173 research outputs found
The local structure of n-Poisson and n-Jacobi manifolds
N-Lie algebra structures on smooth function algebras given by means of
multi-differential operators, are studied. Necessary and sufficient conditions
for the sum and the wedge product of two -Poisson sructures to be again a
multi-Poisson are found. It is proven that the canonical -vector on the dual
of an n-Lie algebra g is n-Poisson iff dim(g) are not greater than n+1. The
problem of compatibility of two n-Lie algebra structures is analyzed and the
compatibility relations connecting hereditary structures of a given n-Lie
algebra are obtained. (n+1)-dimensional n-Lie algebras are classified and their
"elementary particle-like" structure is discovered. Some simple applications to
dynamics are discussed.Comment: 45 pages, latex, no figure
Schur Polynomials and the Yang-Baxter equation
We show that within the six-vertex model there is a parametrized Yang-Baxter
equation with nonabelian parameter group GL(2)xGL(1) at the center of the
disordered regime. As an application we rederive deformations of the Weyl
character formule of Tokuyama and of Hamel and King.Comment: Revised introduction; slightly changed reference
Low-Temperature Thermodynamics of and su(3)-invariant Spin Chains
We formulate the thermodynamic Bethe Ansatz (TBA) equations for the closed
(periodic boundary conditions) quantum spin chain in an external
magnetic field, in the (noncritical) regime where the anisotropy parameter
is real. In the limit , we recover the TBA equations of the
antiferromagnetic su(3)-invariant chain in the fundamental representation. We
solve these equations for low temperature and small field, and calculate the
specific heat and magnetic susceptibility.Comment: 31 pages, UMTG-16
Supercoherent States, Super K\"ahler Geometry and Geometric Quantization
Generalized coherent states provide a means of connecting square integrable
representations of a semi-simple Lie group with the symplectic geometry of some
of its homogeneous spaces. In the first part of the present work this point of
view is extended to the supersymmetric context, through the study of the
OSp(2/2) coherent states. These are explicitly constructed starting from the
known abstract typical and atypical representations of osp(2/2). Their
underlying geometries turn out to be those of supersymplectic OSp(2/2)
homogeneous spaces. Moment maps identifying the latter with coadjoint orbits of
OSp(2/2) are exhibited via Berezin's symbols. When considered within
Rothstein's general paradigm, these results lead to a natural general
definition of a super K\"ahler supermanifold, the supergeometry of which is
determined in terms of the usual geometry of holomorphic Hermitian vector
bundles over K\"ahler manifolds. In particular, the supergeometry of the above
orbits is interpreted in terms of the geometry of Einstein-Hermitian vector
bundles. In the second part, an extension of the full geometric quantization
procedure is applied to the same coadjoint orbits. Thanks to the super K\"ahler
character of the latter, this procedure leads to explicit super unitary
irreducible representations of OSp(2/2) in super Hilbert spaces of
superholomorphic sections of prequantum bundles of the Kostant type. This work
lays the foundations of a program aimed at classifying Lie supergroups'
coadjoint orbits and their associated irreducible representations, ultimately
leading to harmonic superanalysis. For this purpose a set of consistent
conventions is exhibited.Comment: 53 pages, AMS-LaTeX (or LaTeX+AMSfonts
Time of arrival through interacting environments: Tunneling processes
We discuss the propagation of wave packets through interacting environments.
Such environments generally modify the dispersion relation or shape of the wave
function. To study such effects in detail, we define the distribution function
P_{X}(T), which describes the arrival time T of a packet at a detector located
at point X. We calculate P_{X}(T) for wave packets traveling through a
tunneling barrier and find that our results actually explain recent
experiments. We compare our results with Nelson's stochastic interpretation of
quantum mechanics and resolve a paradox previously apparent in Nelson's
viewpoint about the tunneling time.Comment: Latex 19 pages, 11 eps figures, title modified, comments and
references added, final versio
More on quantum groups from the the quantization point of view
Star products on the classical double group of a simple Lie group and on
corresponding symplectic grupoids are given so that the quantum double and the
"quantized tangent bundle" are obtained in the deformation description.
"Complex" quantum groups and bicovariant quantum Lie algebras are discused from
this point of view. Further we discuss the quantization of the Poisson
structure on symmetric algebra leading to the quantized enveloping
algebra as an example of biquantization in the sense of Turaev.
Description of in terms of the generators of the bicovariant
differential calculus on is very convenient for this purpose. Finally
we interpret in the deformation framework some well known properties of compact
quantum groups as simple consequences of corresponding properties of classical
compact Lie groups. An analogue of the classical Kirillov's universal character
formula is given for the unitary irreducible representation in the compact
case.Comment: 18 page
On the 3-particle scattering continuum in quasi one dimensional integer spin Heisenberg magnets
We analyse the three-particle scattering continuum in quasi one dimensional
integer spin Heisenberg antiferromagnets within a low-energy effective field
theory framework. We exactly determine the zero temperature dynamical structure
factor in the O(3) nonlinear sigma model and in Tsvelik's Majorana fermion
theory. We study the effects of interchain coupling in a Random Phase
Approximation. We discuss the application of our results to recent
neutron-scattering experiments on the Haldane-gap material .Comment: 8 pages of revtex, 5 figures, small changes, to appear in PR
Stringing Spins and Spinning Strings
We apply recently developed integrable spin chain and dilatation operator
techniques in order to compute the planar one-loop anomalous dimensions for
certain operators containing a large number of scalar fields in N =4 Super
Yang-Mills. The first set of operators, belonging to the SO(6) representations
[J,L-2J,J], interpolate smoothly between the BMN case of two impurities (J=2)
and the extreme case where the number of impurities equals half the total
number of fields (J=L/2). The result for this particular [J,0,J] operator is
smaller than the anomalous dimension derived by Frolov and Tseytlin
[hep-th/0304255] for a semiclassical string configuration which is the dual of
a gauge invariant operator in the same representation. We then identify a
second set of operators which also belong to [J,L-2J,J] representations, but
which do not have a BMN limit. In this case the anomalous dimension of the
[J,0,J] operator does match the Frolov-Tseytlin prediction. We also show that
the fluctuation spectra for this [J,0,J] operator is consistent with the string
prediction.Comment: 27 pages, 4 figures, LaTex; v2 reference added, typos fixe
Coherent States for Quantum Compact Groups
Coherent states are introduced and their properties are discussed for all
simple quantum compact groups. The multiplicative form of the canonical element
for the quantum double is used to introduce the holomorphic coordinates on a
general quantum dressing orbit and interpret the coherent state as a
holomorphic function on this orbit with values in the carrier Hilbert space of
an irreducible representation of the corresponding quantized enveloping
algebra. Using Gauss decomposition, the commutation relations for the
holomorphic coordinates on the dressing orbit are derived explicitly and given
in a compact R--matrix formulation (generalizing this way the --deformed
Grassmann and flag manifolds). The antiholomorphic realization of the
irreducible representations of a compact quantum group (the analogue of the
Borel--Weil construction) are described using the concept of coherent state.
The relation between representation theory and non--commutative differential
geometry is suggested.}Comment: 25 page
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