Complex deformations of the circle: Group cohomology and Virasoro uniformization

TL;DR

This paper explores complex deformations of the circle's diffeomorphism group, computing cohomology, and establishing a Virasoro uniformization theorem with applications to conformal field theory and moduli spaces.

Contribution

It introduces a framework for complex deformations with Fr"olicher structures, computes relevant cohomology groups, and proves a Virasoro uniformization theorem connecting deformations to moduli space geometry.

Findings

Computed second group cohomology with cocycles extending Bott-Thurston and Gelf'and-Fuks cocycles.

Established that tangent spaces of Segal moduli spaces are spanned by Witt algebra vector fields.

Linked complex deformations to Fenchel-Nielsen coordinates and Schiffer variation.

Abstract

We approach the question of complexification of the diffeomorphism group of the circle by considering real-analytic maps from the circle into the punctured complex plane with winding number +1. Such complex deformations form an infinite-dimensional manifold with partially defined inversion and composition operations, smooth in the sense of Fr\"olicher structures, and with Lie algebra relations at the identity given by the Witt algebra. With applications to conformal field theory in mind, we compute the second group cohomology group with real coefficients, finding cocycles extending the Bott-Thurston cocycle related to the Gelf'and-Fuks cocycle of the Virasoro algebra, and a natural relative cocycle combining the rotation number and conformal radius of a complex deformation. Complex deformations act naturally on the (infinite-dimensional) Segal moduli spaces of Riemann surfaces with analytically parametrized boundary components. These actions equip said moduli spaces with smooth Fr\"olicher structures. We prove a Virasoro uniformization theorem: the tangent spaces of the Segal moduli spaces are spanned by vector fields induced by the Witt algebra. Finally, we relate the actions of complex deformations to Fenchel-Nielsen coordinates and Schiffer variation on finite-dimensional moduli spaces of hyperbolic surfaces with one marked point on each boundary component.