Higher conformal variations and the Virasoro vertex operator algebra

TL;DR

This paper develops a calculus of variations for conformal maps on annular domains, linking derivatives of these maps to the structure of the Virasoro vertex operator algebra, and extending previous work on conformal derivatives and stress-energy tensors.

Contribution

It introduces a new framework connecting conformal derivatives with the Virasoro vertex operator algebra through a three-step construction involving topology, variational equations, and algebraic representations.

Findings

Derivatives of conformal maps form the Virasoro algebra structure.

Series expansions of conformal derivatives match products of vertex operators.

Extension of previous work linking conformal derivatives to stress-energy tensors.

Abstract

We develop a calculus of variations for functionals on certain spaces of conformal maps. Such a space \Omega\ is composed of all maps that are conformal on domains containing a fix compact annular set of the Riemann sphere, and that are "sense-preserving". The calculus of variations is based on describing infinitesimal variations of such maps using vector fields. We show that derivatives of all orders with respect to such conformal maps, upon conjugation by an appropriate functional, give rise to the structure of the Virasoro vertex operator algebra. Our construction proceeds in three steps. We first put a natural topology on \Omega\ and define smooth paths and an operation of differentiation to all orders ("conformal derivatives"). We then study certain second-order variational equations and their solutions. We finally show that such solutions give rise to representations of the Virasoro algebra in terms of conformal derivatives, from which follows constructions for the Virasoro vertex operator algebra. It turns out that series expansions of multiple covariant conformal derivatives are equal to products of multiple vertex operators. This paper extends a recent work by the author, where first-order conformal derivatives were associated with the stress-energy tensor of conformal field theory.