Monodromy Properties of the Energy Momentum Tensor on General Algebraic Curves

TL;DR

This paper introduces a novel decomposition of the energy-momentum tensor on algebraic curves, leading to a new infinite-dimensional Lie algebra that generalizes the Virasoro algebra and encodes the global geometry of Riemann surfaces.

Contribution

It proposes a new method to analyze the energy-momentum tensor on algebraic curves, resulting in a generalized Lie algebra capturing the surface's global properties.

Findings

Decomposition of $T(z)$ into components with distinct monodromy properties.

Explicit structure and central extension of the algebra for hyperelliptic curves.

Identification of a symmetry related to radial quantization of multivalued CFT fields.

Abstract

A new approach to analyze the properties of the energy-momentum tensor $T(z)$ of conformal field theories on generic Riemann surfaces (RS) is proposed. $T(z)$ is decomposed into $N$ components with different monodromy properties, where $N$ is the number of branches in the realization of RS as branch covering over the complex sphere. This decomposition gives rise to new infinite dimensional Lie algebra which can be viewed as a generalization of Virasoro algebra containing information about the global properties of the underlying RS. In the simplest case of hyperelliptic curves the structure of the algebra is calculated in two ways and its central extension is explicitly given. The algebra possess an interesting symmetry with a clear interpretation in the framework of the radial quantization of CFT's with multivalued fields on the complex sphere.