Representations of W-algebras and Frenkel-Kac-Wakimoto conjecture
abstract. We study the representations of W-algebras via the quantized reduction introduced by Feigin-Frenkel. The conjecture of Frenkel-Kac-Wakimoto is (partially) proved as a consequence.
TBA
TBA
qKZ and correlation functions: Algebraic way
Double Affine Hecke Algebras and Verlinde Algebras
abstract. Introduced 13 years ago, DAHA play now a solid role in modern representation theory with wide spectrum of applications. I will begin with general motivation and the definition in the rank one case. The connections to Verlinde algebras will be discussed, as well as recent results on the diagonal coinvariants (the combinatorics of two sets of variables).
Shubert varieties and the fusion products
abstract. We study the fusion product (the deformation of the tensor
product) of
modules. Shubert variety is the closure
of the
![$SL_2(\mathbb{C}[t])$](img4.gif){kind=small}
-orbit of the lowest weight vector in the
projectivization of the fusion product. We study algebro-geometric
properties of the Shubert varieties. For example, we prove that
the fusion product can be reconstructed as a dual space of the sections of
some line bundle on the Shubert variety.
TBA
Zeros of Symmetric Laurent Polynomials of Type
and Self-dual Koornwinder-Macdonald Polynomials Specialized at
abstract. A characterization of the space of
symmetric Laurent polynomials of type which vanish
on a certain set of submanifolds is given by using the self-dual
Koornwinder-Macdonald polynomials. A similar characterization was
given previously for symmetric polynomials of type 
by using the Macdonald polynomials.
We use a new method which exploits the duality relation. The method
simplifies a part of the proof in the case.
Symmetric coinvariant algebra and local Weyl module at a double point
abstract. The symmetric coinvariant algebra
![${\mathbb{C}}[x_1, \dots, x_n]_{S_n}$](img8.gif){kind=small}
is the quotient algebra of the polynomial ring by
the ideal generated by symmetric polynomials vanishing at the
origin. Classically, it is known that the algebra is isomorphic to
the regular representation of 
.
Replacing
![${\mathbb{C}}[x]$](img10.gif){kind=small}
with
![$A = {\mathbb{C}}[x,y]/(xy)$](img11.gif){kind=small}
, we introduce another
symmetric coinvariant algebra
and
determine its -module structure. As a corollary, we calculate
the dimension of the local Weyl module at a double point for
.
Representation theory and quantum integrability
Weyl modules over two-dimensional currents
abstract. Talk is based on joint papers of Boris Feigin and the
speaker. We discuss Weyl modules in the sense of V.Chari and A.Pressley
over the Lie algebra of currents in two variables with values in
. The goal of the talk is to describe how these modules are
related to space of diagonal coinvariants (in the sense of M.Haiman) and
parking functions.
Integrable boundary flows: New face of Baxter's ideas
abstract.
In this talk the following two issues will be discussed:
-Boundary RG flows as
a dynamic framework in the problems of Dissipative Quantum Mechanics
and open String Theory;
-A relation between the integrable boundary states
and Baxter's families of commuting
transfer matrixes.
N. KITANINE (Univ. of Cergy-Pontoise) and
V. TERRAS (Univ. Montpellier II)
Correlation functions of the XXZ Heisenberg chain : recent advances
abstract. We review recent progress in the computation of correlation functions of the XXZ spin-1/2 chain and their long distance asymptotic behavior. Our method is based on the resolution of the quantum inverse scattering problem in the framework of the algebraic Bethe ansatz. It leads to multiple integrals representations of the correlation functions in the context of which we discuss their long distance asymptotic behavior.
Conformal field theories associated to vertex operator operator algebras (JOINT WORK WITH A.TSUCHIYA AND A.MATSUO)
The space of local fields as a module over the ring of local integrals of motion
abstract. The space of local fields of the SU(2)-invariant Thirring model as a module over the ring of local integrals of motion is studied. We construct a free resolution of it in terms of the Fock space of free fermions.
Boundary Liouville field theory and 2D quantum gravity
abstract. Liouville field theory on the disc with Neumann conditions on the Liouville field is considered. We show that the functional equations satisfied by the diverse structure constants (boundary two point function, bulk-boundary function, boundary three point function) have identical structure.
Then we show that the microscopic realization of 2D Quantum Gravity as a
height model on a random surface leads to similar equations. We conclude
that the discrete approach is able to reproduce the dependence on the
boundary parameters of the correlation functions of Liouville theory.
JOINT WORK WITH KOSTOV AND SERBAN.
On local operators in perturbed CFT
abstract. We discuss some new results on local operators in the integrable perturbations of the minimal CFT in the form factor and conformal perturbation theory approaches. We argue that Zamolodchikov-Zamolodchikov relfection relations may bring useful information on the structure of space of local operators in the form-factor desription.
Algebras
abstract. We construct W-algebra generalizations of the
algebra--W algebras generated by two currents 
and
with the highest pole of order 
in their OPE. The 
term in this series is the Bershadsky-Polyakov 
algebra.
We define these algebras as a centralizer (commutant) of the
super quantum group and explicitly construct the
generators via a ``Miura-like'' construction. Another construction of
the 
algebras is in terms of the coset
. The relation between the
two constructions involves the ``duality''
between levels 
and
of two
algebras.
JOINT WORK WITH B. FEIGIN.
TBA
abstract. Q-operator is a one- or multiparametric family of operators sharing the eigenfunctions with commuting Hamiltonians of a given quantum integrable system. The eigenvalues of Q-operator satisfy a differential or difference equation which allows to determine its spectrum. We give a review of applications of Q-operators. In particular, we show that knowing a Q-operator allows to construct a generic separation of variables for the integrable system in question. Our main illustrative example is the Calogero-Sutherland model.
Connecting lattice and relativistic models via conformal field theory
abstract. We consider the quantum group invariant XXZ-model. In infrared limit it describes Conformal Field Theory (CFT) with modified energy-momentum tensor. The correlation functions are related to solutions of level -4 of qKZ equations. We describe these solutions relating them to level 0 solutions. We further consider general matrix elements (form factors) containing local operators and asymptotic states. We explain that the formulae for solutions of qKZ equations suggest a decomposition of these matrix elements with respect to states of corresponding CFT.
TBA
Correlation function of spin
XXZ chain
A monomial basis for the Virasoro minimal series
:
the case 
abstract. We present a basis of the Virasoro minimal series
in the case of .
It is given by a certain set of monomials of
Fourier components of the 
-primary field.
Our basis has good correspondence to
a combinatorial formula of the character of .
This is a joint work with B. Feigin, M. Jimbo, T. Miwa and E. Mukhin.
Nonsemisimple Fusion Algebras and the Verlinde Formula
abstract. A nonsemisimple fusion algebra
associated with
each 
Virasoro model is found. A nonsemisimple
generalization of the Verlinde formula which allows us to derive
from modular transformations of characters is
presented.
TBA