Dualities in 3D Large $N$ Vector Models

TL;DR

This paper explicitly derives and compares dualities in 3D large N vector models using a path integral approach, providing constructive evidence for level/rank duality and analyzing its limitations at subleading order.

Contribution

It introduces a new explicit path integral derivation of 3D bosonization and duality in large N vector models, confirming level/rank duality and exploring its breakdown at subleading order.

Findings

Partition functions of dual theories agree at leading order in large N.

Level/rank duality holds at leading order but not necessarily at subleading order.

Provides a constructive proof of duality using path integral methods.

Abstract

Using an explicit path integral approach we derive non-abelian bosonization and duality of 3D systems in the large $N$ limit. We first consider a fermionic $U(N)$ vector model coupled to level $k$ Chern-Simons theory, following standard techniques we gauge the original global symmetry and impose the corresponding field strength $F_{\mu\nu}$ to vanish introducing a Lagrange multiplier $\Lambda$. Exchanging the order of integrations we obtain the bosonized theory with $\Lambda$ as the propagating field using the large $N$ rather than the previously used large mass limit. Next we follow the same procedure to dualize the scalar $U(N)$ vector model coupled to Chern-Simons and find its corresponding dual theory. Finally, we compare the partition functions of the two resulting theories and find that they agree in the large $N$ limit including a level/rank duality. This provides a constructive evidence for previous proposals on level/rank duality of 3D vector models in the large $N$ limit. We also present a partial analysis at subleading order in large $N$ and find that the duality does not generically hold at this level.