2,069 research outputs found

### Noncommutative Geometric Spaces with Boundary: Spectral Action

We study spectral action for Riemannian manifolds with boundary, and then
generalize this to noncommutative spaces which are products of a Riemannian
manifold times a finite space. We determine the boundary conditions consistent
with the hermiticity of the Dirac operator. We then define spectral triples of
noncommutative spaces with boundary. In particular we evaluate the spectral
action corresponding to the noncommutative space of the standard model and show
that the Einstein-Hilbert action gets modified by the addition of the extrinsic
curvature terms with the right sign and coefficient necessary for consistency
of the Hamiltonian. We also include effects due to the addition of dilaton
field.Comment: 26 page

### Noncommutative Geometry as a Framework for Unification of all Fundamental Interactions including Gravity. Part I

We examine the hypothesis that space-time is a product of a continuous
four-dimensional manifold times a finite space. A new tensorial notation is
developed to present the various constructs of noncommutative geometry. In
particular, this notation is used to determine the spectral data of the
standard model. The particle spectrum with all of its symmetries is derived,
almost uniquely, under the assumption of irreducibility and of dimension 6
modulo 8 for the finite space. The reduction from the natural symmetry group
SU(2)xSU(2)xSU(4) to U(1)xSU(2)xSU(3) is a consequence of the hypothesis that
the two layers of space-time are finite distance apart but is non-dynamical.
The square of the Dirac operator, and all geometrical invariants that appear in
the calculation of the heat kernel expansion are evaluated. We re-derive the
leading order terms in the spectral action. The geometrical action yields
unification of all fundamental interactions including gravity at very high
energies. We make the following predictions: (i) The number of fermions per
family is 16. (ii) The symmetry group is U(1)xSU(2)xSU(3). (iii) There are
quarks and leptons in the correct representations. (iv) There is a doublet
Higgs that breaks the electroweak symmetry to U(1). (v) Top quark mass of
170-175 Gev. (v) There is a right-handed neutrino with a see-saw mechanism.
Moreover, the zeroth order spectral action obtained with a cut-off function is
consistent with experimental data up to few percent. We discuss a number of
open issues. We prepare the ground for computing higher order corrections since
the predicted mass of the Higgs field is quite sensitive to the higher order
corrections. We speculate on the nature of the noncommutative space at
Planckian energies and the possible role of the fundamental group for the
problem of generations.Comment: 56 page

### Higgs mass in Noncommutative Geometry

In the noncommutative geometry approach to the standard model, an extra
scalar field - initially suggested by particle physicist to stabilize the
electroweak vacuum - makes the computation of the Higgs mass compatible with
the 126 GeV experimental value. We give a brief account on how to generate this
field from the Majorana mass of the neutrino, following the principles of
noncommutative geometry.Comment: Proceedings of the Corfou Workshop on noncommutative field theory and
gravity, september 201

### An Invariant Action for Noncommutative Gravity in Four-Dimensions

Two main problems face the construction of noncommutative actions for gravity
with star products: the complex metric and finding an invariant measure. The
only gauge groups that could be used with star products are the unitary groups.
I propose an invariant gravitational action in D=4 dimensions based on the
constrained gauge group U(2,2) broken to $U(1,1)\times U(1,1).$ No metric is
used, thus giving a naturally invariant measure. This action is generalized to
the noncommutative case by replacing ordinary products with star products. The
four dimensional noncommutative action is studied and the deformed action to
first order in deformation parameter is computed.Comment: 11 pages. Paper shortened. Consideration is now limited to gravity in
four-dimension

### Dimensionally Reduced Yang-Mills Theories in Noncommutative Geometry

We study a class of noncommutative geometries that give rise to dimensionally
reduced Yang-Mills theories. The emerging geometries describe sets of copies of
an even dimensional manifold. Similarities to the D-branes in string theory are
discussed.Comment: 12 pages, Latex, minor correction

### The Spectral Action Principle in Noncommutative Geometry and the Superstring

A supersymmetric theory in two-dimensions has enough data to define a
noncommutative space thus making it possible to use all tools of noncommutative
geometry. In particular, we apply this to the N=1 supersymmetric non-linear
sigma model and derive an expression for the generalized loop space Dirac
operator, in presence of a general background, using canonical quantization.
The spectral action principle is then used to determine a spectral action valid
for the fluctuations of the string modes.Comment: Latex file, 13 pages. Correction to equation 47, which should read
Tr| |^2 and not |Tr |^2. Final form to appear in Physics Letters

### Spectral Action for Robertson-Walker metrics

We use the Euler-Maclaurin formula and the Feynman-Kac formula to extend our
previous method of computation of the spectral action based on the Poisson
summation formula. We show how to compute directly the spectral action for the
general case of Robertson-Walker metrics. We check the terms of the expansion
up to a_6 against the known universal formulas of Gilkey and compute the
expansion up to a_{10} using our direct method

### Classical and Quantum Considerations of Two-dimensional Gravity

The two-dimensional theory of gravity describing a graviton-dilaton system is
considered. The graviton-dilaton coupling can be fixed such that the quantum
theory remains free of the conformal anomaly for any conformal dimension of the
coupled matter system, even if the dilaton does not appear as Lagrange
multiplier. Interaction terms are introduced and the system is analyzed and
solutions are given at the classical level and at the quantum level by using
canonical quantization.Comment: 18 page

### BPS black holes in N=2 five dimensional AdS supergravity

BPS black hole solutions of U(1) gauged five-dimensional supergravity are
obtained by solving the Killing spinor equations. These extremal static black
holes live in an asymptotic AdS_5 space time. Unlike black holes in asymptotic
flat space time none of them possess a regular horizon. We also calculate the
influence, of a particular class of these solutions, on the Wilson loops
calculation.Comment: 8 pages, 1 figure, LaTeX, corrected the potentia

### 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 $D$ on an $n$
dimensional compact Riemannian manifold with $n\geq 4$, $n$ even, the Wodzicki
residue Res$(D^{-n+2})$ is the integral of the second coefficient of the heat
kernel expansion of $D^{2}$. 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

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