论文标题
洛伦兹流形的本地索引理论
Local Index Theory for Lorentzian Manifolds
论文作者
论文摘要
洛伦兹狄拉克运营商的索引理论是一个年轻的主题，与椭圆指数理论有显着差异。它基于微局部分析而不是标准椭圆理论，主要特征之一是非平凡指数是由拓扑非平地动力学而不是基本歧管的非平凡拓扑引起的。在本文中，我们在全球双曲机上为洛伦兹狄拉克型操作员建立了局部索引公式。该局部公式意味着在cauchy hypersurfaces上具有atiyah-patodi-singer边界条件的空间紧凑型空间上的一般dirac型操作员的索引定理。这比先前已知的定理需要与Clifford乘法的兼容性和Cauchy Hypersurface上的空间狄拉克操作员相对于正定确定的内部产物，因此是自我相关性的。
Index theory for Lorentzian Dirac operators is a young subject with significant differences to elliptic index theory. It is based on microlocal analysis instead of standard elliptic theory and one of the main features is that a nontrivial index is caused by topologically nontrivial dynamics rather than nontrivial topology of the base manifold. In this paper we establish a local index formula for Lorentzian Dirac-type operators on globally hyperbolic spacetimes. This local formula implies an index theorem for general Dirac-type operators on spatially compact spacetimes with Atiyah-Patodi-Singer boundary conditions on Cauchy hypersurfaces. This is significantly more general than the previously known theorems that require the compatibility of the connection with Clifford multiplication and the spatial Dirac operator on the Cauchy hypersurface to be selfadjoint with respect to a positive definite inner product.
