Higher-Dimensional Generalizations of Affine Kac-Moody and Virasoro Conformal Lie Algebras

Abstract

We discuss the generalizations of the notion of Conformal Algebra and Local Distribution Lie algebras for multi-dimensional bases. We replace the algebra of Laurent polynomials on Open image in new window by an infinite-dimensional representation (with some additional structures) of a simple finite-dimensional Lie algebra Open image in new window in the space of regular functions on the corresponding Grassmann variety Open image in new window that can be described as a ``right'' higher-dimensional generalization of Open image in new window from the point of view of a corresponding group action. For Open image in new window it gives us the usual Vertex Algebra notion. We construct the higher dimensional generalizations of the Virasoro and the Affine Kac-Moody Conformal Lie algebras explicitly and in terms of the Operator Product Expansion.