95 research outputs found
Yang's system of particles and Hecke algebras
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On Nonzero Kronecker Coefficients and their Consequences for Spectra
A triple of spectra (r^A, r^B, r^{AB}) is said to be admissible if there is a
density operator rho^{AB} with (Spec rho^A, Spec rho^B, Spec rho^{AB})=(r^A,
r^B, r^{AB}). How can we characterise such triples? It turns out that the
admissible spectral triples correspond to Young diagrams (mu, nu, lambda) with
nonzero Kronecker coefficient [M. Christandl and G. Mitchison, to appear in
Comm. Math. Phys., quant-ph/0409016; A. Klyachko, quant-ph/0409113]. This means
that the irreducible representation V_lambda is contained in the tensor product
of V_mu and V_nu. Here, we show that such triples form a finitely generated
semigroup, thereby resolving a conjecture of Klyachko. As a consequence we are
able to obtain stronger results than in [M. Ch. and G. M. op. cit.] and give a
complete information-theoretic proof of the correspondence between triples of
spectra and representations. Finally, we show that spectral triples form a
convex polytope.Comment: 13 page
Higgs Bundles, Gauge Theories and Quantum Groups
The appearance of the Bethe Ansatz equation for the Nonlinear Schr\"{o}dinger
equation in the equivariant integration over the moduli space of Higgs bundles
is revisited. We argue that the wave functions of the corresponding
two-dimensional topological U(N) gauge theory reproduce quantum wave functions
of the Nonlinear Schr\"{o}dinger equation in the -particle sector. This
implies the full equivalence between the above gauge theory and the
-particle sub-sector of the quantum theory of Nonlinear Schr\"{o}dinger
equation. This also implies the explicit correspondence between the gauge
theory and the representation theory of degenerate double affine Hecke algebra.
We propose similar construction based on the gauged WZW model leading to
the representation theory of the double affine Hecke algebra. The relation with
the Nahm transform and the geometric Langlands correspondence is briefly
discussed.Comment: 48 pages, typos corrected, one reference adde
Two ways to solve ASEP
The purpose of this article is to describe the two approaches to compute
exact formulas (which are amenable to asymptotic analysis) for the probability
distribution of the current of particles past a given site in the asymmetric
simple exclusion process (ASEP) with step initial data. The first approach is
via a variant of the coordinate Bethe ansatz and was developed in work of Tracy
and Widom in 2008-2009, while the second approach is via a rigorous version of
the replica trick and was developed in work of Borodin, Sasamoto and the author
in 2012.Comment: 10 pages, Chapter in "Topics in percolative and disordered systems
Integral Representations of the Macdonald Symmetric Functions
Multiple-integral representations of the (skew-)Macdonald symmetric functions
are obtained. Some bosonization schemes for the integral representations are
also constructed.Comment: LaTex 21page
The Bose Gas and Asymmetric Simple Exclusion Process on the Half-Line
In this paper we find explicit formulas for: (1) Green's function for a
system of one-dimensional bosons interacting via a delta-function potential
with particles confined to the positive half-line; and (2) the transition
probability for the one-dimensional asymmetric simple exclusion process (ASEP)
with particles confined to the nonnegative integers. These are both for systems
with a finite number of particles. The formulas are analogous to ones obtained
earlier for the Bose gas and ASEP on the line and integers, respectively. We
use coordinate Bethe Ansatz appropriately modified to account for confinement
of the particles to the half-line. As in the earlier work, the proof for the
ASEP is less straightforward than for the Bose gas.Comment: 14 Page
Generalized Calogero-Moser systems from rational Cherednik algebras
We consider ideals of polynomials vanishing on the W-orbits of the
intersections of mirrors of a finite reflection group W. We determine all such
ideals which are invariant under the action of the corresponding rational
Cherednik algebra hence form submodules in the polynomial module. We show that
a quantum integrable system can be defined for every such ideal for a real
reflection group W. This leads to known and new integrable systems of
Calogero-Moser type which we explicitly specify. In the case of classical
Coxeter groups we also obtain generalized Calogero-Moser systems with added
quadratic potential.Comment: 36 pages; the main change is an improvement of section 7 so that it
now deals with an arbitrary complex reflection group; Selecta Math, 201
Active Galaxies in the UV
In this article we present different aspects of AGN studies demonstrating the
importance of the UV spectral range. Most important diagnostic lines for
studying the general physical conditions as well as the metalicities in the
central broad line region in AGN are emitted in the UV. The UV/FUV continuum in
AGN excites not only the emission lines in the immediate surrounding but it is
responsible for the ionization of the intergalactic medium in the early stages
of the universe. Variability studies of the emission line profiles of AGN in
the UV give us information on the structure and kinematics of the immediate
surrounding of the central supermassive black hole as well as on its mass
itself.Comment: 29 pages, 13 figures, Ap&SS in pres
The Seyfert-LINER Galaxy NGC 7213: An XMM-Newton Observation
We examine the XMM X-ray spectrum of the LINER-AGN NGC 7213, which is best
fit with a power law, K-alpha emission lines from Fe I, Fe XXV and Fe XXVI and
a soft X-ray collisionally ionised thermal plasma with kT=0.18 +0.03/-0.01 keV.
We find a luminosity of 7x10^(-4) L_Edd, and a lack of soft X-ray excess
emission, suggesting a truncated accretion disc. NGC 7213 has intermediate
X-ray spectral properties, between those of the weak AGN found in the LINER M81
and higher luminosity Seyfert galaxies. This supports the notion of a
continuous sequence of X-ray properties from the Galactic Centre through LINER
galaxies to Seyferts, likely determined by the amount of material available for
accretion in the central regions.Comment: 7 pages, 2 figures. To appear in From X-ray Binaries to Quasars:
Black Hole Accretion on All Mass Scales, ed. T. J. Maccarone, R. P. Fender,
and L. C. Ho (Dordrecht: Kluwer
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