Infinite-dimensional Geometry of the Universal Deformation of the Complex Disk

The universal deformation of the complex disk is studied from the viewpoint of infinite-dimensional geometry. The structure of a subsymmetric space on the universal deformation is described. The foliation of the universal deformation by subsymmetry mirrors is shown to determine a real polarization. The subject of this paper may be of interest to specialists in algebraic geometry and representation theory as well as to researchers dealing with mathematical problems of modern quantum field theory. The universal deformation of the complex disk is one of the crucial concepts used in the geometric statement of quantum conformal field theory [1] and quantum-field theory of strings [2] (see also [3]). The characteristic feature of the approach developed in the present paper is that the universal deformation of the complex disk is studied in terms of infinite-dimensional geometry. On this way the structure of the subsymmetric space [4–6] on the universal deformation is described. The foliation of the universal deformation defined by the mirrors of subsymmetries determines a real polarization. For a long time real polarizations on complex manifolds and their quantization have been attracting the attention of mathematicians dealing with algebraic geometry and representation theory and of specialists in mathematical physics [44–47]. The results of this paper confirm the importance of studying such polarizations and expose a connection between the traditions of classical synthetic geometry and recent trends in algebraic geometry, representation theory, and modern quantum field theory. 1. The infinite-dimensional geometry of the flag manifold of the Virasoro-Bott group (the base of the universal deformation of the complex disk). 1.1. The Virasoro algebra, the Virasoro-Bott group, and the Neretin semigroup. Let Diff(S1) denote the group of diffeomorphisms of the unit circle S1. The group manifold Diff(S1) splits into two connected components, the subgroup Diff+(S) and the coset Diff−(S). The diffeomorphisms in Diff+(S) preserve the orientation on the circle S1 and those in Diff−(S) reverse it. The Lie algebra of Diff+(S) can be identified with the linear space Vect(S1) of smooth vector fields on the circle equipped with the commutator (1) [v(t)d/dt, u(t)d/dt] = (v(t)u(t) − v(t)u(t))d/dt.