# Quantum line operators from Lax pairs

## TL;DR

This paper explores embedding 2d Toda integrable models into 4d gauge theory using Lax pairs, constructing quantum line operators, and analyzing their interactions, advancing the understanding of integrability within gauge theories.

## Contribution

It introduces a novel approach to embed 2d Toda models into 4d CWY gauge theory via Lax pairs and constructs quantum line operators with one-loop interaction calculations.

## Key findings

- Successful embedding of 2d Toda models into CWY theory.
- Construction of quantum line operators and one-loop gluon exchange analysis.
- Discussion of symmetries and future refinement of the embedding method.

## Abstract

Motivated by the realisation of Yang-Baxter equation of 2d Integrable models in the 4d gauge theory of Costello-Witten-Yamazaki (CWY), we study the embedding of integrable 2d Toda field models inside this construction. This is done by using the Lax formulation of 2d integrable systems and by thinking of the standard Lax pair $L_{\pm }$ in terms of components of CWY gauge connection propagating along particular directions in the gauge bundle. We also use results of the CWY theory to build quantum line operators for 2d Toda models and compute the one loop contribution of two intersecting lines exchanging one gluon. Other features like local symmetries and comments on extension of our method to other 2d integrable models are also discussed. We also comment on\textrm{\ }some basic points that still need a refinement before talking about a fully consistent embedding of Lax pairs into CWY theory.