AdS Description of Induced Higher-Spin Gauge Theory

TL;DR

This work establishes a concrete holographic framework for deformations of 3d large-$N$ CFTs by double-trace operators built from spin-$s$ currents, linking RG flows to modified AdS$_4$ boundary conditions for the dual spin-$s$ fields. It derives a universal expression for the sphere free-energy shift $oxed{oxed{\delta F^{(s)}_ abla}}$ and shows that, at large $N$, conserved currents with spin $s$ induce parity-invariant conformal higher-spin gauge theories in 3d, dual to Vasiliev-like bulk theories with alternate boundary conditions. The paper also explains the appearance of a $oxed{(1/2)\log N}$ contribution governed by the number $n_{s-1}$ of trivial spin-$s$ gauge transformations (conformal Killing tensors), and extends the analysis to half-integer spins, Chern–Simons terms, and Weyl anomalies in even dimensions, providing efficient AdS-based prescriptions to compute anomaly coefficients $a$ in $d=4$ and central charges in $d=2$. Together, these results solidify the AdS/induced-gauge-theory correspondence for higher spins and illuminate the structure of induced conformal higher-spin theories across dimensions.

Abstract

We study deformations of three-dimensional large N CFTs by double-trace operators constructed from spin s single-trace operators of dimension Δ. These theories possess UV fixed points, and we calculate the change of the 3-sphere free energy δF= F_{UV}- F_{IR}. To describe the UV fixed point using the dual AdS_4 space we modify the boundary conditions on the spin s field in the bulk; this approach produces δF in agreement with the field theory calculations. If the spin s operator is a conserved current, then the fixed point is described by an induced parity invariant conformal spin s gauge theory. The low spin examples are QED_3 (s=1) and the 3-d induced conformal gravity (s=2). When the original CFT is that of N conformal complex scalar or fermion fields, the U(N) singlet sector of the induced 3-d gauge theory is dual to Vasiliev's theory in AdS_4 with alternate boundary conditions on the spin s massless gauge field. We test this correspondence by calculating the leading term in δF for large N. We show that the coefficient of (1/2)\log N in δF is equal to the number of spin s-1 gauge parameters that act trivially on the spin s gauge field. We discuss generalizations of these results to 3-d gauge theories including Chern-Simons terms and to theories where s is half-integer. We also argue that the Weyl anomaly a-coefficients of conformal spin s theories in even dimensions d, such as that of the Weyl-squared gravity in d=4, can be efficiently calculated using massless spin s fields in AdS_{d+1} with alternate boundary conditions. Using this method we derive a simple formula for the Weyl anomaly a-coefficients of the d=4 Fradkin-Tseytlin conformal higher-spin gauge fields. Similarly, using alternate boundary conditions in AdS_3 we reproduce the well-known central charge c=-26 of the bc ghosts in 2-d gravity, as well as its higher-spin generalizations.

Figures (1)

• Figure 1: $\delta F_\Delta^{(s)}$ plotted as a function of $\Delta$ for $s =0$, 1, 2, and 3. When $s = 0$ this quantity is required by the $F$-theorem to be positive for $3/2 < \Delta < 5/2$, but outside of this range and also for higher-spin, the $F$-theorem does not apply since one or both of the fixed points is non-unitary. The exception is when $\Delta = s+1$, since in this case the naive unitarity arguments are not valid.