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

Abstract

Abstract:We describe two constructions giving rise to curved A ∞ -algebras. The first consists of deforming A ∞ -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 ∞ -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.

Curved $$A_{\infty }$$-algebras and gauge theory

“…It was recently shown in [11] that under certain technical conditions one can transfer along chain contractions curved dg structures to curved A ∞ -structures, building on previous transfer results (see e.g. [7] and [8]).…”

“…The perturbation lemma. We recall the coalgebra homological perturbation lemma from [7] as stated in [11]. Given a chain complex (C, d), we say that a map δ : (C 1 , d 1 ) be a chain complex and δ 1 be a perturbation of d 1 .…”

“…Let (p, i, H) be a special contraction from A to a chain complex B and assume that the conditions stated in [11,Proposition 3.3] hold with respect to a fixed norm on A and the norm on B induced by the inclusion i. Thus according to [11,Proposition 3.3] there exists a subset Γ A of A 1 such that for each γ ∈ Γ A the curved dg structure D γ on A may be transferred to a curved A ∞ -structure D γ on B. Moreover, by the homological perturbation lemma, for every γ ∈ Γ A the special contraction (p, i, H) induces a special contraction (P γ , I γ , H γ ) from T (sA) to T (sB) such that P γ and I γ are morphisms of curved A ∞ -algebras.…”

“…We define an L 2 -inner product on Ω • (K, M l ) as in Subsection 3.1, fix e > 0 and set Γ e = γ ∈ Ω 1 (K, M l ) | γ ≤ e , where · is the norm induced by the inner product. It is shown in [11,Example 3.9] that there is a fine enough subdivision K e of K such that every γ ∈ Γ e defines a transferred curved A ∞ -structure D e γ on C • (K e , M l ) as in subsection 4.1, using Dupont's contraction.…”

Strand algebras and contact categories

“…The general construction of A ∞ structures on differential graded algebras by Kadeishvili in [8] is part of a much larger subject, not one to which the author claims much expertise. There are other methods, such as those of Kontsevich-Soibelman [10], Nikolov-Zahariev [24] and Huebschmann [7]. We do not know of examples where Kadeishvili's construction has been made as absolutely explicit as by the "creation" operators here.…”