Infinite dimensional geometry of $M_1=Diff_+(S^1)/PSL(2,R)$ and $q_R$--conformal symmetries

Abstract

A geometric interpretation of approximate ($HS$-projective or $TC$-projective) representations of the Witt algebra $w^C$ by $q_R$-conformal symmetries in the Verma modules $V_h$ over the Lie algebra $sl(2,C)$ is established and some their characteristics are calculated. It is shown that the generators of representations coincide with the Nomizu operators of holomorphic $w^C$-invariant hermitean connections in the deformed holomorphic tangent bundles $T_h(M_1)$ over the infinite-dimensional K\"ahler manifold $M_1=\Diff_+(S^1)/PSL(2,R)$, whereas the deviations $A_{XY}$ of the approximate representations coincide with the curvature operators for these connections, which supply the determinant bundle $det T_h(M_1)$ by a structure of the prequantization bundle over $M_1$. At $h=2$ the geometric picture reduces to one considered by A.A.Kirillov and the author [Funct.Anal.Appl. 21(4) (1987) 284-293] (without any relation to the approximate representations) for ordinary tangent bundles.