带角流形上混合边界条件的指数公式
Index formulae for mixed boundary conditions on manifolds with corners
摘要：
我们研究的问题计算Fredholm指数的几何Dirac算子的局部（如Dirichlet和Neumann）和非局部（APS）的边界条件上的层的流形的角落。带角点流形的边界层可以在高余维相交。为了计算指数，我们引入了一个胶合结构和相应的李群胚。通过李群胚纤维上的等变Dirac算子族描述了满足混合边界条件的Dirac算子。利用重新标度的热核方法，我们得到了Atiyah-Singer型的一般指数公式。
英文摘要：
We investigate the problem of calculating the Fredholm index of a geometric Dirac operator subject to local (e.g. Dirichlet and Neumann) and non-local (APS) boundary conditions posed on the strata of a manifold with corners. The boundary strata of the manifold with corners can intersect in higher codimension. To calculate the index we introduce a glueing construction and a corresponding Lie groupoid. We describe the Dirac operator subject to mixed boundary conditions via an equivariant family of Dirac operators on the fibers of the Lie groupoid. Using a heat kernel method with rescaling we derive a general index formula of the Atiyah-Singer type.
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