19,800 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
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.Comment: Invited lecture for JMP 2000, 45
Modular Theory, Non-Commutative Geometry and Quantum Gravity
This paper contains the first written exposition of some ideas (announced in
a previous survey) on an approach to quantum gravity based on Tomita-Takesaki
modular theory and A. Connes non-commutative geometry aiming at the
reconstruction of spectral geometries from an operational formalism of states
and categories of observables in a covariant theory. Care has been taken to
provide a coverage of the relevant background on modular theory, its
applications in non-commutative geometry and physics and to the detailed
discussion of the main foundational issues raised by the proposal.Comment: Special Issue "Noncommutative Spaces and Fields
Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology
We introduce some algebraic geometric models in cosmology related to the
"boundaries" of space-time: Big Bang, Mixmaster Universe, Penrose's crossovers
between aeons. We suggest to model the kinematics of Big Bang using the
algebraic geometric (or analytic) blow up of a point . This creates a
boundary which consists of the projective space of tangent directions to
and possibly of the light cone of . We argue that time on the boundary
undergoes the Wick rotation and becomes purely imaginary. The Mixmaster
(Bianchi IX) model of the early history of the universe is neatly explained in
this picture by postulating that the reverse Wick rotation follows a hyperbolic
geodesic connecting imaginary time axis to the real one. Penrose's idea to see
the Big Bang as a sign of crossover from "the end of previous aeon" of the
expanding and cooling Universe to the "beginning of the next aeon" is
interpreted as an identification of a natural boundary of Minkowski space at
infinity with the Big Bang boundary
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
Phase Diagram of a Spin Ladder with Cyclic Four Spin Exchange
We present the phase diagram of the Heisenberg model on the two leg
ladder with cyclic four spin exchange, determined by a combination of Exact
Diagonalization and Density Matrix Renormalization Group techniques. We find
six different phases and regimes: the rung singlet phase, a ferromagnetic
phase, two symmetry broken phases with staggered dimers and staggered scalar
chiralities, and a gapped region with dominant vector chirality or collinear
spin correlations. We localize the phase transitions and investigate their
nature.Comment: 4 pages, 6 figures, REVTeX 4, published versio
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