Characteristic classes of A-infinity algebras

TL;DR

This paper introduces a noncommutative geometric approach to characteristic classes of A-infinity algebras, proving invariance under homotopy and relating graph homology to Lie algebra homology, with applications to topological field theories.

Contribution

It provides an alternative construction of characteristic classes using noncommutative geometry and proves their invariance under homotopy, extending Kontsevich's theorem.

Findings

Homotopy equivalent A-infinity algebras produce the same cohomology classes.

Re-proved Kontsevich's theorem linking graph homology to Lie algebra homology.

Applied the theory to topological conformal field theories.

Abstract

A standard combinatorial construction, due to Kontsevich, associates to any A-infinity algebra with an invariant inner product, an inhomogeneous class in the cohomology of the moduli spaces of Riemann surfaces with marked points. We describe an alternative version of this construction based on noncommutative geometry and use it to prove that homotopy equivalent algebras give rise to the same cohomology classes. Along the way we re-prove Kontsevich's theorem relating graph homology to the homology of certain infinite-dimensional Lie algebras. An application to topological conformal field theories is given.