Geometric approach to Kac–Moody and Virasoro algebras

Abstract

a b s t r a c tIn this paper we show the existence of a group acting infinitesimally transitively on the moduli space of pointed-curves and vector bundles (with formal trivialization data) and whose Lie algebra is an algebra of differential operators. The central extension of this Lie algebra induced by the determinant bundle on the Sato Grassmannian is precisely a semidirect product of a Kac-Moody algebra and the Virasoro algebra. As an application of this geometric approach, we give a local Mumford-type formula in terms of the cocycle associated with this central extension. Finally, using the original Mumford formula we show that this local formula is an infinitesimal version of a general relation in the Picard group of the moduli of vector bundles on a family of curves (without any formal trivialization).

Algebro-Geometric Solutions of the Generalized Virasoro Constraints

“…From a more theoretical point of view, the equivalence of categories between Atiyah algebras and differential operator algebras [5], means that D 1 C((z))/C (V, V ) is the natural candidate to begin with. Consequently, one expect a link with he moduli space of pairs consisting of a curve and a line bundle on it, since the Lie group associated with D 1 C((z))/C (V, V ) uniformizes this moduli space [18]. By studying higher order differential operators, one may relate this with the study W -algebras (and their representation theory, etc.…”

“…In particular, conjugation is an instance of gauge transformation. Let us recall from [18] the definition of the group of semilinear transformations and some of its properties. The group of semilinear transformations of a finite-dimensional C((z))-vector space V , denoted by SGl C((z)) (V ), consists of C-linear automorphisms γ : V → V such that there exists a C-algebra automorphism of C((z)), g, satisfying…”

“…and ρ induces a splitting of it. To prove this, one uses the sequence [18,Remark 3.9] and the explicit expressions of the operators ρ(L i ) computed in Theorem 2.1.…”