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Regular version of the site
Of all publications in the section: 53
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Article
Covolo T., Grabowski J., Poncin N. Journal of Geometry and Physics. 2016. Vol. 110. P. 393-401.

Smooth -supermanifolds have been introduced and studied recently. The corresponding sign rule is given by the ‘scalar product’ of the involved -degrees. It exhibits interesting changes in comparison with the sign rule using the parity of the total degree. With the new rule, nonzero degree even coordinates are not nilpotent, and even (resp., odd) coordinates do not necessarily commute (resp., anticommute) pairwise. The classical Batchelor–Gawȩdzki theorem says that any smooth supermanifold is diffeomorphic to the ‘superization’ ΠE of a vector bundle E. It is also known that this result fails in the complex analytic category. Hence, it is natural to ask whether an analogous statement goes through in the category of -supermanifolds with its local model made of formal power series. We give a positive answer to this question.

Added: Nov 12, 2016
Article
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 more intrinsic as compared to scalar conservation laws derived from it and, as a rule, it is “simpler”. Thus it is natural to ask: can the dynamics of a bihamiltonian system be understood by studying its hamiltonian pair, without studying the associated first integrals?In this paper, the problem of stability of equilibria in bihamiltonian systems is considered and it is shown that the conditions for nonlinear stability can be expressed in algebraic terms of linearization of the underlying Poisson pencil. This is used to study stability of stationary rotations of a free multidimensional rigid body.

Added: Nov 18, 2013
Article
Soldatenkov A. O., Verbitsky M. Journal of Geometry and Physics. 2012. Vol. 62. No. 11. P. 2234-2240.

A hypercomplex manifold M is a manifold with a triple I,J,K of complex structure operators satisfying quaternionic relations. For each quaternion L=aI+bJ+cK, L2=−1, L is also a complex structure operator on M, called an induced complex structure. We study compact complex subvarieties of (M,L), for L a generic induced complex structure. Under additional assumptions (Obata holonomy contained in SL(n,H), the existence of an HKT-metric), we prove that (M,L) contains no divisors, and all complex subvarieties of codimension 2 are trianalytic (that is, also hypercomplex).

Added: Aug 30, 2012
Article
Amerik E., Verbitsky M. Journal of Geometry and Physics. 2015. Vol. 97. P. 44-50.

Let S be an infinite-dimensional manifold of all symplectic, or hyperkähler, structures on a compact manifold M, and Diff0 the connected component of its diffeomorphism group. The quotient S/Diff0 is called the Teichmüller space of symplectic (or hyperkähler) structures on M. MBM classes on a hyperkähler manifold M are cohomology classes which can be represented by a minimal rational curve on a deformation of M. We determine the Teichmüller space of hyperkähler structures on a hyperkähler manifold, identifying any of its connected components with an open subset of the Grassmannian variety SO(b2-3, 3)/SO(3)×SO(b2-3) consisting of all Beauville-Bogomolov positive 3-planes in H2(M,R) which are not orthogonal to any of the MBM classes. This is used to determine the Teichmüller space of symplectic structures of Kähler type on a hyperkähler manifold of maximal holonomy. We show that any connected component of this space is naturally identified with the space of cohomology classes v∈H2(M,R) with q(v,v)>0, where q is the Bogomolov-Beauville-Fujiki form on H2(M,R). © 2015 Elsevier B.V.

Added: Sep 8, 2015
Article
Kharchev S., Zabrodin A. Journal of Geometry and Physics. 2015. Vol. 94. P. 19-31.

This paper is an annotated list of transformation properties and identities satisfied by the four theta functions of one complex variable, presented in a ready-to-use form. An attempt is made to reveal a pattern behind various identities for the theta-functions. It is shown that all possible 3, 4 and 5-term identities of degree four emerge as algebraic consequences of the six fundamental bilinear 3-term identities connecting the theta-functions with different modular parameters 2τ  

Added: Oct 12, 2015
Article
Kharchev S., Zabrodin A. Journal of Geometry and Physics. 2016. Vol. 104. P. 112-120.

It is shown that the Jacobi and Riemann identities of degree four for the multidimensional theta functions as well as the Weierstrass identities emerge as algebraic consequences of the fundamental multidimensional binary identities connecting the theta functions with Riemann matrices tau and 2tau .  

Added: Sep 30, 2016
Article
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: Sep 26, 2012
Article
Kazakov A., Bolsinov A. V., Kilin A. A. Journal of Geometry and Physics. 2015. Vol. 87. P. 61-75.

The phenomenon of a topological monodromy in integrable Hamiltonian and nonholonomic systems is discussed. An efficient method for computing and visualizing the monodromy is developed. The comparative analysis of the topological monodromy is given for the rolling ellipsoid of revolution problem in two cases, namely, on a smooth and on a rough plane. The first of these systems is Hamiltonian, the second is nonholonomic. We show that, from the viewpoint of monodromy, there is no difference between the two systems, and thus disprove the conjecture by Cushman and Duistermaat stating that the topological monodromy gives a topological obstruction for Hamiltonization of the rolling ellipsoid of revolution on a rough plane

Added: Mar 29, 2015
Article
Losev A. S., Mnev P., Youmans D. R. Journal of Geometry and Physics. 2018. Vol. 131. P. 122-137.

We study the two-dimensional topological abelian BF theory in the Lorenz gauge and, surprisingly, we find that the gauged-fixed theory is a free type B twisted N = (2, 2) superconformal theory with odd linear target space, with the ghost field c being the pullback of the linear holomorphic coordinate on the target. The Q(BRST) of the gauge-fixed theory equals the total Q of type B twisted theory. This unexpected identification of two different theories opens a way for nontrivial deformations of both of these theories. (C) 2018 Elsevier B.V. All rights reserved.

Added: Oct 5, 2018
Article
Poberezhny V. A., Matveeva A. Journal of Geometry and Physics. 2017. Vol. 114. P. 384-393.

We consider a generalization of Riemann–Hilbert problem on elliptic curves. For a given elliptic curve and irreducible representation of free group with two generators we construct explicitly a semistable vector bundle of degree zero obeying a logarithmic connection such that its monodromy over fundamental parallelogram is equivalent to given free group representation, monodromy along a−cycle is trivial and monodromy along b−cycle belong to certain orbit.

Added: Oct 26, 2016
Article
Matveev V. S., Shevchishin V. Journal of Geometry and Physics. 2011. Vol. 61. No. 8. P. 1353-1377.
Added: Mar 18, 2013