On higher-dimensional loop algebras, pseudodifferential operators and Fock space realizations

Abstract

We discuss a previously discovered [1] extension of the infinite-dimensional Lie algebras Map(M, g) which generalizes the Kac-Moody algebras in 1+1 dimensions and the Mickelsson-Faddeev algebras in 3+1 dimensions to manifolds M of general dimensions. Furthermore, we review the method of regularizing current algebras in higher dimensions using pseudodifferential operator (PSDO) symbol calculus. In particular, we discuss the issue of Lie algebra cohomology of PSDOs and its relation to the Schwinger terms arising in the quantization process. Finally, we apply this regularization method to the algebra of ref.[1] with partial success, and discuss the remaining obstacles to the construction of a Fock space representation.