Correlators of Large N Fermionic Chern-Simons Vector Models

TL;DR

This work computes planar correlators in the large $N$ limit of the $U(N)_k$ Chern-Simons theory with a fundamental Dirac fermion and demonstrates exact agreement with the bosonization duality to the Legendre-transformed bosonic theory for all $\lambda=N/k$ at the level of three-point functions. By solving Schwinger-Dyson equations in light-cone gauge, the authors obtain closed-form exact vertices for $J^{(0)}$ and $J^{(1)}$, enabling precise momentum-space two- and three-point functions that determine the Maldacena-Zhiboedov parameters $\tilde{N}$ and $\tilde{\lambda}$. The duality map to the critical bosonic theory is fixed, including the sign of the CS level and the relation between triple-trace deformations, via explicit matching of $\langle J^{(s)} J^{(s')} J^{(s'')}\rangle$ structures. The critical (Legendre-transformed) fermionic model is analyzed, revealing the same duality structure and a consistent mapping of $\lambda_6$ couplings, enriching the web of high-spin/vector-model dualities. Overall, the results provide robust planar evidence for non-supersymmetric bosonization in three dimensions with concrete, testable predictions for correlation functions and deformation parameters.

Abstract

We consider the large N limit of three-dimensional U(N)_k Chern-Simons theory coupled to a Dirac fermion in the fundamental representation. In this limit, we compute several correlators to all orders in the `t Hooft coupling N/k. It was suggested recently that this theory is dual to the Legendre-transformed theory of scalar fields coupled to Chern-Simons gauge interactions. Our results show that this duality holds for any value of the `t Hooft coupling, at least at the level of the planar 3-point functions. In addition, we determine the sign in the duality transformation of the Chern-Simons level, as well as the relation between the "triple-trace" deformation which exists in the bosonic Chern-Simons theory and in the Legendre-transformed fermionic theory.

Figures (3)

• Figure 1: Schematic form of the Schwinger-Dyson equations for the vertices. The circled cross denotes an insertion of an exact $J^{(0)}$ or $J^{(1)}$ vertex. The un-circled cross denotes an insertion of the free vertex. A filled circle denotes the exact fermion propagator.
• Figure 2: Diagrams contributing to $\langle J^{(s)} J^{(s)} \rangle$.
• Figure 3: Diagrams contributing to $\langle J^{(0)} J^{(0)} J^{(0)} \rangle$ and $\langle J^{(0)} J^{(1)}_+ J^{(1)}_- \rangle$.