论文标题
APS索引和光谱流
The APS-index and the spectral flow
论文作者
论文摘要
我们以抽象的功能分析设置研究了Atiyah-Patodi-Singer（APS）指数及其在光谱流量的平等。更确切地说，我们考虑了一个（合适的连续或微分）家庭在希尔伯特空间上的自我伴侣弗雷姆操作员$ a（t）$，在有限间隔内由$ t $参数。然后，我们考虑两个不同的操作员，即$ d：= \ frac {d} {dt}+a $（Riemannian Dirac Operator的抽象模拟）和$ d：= \ frac {d} {dt} {dt} -ia $（lorentzian dirac operator的抽象类似物）。后一种情况的灵感来自Bärand Strohmaier（Amer。\ J. \Math。141（2019），1421---1455）的最新索引定理，该案例具有APS边界条件。在这两种情况下，我们都证明，配备APS边界条件的操作员的Fredholm指数等于家庭$ a（t）$的光谱流。
We study the Atiyah-Patodi-Singer (APS) index, and its equality to the spectral flow, in an abstract, functional analytic setting. More precisely, we consider a (suitably continuous or differentiable) family of self-adjoint Fredholm operators $A(t)$ on a Hilbert space, parametrised by $t$ in a finite interval. We then consider two different operators, namely $D := \frac{d}{dt}+A$ (the abstract analogue of a Riemannian Dirac operator) and $D := \frac{d}{dt}-iA$ (the abstract analogue of a Lorentzian Dirac operator). The latter case is inspired by a recent index theorem by Bär and Strohmaier (Amer.\ J.\ Math. 141 (2019), 1421--1455) for a Lorentzian Dirac operator equipped with APS boundary conditions. In both cases, we prove that Fredholm index of the operator $D$ equipped with APS boundary conditions is equal to the spectral flow of the family $A(t)$.
