arXiv math/06123992006-12math.SGmath.RT

微局部膜是可构造的层

Microlocal branes are constructible sheaves

David Nadler

备注：: 49 pages; minor expository changes

摘要：

设$X$是一个实解析流形，$T^*X$是它的余切丛.在最近与E. Zaslow \cite{NZ}，我们证明了$X$上的可构造层的dg范畴$Sh_c(X)$拟嵌入到$T^*X$的可构造层范畴的三角包络$F(T^*X)$中.我们在这里证明了拟嵌入实际上是一个拟等价。当$X$是复数时，人们可以将其解释为在$T^*X$和完整$D_X$中识别拉格朗日膜的拓扑类似物--由Kapustin和Kapustin-Witten从物理角度开发的模块。作为一个具体的应用，我们表明，紧连接的精确拉格朗日在$T^*X$（与一些适度的同调假设）是等价的，在类的零节。特别地，这确定了它们在$T^*X$中的（复）上同调环和同调类，并提供了它们相交点数目的同调界。$T^*X$中紧膜的一个独立的特征最近由Escheraya-Seidel-Smith得到。

英文摘要：

Let $X$ be a real analytic manifold, and let $T^*X$ be its cotangent bundle. In a recent paper with E. Zaslow \cite{NZ}, we showed that the dg category $Sh_c(X)$ of constructible sheaves on $X$ quasi-embeds into the triangulated envelope $F(T^*X)$ of the Fukaya category of $T^*X$. We prove here that the quasi-embedding is in fact a quasi-equivalence. When $X$ is complex, one may interpret this as a topological analogue of the identification of Lagrangian branes in $T^*X$ and holonomic $D_X$-modules developed by Kapustin and Kapustin-Witten from a physical perspective. As a concrete application, we show that compact connected exact Lagrangians in $T^*X$ (with some modest homological assumptions) are equivalent in the Fukaya category to the zero section. In particular, this determines their (complex) cohomology ring and homology class in $T^*X$, and provides a homological bound on their number of intersection points. An independent characterization of compact branes in $T^*X$ has recently been obtained by Fukaya-Seidel-Smith.

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