34 research outputs found
Fuzzy Nambu-Goldstone Physics
In spacetime dimensions larger than 2, whenever a global symmetry G is
spontaneously broken to a subgroup H, and G and H are Lie groups, there are
Nambu-Goldstone modes described by fields with values in G/H. In
two-dimensional spacetimes as well, models where fields take values in G/H are
of considerable interest even though in that case there is no spontaneous
breaking of continuous symmetries. We consider such models when the world sheet
is a two-sphere and describe their fuzzy analogues for G=SU(N+1),
H=S(U(N-1)xU(1)) ~ U(N) and G/H=CP^N. More generally our methods give fuzzy
versions of continuum models on S^2 when the target spaces are Grassmannians
and flag manifolds described by (N+1)x(N+1) projectors of rank =< (N+1)/2.
These fuzzy models are finite-dimensional matrix models which nevertheless
retain all the essential continuum topological features like solitonic sectors.
They seem well-suited for numerical work.Comment: Latex, 18 pages; references added, typos correcte
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
On the structure of the space of generalized connections
We give a modern account of the construction and structure of the space of
generalized connections, an extension of the space of connections that plays a
central role in loop quantum gravity.Comment: 30 pages, added references, minor changes. To appear in International
Journal of Geometric Methods in Modern Physic
A separability criterion for density operators
We give a necessary and sufficient condition for a mixed quantum mechanical
state to be separable. The criterion is formulated as a boundedness condition
in terms of the greatest cross norm on the tensor product of trace class
operators.Comment: REVTeX, 5 page
Full regularity for a C*-algebra of the Canonical Commutation Relations. (Erratum added)
The Weyl algebra,- the usual C*-algebra employed to model the canonical
commutation relations (CCRs), has a well-known defect in that it has a large
number of representations which are not regular and these cannot model physical
fields. Here, we construct explicitly a C*-algebra which can reproduce the CCRs
of a countably dimensional symplectic space (S,B) and such that its
representation set is exactly the full set of regular representations of the
CCRs. This construction uses Blackadar's version of infinite tensor products of
nonunital C*-algebras, and it produces a "host algebra" (i.e. a generalised
group algebra, explained below) for the \sigma-representation theory of the
abelian group S where \sigma(.,.):=e^{iB(.,.)/2}.
As an easy application, it then follows that for every regular representation
of the Weyl algebra of (S,B) on a separable Hilbert space, there is a direct
integral decomposition of it into irreducible regular representations (a known
result).
An Erratum for this paper is added at the end.Comment: An erratum was added to the original pape
Quantum line bundles on noncommutative sphere
Noncommutative (NC) sphere is introduced as a quotient of the enveloping
algebra of the Lie algebra su(2). Using the Cayley-Hamilton identities we
introduce projective modules which are analogues of line bundles on the usual
sphere (we call them quantum line bundles) and define a multiplicative
structure in their family. Also, we compute a pairing between certain quantum
line bundles and finite dimensional representations of the NC sphere in the
spirit of the NC index theorem. A new approach to constructing the differential
calculus on a NC sphere is suggested. The approach makes use of the projective
modules in question and gives rise to a NC de Rham complex being a deformation
of the classical one.Comment: LaTeX file, 15 pp, no figures. Some clarifying remarks are added at
the beginning of section 2 and into section
Barycentric decomposition of quantum measurements in finite dimensions
We analyze the convex structure of the set of positive operator valued
measures (POVMs) representing quantum measurements on a given finite
dimensional quantum system, with outcomes in a given locally compact Hausdorff
space. The extreme points of the convex set are operator valued measures
concentrated on a finite set of k \le d^2 points of the outcome space, d<
\infty being the dimension of the Hilbert space. We prove that for second
countable outcome spaces any POVM admits a Choquet representation as the
barycenter of the set of extreme points with respect to a suitable probability
measure. In the general case, Krein-Milman theorem is invoked to represent
POVMs as barycenters of a certain set of POVMs concentrated on k \le d^2 points
of the outcome space.Comment: !5 pages, no figure
Homology and K--Theory Methods for Classes of Branes Wrapping Nontrivial Cycles
We apply some methods of homology and K-theory to special classes of branes
wrapping homologically nontrivial cycles. We treat the classification of
four-geometries in terms of compact stabilizers (by analogy with Thurston's
classification of three-geometries) and derive the K-amenability of Lie groups
associated with locally symmetric spaces listed in this case. More complicated
examples of T-duality and topology change from fluxes are also considered. We
analyse D-branes and fluxes in type II string theory on with torsion flux and demonstrate in details
the conjectured T-duality to with no flux. In the
simple case of , T-dualizing the circles reduces to
duality between with
flux and with no flux.Comment: 27 pages, tex file, no figure
Fluxes, Brane Charges and Chern Morphisms of Hyperbolic Geometry
The purpose of this paper is to provide the reader with a collection of
results which can be found in the mathematical literature and to apply them to
hyperbolic spaces that may have a role in physical theories. Specifically we
apply K-theory methods for the calculation of brane charges and RR-fields on
hyperbolic spaces (and orbifolds thereof). It is known that by tensoring
K-groups with the rationals, K-theory can be mapped to rational cohomology by
means of the Chern character isomorphisms. The Chern character allows one to
relate the analytic Dirac index with a topological index, which can be
expressed in terms of cohomological characteristic classes. We obtain explicit
formulas for Chern character, spectral invariants, and the index of a twisted
Dirac operator associated with real hyperbolic spaces. Some notes for a
bivariant version of topological K-theory (KK-theory) with its connection to
the index of the twisted Dirac operator and twisted cohomology of hyperbolic
spaces are given. Finally we concentrate on lower K-groups useful for
description of torsion charges.Comment: 26 pages, no figures, LATEX. To appear in the Classical and Quantum
Gravit
Monoids of intervals of simple refinement monoids and non-stable K-Theory of multiplier algebras
We show that the representation of the monoid of intervals of a simple refinement monoid in terms of affine semicontinuous functions, given by Perera in 2001, fails to be faithful in the case of strictly perforated monoids. We give some potential applications of this result in the context of monoids of intervals and K-Theory of multiplier rings