478 research outputs found
Constructive spherical codes on layers of flat tori
A new class of spherical codes is constructed by selecting a finite subset of
flat tori from a foliation of the unit sphere S^{2L-1} of R^{2L} and designing
a structured codebook on each torus layer. The resulting spherical code can be
the image of a lattice restricted to a specific hyperbox in R^L in each layer.
Group structure and homogeneity, useful for efficient storage and decoding, are
inherited from the underlying lattice codebook. A systematic method for
constructing such codes are presented and, as an example, the Leech lattice is
used to construct a spherical code in R^{48}. Upper and lower bounds on the
performance, the asymptotic packing density and a method for decoding are
derived.Comment: 9 pages, 5 figures, submitted to IEEE Transactions on Information
Theor
Hexagonal dielectric resonators and microcrystal lasers
We study long-lived resonances (lowest-loss modes) in hexagonally shaped
dielectric resonators in order to gain insight into the physics of a class of
microcrystal lasers. Numerical results on resonance positions and lifetimes,
near-field intensity patterns, far-field emission patterns, and effects of
rounding of corners are presented. Most features are explained by a
semiclassical approximation based on pseudointegrable ray dynamics and boundary
waves. The semiclassical model is also relevant for other microlasers of
polygonal geometry.Comment: 12 pages, 17 figures (3 with reduced quality
Noncommutative Field Theory
We review the generalization of field theory to space-time with noncommuting
coordinates, starting with the basics and covering most of the active
directions of research. Such theories are now known to emerge from limits of M
theory and string theory, and to describe quantum Hall states. In the last few
years they have been studied intensively, and many qualitatively new phenomena
have been discovered, both on the classical and quantum level.
To appear in Reviews of Modern Physics.Comment: Revtex, 56 pp, 6 figures. Final versio
Seiberg-Witten geometry of four dimensional N=2 quiver gauge theories
Seiberg-Witten geometry of mass deformed N=2 superconformal ADE quiver gauge
theories in four dimensions is determined. We solve the limit shape equations
derived from the gauge theory and identify the space M of vacua of the theory
with the moduli space of the genus zero holomorphic (quasi)maps to the moduli
space of holomorphic G-bundles on a (possibly degenerate) elliptic curve
defined in terms of the microscopic gauge couplings, for the corresponding
simple ADE Lie group G. The integrable systems underlying, or, rather,
overlooking the special geometry of M are identified. The moduli spaces of
framed G-instantons on R^2xT^2, of G-monopoles with singularities on R^2xS^1,
the Hitchin systems on curves with punctures, as well as various spin chains
play an important role in our story. We also comment on the higher dimensional
theories. In the companion paper the quantum integrable systems and their
connections to the representation theory of quantum affine algebras will be
discussedComment: 197 page
Evidence for F(uzz) Theory
We show that in the decoupling limit of an F-theory compactification, the
internal directions of the seven-branes must wrap a non-commutative four-cycle
S. We introduce a general method for obtaining fuzzy geometric spaces via toric
geometry, and develop tools for engineering four-dimensional GUT models from
this non-commutative setup. We obtain the chiral matter content and Yukawa
couplings, and show that the theory has a finite Kaluza-Klein spectrum. The
value of 1/alpha_(GUT) is predicted to be equal to the number of fuzzy points
on the internal four-cycle S. This relation puts a non-trivial restriction on
the space of gauge theories that can arise as a limit of F-theory. By viewing
the seven-brane as tiled by D3-branes sitting at the N fuzzy points of the
geometry, we argue that this theory admits a holographic dual description in
the large N limit. We also entertain the possibility of constructing string
models with large fuzzy extra dimensions, but with a high scale for quantum
gravity.Comment: v2: 66 pages, 3 figures, references and clarifications adde
M(atrix) Theory: Matrix Quantum Mechanics as a Fundamental Theory
A self-contained review is given of the matrix model of M-theory. The
introductory part of the review is intended to be accessible to the general
reader. M-theory is an eleven-dimensional quantum theory of gravity which is
believed to underlie all superstring theories. This is the only candidate at
present for a theory of fundamental physics which reconciles gravity and
quantum field theory in a potentially realistic fashion. Evidence for the
existence of M-theory is still only circumstantial---no complete
background-independent formulation of the theory yet exists. Matrix theory was
first developed as a regularized theory of a supersymmetric quantum membrane.
More recently, the theory appeared in a different guise as the discrete
light-cone quantization of M-theory in flat space. These two approaches to
matrix theory are described in detail and compared. It is shown that matrix
theory is a well-defined quantum theory which reduces to a supersymmetric
theory of gravity at low energies. Although the fundamental degrees of freedom
of matrix theory are essentially pointlike, it is shown that higher-dimensional
fluctuating objects (branes) arise through the nonabelian structure of the
matrix degrees of freedom. The problem of formulating matrix theory in a
general space-time background is discussed, and the connections between matrix
theory and other related models are reviewed.Comment: 56 pages, 3 figures, LaTeX, revtex style; v2: references adde
Rational Maps, Monopoles and Skyrmions
We discuss the similarities between BPS monopoles and Skyrmions, and point to
an underlying connection in terms of rational maps between Riemann spheres.
This involves the introduction of a new ansatz for Skyrme fields. We use this
to construct good approximations to several known Skyrmions, including all the
minimal energy configurations up to baryon number nine, and some new solutions
such as a baryon number seventeen Skyrme field with the truncated icosahedron
structure of a buckyball.
The new approach is also used to understand the low-lying vibrational modes
of Skyrmions, which are required for quantization. Along the way we discover an
interesting Morse function on the space of rational maps which may be of use in
understanding the Sen forms on the monopole moduli spaces.Comment: 35pp including four figures, typos corrected, appearing in Nuclear
Physics
Self-duality and vacuum selection
I propose that self-duality in quantum phase-space provides the criteria for
the selection of the quantum gravity vacuum. The evidence for this assertion
arises from two independent considerations. The first is the phenomenological
success of the free fermionic heterotic-string models, which are constructed in
the vicinity of the self-dual point under T-duality. The relation between the
free fermionic models and the underlying Z2 X Z2 toroidal orbifolds is
discussed. Recent analysis revealed that the Z2 X Z2 free fermionic orbifolds
utilize an asymmetric shift in the reduction to three generations, which
indicates that the untwisted geometrical moduli are fixed near the self-dual
point. The second consideration arises from the recent formulation of quantum
mechanics from an equivalence postulate and its relation to phase-space
duality. In this context it is demonstrated that the trivial state, with
V(q)=E=0, is identified with the self-dual state under phase-space duality.
These observations suggest a more general mathematical principle in operation.
In physical systems that exhibit a duality structure, the self-dual states
under the given duality transformations correspond to critical points.Comment: 40 pages. Standard Latex. 1 figur
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