101,682 research outputs found
The Parthasarathy formula and a spectral triple for the quantum Lagrangian Grassmannian of rank two
We show that the Dolbeault--Dirac operator on the quantum Lagrangian
Grassmannian of rank two, an example of a quantum irreducible flag manifold,
satisfies an appropriate version of the Parthasarathy formula. We use this
result to complete the proof that the candidate spectral triple for this space,
as defined by Kr\"ahmer, is a spectral triple.Comment: v2: General rewriting due to mix of conventions, main results
unchange
Clifford-based spectral action and renormalization group analysis of the gauge couplings
The Spectral Action Principle in noncommutative geometry derives the actions
of the Standard Model and General Relativity (along with several other
gravitational terms) by reconciling them in a geometric setting, and hence
offers an explanation for their common origin. However, one of the requirements
in the minimal formalism, unification of the gauge coupling constants, is not
satisfied, since the basic construction does not introduce anything new that
can change the renormalization group (RG) running of the Standard Model. On the
other hand, it has been recently argued that incorporating structure of the
Clifford algebra into the finite part of the spectral triple, the main object
that encodes the complete information of a noncommutative space, gives rise to
five additional scalar fields in the basic framework. We investigate whether
these scalars can help to achieve unification. We perform a RG analysis at the
one-loop level, allowing possible mass values of these scalars to float from
the electroweak scale to the putative unification scale. We show that out of
twenty configurations of mass hierarchy in total, there does not exist even a
single case that can lead to unification. In consequence, we confirm that the
spectral action formalism requires a model-construction scheme beyond the
(modified) minimal framework.Comment: 20 pages, 1 figure, 1 table of results; matches the published versio
On a Classification of Irreducible Almost-Commutative Geometries IV
In this paper we will classify the finite spectral triples with KO-dimension
six, following the classification found in [1,2,3,4], with up to four summands
in the matrix algebra. Again, heavy use is made of Kra jewski diagrams [5].
Furthermore we will show that any real finite spectral triple in KO-dimension 6
is automatically S 0 -real. This work has been inspired by the recent paper by
Alain Connes [6] and John Barrett [7].
In the classification we find that the standard model of particle physics in
its minimal version fits the axioms of noncommutative geometry in the case of
KO-dimension six. By minimal version it is meant that at least one neutrino has
to be massless and mass-terms mixing particles and antiparticles are prohibitedComment: Revised version for publication in the Journal of Mathematical
Physic
Noncommutative geometry, topology and the standard model vacuum
As a ramification of a motivational discussion for previous joint work, in
which equations of motion for the finite spectral action of the Standard Model
were derived, we provide a new analysis of the results of the calculations
herein, switching from the perspective of Spectral triple to that of Fredholm
module and thus from the analogy with Riemannian geometry to the pre-metrical
structure of the Noncommutative geometry. Using a suggested Noncommutative
version of Morse theory together with algebraic -theory to analyse the
vacuum solutions, the first two summands of the algebra for the finite triple
of the Standard Model arise up to Morita equivalence. We also demonstrate a new
vacuum solution whose features are compatible with the physical mass matrix.Comment: 24 page
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