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

TL;DR

This paper constructs flat connections from minimal models of $C_{ fty}$-algebras associated with manifolds under group actions, extending Chen's theory to equivariant and holomorphic contexts.

Contribution

It introduces a method to derive unique flat connections from $C_{ fty}$-algebra minimal models, linking algebraic structures to geometric flat connections, including equivariant and holomorphic cases.

Findings

Each $C_{ infty}$-algebra 1-minimal model yields a flat connection on a trivial bundle.

The constructed connections are unique up to isomorphism and coincide with Chen's flat connection.

In the holomorphic case, the connections are holomorphic with logarithmic singularities.

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 $\operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right)$ 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.