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

TL;DR

This paper explores higher-dimensional loop algebras, their regularization via pseudodifferential operators, and the challenges in constructing Fock space representations, extending known algebraic structures to general manifolds.

Contribution

It generalizes infinite-dimensional Lie algebras to higher dimensions and reviews a PSDO-based regularization method, highlighting obstacles in Fock space realization.

Findings

Extension of Lie algebras to arbitrary manifolds.

Analysis of PSDO symbol calculus in current algebra regularization.

Partial success and remaining challenges in Fock space construction.

Abstract

We discuss a previously discovered [hep-th/9401027] 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 the above reference with partial success, and discuss the remaining obstacles to the construction of a Fock space representation.