116 research outputs found
Geometry of q-Hypergeometric Functions, Quantum Affine Algebras and Elliptic Quantum Groups
The trigonometric quantized Knizhnik-Zamolodchikov equation (qKZ equation) associated with the quantum group is a system of linear difference equations with values in a tensor product of Verma modules. We solve the equation in terms of multidimensional -hypergeometric functions and define a natural isomorphism between the space of solutions and the tensor product of the corresponding evaluation Verma modules over the elliptic quantum group , where parameters and are related to the parameter of the quantum group and the step of the qKZ equation via p=e^{2\pii\rho} and q=e^{-2\pii\gamma}. We construct asymptotic solutions associated with suitable asymptotic zones and compute the transition functions between the asymptotic solutions in terms of the dynamical elliptic -matrices. This description of the transition functions gives a connection between representation theories of the quantum loop algebra and the elliptic quantum group and is analogous to the Kohno-Drinfeld theorem on the monodromy group of the differential Knizhnik-Zamolodchikov equation. In order to establish these results we construct a discrete Gauss-Manin connection, in particular, a suitable discrete local system, discrete homology and cohomology groups with coefficients in this local system, and identify an associated difference equation with the qKZ equation
Spaces of quasi-exponentials and representations of gl_N
We consider the action of the Bethe algebra B_K on (\otimes_{s=1}^k
L_{\lambda^{(s)}})_\lambda, the weight subspace of weight of the
tensor product of k polynomial irreducible gl_N-modules with highest weights
\lambda^{(1)},...,\lambda^{(k)}, respectively. The Bethe algebra depends on N
complex numbers K=(K_1,...,K_N). Under the assumption that K_1,...,K_N are
distinct, we prove that the image of B_K in the endomorphisms of
(\otimes_{s=1}^k L_{\lambda^{(s)}})_\lambda is isomorphic to the algebra of
functions on the intersection of k suitable Schubert cycles in the Grassmannian
of N-dimensional spaces of quasi-exponentials with exponents K. We also prove
that the B_K-module (\otimes_{s=1}^k L_{\lambda^{(s)}})_\lambda is isomorphic
to the coregular representation of that algebra of functions. We present a
Bethe ansatz construction identifying the eigenvectors of the Bethe algebra
with points of that intersection of Schubert cycles.Comment: Latex, 29 page
Bethe eigenvectors of higher transfer matrices
We consider the XXX-type and Gaudin quantum integrable models associated with
the Lie algebra . The models are defined on a tensor product irreducible
-modules. For each model, there exist one-parameter families of
commuting operators on the tensor product, called the transfer matrices. We
show that the Bethe vectors for these models, given by the algebraic nested
Bethe ansatz are eigenvectors of higher transfer matrices and compute the
corresponding eigenvalues.Comment: 48 pages, amstex.tex (ver 2.2), misprints correcte
Defect and Hodge numbers of hypersurfaces
We define defect for hypersurfaces with A-D-E singularities in complex
projective normal Cohen-Macaulay fourfolds having some vanishing properties of
Bott-type and prove formulae for Hodge numbers of big resolutions of such
hypersurfaces. We compute Hodge numbers of Calabi-Yau manifolds obtained as
small resolutions of cuspidal triple sextics and double octics with higher A_j
singularities.Comment: 25 page
On algebraic equations satisfied by hypergeometric correlators in WZW models. II
We give an explicit description of "bundles of conformal blocks" in
Wess-Zumino-Witten models of Conformal field theory and prove that integral
representations of Knizhnik-Zamolodchikov equations constructed earlier by the
second and third authors are in fact sections of these bundles.Comment: 32 pp., amslate
Highest coefficient of scalar products in SU(3)-invariant integrable models
We study SU(3)-invariant integrable models solvable by nested algebraic Bethe
ansatz. Scalar products of Bethe vectors in such models can be expressed in
terms of a bilinear combination of their highest coefficients. We obtain
various different representations for the highest coefficient in terms of sums
over partitions. We also obtain multiple integral representations for the
highest coefficient.Comment: 17 page
On the Bethe Ansatz for the Jaynes-Cummings-Gaudin model
We investigate the quantum Jaynes-Cummings model - a particular case of the
Gaudin model with one of the spins being infinite. Starting from the Bethe
equations we derive Baxter's equation and from it a closed set of equations for
the eigenvalues of the commuting Hamiltonians. A scalar product in the
separated variables representation is found for which the commuting
Hamiltonians are Hermitian. In the semi classical limit the Bethe roots
accumulate on very specific curves in the complex plane. We give the equation
of these curves. They build up a system of cuts modeling the spectral curve as
a two sheeted cover of the complex plane. Finally, we extend some of these
results to the XXX Heisenberg spin chain.Comment: 16 page
Laplacian Growth, Elliptic Growth, and Singularities of the Schwarz Potential
The Schwarz function has played an elegant role in understanding and in
generating new examples of exact solutions to the Laplacian growth (or "Hele-
Shaw") problem in the plane. The guiding principle in this connection is the
fact that "non-physical" singularities in the "oil domain" of the Schwarz
function are stationary, and the "physical" singularities obey simple dynamics.
We give an elementary proof that the same holds in any number of dimensions for
the Schwarz potential, introduced by D. Khavinson and H. S. Shapiro [17]
(1989). A generalization is also given for the so-called "elliptic growth"
problem by defining a generalized Schwarz potential. New exact solutions are
constructed, and we solve inverse problems of describing the driving
singularities of a given flow. We demonstrate, by example, how \mathbb{C}^n -
techniques can be used to locate the singularity set of the Schwarz potential.
One of our methods is to prolong available local extension theorems by
constructing "globalizing families". We make three conjectures in potential
theory relating to our investigation
Gaudin model and its associated Knizhnik-Zamolodchikov equation
The semiclassical limit of the algebraic Bethe Ansatz for the Izergin-Korepin
19-vertex model is used to solve the theory of Gaudin models associated with
the twisted R-matrix. We find the spectra and eigenvectors of the
independents Gaudin Hamiltonians. We also use the off-shell Bethe Ansatz
method to show how the off-shell Gaudin equation solves the associated
trigonometric system of Knizhnik-Zamolodchikov equations.Comment: 20 pages,no figure, typos corrected, LaTe
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