651 research outputs found
A walk in the noncommutative garden
This text is written for the volume of the school/conference "Noncommutative
Geometry 2005" held at IPM Tehran. It gives a survey of methods and results in
noncommutative geometry, based on a discussion of significant examples of
noncommutative spaces in geometry, number theory, and physics. The paper also
contains an outline (the ``Tehran program'') of ongoing joint work with Consani
on the noncommutative geometry of the adeles class space and its relation to
number theoretic questions.Comment: 106 pages, LaTeX, 23 figure
Spin Foams and Noncommutative Geometry
We extend the formalism of embedded spin networks and spin foams to include
topological data that encode the underlying three-manifold or four-manifold as
a branched cover. These data are expressed as monodromies, in a way similar to
the encoding of the gravitational field via holonomies. We then describe
convolution algebras of spin networks and spin foams, based on the different
ways in which the same topology can be realized as a branched covering via
covering moves, and on possible composition operations on spin foams. We
illustrate the case of the groupoid algebra of the equivalence relation
determined by covering moves and a 2-semigroupoid algebra arising from a
2-category of spin foams with composition operations corresponding to a fibered
product of the branched coverings and the gluing of cobordisms. The spin foam
amplitudes then give rise to dynamical flows on these algebras, and the
existence of low temperature equilibrium states of Gibbs form is related to
questions on the existence of topological invariants of embedded graphs and
embedded two-complexes with given properties. We end by sketching a possible
approach to combining the spin network and spin foam formalism with matter
within the framework of spectral triples in noncommutative geometry.Comment: 48 pages LaTeX, 30 PDF figure
Logic and operator algebras
The most recent wave of applications of logic to operator algebras is a young
and rapidly developing field. This is a snapshot of the current state of the
art.Comment: A minor chang
Noncommutative geometry and motives: the thermodynamics of endomotives
We combine aspects of the theory of motives in algebraic geometry with
noncommutative geometry and the classification of factors to obtain a
cohomological interpretation of the spectral realization of zeros of
-functions. The analogue in characteristic zero of the action of the
Frobenius on l-adic cohomology is the action of the scaling group on the cyclic
homology of the cokernel (in a suitable category of motives) of a restriction
map of noncommutative spaces. The latter is obtained through the thermodynamics
of the quantum statistical system associated to an endomotive (a noncommutative
generalization of Artin motives). Semigroups of endomorphisms of algebraic
varieties give rise canonically to such endomotives, with an action of the
absolute Galois group. The semigroup of endomorphisms of the multiplicative
group yields the Bost-Connes system, from which one obtains, through the above
procedure, the desired cohomological interpretation of the zeros of the Riemann
zeta function. In the last section we also give a Lefschetz formula for the
archimedean local L-factors of arithmetic varieties.Comment: 52 pages, amslatex, 1 eps figure, v2: final version to appea
Arithmetic Spacetime Geometry from String Theory
An arithmetic framework to string compactification is described. The approach
is exemplified by formulating a strategy that allows to construct geometric
compactifications from exactly solvable theories at . It is shown that the
conformal field theoretic characters can be derived from the geometry of
spacetime, and that the geometry is uniquely determined by the two-dimensional
field theory on the world sheet. The modular forms that appear in these
constructions admit complex multiplication, and allow an interpretation as
generalized McKay-Thompson series associated to the Mathieu and Conway groups.
This leads to a string motivated notion of arithmetic moonshine.Comment: 36 page
Crystals, instantons and quantum toric geometry
We describe the statistical mechanics of a melting crystal in three
dimensions and its relation to a diverse range of models arising in
combinatorics, algebraic geometry, integrable systems, low-dimensional gauge
theories, topological string theory and quantum gravity. Its partition function
can be computed by enumerating the contributions from noncommutative instantons
to a six-dimensional cohomological gauge theory, which yields a dynamical
realization of the crystal as a discretization of spacetime at the Planck
scale. We describe analogous relations between a melting crystal model in two
dimensions and N=4 supersymmetric Yang-Mills theory in four dimensions. We
elaborate on some mathematical details of the construction of the quantum
geometry which combines methods from toric geometry, isospectral deformation
theory and noncommutative geometry in braided monoidal categories. In
particular, we relate the construction of noncommutative instantons to deformed
ADHM data, torsion-free modules and a noncommutative twistor correspondence.Comment: 33 pages, 5 figures; Contribution to the proceedings of "Geometry and
Physics in Cracow", Jagiellonian University, Cracow, Poland, September 21-25,
2010. To be published in Acta Physica Polonica Proceedings Supplemen
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