Associative algebra twisted bundles over compact topological spaces

TL;DR

This paper introduces a new class of twisted fiber bundles over compact spaces with fibers from algebraic completions of formal series, relevant for Lie algebra cohomology, K-theory, and conformal field theory.

Contribution

It defines twisted $A(rak g)$-bundles over compact spaces, proves their homotopical invariance and covariance, and explores applications in various mathematical and physical theories.

Findings

Homotopical invariance of twisted bundles established

Covariance under trivial bundle transformations proven

Applications outlined for cohomology, K-theory, and conformal field theory

Abstract

For the associative algebra $A(\mathfrak g)$ of an infinite-dimensional Lie algebra $\mathfrak g$, we introduce twisted fiber bundles over arbitrary compact topological spaces. Fibers of such bundles are given by elements of algebraic completion of the space of formal series in complex parameters, sections are provided by rational functions with prescribed analytic properties. Homotopical invariance as well as covariance in terms of trivial bundles of twisted $A(\mathfrak g)$-bundles is proven. Further applications of the paper's results useful for studies of the cohomology of infinite-dimensional Lie algebras on smooth manifolds, $K$-theory, as well as for purposes of conformal field theory, deformation theory, and the theory of foliations are mentioned.