Geometry of unimodular systems

A collection of vectors in a real vector space is called a unimodular system if any of its maximal linearly independent subsets generates the same free abelian group. This notion is closely connected with totally unimodular matrices: rows or columns of a totally unimodular matrix form a unimodular system and the matrix of coefficients of expansions of all vectors of a unimodular system with respect to its maximal linearly independent subset is totally unimodular. In this paper we show that a unimodular system defines the following geometric data: a Euclidean space, an integral lattice in it, and a reflexive lattice zonotope. The discriminant of the lattice is equal to the number of maximal linearly independent subsystems, and we call this number the complexity of the unimodular system. For a unimodular system $\Omega$ we also define the Gale dual unimodular system $\Omega ^{\bot}$ which has the same complexity. These notions may be illustrated by the well-known graphic and cographic unimodular systems of a graph. Both graphic and cographic unimodular systems have the same complexity which is equal to the complexity of the graph. For graphs without loops and bridges the graphic and the cographic unimodular systems are Gale dual to each other. We describe this geometric data for certain examples: for the graphic and the cographic unimodular systems of a generalized theta-graph, consisting of two vertices connected by $N$ edges, for the cographic system of the complete graph $K_N$, and for the famous Bixby-Seymour unimodular system, which is neither graphic nor cographic

Zero dimensional Donaldson-Thomas invariants of threefolds

Using a homotopy approach, we prove in this paper a conjecture of Maulik, Nekrasov, Okounkov and Pandharipande on the dimension zero Donaldson-Thomas invariants of all smooth complex threefolds.Comment: This is the version published by Geometry & Topology on 29 November 200

Geometry of tropical moduli spaces and linkage of graphs

We prove the following "linkage" theorem: two p-regular graphs of the same genus can be obtained from one another by a finite alternating sequence of one-edge-contractions; moreover this preserves 3-edge-connectivity. We use the linkage theorem to prove that various moduli spaces of tropical curves are connected through codimension one.Comment: Final version incorporating the referees correction

Spectral Bundles and the DRY-Conjecture

Supersymmetric heterotic string models, built from a Calabi-Yau threefold $X$ endowed with a stable vector bundle $V$, usually start from a phenomenologically motivated choice of a bundle $V_v$ in the visible sector, the spectral cover construction on an elliptically fibered $X$ being a prominent example. The ensuing anomaly mismatch between $c_2(V_v)$ and $c_2(X)$, or rather the corresponding differential forms, is often 'solved', on the cohomological level, by including a fivebrane. This leads to the question whether the difference can be alternatively realized by a further stable bundle. The 'DRY'-conjecture of Douglas, Reinbacher and Yau in math.AG/0604597 gives a sufficient condition on cohomology classes on $X$ to be realized as the Chern classes of a stable sheaf. In arXiv:1010.1644 we showed that infinitely many classes on $X$ exist for which the conjecture ist true. In this note we give the sufficient condition for the mentioned fivebrane classes to be realized by a further stable bundle in the hidden sector. Using a result obtained in arXiv:1011.6246 we show that corresponding bundles exist, thereby confirming this version of the DRY-Conjecture.Comment: 6 page

Использование дистанционных образовательных технологий на курсах повышения квалификации

В статье описан опыт использования дистанционных образовательных технологий на курсах повышения квалификации специалистов предприятий — членов саморегулируемых организаций (СРО) в области строительства, энергетики и теплоснабжения, посредством сети Интернет с использованием системы дистанционного обучения «WebSET»

Heterotic Non-Kahler Geometries via Polystable Bundles on Calabi-Yau Threefolds

In arXiv:1008.1018 it is shown that a given stable vector bundle $V$ on a Calabi-Yau threefold $X$ which satisfies $c_2(X)=c_2(V)$ can be deformed to a solution of the Strominger system and the equations of motion of heterotic string theory. In this note we extend this result to the polystable case and construct explicit examples of polystable bundles on elliptically fibered Calabi-Yau threefolds where it applies. The polystable bundle is given by a spectral cover bundle, for the visible sector, and a suitably chosen bundle, for the hidden sector. This provides a new class of heterotic flux compactifications via non-Kahler deformation of Calabi-Yau geometries with polystable bundles. As an application, we obtain examples of non-Kahler deformations of some three generation GUT models.Comment: 12 pages, late

Fixed point loci of moduli spaces of sheaves on toric varieties

Extending work of Klyachko and Perling, we develop a combinatorial description of pure equivariant sheaves of any dimension on an arbitrary nonsingular toric variety $X$. Using geometric invariant theory (GIT), this allows us to construct explicit moduli spaces of pure equivariant sheaves on $X$ corepresenting natural moduli functors (similar to work of Payne in the case of equivariant vector bundles). The action of the algebraic torus on $X$ lifts to the moduli space of all Gieseker stable sheaves on $X$ and we express its fixed point locus explicitly in terms of moduli spaces of pure equivariant sheaves on $X$. One of the problems arising is to find an equivariant line bundle on the side of the GIT problem, which precisely recovers Gieseker stability. In the case of torsion free equivariant sheaves, we can always construct such equivariant line bundles. As a by-product, we get a combinatorial description of the fixed point locus of the moduli space of $\mu$-stable reflexive sheaves on $X$. As an application, we show in a sequel how these methods can be used to compute generating functions of Euler characteristics of moduli spaces of $\mu$-stable torsion free sheaves on nonsingular complete toric surfaces.Comment: 55 pages. Published versio

"Apparent PT-symmetric terahertz photoconductivity in the topological phase of Hg1−xCdxTe-based structures"

We show that the terahertz (THz) photoconductivity in the topological phase of Hg1-xCdxTe-based structures exhibits the apparent PT- (parity-time) symmetry whereas the P-symmetry and the T-symmetry, separately, are not conserved. Moreover, it is demonstrated that the P- and T-symmetry breaking may not be related to any type of the sample anisotropy. This result contradicts the apparent symmetry arguments and means that there exists an external factor that interacts with the sample electronic system and breaks the symmetry. We show that deviations from the ideal experimental geometry may not be such a factor