The A∞ de Rham Theorem and Integration of Representations up to Homotopy

Abstract

We use Chen's iterated integrals to integrate representations up to homotopy. That is, we construct an functor from the representations up to homotopy of a Lie algebroid A to those of its infinity groupoid. This construction extends the usual integration of representations in Lie theory. We discuss several examples including Lie algebras and Poisson manifolds. The construction is based on an version of de Rham's theorem due to Gugenheim [15]. The integration procedure we explain here amounts to extending the construction of parallel transport for superconnections, introduced by Igusa [17] and Block-Smith [6], to the case of certain differential graded manifolds

Higher holonomies: Comparing two constructions

“…In [1], an equivalent description of representations up to homotopy in terms of endomorphism-valued cochains is given. As a special case, we obtain the following description of a representation up to homotopy of X • on (V, ∂).…”

“…We will not describe the details of the construction of the A ∞ -functor here, the interested reader can find them in [7,4,1]. However, we briefly discuss those parts of the construction that will be needed later on.…”

“…the invere of the usual holonomy along γ. The interested reader can consult Remark 4.13 of [1], for instance.…”

“…Let γ be a path in M . By Remark 4.13. in [1], β 1 (γ) equals the ordinary holonomy of α 1 along the reversal of γ. Hence (T 2 β)(γ) = β 1 (γ −1 ) is the ordinary holonomy of γ.…”

“…On the other hand, there is the parallel transport of flat superconnections introduced recently by Igusa [7] and subsequently studied by Block-Smith [4] and Arias Abad-Schätz [1]. The fundamental result in this direction is that there is a weak equivalence of dg-categories between the category Rep ∞ (T M ) of flat superconnections on M and the category of ∞-representations of the ∞-groupoid Rep ∞ (π ∞ (M )) of M .…”