Given a smooth manifold $M$ equipped with a properly and discontinuous smooth action of a discrete group $G$, the nerve $M_{ \bullet}G$ is a simplicial manifold and its vector space of differential forms $\operatorname{ Tot}_{ N}\left(A_{ DR}(M_{ \bullet}G)\right)$ carry a $C_{ \infty}$-algebra structure $m_{ \bullet}$. We show that each $C_{ \infty}$-algebra $1$-minimal model $g_{ \bullet}\: : \: \left(W, { m'}_{ \bullet} \right) \to \left( \operatorname{ Tot}_{ N}\left(A_{ DR}(M_{ \bullet}G)\right),m_{ \bullet}\right) $ gives a flat connection $\nabla$ on a smooth trivial bundle $E$ on $M$ where the fiber is the Malcev Lie algebra of $\pi_{ 1}(M/G)$ and its monodromy representation is the Malcev completion of $\pi_{ 1}(M/G)$. This connection is unique in the sense that different $1$-models give isomorphic connections. In particular, the resulting connections are isomorphic to Chen's flat connection on $M/G$. If the action is holomorphic and $g_{ \bullet}$ has holomorphic image (with logarithmic singularities), $(\nabla,E)$ is holomorphic and different $1$-models give (holomorphically) isomorphic connections (with logarithmic singularities). These results are the equivariant and holomorphic version of Chen's theory of flat connections.
Categories: math.AT math.DG