9,663 research outputs found
Localization for Yang-Mills Theory on the Fuzzy Sphere
We present a new model for Yang-Mills theory on the fuzzy sphere in which the
configuration space of gauge fields is given by a coadjoint orbit. In the
classical limit it reduces to ordinary Yang-Mills theory on the sphere. We find
all classical solutions of the gauge theory and use nonabelian localization
techniques to write the partition function entirely as a sum over local
contributions from critical points of the action, which are evaluated
explicitly. The partition function of ordinary Yang-Mills theory on the sphere
is recovered in the classical limit as a sum over instantons. We also apply
abelian localization techniques and the geometry of symmetric spaces to derive
an explicit combinatorial expression for the partition function, and compare
the two approaches. These extend the standard techniques for solving gauge
theory on the sphere to the fuzzy case in a rigorous framework.Comment: 55 pages. V2: references added; V3: minor corrections, reference
added; Final version to be published in Communications in Mathematical
Physic
Theoretical Interpretations and Applications of Radial Basis Function Networks
Medical applications usually used Radial Basis Function Networks just as Artificial Neural Networks. However, RBFNs are Knowledge-Based Networks that can be interpreted in several way: Artificial Neural Networks, Regularization Networks, Support Vector Machines, Wavelet Networks, Fuzzy Controllers, Kernel Estimators, Instanced-Based Learners. A survey of their interpretations and of their corresponding learning algorithms is provided as well as a brief survey on dynamic learning algorithms. RBFNs' interpretations can suggest applications that are particularly interesting in medical domains
Noncommutative Vortices and Flux-Tubes from Yang-Mills Theories with Spontaneously Generated Fuzzy Extra Dimensions
We consider a U(2) Yang-Mills theory on M x S_F^2 where M is an arbitrary
noncommutative manifold and S_F^2 is a fuzzy sphere spontaneously generated
from a noncommutative U(N) Yang-Mills theory on M, coupled to a triplet of
scalars in the adjoint of U(N). Employing the SU(2)-equivariant gauge field
constructed in arXiv:0905.2338, we perform the dimensional reduction of the
theory over the fuzzy sphere. The emergent model is a noncommutative U(1) gauge
theory coupled adjointly to a set of scalar fields. We study this model on the
Groenewald-Moyal plane and find that, in certain limits, it admits
noncommutative, non-BPS vortex as well as flux-tube (fluxon) solutions and
discuss some of their properties.Comment: 18+1 pages, typos corrected, published versio
The beat of a fuzzy drum: fuzzy Bessel functions for the disc
The fuzzy disc is a matrix approximation of the functions on a disc which
preserves rotational symmetry. In this paper we introduce a basis for the
algebra of functions on the fuzzy disc in terms of the eigenfunctions of a
properly defined fuzzy Laplacian. In the commutative limit they tend to the
eigenfunctions of the ordinary Laplacian on the disc, i.e. Bessel functions of
the first kind, thus deserving the name of fuzzy Bessel functions.Comment: 30 pages, 8 figure
Noncommutative Harmonic Analysis, Sampling Theory and the Duflo Map in 2+1 Quantum Gravity
We show that the -product for , group Fourier transform and
effective action arising in [1] in an effective theory for the integer spin
Ponzano-Regge quantum gravity model are compatible with the noncommutative
bicovariant differential calculus, quantum group Fourier transform and
noncommutative scalar field theory previously proposed for 2+1 Euclidean
quantum gravity using quantum group methods in [2]. The two are related by a
classicalisation map which we introduce. We show, however, that noncommutative
spacetime has a richer structure which already sees the half-integer spin
information. We argue that the anomalous extra `time' dimension seen in the
noncommutative geometry should be viewed as the renormalisation group flow
visible in the coarse-graining in going from to . Combining our
methods we develop practical tools for noncommutative harmonic analysis for the
model including radial quantum delta-functions and Gaussians, the Duflo map and
elements of `noncommutative sampling theory'. This allows us to understand the
bandwidth limitation in 2+1 quantum gravity arising from the bounded
momentum and to interpret the Duflo map as noncommutative compression. Our
methods also provide a generalised twist operator for the -product.Comment: 53 pages latex, no figures; extended the intro for this final versio
Dynamical generation of fuzzy extra dimensions, dimensional reduction and symmetry breaking
We present a renormalizable 4-dimensional SU(N) gauge theory with a suitable
multiplet of scalar fields, which dynamically develops extra dimensions in the
form of a fuzzy sphere S^2. We explicitly find the tower of massive
Kaluza-Klein modes consistent with an interpretation as gauge theory on M^4 x
S^2, the scalars being interpreted as gauge fields on S^2. The gauge group is
broken dynamically, and the low-energy content of the model is determined.
Depending on the parameters of the model the low-energy gauge group can be
SU(n), or broken further to SU(n_1) x SU(n_2) x U(1), with mass scale
determined by the size of the extra dimension.Comment: 27 pages. V2: discussion and references added, published versio
2D fuzzy Anti-de Sitter space from matrix models
We study the fuzzy hyperboloids AdS^2 and dS^2 as brane solutions in matrix
models. The unitary representations of SO(2,1) required for quantum field
theory are identified, and explicit formulae for their realization in terms of
fuzzy wavefunctions are given. In a second part, we study the (A)dS^2 brane
geometry and its dynamics, as governed by a suitable matrix model. In
particular, we show that trace of the energy-momentum tensor of matter induces
transversal perturbations of the brane and of the Ricci scalar. This leads to a
linearized form of Henneaux-Teitelboim-type gravity, illustrating the mechanism
of emergent gravity in matrix models.Comment: 25 page
The fuzzy S^2 structure of M2-M5 systems in ABJM membrane theories
We analyse the fluctuations of the ground-state/funnel solutions proposed to
describe M2-M5 systems in the level-k mass-deformed/pure Chern-Simons-matter
ABJM theory of multiple membranes. We show that in the large N limit the
fluctuations approach the space of functions on the 2-sphere rather than the
naively expected 3-sphere. This is a novel realisation of the fuzzy 2-sphere in
the context of Matrix Theories, which uses bifundamental instead of adjoint
scalars. Starting from the multiple M2-brane action, a U(1) Yang-Mills theory
on R^{2,1} x S^2 is recovered at large N, which is consistent with a single
D4-brane interpretation in Type IIA string theory. This is as expected at large
k, where the semiclassical analysis is valid. Several aspects of the
fluctuation analysis, the ground-state/funnel solutions and the
mass-deformed/pure ABJM equations can be understood in terms of a discrete
noncommutative realisation of the Hopf fibration. We discuss the implications
for the possibility of finding an M2-brane worldvolume derivation of the
classical S^3 geometry of the M2-M5 system. Using a rewriting of the equations
of the SO(4)-covariant fuzzy 3-sphere construction, we also directly compare
this fuzzy 3-sphere against the ABJM ground-state/funnel solutions and show
them to be different.Comment: 60 pages, Latex; v2: references added; v3: typos corrected and
references adde
Quantized Nambu-Poisson Manifolds and n-Lie Algebras
We investigate the geometric interpretation of quantized Nambu-Poisson
structures in terms of noncommutative geometries. We describe an extension of
the usual axioms of quantization in which classical Nambu-Poisson structures
are translated to n-Lie algebras at quantum level. We demonstrate that this
generalized procedure matches an extension of Berezin-Toeplitz quantization
yielding quantized spheres, hyperboloids, and superspheres. The extended
Berezin quantization of spheres is closely related to a deformation
quantization of n-Lie algebras, as well as the approach based on harmonic
analysis. We find an interpretation of Nambu-Heisenberg n-Lie algebras in terms
of foliations of R^n by fuzzy spheres, fuzzy hyperboloids, and noncommutative
hyperplanes. Some applications to the quantum geometry of branes in M-theory
are also briefly discussed.Comment: 43 pages, minor corrections, presentation improved, references adde
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