Yangian for cotangent Lie algebras and spectral $R$-matrices

TL;DR

This paper develops a canonical quantization of Lie bialgebra structures on cotangent Lie algebras, leading to twisted Yangians with spectral R-matrices, motivated by 4d gauge theory and line operator representations.

Contribution

It introduces a new quantization framework for cotangent Lie algebras, constructing twisted Yangians and spectral R-matrices with applications to gauge theory.

Findings

Quantization of Lie bialgebras on cotangent Lie algebras.

Construction of spectral R-matrices for twisted Yangians.

Connection to 4d holomorphic-topological gauge theory and line operators.

Abstract

In this paper, we present a canonical quantization of Lie bialgebra structures on the formal power series $\mathfrak{d}[\![t]\!]$ with coefficients in the cotangent Lie algebra $\mathfrak{d} = T^*\mathfrak{g} = \mathfrak{g} \ltimes \mathfrak{g}^*$ to a simple complex Lie algebra $\mathfrak{g}$. We prove that these quantizations produce twists to the natural analog of the Yangian for $\mathfrak{d}$. Moreover, we construct spectral $R$-matrices for these twisted Yangians as compositions of twisting matrices. The motivation for the construction of these twisted Yangians over $\mathfrak{d}$ comes from certain 4d holomorphic-topological gauge theory. More precisely, we show that pertubative line operators in this theory can be realized as representations of these Yangians. Moreover, the comultiplications of these Yangians correspond to the monodial structure of the category of line operators.