# An all order exact result for the anomalous dimension of the scalar primary in Chern Simons Vector Models

## TL;DR

This paper conjectures an all-order formula for the leading 1/N anomalous dimension of the scalar primary in Chern-Simons vector models, confirming it through perturbative calculations and duality constraints.

## Contribution

It proposes a universal all-order conjecture for the scalar anomalous dimension in Chern-Simons theories, supported by perturbative checks and duality consistency.

## Key findings

- Perturbative two-loop results match the conjecture.
- The conjecture is consistent with known dualities.
- It passes a non-trivial all-loop consistency test.

## Abstract

We present a conjecture for the leading $1/N$ anomalous dimension of the scalar primary operator in $U(N)_k$ Chern-Simons theories coupled to a single fundamental field, to all orders in the t'Hooft coupling $\lambda=\frac{N}{k}$. Following this we compute the anomalous dimension of the scalar in a Regular Bosonic theory perturbatively at two-loop order and demonstrate that matches exactly with the result predicted by our conjecture. We also show that our proposed expression for the anomalous dimension is consistent with all other existing two-loop perturbative results, which constrain its form at both weak and strong coupling thanks to the bosonization duality. Furthermore, our conjecture passes a novel non-trivial all loop test which provides a strong evidence for its consistency.