1-$C_{\infty}$ -代数与平坦联络的极小模型
1-minimal models for $C_{\infty}$-algebras and flat connections
摘要：
给定一个光滑流形$M$，它具有离散群$G$的一个适当的和不连续的光滑作用，神经$M_{\bullet}G$是一个单纯流形，它的微分形式的向量空间$\operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right)$带有一个$C_{\infty}$ -代数结构$m_{\bullet}$。证明了每个$C_{\infty}$ -代数$1$ -极小模型$g_{\bullet}\: : \: \left(W, {m'}_{\bullet} \right) \to \left( \operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right),m_{\bullet}\right) $在$M$上的光滑平凡丛$E$上给出一个平坦连接$\nabla$，其中纤维是$π_{1}(M/G)$的Malcev李代数，其单值表示是$π_{1}(M/G)$的Malcev完备化.这种连接是唯一的意义上说，不同的$1$模型给出同构的连接。特别地，所得到的联络同构于陈在$M/G$上的平坦联络。如果作用是全纯的，并且$g_{\bullet}$有全纯像（具有对数奇点），则$(\nabla,E)$是全纯的，并且不同的$1$ -模型给出（全纯）同构联系（具有对数奇点）。这些结果是Chen平坦联络理论的等变和全纯形式。
英文摘要：
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 $π_{1}(M/G)$ and its monodromy representation is the Malcev completion of $π_{1}(M/G)$. 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.
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