1-minimal models for $C_{\infty }$-algebras and flat connections

Abstract

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 ${\mathrm{Tot}}_N\left(A_{DR}(M_{\bullet }G)\right)$ carry a $C_{\infty }$-algebra structure $m_{\bullet }$. We show that each $C_{\infty }$-algebra $1$-minimal model $g_{\bullet }:\left(W,{m^{\prime }}_{\bullet }\right)\to \left({\mathrm{Tot}}_N\left(A_{DR}(M_{\bullet }G)\right),m_{\bullet }\right)$ 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.

45 pages. First version, comments are welcome!

References in corpus (1)

• Maurer-Cartan spaces of filtered L-infinity algebras