Multiple Mellin-Barnes Integrals as Periods of Calabi-Yau Manifolds With Several Moduli

We give a representation, in terms of iterated Mellin-Barnes integrals, of periods on multi-moduli Calabi-Yau manifolds arising in superstring theory. Using this representation and the theory of multidimensional residues, we present a method for analytic continuation of the fundamental period in the form of Horn series.Comment: 18 pages, AMS-tex + 3 postscript figures, to be published in Theor. Math. Phys. Russi

Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom

We consider natural complex Hamiltonian systems with $n$ degrees of freedom given by a Hamiltonian function which is a sum of the standard kinetic energy and a homogeneous polynomial potential $V$ of degree $k>2$. The well known Morales-Ramis theorem gives the strongest known necessary conditions for the Liouville integrability of such systems. It states that for each $k$ there exists an explicitly known infinite set \scM_k\subset\Q such that if the system is integrable, then all eigenvalues of the Hessian matrix V''(\vd) calculated at a non-zero \vd\in\C^n satisfying V'(\vd)=\vd, belong to \scM_k. The aim of this paper is, among others, to sharpen this result. Under certain genericity assumption concerning $V$ we prove the following fact. For each $k$ and $n$ there exists a finite set \scI_{n,k}\subset\scM_k such that if the system is integrable, then all eigenvalues of the Hessian matrix V''(\vd) belong to \scI_{n,k}. We give an algorithm which allows to find sets \scI_{n,k}. We applied this results for the case $n=k=3$ and we found all integrable potentials satisfying the genericity assumption. Among them several are new and they are integrable in a highly non-trivial way. We found three potentials for which the additional first integrals are of degree 4 and 6 with respect to the momenta.Comment: 54 pages, 1 figur

Residues and World-Sheet Instantons

We reconsider the question of which Calabi-Yau compactifications of the heterotic string are stable under world-sheet instanton corrections to the effective space-time superpotential. For instance, compactifications described by (0,2) linear sigma models are believed to be stable, suggesting a remarkable cancellation among the instanton effects in these theories. Here, we show that this cancellation follows directly from a residue theorem, whose proof relies only upon the right-moving world-sheet supersymmetries and suitable compactness properties of the (0,2) linear sigma model. Our residue theorem also extends to a new class of "half-linear" sigma models. Using these half-linear models, we show that heterotic compactifications on the quintic hypersurface in CP^4 for which the gauge bundle pulls back from a bundle on CP^4 are stable. Finally, we apply similar ideas to compute the superpotential contributions from families of membrane instantons in M-theory compactifications on manifolds of G_2 holonomy.Comment: 47 page

On the Classification of Residues of the Grassmannian

We study leading singularities of scattering amplitudes which are obtained as residues of an integral over a Grassmannian manifold. We recursively do the transformation from twistors to momentum twistors and obtain an iterative formula for Yangian invariants that involves a succession of dualized twistor variables. This turns out to be useful in addressing the problem of classifying the residues of the Grassmannian. The iterative formula leads naturally to new coordinates on the Grassmannian in terms of which both composite and non-composite residues appear on an equal footing. We write down residue theorems in these new variables and classify the independent residues for some simple examples. These variables also explicitly exhibit the distinct solutions one expects to find for a given set of vanishing minors from Schubert calculus.Comment: 20 page

Multidimensional residues and their applications

The technique of residues is known for its many applications in different branches of mathematics. Tsikh's book presents a systematic account of residues associated with holomorphic mappings and indicates many applications. The book begins with preliminaries from the theory of analytic sets, together with material from algebraic topology that is necessary for the integration of differential forms over chains. Tsikh then presents a detailed study of residues associated with mappings that preserve dimension (local residues). Local residues are applied to algebraic geometry and to problems connected with the investigation and calculation of double series and integrals. There is also a treatment of residues associated with mappings that reduce dimension--that is, residues of semimeromorphic forms, connected with integration over tubes around nondiscrete analytic sets

Convergence of two-dimensional hypergeometric series for algebraic functions

Description of convergence domains for multiple power series is a quite difficult problem. In 1889 J.Horn showed that the case of hypergeomteric series is more favourable. He found a parameterization formula for surfaces of conjugative radii of such series. But until recently almost nothing was known about the description of convergence domains in terms of functional inequalities ρj(|a1|, . . . , |am|) < 0 relatively moduli |ai| of series variables. In this paper we give a such description for hypergeometric series representing solutions to tetranomial algebraic equations. In our study we use the remarkable observation by M. Kapranov (16) consisting in the fact that the Horn’s formulae give a parameterization of discriminant locus for a corresponding A-discriminant. We prove that usually the considered convergence domains are determined by a signle or two inequalities ρ(|at |, |as|) ≶ 0, where ρ is a reduced discriminant

Об асимптотике гомологических решений многомерных линейных разностных уравнений

Given a linear homogeneous multidimensional difference equation with constant coefficients, we choose a pair ( , !), where is a homological k-dimensional cycle on the characteristic set of the equation and ! is a holomorphic form of degree k. This pair defines a so called homological solution by the integral over of the form ! multiplied by an exponential kernel. A multidimensional variant of Perron’s theorem in the class of homological solutions is illustrated by an example of the first order equationРассматривается многомерное линейное разностное уравнение с постоянными коэффициента- ми и пара ( , !), где гомологический k-мерный цикл на характеристическом множестве уравнения, а ! голоморфная форма степени k. Интеграл по формы !, умноженной на экспо- ненциальное ядро, называется гомологическим решением. На примере уравнения первого порядка иллюстрируется многомерный вариант теоремы Перрона в классе гомологических решени