Non-highest weight representations of the current algebra $\hat{so}(1,n)$, and Laplace Operators

TL;DR

This paper constructs and analyzes non-highest weight representations of the current algebra so(1,n), introduces associated Laplace operators, and explores their potential connection to Yang-Mills loop operators.

Contribution

It introduces canonical non-highest weight representations of so(1,n) current algebra and constructs Laplacian operators within this framework.

Findings

Constructed unitary and non-unitary non-highest weight representations.

Defined Laplacian operators as elements of the universal enveloping algebra.

Speculated on the relation between these Laplacians and Yang-Mills loop operators.

Abstract

We constructed canonical non-highest weight unitary irreducible representation of $\hat{so}(1,n)$ current algebra as well as canonical non-highest weight non-unitary representations, We constructed certain Laplacian operators as elements of the universal enveloping algebra, acting in representation space. We speculated about a possible relation of those Laplacians with the loop operator for the Yang-Mills.