Revisiting the computation of cohomology classes of the Witt algebra using conformal field theory and aspects of conformal algebra

TL;DR

This paper explores the computation of the cohomology class of the Witt algebra using conformal field theory and conformal algebra, providing a reformulation of the operator product expansion to understand the central extension to the Virasoro algebra.

Contribution

It introduces a modified conformal algebra approach to compute the cohomology class of the Witt algebra, connecting conformal field theory with algebraic cohomology.

Findings

Reformulation of operator product expansion using modified conformal algebra

Explicit computation of the cohomology class for the Witt algebra

Insight into the central extension to the Virasoro algebra

Abstract

In this article, we revisit some aspects of the computation of the cohomology class of $H^2 ( \text{Witt}, \mathbb{C})$ using some methods in two-dimensional conformal field theory and conformal algebra to obtain the one-dimensional central extension of the Witt algebra to the Virasoro algebra. Even though this is well-known in the context of standard mathematical physics literature, the operator product expansion of the energy-momentum tensor in two-dimensional conformal field theory is presented almost axiomatically. In this paper, we attempt to reformulate it with the help of a suitable modification of conformal algebra (as developed by V. Kac), and apply it to compute the representative element of the cohomology class which gives the desired central extension. This paper was written in the scope of an undergraduate's exploration of conformal field theory and his attempt to gain insight on the subject from a mathematical perspective.