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

TL;DR

This paper uses analytic bootstrap techniques to analyze four-point functions in large N Chern-Simons vector models, revealing new insights into operator dimensions, OPE data, and crossing symmetry constraints up to order 1/N^2.

Contribution

It develops a systematic method to determine corrections to OPE data and correlators in large N Chern-Simons vector models, including effects of scalar exchanges and operator mixing.

Findings

Crossing symmetry fixes contributions from current towers at order 1/N.

Identifies odd-twist double-trace operators and computes their OPE coefficients.

Finds non-trivial anomalous dimensions for even-twist double-trace operators at order 1/N^2.

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 $\sigma$ of approximate twist $1$ or $2$. We study the consequences of crossing symmetry for the four-point correlator of $\sigma$ 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 $\sigma$ has twist $1$ its OPE receives a contribution from the exchange of $\sigma$ 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.