闭路空间和连接
Loop Spaces and Connections
摘要：
我们研究几何的闭路空间在派生代数几何和延伸在几个方向上众所周知的连接之间的旋转循环和德拉姆微分。我们的主要结果是对循环同调的几何描述的一个分类，它将特征为零的光滑概形或几何堆叠X的闭路空间上的S^1-等变拟同调层与X上的平坦联络层或等价的D_X-模联系起来。通过由圆上上链的形式推导出de Rham模上的Hodge滤子，我们可以精确地恢复D_X-模而不是周期模.更一般地说，我们考虑旋转的Hopf纤维化Ω S^3 --> Ω S^2 --> S^1，并将闭路空间上的Ω S^2-等变层与X上具有任意连接的层联系起来，曲率由它们的Ω S^3-等变给出。
英文摘要：
We examine the geometry of loop spaces in derived algebraic geometry and extend in several directions the well known connection between rotation of loops and the de Rham differential. Our main result, a categorification of the geometric description of cyclic homology, relates S^1-equivariant quasicoherent sheaves on the loop space of a smooth scheme or geometric stack X in characteristic zero with sheaves on X with flat connection, or equivalently D_X-modules. By deducing the Hodge filtration on de Rham modules from the formality of cochains on the circle, we are able to recover D_X-modules precisely rather than a periodic version. More generally, we consider the rotated Hopf fibration Omega S^3 --> Omega S^2 --> S^1, and relate Omega S^2-equivariant sheaves on the loop space with sheaves on X with arbitrary connection, with curvature given by their Omega S^3-equivariance.
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