11 research outputs found
Representations of affine Lie algebras, elliptic r-matrix systems, and special functions
There were some errors in paper hep-th/9303018 in formulas 6.1, 6.6, 6.8,
6.11. These errors have been corrected in the present version of this paper.
There are also some minor changes in the introduction.Comment: 33 pages, no figure
Geometry of q-Hypergeometric Functions as a Bridge between Yangians and Quantum Affine Algebras
The rational quantized Knizhnik-Zamolodchikov equation (qKZ equation)
associated with the Lie algebra 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 quantum group Verma modules, where the parameter
is related to the step of the qKZ equation via .
We construct asymptotic solutions associated with suitable asymptotic zones
and compute the transition functions between the asymptotic solutions in terms
of the trigonometric -matrices. This description of the transition functions
gives a new connection between representation theories of Yangians and quantum
loop algebras 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.Comment: 66 pages, amstex.tex (ver. 2.1) and amssym.tex are required;
misprints are correcte
New Formula for the Eigenvectors of the Gaudin Model in the sl(3) Case
We propose new formulas for eigenvectors of the Gaudin model in the \sl(3)
case. The central point of the construction is the explicit form of some
operator P, which is used for derivation of eigenvalues given by the formula , where , fulfil
the standard well-know Bethe Ansatz equations
SU(2) WZW Theory at Higher Genera
We compute, by free field techniques, the scalar product of the SU(2)
Chern-Simons states on genus > 1 surfaces. The result is a finite-dimensional
integral over positions of ``screening charges'' and one complex modular
parameter. It uses an effective description of the CS states closely related to
the one worked out by Bertram. The scalar product formula allows to express the
higher genus partition functions of the WZW conformal field theory by
finite-dimensional integrals. It should provide the hermitian metric preserved
by the Knizhnik-Zamolodchikov-Bernard connection describing the variations of
the CS states under the change of the complex structure of the surface.Comment: 44 pages, IHES/P/94/10, Latex fil
On operad structures of moduli spaces and string theory
Recent algebraic structures of string theory, including homotopy Lie
algebras, gravity algebras and Batalin-Vilkovisky algebras, are deduced from
the topology of the moduli spaces of punctured Riemann spheres. The principal
reason for these structures to appear is as simple as the following. A
conformal field theory is an algebra over the operad of punctured Riemann
surfaces, this operad gives rise to certain standard operads governing the
three kinds of algebras, and that yields the structures of such algebras on the
(physical) state space naturally.Comment: 33 pages (An elaboration of minimal area metrics and new references
are added
Boundary quantum Knizhnik-Zamolodchikov equations and Bethe vectors
Solutions to boundary quantum Knizhnik-Zamolodchikov equations are constructed as bilateral sums involving "off-shell" Bethe vectors in case the reflection matrix is diagonal and only the 2-dimensional representation of is involved. We also consider their rational and classical degenerations
Estimating the locations and number of change points by the sample-splitting method
Multiple Structural Changes, Change-Point Estimator, Brownian Motion, Bessel Process,