical $O(N)$ vector models in three dimensions, depending on the bulk scalar boundary condition. The authors construct boundary-to-bulk propagators for the master fields, solve the Vasiliev equations perturbatively to second order, and extract the coefficients $C(s_1,s_2;s_3)$ of the three-point functions, including cases with one scalar and two higher-spin currents. They show that, with appropriate normalization, the spin-$s$ correlators reproduce the free theory results for Δ=1 and the Δ=2 critical model results, and they discuss subtle regularization issues arising in certain correlators (notably $C(0,0;s)$ and $C(s,s;0)$). The results strengthen the program of higher-spin holography, indicate that bulk nonlocalities can be tamed in the correlator limit, and provide a concrete map between bulk couplings and the $1/N$ expansion of the boundary CFTs. The paper also outlines future directions, including resolving remaining puzzles and extending to general spin configurations $C(s_1,s_2;s_3)$, which could fix the bulk interaction function $f(\\Psi)$ in the dual theory.

Abstract

In this paper we calculate the tree level three-point functions of Vasiliev's higher spin gauge theory in AdS4 and find agreement with the correlators of the free field theory of N massless scalars in three dimensions in the O(N) singlet sector. This provides substantial evidence that Vasiliev theory is dual to the free field theory, thus verifying a conjecture of Klebanov and Polyakov. We also find agreement with the critical O(N) vector model, when the bulk scalar field is subject to the alternative boundary condition such that its dual operator has classical dimension 2.

Figures (3)

• Figure 1: $C(s_1,s_2;s_3)$ will be computed by sewing two boundary-to-bulk propagators, corresponding to sources of currents of spin $s_1$ and $s_2$, into a spin-$s_3$ field via the nonlinear equations of motion.
• Figure 2: The 1-loop contribution to $\langle\alpha\alpha\alpha\rangle$. The dotted lines represent the effective propagators $G(x)$ of $\alpha$, while the solid lines are propagators of $\vec{S}$.
• Figure 3: The leading $1/N$ contributions to $\langle SSSS\alpha\rangle$, from which we will extract $\langle J_s J_{s'} \alpha\rangle$.