Central extensions of current groups in two dimensions

TL;DR

This paper extends the theory of loop groups and the Virasoro algebra to two-dimensional surfaces, exploring their structure, coadjoint orbits, and potential for rich representation theory, with applications to complex curves and holomorphic bundles.

Contribution

It introduces and studies central extensions of two-dimensional current groups and a Virasoro-like algebra, connecting their structure to complex geometry and moduli spaces.

Findings

Coadjoint orbits are labeled by moduli of holomorphic G-bundles.

In genus one, orbits relate to conjugacy classes of twisted loop groups.

Invariants of the two-dimensional Virasoro analogue match those of the one-dimensional case.

Abstract

In this paper we generalize some of these results for loop algebras and groups as well as for the Virasoro algebra to the two-dimensional case. We define and study a class of infinite dimensional complex Lie groups which are central extensions of the group of smooth maps from a two dimensional orientable surface without boundary to a simple complex Lie group G. These extensions naturally correspond to complex curves. The kernel of such an extension is the Jacobian of the curve. The study of the coadjoint action shows that its orbits are labelled by moduli of holomorphic principal G-bundles over the curve and can be described in the language of partial differential equations. In genus one it is also possible to describe the orbits as conjugacy classes of the twisted loop group, which leads to consideration of difference equations for holomorphic functions. This gives rise to a hope that the described groups should possess a counterpart of the rich representation theory that has been developed for loop groups. We also define a two-dimensional analogue of the Virasoro algebra associated with a complex curve. In genus one, a study of a complex analogue of Hill's operator yields a description of invariants of the coadjoint action of this Lie algebra. The answer turns out to be the same as in dimension one: the invariants coincide with those for the extended algebra of currents in sl(2).