Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The APS-index and the spectral flow   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 B\"ar 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 $A(t)$.

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The categories of this paper are: q-fin
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The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
11
The categories of this paper are: hep-lat
6
The categories of this paper are: astro-ph
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
32
The categories of this paper are: atom-ph
20
The categories of this paper are: acc-phys
36
The categories of this paper are: cmp-lg
21
The categories of this paper are: adap-org
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int
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The categories of this paper are: math-ph