Wilson Loops on Riemann Surfaces, Liouville Theory and Covariantization of the Conformal Group

TL;DR

This paper introduces a general covariantization method for differential operators, especially those generating conformal transformations, revealing a deep link between gauge theories on Riemann surfaces and Liouville theory.

Contribution

It presents a universal covariantization procedure for differential operators related to the conformal group, applicable to Riemann surfaces and AdS/CFT contexts.

Findings

Covariantized conformal operators are expressed via Wilson loops around geodesics.

Deep connection established between gauge theories on Riemann surfaces and Liouville theory.

Covariantization of the conformal group is always well-defined.

Abstract

The covariantization procedure is usually referred to the translation operator, that is the derivative. Here we introduce a general method to covariantize arbitrary differential operators, such as the ones defining the fundamental group of a given manifold. We focus on the differential operators representing the sl(2,R) generators, which in turn, generate, by exponentiation, the two-dimensional conformal transformations. A key point of our construction is the recent result on the closed forms of the Baker-Campbell-Hausdorff formula. In particular, our covariantization receipt is quite general. This has a deep consequence since it means that the covariantization of the conformal group is {\it always definite}. Our covariantization receipt is quite general and apply in general situations, including AdS/CFT. Here we focus on the projective unitary representations of the fundamental group of a Riemann surface, which may include elliptic points and punctures, introduced in the framework of noncommutative Riemann surfaces. It turns out that the covariantized conformal operators are built in terms of Wilson loops around Poincar\'e geodesics, implying a deep relationship between gauge theories on Riemann surfaces and Liouville theory.