2,159 research outputs found

### Curved noncommutative torus and Gauss--Bonnet

We study perturbations of the flat geometry of the noncommutative
two-dimensional torus T^2_\theta (with irrational \theta). They are described
by spectral triples (A_\theta, \H, D), with the Dirac operator D, which is a
differential operator with coefficients in the commutant of the (smooth)
algebra A_\theta of T_\theta. We show, up to the second order in perturbation,
that the zeta-function at 0 vanishes and so the Gauss-Bonnet theorem holds. We
also calculate first two terms of the perturbative expansion of the
corresponding local scalar curvature.Comment: 13 pages, LaTe

### Noncommutative Geometry and The Ising Model

The main aim of this work is to present the interpretation of the Ising type
models as a kind of field theory in the framework of noncommutative geometry.
We present the method and construct sample models of field theory on discrete
spaces using the introduced tools of discrete geometry. We write the action for
few models, then we compare them with various models of statistical physics. We
construct also the gauge theory with a discrete gauge group.Comment: 12 pages, LaTeX, TPJU - 18/92, December 199

### Black hole evaporation based upon a q-deformation description

A toy model based upon the $q$-deformation description for studying the
radiation spectrum of black hole is proposed. The starting point is to make an
attempt to consider the spacetime noncommutativity in the vicinity of black
hole horizon. We use a trick that all the spacetime noncommutative effects are
ascribed to the modification of the behavior of the radiation field of black
hole and a kind of q-deformed degrees of freedom are postulated to mimic the
radiation particles that live on the noncommutative spacetime, meanwhile the
background metric is preserved as usual. We calculate the radiation spectrum of
Schwarzschild black hole in this framework. The new distribution deviates from
the standard thermal spectrum evidently. The result indicates that some
correlation effect will be introduced to the system if the noncommutativity is
taken into account. In addition, an infrared cut-off of the spectrum is the
prediction of the model.Comment: 13 pages, 2 figure

### Gravity in Non-Commutative Geometry

We study general relativity in the framework of non-commutative differential
geometry. In particular, we introduce a gravity action for a space-time which
is the product of a four dimensional manifold by a two-point space. In the
simplest situation, where the Riemannian metric is taken to be the same on the
two copies of the manifold, one obtains a model of a scalar field coupled to
Einstein gravity. This field is geometrically interpreted as describing the
distance between the two points in the internal space.Comment: ZU-TH-30/1992 and ETH/TH/92/44, 11 pages. (The earlier version of
this paper was the incomplete and unedited file which accidently replaced the
corrected file

### On a Classification of Irreducible Almost Commutative Geometries, A Second Helping

We complete the classification of almost commutative geometries from a
particle physics point of view given in hep-th/0312276. Four missing Krajewski
diagrams will be presented after a short introduction into irreducible,
non-degenerate spectral triples.Comment: 11 page

### Gravity coupled with matter and foundation of non-commutative geometry

We first exhibit in the commutative case the simple algebraic relations
between the algebra of functions on a manifold and its infinitesimal length
element $ds$. Its unitary representations correspond to Riemannian metrics and
Spin structure while $ds$ is the Dirac propagator ds = \ts \!\!---\!\! \ts =
D^{-1} where $D$ is the Dirac operator. We extend these simple relations to
the non commutative case using Tomita's involution $J$. We then write a
spectral action, the trace of a function of the length element in Planck units,
which when applied to the non commutative geometry of the Standard Model will
be shown (in a joint work with Ali Chamseddine) to give the SM Lagrangian
coupled to gravity. The internal fluctuations of the non commutative geometry
are trivial in the commutative case but yield the full bosonic sector of SM
with all correct quantum numbers in the slightly non commutative case. The
group of local gauge transformations appears spontaneously as a normal subgroup
of the diffeomorphism group.Comment: 30 pages, Plain Te

### Noncommutative Geometry and the standard model with neutrino mixing

We show that allowing the metric dimension of a space to be independent of
its KO-dimension and turning the finite noncommutative geometry F-- whose
product with classical 4-dimensional space-time gives the standard model
coupled with gravity--into a space of KO-dimension 6 by changing the grading on
the antiparticle sector into its opposite, allows to solve three problems of
the previous noncommutative geometry interpretation of the standard model of
particle physics:
The finite geometry F is no longer put in "by hand" but a conceptual
understanding of its structure and a classification of its metrics is given.
The fermion doubling problem in the fermionic part of the action is resolved.
The spectral action of our joint work with Chamseddine now automatically
generates the full standard model coupled with gravity with neutrino mixing and
see-saw mechanism for neutrino masses. The predictions of the Weinberg angle
and the Higgs scattering parameter at unification scale are the same as in our
joint work but we also find a mass relation (to be imposed at unification
scale).Comment: Typos removed, to appear in JHE

### The Gravitational Sector in the Connes-Lott Formulation of the Standard Model

We study the Riemannian aspect and the Hilbert-Einstein gravitational action
of the non-commutative geometry underlying the Connes-Lott construction of the
action functional of the standard model. This geometry involves a two-sheeted,
Euclidian space-time. We show that if we require the space of forms to be
locally isotropic and the Higgs scalar to be dynamical, then the Riemannian
metrics on the two sheets of Euclidian space-time must be identical. We also
show that the distance function between the two sheets is determined by a
single, real scalar field whose VEV sets the weak scale.Comment: Latex file, 29 page

### The Moyal Sphere

We construct a family of constant curvature metrics on the Moyal plane and
compute the Gauss-Bonnet term for each of them. They arise from the conformal
rescaling of the metric in the orthonormal frame approach. We find a particular
solution, which corresponds to the Fubini-Study metric and which equips the
Moyal algebra with the geometry of a noncommutative sphere.Comment: 16 pages, 3 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

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