On Kähler Geometry of Infinite-dimensional Complex Manifolds Diff+(S1)/S1 and Diff+(S1)/Möb(S1)

Abstract

The infinite-dimensional complex Frechet manifolds $${\\cal R}: = {\\rm{Dif}}{{\\rm{f}}_ + }({S^1})/{S^1}$$ℛ:=Diff+(S1)/S1 and $${\\cal S}: = {\\rm{Dif}}{{\\rm{f}}_ + }({S^1})/{\\rm{M\\ddot ob}}({S^1})$$S:=Diff+(S1)/Mo¨b(S1) are the quotients of the group Diff+(S1) of orientation-preserving diffeomorphisms of the unit circle S1 modulo subgroups of rotations and fractional-linear transformations respectively. These manifolds are the coadjoint orbits of the Virasoro group and the only ones having a Kähler structure. It motivates the study of their complex geometry. These manifolds are also closely related to string theory because they can be realized as the spaces of complex structures on loop spaces.