Pure and Applied Mathematics Quarterly

Volume 9 (2013)

Number 2

Special Issue: In Honor of Dennis Sullivan, Part 2 of 2

Curved $A_{\infty}$-algebras and Chern classes

Pages: 333 – 369

DOI: https://dx.doi.org/10.4310/PAMQ.2013.v9.n2.a3

Authors

Nikolay M. Nikolov (INRNE, Bulgarian Academy of Sciences, Sofia, Bulgaria)

Svetoslav Zahariev (MEC Department, LaGuardia Community College, City University of New York, Long Island City, N.Y., U.S.A.)

Abstract

We describe two constructions giving rise to curved $A_{\infty}$-algebras. The first consists of deforming $A_{\infty}$-algebras, while the second involves transferring curved dg structures that are deformations of (ordinary) dg structures along chain contractions. As an application of the second construction, given a vector bundle on a polyhedron $X$, we exhibit a curved $A_{\infty}$-structure on the complex of matrix-valued cochains of sufficiently fine triangulations of $X$. We use this structure as a motivation to develop a homotopy associative version of Chern-Weil theory.

Keywords

$A_{\infty}$-algebra, Chern-Weil theory

2010 Mathematics Subject Classification

55Uxx, 57R20

Published 7 November 2013