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Local geometry of bihamiltonian structures and invariant volume forms
Cornell University
,
2013.
Izosimov A.
It is shown that a generic bihamiltonian structure on an odd-dimensional manifold is flat if and only if it is locally unimodular.
Language:
English
Izosimov A., Differential Geometry and its Application 2013 Vol. 31 P. 557-567
A Poisson pencil is called flat if all brackets of the pencil can be simultaneously locally brought to a constant form. Given a Poisson pencil on a 3-manifold, we study under which conditions it is flat. Since the works of Gelfand and Zakharevich, it is known that a pencil is flat if and only if ...
Added: November 18, 2013
Aminov S., Arthamonov S., A. Levin et al., / Cornell University. Series math "arxiv.org". 2013.
We propose multidimensional versions of the Painleve VI equation and its degenerations. These field theories are related to the isomonodromy problems on flat holomorphic infinite rank bundles over elliptic curves and take the form of non-autonomous Hamiltonian equations. The modular parameter of curves plays the role of "time". Reduction of the field equations to the ...
Added: December 27, 2013
Grekov A., A. Zabrodin, A. Zotov, Nuclear Physics B 2019 Vol. 939 P. 174-190
We describe the correspondence of the Matsuo-Cherednik type between the quantum nn -body Ruijsenaars-Schneider model and the quantum Knizhnik-Zamolodchikov equations related to supergroup GL(N|M)GL(N|M) . The spectrum of the Ruijsenaars-Schneider Hamiltonians is shown to be independent of the {\mathbb Z}_2 -grading for a fixed value of N+M , so that N+M+1 different qKZ systems of ...
Added: May 24, 2019
Levin A., Olshanetsky M., Zotov A., / Cornell University. Series math "arxiv.org". 2014.
We construct special rational ${\rm gl}_N$ Knizhnik-Zamolodchikov-Bernard
(KZB) equations with $\tilde N$ punctures by deformation of the corresponding
quantum ${\rm gl}_N$ rational $R$-matrix. They have two parameters. The limit
of the first one brings the model to the ordinary rational KZ equation. Another
one is $\tau$. At the level of classical mechanics the deformation parameter
$\tau$ allows to extend the ...
Added: January 23, 2015
Kazaryan M., Uribe-Vargas R., Moscow Mathematical Journal 2020 Vol. 20 No. 3 P. 511-530
We define local indices for projective umbilics and godrons (also called cusps of Gauss) on generic smooth surfaces in projective 3-space. By means of these indices, we provide formulas that relate the algebraic numbers of those characteristic points on a surface (and on domains of the surface) with the Euler characteristic of that surface (resp. ...
Added: August 24, 2020
Kurnosov N., / Cornell University. Series math "arxiv.org". 2015.
We prove that a generic complex deformation of a generalized Kummer variety contains no complex analytic tori. ...
Added: October 16, 2015
Ivan Cheltsov, Martinez-Garcia J., / Cornell University. Series math "arxiv.org". 2014.
For every smooth del Pezzo surface $S$, smooth curve $C\in|-K_{S}|$ and $\beta\in(0,1]$, we compute the $\alpha$-invariant of Tian $\alpha(S,(1-\beta)C)$ and prove the existence of K\"ahler--Einstein metrics on $S$ with edge singularities along $C$ of angle $2\pi\beta$ for $\beta$ in certain interval. In particular we give lower bounds for the invariant $R(S,C)$, introduced by Donaldson as ...
Added: February 5, 2015
Khoroshkin S. M., Shapiro A., Journal of Geometry and Physics 2010 Vol. 60 No. 11 P. 1833-1851
In this article, we give an explicit formula for the universal weight function of the quantum twisted affine algebra Uq(A(2)2 ). The calculations use the technique of projecting products of Drinfeld currents onto the intersection of Borel subalgebras of different types. ...
Added: September 26, 2012
Kamenova L., Lu S., Verbitsky M., / Cornell University. Series math "arxiv.org". 2013.
The Kobayashi pseudometric on a complex manifold $M$ is the maximal pseudometric such that any holomorphic map from the Poincare disk to $M$ is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this result for any hyperkaehler manifold if it admits a deformation with a ...
Added: August 28, 2013
Bolsinov A., Izosimov A., / Cornell University. Series math "arxiv.org". 2013.
We study the relationship between singularities of bi-Hamiltonian systems and algebraic properties of compatible Poisson brackets. As the main tool, we introduce the notion of linearization of a Poisson pencil. From the algebraic viewpoint, a linearized Poisson pencil can be understood as a Lie algebra with a fixed 2-cocycle. In terms of such linearizations, we ...
Added: November 19, 2013
Izosimov A., / Cornell University. Series math "arxiv.org". 2013.
In 1970s, a method was developed for integration of nonlinear equations by means of algebraic geometry. Starting from a Lax representation with a spectral parameter, the algebro-geometric method allows to solve the system explicitly in terms of Theta functions of Riemann surfaces. However, the explicit formulas obtained in this way fail to answer such natural ...
Added: November 19, 2013
Levin A., Olshanetsky M., Zotov A., / Cornell University. Series math "arxiv.org". 2014.
e describe classical top-like integrable systems arising from the quantum
exchange relations and corresponding Sklyanin algebras. The Lax operator is
expressed in terms of the quantum non-dynamical $R$-matrix even at the
classical level, where the Planck constant plays the role of the relativistic
deformation parameter in the sense of Ruijsenaars and Schneider (RS). The
integrable systems (relativistic tops) are described ...
Added: January 23, 2015
Verbitsky M., / Cornell University. Series math "arxiv.org". 2013.
Let M be a hyperkaehler manifold, and η a closed, positive (1,1)-form which is degenerate everywhere on M. We associate to η a family of complex structures on M, called a degenerate twistor family, and parametrized by a complex line. When η is a pullback of a Kaehler form under a Lagrangian fibration L, all ...
Added: December 27, 2013
Izosimov A., Journal of Geometry and Physics 2012 Vol. 62 No. 12 P. 2414-2423
The presence of two compatible hamiltonian structures is known to be one of the main, and the most natural, mechanisms of integrability. For every pair of hamiltonian structures, there are associated conservation laws (first integrals). Another approach is to consider the second hamiltonian structure on its own as a tensor conservation law. The latter is ...
Added: November 18, 2013
Zotov A., Atalikov K., JETP Letters 2022 Vol. 115 No. 12 P. 809-810
We propose a construction of 1 + 1 integrable Heisenberg–Landau–Lifshitz type equations in the glN case. The dynamical variables are matrix elements of N × N matrix S with the property S2 = const · S. The Lax pair with spectral parameter is constructed by means of a quantum R-matrix satisfying the associative Yang–Baxter equation. Equations of motion for glN Landau–Lifshitz model are derived from the Zakharov–Shabat ...
Added: June 20, 2022
Pushkar P. E., / Cornell University. Series arXiv "math". 2016. No. arXiv:1602.07948.
We construct counterexamples to lifting properties of Hamiltonian and contact isotopies ...
Added: December 7, 2016
A. Levin, Olshanetsky M., Zotov A., / Cornell University. Series math "arxiv.org". 2013.
We consider the isomonodromy problems for flat $G$-bundles over punctured
elliptic curves $\Sigma_\tau$ with regular singularities of connections at
marked points. The bundles are classified by their characteristic classes.
These classes are elements of the second cohomology group
$H^2(\Sigma_\tau,{\mathcal Z}(G))$, where ${\mathcal Z}(G)$ is the center of
$G$. For any complex simple Lie group $G$ and arbitrary class we define ...
Added: December 27, 2013
Verbitsky M., Grantcharov G., Lejmi M., / Cornell University. Series math "arxiv.org". 2014.
A hypercomplex manifold M is a manifold equipped with three complex structures satisfying quaternionic relations. Such a manifold admits a canonical torsion-free connection preserving the quaternion action, called Obata connection. A quaternionic Hermitian metric is a Riemannian metric on which is invariant with respect to unitary quaternions. Such a metric is called HKT if it ...
Added: September 19, 2014
Pushkar P. E., / Cornell University. Series arXiv "math". 2016. No. arXiv:1602.08743.
We prove a Chekanov-type theorem for the spherization of the cotangent bundle ST∗B of a closed manifold B. It claims that for Legendrian submanifolds in ST∗B the property "to be given by a generating family quadratic at infinity" persists under Legendrian isotopies. ...
Added: December 7, 2016
Gavrylenko P., Iorgov N., Lisovyy O., Journal of Physics A: Mathematical and Theoretical 2012 Vol. 45 No. 2 P. 025402
We study U(1) twist fields in a two-dimensional lattice theory of massive Dirac fermions. Factorized formulas for finite-lattice form factors of these fields are derived using elliptic parametrization of the spectral curve of the model, elliptic determinant identities and theta functional interpolation. We also investigate the thermodynamic and infinite-volume scaling limit, where the corresponding expressions ...
Added: October 20, 2014
Déev R. N., / Cornell University. Series arXiv "math". 2016.
Essential dimension of a family of complex manifolds is the dimension of the image of its base in the Kuranishi space of the fiber. We prove that any family of hyperk\"ahler manifolds over a compact simply connected base has essential dimension not greater than 1. A similar result about families of complex tori is also ...
Added: September 23, 2016
Derbyshev A. E., Poghosyan S. S., Povolotsky A. M. et al., Journal of Statistical Mechanics: Theory and Experiment 2012 Vol. P05014 P. 1-13
We consider the totally asymmetric exclusion process in discrete time with generalized updating rules. We introduce a control parameter into the interaction between particles. Two particular values of the parameter correspond to known parallel and sequential updates. In the whole range of its values the interaction varies from repulsive to attractive. In the latter case ...
Added: February 12, 2013
Miolane N., Caorsi M., Lupo U. et al., / Cornell University. Series CS "arxiv.org". 2021. No. 2108.09810.
This paper presents the computational challenge on differential geometry and topology that happened within the ICLR 2021 workshop "Geometric and Topological Representation Learning". The competition asked participants to provide creative contributions to the fields of computational geometry and topology through the open-source repositories Geomstats and Giotto-TDA. The challenge attracted 16 teams in its two month ...
Added: October 16, 2021
A. Levin, Olshanetsky M., Zotov A., Nuclear Physics B 2014 Vol. 887 P. 400-422
In our recent paper we suggested a natural construction of the classical relativistic integrable tops in terms of the quantum R -matrices. Here we study the simplest case – the 11-vertex R -matrix and related gl2 rational models. The corresponding top is equivalent to the 2-body Ruijsenaars–Schneider (RS) or the 2-body Calogero–Moser (CM) model depending ...
Added: January 22, 2015