259 research outputs found
Cosmological and Black Hole Spacetimes in Twisted Noncommutative Gravity
We derive noncommutative Einstein equations for abelian twists and their
solutions in consistently symmetry reduced sectors, corresponding to twisted
FRW cosmology and Schwarzschild black holes. While some of these solutions must
be rejected as models for physical spacetimes because they contradict
observations, we find also solutions that can be made compatible with low
energy phenomenology, while exhibiting strong noncommutativity at very short
distances and early times.Comment: LaTeX 12 pages, JHEP.st
Noncommutative Symmetries and Gravity
Spacetime geometry is twisted (deformed) into noncommutative spacetime
geometry, where functions and tensors are now star-multiplied. Consistently,
spacetime diffeomorhisms are twisted into noncommutative diffeomorphisms. Their
deformed Lie algebra structure and that of infinitesimal Poincare'
transformations is defined and explicitly constructed.
This allows to construct a noncommutative theory of gravity.Comment: 26 pages. Lectures given at the workshop `Noncommutative Geometry in
Field and String Theories', Corfu Summer Institute on EPP, September 2005,
Corfu, Greece. Version 2: Marie Curie European Reintegration Grant
MERG-CT-2004-006374 acknowledge
Noncommutative gravity coupled to fermions: second order expansion via Seiberg-Witten map
We use the Seiberg-Witten map (SW map) to expand noncommutative gravity
coupled to fermions in terms of ordinary commuting fields. The action is
invariant under general coordinate transformations and local Lorentz rotations,
and has the same degrees of freedom as the commutative gravity action. The
expansion is given up to second order in the noncommutativity parameter
{\theta}. A geometric reformulation and generalization of the SW map is
presented that applies to any abelian twist. Compatibility of the map with
hermiticity and charge conjugation conditions is proven. The action is shown to
be real and invariant under charge conjugation at all orders in {\theta}. This
implies the bosonic part of the action to be even in {\theta}, while the
fermionic part is even in {\theta} for Majorana fermions.Comment: 27 pages, LaTeX. Revised version with proof of charge conjugation
symmetry of the NC action and its parity under theta --> - theta (see new
sect. 2.6, sect. 6 and app. B). References added. arXiv admin note:
substantial text overlap with arXiv:0902.381
Reality in Noncommutative Gravity
We study the problem of reality in the geometric formalism of the 4D
noncommutative gravity using the known deformation of the diffeomorphism group
induced by the twist operator with the constant deformation parameters
\vt^{mn}. It is shown that real covariant derivatives can be constructed via
-anticommutators of the real connection with the corresponding fields.
The minimal noncommutative generalization of the real Riemann tensor contains
only \vt^{mn}-corrections of the even degrees in comparison with the
undeformed tensor. The gauge field describes a gravitational field on
the flat background. All geometric objects are constructed as the perturbation
series using -polynomial decomposition in terms of . We consider
the nonminimal tensor and scalar functions of of the odd degrees in
\vt^{mn} and remark that these pure noncommutative objects can be used in the
noncommutative gravity.Comment: Latex file, 14 pages, corrected version to be publised in CQ
Star Product Geometries
We consider noncommutative geometries obtained from a triangular Drinfeld
twist. This allows to construct and study a wide class of noncommutative
manifolds and their deformed Lie algebras of infinitesimal diffeomorphisms.
This way symmetry principles can be implemented. We review two main examples
[15]-[18]: a) general covariance in noncommutative spacetime. This leads to a
noncommutative gravity theory. b) Symplectomorphims of the algebra of
observables associated to a noncommutative configuration space. This leads to a
geometric formulation of quantization on noncommutative spacetime, i.e., we
establish a noncommutative correspondence principle from *-Poisson brackets to
*-commutators.
New results concerning noncommutative gravity include the Cartan structural
equations for the torsion and curvature tensors, and the associated Bianchi
identities. Concerning scalar field theories the deformed algebra of classical
and quantum observables has been understood in terms of a twist within the
algebra.Comment: 27 pages. Based on the talk presented at the conference "Geometry and
Operators Theory," Ancona (Italy), September 200
Unified Theories from Fuzzy Extra Dimensions
We combine and exploit ideas from Coset Space Dimensional Reduction (CSDR)
methods and Non-commutative Geometry. We consider the dimensional reduction of
gauge theories defined in high dimensions where the compact directions are a
fuzzy space (matrix manifold). In the CSDR one assumes that the form of
space-time is M^D=M^4 x S/R with S/R a homogeneous space. Then a gauge theory
with gauge group G defined on M^D can be dimensionally reduced to M^4 in an
elegant way using the symmetries of S/R, in particular the resulting four
dimensional gauge is a subgroup of G. In the present work we show that one can
apply the CSDR ideas in the case where the compact part of the space-time is a
finite approximation of the homogeneous space S/R, i.e. a fuzzy coset. In
particular we study the fuzzy sphere case.Comment: 6 pages, Invited talk given by G. Zoupanos at the 36th International
Symposium Ahrenshoop, Wernsdorf, Germany, 26-30 Aug 200
Twist as a Symmetry Principle and the Noncommutative Gauge Theory Formulation
Based on the analysis of the most natural and general ansatz, we conclude
that the concept of twist symmetry, originally obtained for the noncommutative
space-time, cannot be extended to include internal gauge symmetry. The case is
reminiscent of the Coleman-Mandula theorem. Invoking the supersymmetry may
reverse the situation.Comment: 13 pages, more accurate motivation adde
Fuzzy Extra Dimensions: Dimensional Reduction, Dynamical Generation and Renormalizability
We examine gauge theories defined in higher dimensions where theextra
dimensions form a fuzzy (finite matrix) manifold. First we reinterpret these
gauge theories as four-dimensional theories with Kaluza-Klein modes and then we
perform a generalized \`a la Forgacs-Manton dimensional reduction. We emphasize
some striking features emerging in the later case such as (i) the appearance of
non-abelian gauge theories in four dimensions starting from an abelian gauge
theory in higher dimensions, (ii) the fact that the spontaneous symmetry
breaking of the theory takes place entirely in the extra dimensions and (iii)
the renormalizability of the theory both in higher as well as in four
dimensions. Then reversing the above approach we present a renormalizable four
dimensional SU(N) gauge theory with a suitable multiplet of scalar fields,
which via spontaneous symmetry breaking dynamically develops extra dimensions
in the form of a fuzzy sphere. We explicitly find the tower of massive
Kaluza-Klein modes consistent with an interpretation as gauge theory on , the scalars being interpreted as gauge fields on . Depending
on the parameters of the model the low-energy gauge group can be of the form
.Comment: 18 pages, Based on invited talks presented at various conferences,
Minor corrections, Acknowledgements adde
Algebraic approach to quantum field theory on a class of noncommutative curved spacetimes
In this article we study the quantization of a free real scalar field on a
class of noncommutative manifolds, obtained via formal deformation quantization
using triangular Drinfel'd twists. We construct deformed quadratic action
functionals and compute the corresponding equation of motion operators. The
Green's operators and the fundamental solution of the deformed equation of
motion are obtained in terms of formal power series. It is shown that, using
the deformed fundamental solution, we can define deformed *-algebras of field
observables, which in general depend on the spacetime deformation parameter.
This dependence is absent in the special case of Killing deformations, which
include in particular the Moyal-Weyl deformation of the Minkowski spacetime.Comment: LaTeX 14 pages, no figures, svjour3.cls style; v2: clarifications and
references added, compatible with published versio
Hidden Quantum Group Structure in Einstein's General Relativity
A new formal scheme is presented in which Einstein's classical theory of
General Relativity appears as the common, invariant sector of a one-parameter
family of different theories. This is achieved by replacing the Poincare` group
of the ordinary tetrad formalism with a q-deformed Poincare` group, the usual
theory being recovered at q=1. Although written in terms of noncommuting
vierbein and spin-connection fields, each theory has the same metric sector
leading to the ordinary Einstein-Hilbert action and to the corresponding
equations of motion. The Christoffel symbols and the components of the Riemann
tensor are ordinary commuting numbers and have the usual form in terms of a
metric tensor built as an appropriate bilinear in the vierbeins. Furthermore we
exhibit a one-parameter family of Hamiltonian formalisms for general
relativity, by showing that a canonical formalism a` la Ashtekar can be built
for any value of q. The constraints are still polynomial, but the Poisson
brackets are not skewsymmetric for q different from 1.Comment: LaTex file, 21 pages, no figure
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