The Analytic Bootstrap for Large $N$ Chern-Simons Vector Models

TL;DR

This work develops an analytic bootstrap program for large $N$ CS-matter theories in three dimensions, leveraging approximate higher-spin symmetry to constrain four-point functions and extract CFT data order by order in $1/N$. By combining crossing symmetry, twist conformal blocks, and the inversion formula, the authors fix leading single-trace exchanges, compute $O(1/N)$ corrections to double-trace data, and reconstruct the four-point function up to a finite set of truncations; at $O(1/N^2)$ they show the appearance of odd-twist double-trace operators and uncover operator-mixing effects that induce nontrivial $O(1/N)$ anomalous dimensions for even-twist double-trace operators. They provide explicit results for CS-Scalar and σ theories, as well as the parity-preserving bosonic $ ext{φ}^6$ model, including large-spin sums and exact expressions for anomalous dimensions and OPE coefficients. The findings illuminate one-loop (AdS$_4$) corrections in higher-spin holography and establish a concrete bootstrap framework for CS-matter theories beyond leading order, with potential extensions to more general CS-matter systems and supersymmetric cases.

Abstract

Three-dimensional Chern-Simons vector models display an approximate higher spin symmetry in the large $N$ limit. Their single-trace operators consist of a tower of weakly broken currents, as well as a scalar $σ$ of approximate twist $1$ or $2$. We study the consequences of crossing symmetry for the four-point correlator of $σ$ in a $1/N$ expansion, using analytic bootstrap techniques. To order $1/N$ we show that crossing symmetry fixes the contribution from the tower of currents, providing an alternative derivation of well-known results by Maldacena and Zhiboedov. When $σ$ has twist $1$ its OPE receives a contribution from the exchange of $σ$ itself with an arbitrary coefficient, due to the existence of a marginal sextic coupling. We develop the machinery to determine the corrections to the OPE data of double-trace operators due to this, and to similar exchanges. This in turns allows us to fix completely the correlator up to three known truncated solutions to crossing. We then proceed to study the problem to order $1/N^2$. We find that crossing implies the appearance of odd-twist double-trace operators, and calculate their OPE coefficients in a large spin expansion. Also, surprisingly, crossing at order $1/N^2$, implies non-trivial $O(1/N)$ anomalous dimensions for even-twist double-trace operators, even though such contributions do not appear in the four-point function at order $1/N$ (in the case where there is no scalar exchange). We argue that this phenomenon arises due to operator mixing. Finally, we analyse the bosonic vector model with a sextic coupling without gauge interactions, and determine the order $1/N^2$ corrections to the dimensions of twist-$2$ double-trace operators.