Schwarzian Derivative - Diffeomorphism Group of The Circle

Diffeomorphism Group of The Circle

Let be the space of tensor densities of degree on S1. The group of orientation-preserving diffeomorphisms of S1, Diff(S1), acts on via pushforwards. If f is an element of Diff(S1) then consider the mapping

In the language of group cohomology the chain-like rule above says that this mapping is a 1-cocycle on with coefficients in . In fact

and the 1-cocycle generating the cohomology is f → S(f−1).

There is an infinitesimal version of this result giving a 1-cocycle for the Lie algebra of vector fields. This in turn gives the unique non-trivial central extension of, the Virasoro algebra.

The group Diff(S1) and its central extension also appear naturally in the context of Teichmüller theory and string theory. In fact the homeomorphisms of the circle induced by quasiconformal self-maps of the unit disc are precisely the quasisymmetric homeomorphisms of the circle; these are exactly homeomorphisms which do not send four points with cross ratio 1/2 to points with cross ratio near 1 or 0. Taking boundary values, universal Teichmüller can be identified with the quotient of the group of quasisymmetric homoemorphisms QS(S1) by the subgroup of Möbius transformations Moeb(S1). (It can also be realized naturally as the space of quasicircles in C.) Since

the homogeneous space Diff(S1) / Moeb(S1) is naturally a subspace of universal Teichmüller space. It is also naturally a complex manifold and this and other natural geometric structures are compatible with those on Teichmüller space. The dual of the Lie algebra of Diff(S1) can be identified with the space of Hill's operators on S1

and the coadjoint action of Diff(S1) invokes the Schwarzian derivative. The inverse of the diffeomorphism f sends the Hill's operator to