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

We use Chen’s iterated integrals to integrate representations up to homotopy. That is, we construct an ${\mathrm A}_{\infty}$ functor $\int:\mathcal R{\mathrm ep}^{\infty}(A)\rightarrow \hat {\mathcal R}{\mathrm ep}^{\infty}(\Pi_{\infty}(A))$ 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 ${\mathrm A}_{\infty}$ 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.