83 research outputs found
A remark on deformations of Hurwitz Frobenius manifolds
In this note we use the formalism of multi-KP hierarchies in order to give
some general formulas for infinitesimal deformations of solutions of the
Darboux-Egoroff system. As an application, we explain how Shramchenko's
deformations of Frobenius manifold structures on Hurwitz spaces fit into the
general formalism of Givental-van de Leur twisted loop group action on the
space of semi-simple Frobenius manifolds.Comment: 10 page
--Geometry and Associated Continuous Toda System
We discuss an infinite--dimensional k\"ahlerian manifold associated with the
area--preserving diffeomorphisms on two--dimensional torus, and,
correspondingly, with a continuous limit of the --Toda system. In
particular, a continuous limit of the --Grassmannians and a related
Pl\"ucker type formula are introduced as relevant notions for
--geometry of the self--dual Einstein space with the rotational
Killing vector.Comment: 6 pages, no figure report\# ETH-TH/93-2
Modules of Abelian integrals and Picard-Fuchs systems
We give a simple proof of an isomorphism between the two
-modules: the module of relative cohomologies and the module of Abelian integrals corresponding to a regular at
infinity polynomial in two variables. Using this isomorphism, we prove
existence and deduce some properties of the corresponding Picard-Fuchs system.Comment: A separate section discusses Fuchsian properties of the Picard-Fuchs
system, Morse condition exterminated. Few errors were correcte
Transition function for the Toda chain model
The method of Lambda-operators developed by S. Derkachov, G. Korchemsky, A.
Manashov is applied to a derivation of eigenfunctions for the open Toda chain.
The Sklyanin measure is reproduced using diagram technique developed for these
Lambda-operators. The properties of the Lambda-operators are studied. This
approach to the open Toda chain eigenfunctions reproduces Gauss-Givental
representation for these eigenfunctions
From Hurwitz numbers to Kontsevich-Witten tau-function: a connection by Virasoro operators
In this letter,we present our conjecture on the connection between the
Kontsevich--Witten and the Hurwitz tau-functions. The conjectural formula
connects these two tau-functions by means of the group element. An
important feature of this group element is its simplicity: this is a group
element of the Virasoro subalgebra of . If proved, this conjecture
would allow to derive the Virasoro constraints for the Hurwitz tau-function,
which remain unknown in spite of existence of several matrix model
representations, as well as to give an integrable operator description of the
Kontsevich--Witten tau-function.Comment: 13 page
Homology class of a Lagrangian Klein bottle
It is shown that an embedded Lagrangian Klein bottle represents a non-trivial
mod 2 homology class in a compact symplectic four-manifold with
. (In versions 1 and 2, the last assumption was missing.
A counterexample to this general claim and the first proof of the corrected
result have been found by Vsevolod Shevchishin.) As a corollary one obtains
that the Klein bottle does not admit a Lagrangian embedding into the standard
symplectic four-space.Comment: Version 3 - completely rewritten to correct a mistake; Version 4 -
minor edits, added references; AMSLaTeX, 6 page
--geometry of the Toda systems associated with non-exceptional simple Lie algebras
The present paper describes the --geometry of the Abelian finite
non-periodic (conformal) Toda systems associated with the and series
of the simple Lie algebras endowed with the canonical gradation. The principal
tool here is a generalization of the classical Pl\"ucker embedding of the
-case to the flag manifolds associated with the fundamental representations
of , and , and a direct proof that the corresponding K\"ahler
potentials satisfy the system of two--dimensional finite non-periodic
(conformal) Toda equations. It is shown that the --geometry of the type
mentioned above coincide with the differential geometry of special holomorphic
(W) surfaces in target spaces which are submanifolds (quadrics) of with
appropriate choices of . In addition, these W-surfaces are defined to
satisfy quadratic holomorphic differential conditions that ensure consistency
of the generalized Pl\"ucker embedding. These conditions are automatically
fulfiled when Toda equations hold.Comment: 30 pages, no figur
A note on quantization operators on Nichols algebra model for Schubert calculus on Weyl groups
We give a description of the (small) quantum cohomology ring of the flag
variety as a certain commutative subalgebra in the tensor product of the
Nichols algebras. Our main result can be considered as a quantum analog of a
result by Y. Bazlov
Mirror symmetry and quantization of abelian varieties
The paper consists of two sections. The first section provides a new
definition of mirror symmetry of abelian varieties making sense also over
-adic fields. The second section introduces and studies quantized
theta-functions with two-sided multipliers, which are functions on
non-commutative tori. This is an extension of an earlier work by the author. In
the Introduction and in the Appendix the constructions of this paper are put
into a wider context.Comment: 24 pp., amstex file, no figure
Quantum deformations of associative algebras and integrable systems
Quantum deformations of the structure constants for a class of associative
noncommutative algebras are studied. It is shown that these deformations are
governed by the quantum central systems which has a geometrical meaning of
vanishing Riemann curvature tensor for Christoffel symbols identified with the
structure constants. A subclass of isoassociative quantum deformations is
described by the oriented associativity equation and, in particular, by the
WDVV equation. It is demonstrated that a wider class of weakly (non)associative
quantum deformations is connected with the integrable soliton equations too. In
particular, such deformations for the three-dimensional and
infinite-dimensional algebras are described by the Boussinesq equation and KP
hierarchy, respectively.Comment: Numeration of the formulas is correcte
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