Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Short Survey of Noncommutative Geometry   We give a survey of selected topics in noncommutative geometry, with some
emphasis on those directly related to physics, including our recent work with
Dirk Kreimer on renormalization and the Riemann-Hilbert problem. We discuss at
length two issues. The first is the relevance of the paradigm of geometric
space, based on spectral considerations, which is central in the theory. As a
simple illustration of the spectral formulation of geometry in the ordinary
commutative case, we give a polynomial equation for geometries on the four
dimensional sphere with fixed volume. The equation involves an idempotent e,
playing the role of the instanton, and the Dirac operator D. It expresses the
gamma five matrix as the pairing between the operator theoretic chern
characters of e and D. It is of degree five in the idempotent and four in the
Dirac operator which only appears through its commutant with the idempotent. It
determines both the sphere and all its metrics with fixed volume form.
  We also show using the noncommutative analogue of the Polyakov action, how to
obtain the noncommutative metric (in spectral form) on the noncommutative tori
from the formal naive metric. We conclude on some questions related to string
theory.

29
The categories of this paper are: funct-an
31
The categories of this paper are: ao-sci
26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
18
The categories of this paper are: econ
14
The categories of this paper are: math-ph
22
The categories of this paper are: chao-dyn
12
The categories of this paper are: nucl-th
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The categories of this paper are: chem-ph
13
The categories of this paper are: quant-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ex
5
The categories of this paper are: gr-qc
10
The categories of this paper are: q-bio
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The categories of this paper are: eess
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The categories of this paper are: comp-gas