41 research outputs found
Gravity, Non-Commutative Geometry and the Wodzicki Residue
We derive an action for gravity in the framework of non-commutative geometry
by using the Wodzicki residue. We prove that for a Dirac operator on an
dimensional compact Riemannian manifold with , even, the Wodzicki
residue Res is the integral of the second coefficient of the heat
kernel expansion of . We use this result to derive a gravity action for
commutative geometry which is the usual Einstein Hilbert action and we also
apply our results to a non-commutative extension which, is given by the tensor
product of the algebra of smooth functions on a manifold and a finite
dimensional matrix algebra. In this case we obtain gravity with a cosmological
constant.Comment: 17p., MZ-TH/93-3
Differential Algebras in Non-Commutative Geometry
We discuss the differential algebras used in Connes' approach to Yang-Mills
theories with spontaneous symmetry breaking. These differential algebras
generated by algebras of the form functions matrix are shown to be
skew tensorproducts of differential forms with a specific matrix algebra. For
that we derive a general formula for differential algebras based on tensor
products of algebras. The result is used to characterize differential algebras
which appear in models with one symmetry breaking scale.Comment: 21 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 . Its unitary representations correspond to Riemannian metrics and
Spin structure while is the Dirac propagator ds = \ts \!\!---\!\! \ts =
D^{-1} where is the Dirac operator. We extend these simple relations to
the non commutative case using Tomita's involution . 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
Supersymmetry and Noncommutative Geometry
The purpose of this article is to apply the concept of the spectral triple,
the starting point for the analysis of noncommutative spaces in the sense of
A.~Connes, to the case where the algebra \cA contains both bosonic and
fermionic degrees of freedom. The operator \cD of the spectral triple under
consideration is the square root of the Dirac operator und thus the forms of
the generalized differential algebra constructed out of the spectral triple are
in a representation of the Lorentz group with integer spin if the form degree
is even and they are in a representation with half-integer spin if the form
degree is odd. However, we find that the 2-forms, obtained by squaring the
connection, contains exactly the components of the vector multiplet
representation of the supersymmetry algebra. This allows to construct an action
for supersymmetric Yang-Mills theory in the framework of noncommutative
geometry.Comment: 26pp., LaTe
Noncommutative geometry and lower dimensional volumes in Riemannian geometry
In this paper we explain how to define "lower dimensional'' volumes of any
compact Riemannian manifold as the integrals of local Riemannian invariants.
For instance we give sense to the area and the length of such a manifold in any
dimension. Our reasoning is motivated by an idea of Connes and involves in an
essential way noncommutative geometry and the analysis of Dirac operators on
spin manifolds. However, the ultimate definitions of the lower dimensional
volumes don't involve noncommutative geometry or spin structures at all.Comment: 12 page
Euclidean Supergravity in Terms of Dirac Eigenvalues
It has been recently shown that the eigenvalues of the Dirac operator can be
considered as dynamical variables of Euclidean gravity. The purpose of this
paper is to explore the possiblity that the eigenvalues of the Dirac operator
might play the same role in the case of supergravity. It is shown that for this
purpose some primary constraints on covariant phase space as well as secondary
constraints on the eigenspinors must be imposed. The validity of primary
constraints under covariant transport is further analyzed. It is show that in
the this case restrictions on the tanget bundle and on the spinor bundle of
spacetime arise. The form of these restrictions is determined under some
simplifying assumptions. It is also shown that manifolds with flat curvature of
tangent bundle and spinor bundle and spinor bundle satisfy these restrictons
and thus they support the Dirac eigenvalues as global observables.Comment: Misprints and formulae corrected; to appear in Phys. Rev.
Wodzicki Residue for Operators on Manifolds with Cylindrical Ends
We define the Wodzicki Residue TR(A) for A in a space of operators with
double order (m_1,m_2). Such operators are globally defined initially on R^n
and then, more generally, on a class of non-compact manifolds, namely, the
manifolds with cylindrical ends. The definition is based on the analysis of the
associate zeta function. Using this approach, under suitable ellipticity
assumptions, we also compute a two terms leading part of the Weyl formula for a
positive selfadjoint operator belonging the mentioned class in the case
m_1=m_2.Comment: 24 pages, picture changed, added references, corrected typo
Linear Connections in Non-Commutative Geometry
A construction is proposed for linear connections on non-commutative
algebras. The construction relies on a generalisation of the Leibnitz rules of
commutative geometry and uses the bimodule structure of . A special
role is played by the extension to the framework of non-commutative geometry of
the permutation of two copies of . The construction of the linear
connection as well as the definition of torsion and curvature is first proposed
in the setting of the derivations based differential calculus of Dubois-
Violette and then a generalisation to the framework proposed by Connes as well
as other non-commutative differential calculi is suggested. The covariant
derivative obtained admits an extension to the tensor product of several copies
of . These constructions are illustrated with the example of the
algebra of matrices.Comment: 15 pages, LMPM ../94 (uses phyzzx
Noncommutative gravity: fuzzy sphere and others
Gravity on noncommutative analogues of compact spaces can give a finite mode
truncation of ordinary commutative gravity. We obtain the actions for gravity
on the noncommutative two-sphere and on the noncommutative in
terms of finite dimensional -matrices. The commutative large
limit is also discussed.Comment: LaTeX, 13 pages, section on CP^2 added + minor change
Chiral spinors and gauge fields in noncommutative curved space-time
The fundamental concepts of Riemannian geometry, such as differential forms,
vielbein, metric, connection, torsion and curvature, are generalized in the
context of non-commutative geometry. This allows us to construct the
Einstein-Hilbert-Cartan terms, in addition to the bosonic and fermionic ones in
the Lagrangian of an action functional on non-commutative spaces. As an
example, and also as a prelude to the Standard Model that includes
gravitational interactions, we present a model of chiral spinor fields on a
curved two-sheeted space-time with two distinct abelian gauge fields. In this
model, the full spectrum of the generalized metric consists of pairs of tensor,
vector and scalar fields. They are coupled to the chiral fermions and the gauge
fields leading to possible parity violation effects triggered by gravity.Comment: 50 pages LaTeX, minor corrections and references adde