Showing papers on "Monodromy published in 2007"

Multivalued functions and functionals. an analogue of the morse theory

Abstract: I. LetM be a finite or infinite dimensional manifold and ω a closed 1-form, dω = 0. Integrating ω over paths in M defines a “multivalued function” S which becomes single valued on some covering M π −→ M with a free abelian monodromy group: dS = π∗ω. The number of generators of the monodromy group is equal to the number of rationally independent integrals of the 1-form ω over integral cycles in M .

Mirror Symmetry via Logarithmic Degeneration Data I

Abstract: This paper continues the authors' program of studying mirror symmetry via log geometry and toric degenerations, relating affine manifolds with singularities, log Calabi-Yau spaces, and toric degenerations of Calabi-Yaus. The main focus of this paper is the calculation of the cohomology of a Calabi-Yau variety associated to a given affine manifold with singularities B. We show that the Dolbeault cohomology groups of the Calabi-Yau associated to B are described in terms of some cohomology groups of sheaves on B, as expected. This is proved first by calculating the log de Rham and log Dolbeault cohomology groups on the log Calabi-Yau space associated to B, and then proving a base-change theorem for cohomology in our logarithmic setting. As applications, this shows that our mirror symmetry construction via Legendre duality of affine manifolds results in the usual interchange of Hodge numbers expected in mirror symmetry, and gives an explicit description of the monodromy of a smoothing.

Seven-branes and supersymmetry

Abstract: We re-investigate the construction of half-supersymmetric 7-brane solutions of IIB supergravity. Our method is based on the requirement of having globally well-defined Killing spinors and the inclusion of SL(2,)-invariant source terms. In addition to the well-known solutions going back to Greene, Shapere, Vafa and Yau we find new supersymmetric configurations, containing objects whose monodromies are not related to the monodromy of a D7-brane by an SL(2,) transformation.

Quasi-Hamiltonian geometry of meromorphic connections

Abstract: For each connected complex reductive group G, we find a family of new examples of complex quasi-Hamiltonian G-spaces with G-valued moment maps. These spaces arise naturally as moduli spaces of (suitably framed) meromorphic connections on principal G-bundles over a disk, and they generalise the conjugacy class example of Alekseev, Malkin, and Meinrenken [3] (which appears in the simple pole case). Using the “fusion product” in the theory, this gives a finite-dimensional construction of the natural symplectic structures on the spaces of monodromy/Stokes data of meromorphic connections over arbitrary genus Riemann surfaces, together with a new proof of the symplectic nature of isomonodromic deformations of such connections

Motivic Serre invariants, ramification, and the analytic Milnor fiber

Abstract: We show how formal and rigid geometry can be used in the theory of complex singularities, and in particular in the study of the Milnor fibration and the motivic zeta function. We introduce the so-called analytic Milnor fiber associated to the germ of a morphism f from a smooth complex algebraic variety X to the affine line. This analytic Milnor fiber is a smooth rigid variety over the field of Laurent series C((t)). Its etale cohomology coincides with the singular cohomology of the classical topological Milnor fiber of f; the monodromy transformation is given by the Galois action. Moreover, the points on the analytic Milnor fiber are closely related to the motivic zeta function of f, and the arc space of X. We show how the motivic zeta function can be recovered as some kind of Weil zeta function of the formal completion of X along the special fiber of f, and we establish a corresponding Grothendieck trace formula, which relates, in particular, the rational points on the analytic Milnor fiber over finite extensions of C((t)), to the Galois action on its etale cohomology. The general observation is that the arithmetic properties of the analytic Milnor fiber reflect the structure of the singularity of the germ f.

Gauss’ Hypergeometric Function

Abstract: We give a basic introduction to the properties of Gauss’ hypergeometric functions, with an emphasis on the determination of the monodromy group of the Gaussian hypergeometric equation.

Planar Ising Correlations

Abstract: Preface.- I. Ising Model on a Finite Square Lattice.- II. Infinite Volume Limits.- III. Scaling Limits.- IV. Monodromy Preserving Deformations of the Euclidean Dirac Equation.- V. Analysis of Tau Functions.- VI. Holonomic Quantum Fields.- Appendix: Infinite Dimensional Spin Groups.- Bibliography.- Index.

Canonical structure and symmetries of the Schlesinger equations

Abstract: The Schlesinger equations S (n,m) describe monodromy preserving deformations of order m Fuchsian systems with n + 1 poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of n copies of m × m matrix algebras equipped with the standard linear Poisson bracket. In this paper we present a new canonical Hamiltonian formulation of the general Schlesinger equations S (n,m) for all n, m and we compute the action of the symmetries of the Schlesinger equations in these coordinates.

Factorization of the finite temperature correlation functions of the XXZ chain in a magnetic field

Abstract: We present a conjecture for the density matrix of a finite segment of the XXZ chain coupled to a heat bath and to a constant longitudinal magnetic field. It states that the inhomogeneous density matrix, conceived as a map which associates with every local operator its thermal expectation value, can be written as the trace of the exponential of an operator constructed from weighted traces of the elements of certain monodromy matrices related to U q (5l 2 ) and only two transcendental functions pertaining to the one-point function and the neighbour correlators, respectively. Our conjecture implies that all static correlation functions of the XXZ chain are polynomials in these two functions and their derivatives with coefficients of purely algebraic origin.

Chern-Simons theory, analytic continuation and arithmetic

Abstract: The purpose of the paper is to introduce some conjectures re- garding the analytic continuation and the arithmetic properties of quantum invariants of knotted objects. More precisely, we package the perturbative and nonperturbative invariants of knots and 3-manifolds into two power series of type P and NP, convergent in a neighborhood of zero, and we postulate their arithmetic resurgence. By the latter term, we mean analytic continuation as a multivalued analytic function in the complex numbers minus a discrete set of points, with restricted singularities, local and global monodromy. We point out some key features of arithmetic resurgence in connection to various problems of asymptotic expansions of exact and perturbative Chern-Simons theory with compact or complex gauge group. Finally, we discuss theoretical and experimental evidence for our conjecture.

Geometry of KAM tori for nearly integrable Hamiltonian systems

Abstract: We obtain a global version of the Hamiltonian KAM theorem for invariant Lagrangian tori by gluing together local KAM conjugacies with the help of a partition of unity. In this way we find a global Whitney smooth conjugacy between a nearly integrable system and an integrable one. This leads to the preservation of geometry, which allows us to define all non-trivial geometric invariants of an integrable Hamiltonian system (like monodromy) for a nearly integrable one.

Unitarization of monodromy representations and constant mean curvature trinoids in 3-dimensional space forms

Abstract: We present a theorem on the unitarizability of loop group valued monodromy representations and apply this to show the existence of new families of constant mean curvature surfaces homeomorphic to a thrice-punctured sphere in the simply-connected 3-dimensional space forms $\R^3$, $\bbS^3 $ and $\bbH^3$. Additionally, we compute the extended frame for any associated family of Delaunay surfaces.

The integral monodromy of hyperelliptic and trielliptic curves

Abstract: We compute the $$\mathbb{Z}/\ell$$ and $$\mathbb{Z}_{\ell}$$ monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves In particular, we provide a proof that the $$\mathbb{Z}/\ell$$ monodromy of the moduli space of hyperelliptic curves of genus g is the symplectic group $${\rm S}_{P2g}(\mathbb{Z}/\ell)$$ We prove that the $$\mathbb{Z}/\ell$$ monodromy of the moduli space of trielliptic curves with signature (r,s) is the special unitary group $${\rm SU}_{(r,s)}(\mathbb{Z}/\ell\otimes\mathbb{Z}[\zeta_{3}])$$

Factorization of the finite temperature correlation functions of the XXZ chain in a magnetic field

Abstract: We present a conjecture for the density matrix of a finite segment of the XXZ chain coupled to a heat bath and to a constant longitudinal magnetic field. It states that the inhomogeneous density matrix, conceived as a map which associates with every local operator its thermal expectation value, can be written as the trace of the exponential of an operator constructed from weighted traces of the elements of certain monodromy matrices related to $U_q (\hat{\mathfrak{sl}}_2)$ and only two transcendental functions pertaining to the one-point function and the neighbour correlators, respectively. Our conjecture implies that all static correlation functions of the XXZ chain are polynomials in these two functions and their derivatives with coefficients of purely algebraic origin.

Discrete Monodromy, Pentagrams, and the Method of Condensation

Abstract: This paper considers a simple geometric construction, called the Pentagram map. The pentagram map, performed on N-gons, gives rise to a birational mapping on the space of all N-gons. This paper finds what conjecturally are all the invariants for this map, and along the way relates the construction to the monodromy of 3rd order differential equations, and also to Dodgson's method of condensation for computing determinants.

Polynomials and vanishing cycles

Abstract: Preface Part I. Singularities at Infinity of Polynomial Functions: 1. Regularity conditions at infinity 2. Detecting atypical values via singularities at infinity 3. Local and global fibrations 4. Families of complex polynomials 5. Topology of a family and contact structures Part II. The Impact of Global Polar Varieties: 6. Polar invariants and topology of affine varieties 7. Relative polar curves and families of affine hypersurfaces 8. Monodromy of polynomials Part III. Vanishing Cycles of Non-Generic Pencils: 9. Topology of meromorphic functions 10. Slicing by pencils of hypersurfaces 11. Higher Zariski-Lefschetz theorems Notes References Bibliography Appendix 1. Stratified singularities Appendix 2. Hints to exercises Index.

Integral ratios of factorials and algebraic hypergeometric functions

Abstract: Sketch of proof of a theorem relating the two subjects of the title. It can be thought as an extension of results of Landau for the classical hypergeometric function. It relies on the characterization of algebraic hypergeometric functions of Beukers and Heckman. In the process we also show that a variant of a classical construction of Bezout (producing a quadratic form, the Bezoutian, out of two polynomials in one variable) gives the Hermitian form fixed by the monodromy group, up to scaling.

Duality and monodromy reducibility of A -hypergeometric systems

Abstract: We study hypergeometric systems H A (β) in the sense of Gelfand, Kapranov and Zelevinsky under two aspects: the structure of their holonomically dual system, and reducibility of their rank module. We prove in the first part that rank-jumping parameters always correspond to reducible systems. We show further that the property of being reducible is “invariant modulo the lattice”, and obtain as a special instance a theorem of Alicia Dickenstein and Timur Sadykov on reducibility of Mellin systems. In the second part we study a conjecture of Nobuki Takayama which states that the holonomic dual of H A (β) is of the form H A (β′) for suitable β′. We prove the conjecture for all matrices A and generic parameter β, exhibit an example that shows that in general the conjecture cannot hold, and present a refined version of the conjecture. Questions on both duality and reducibility have been quite difficult to answer with classical methods. This paper may be seen as an example of the usefulness, and scope of applications, of the homological tools for A-hypergeometric systems developed in Matusevich et al. (J. Amer. Math. Soc. 18:919–941, 2005)

D-brane monodromies from a matrix-factorization perspective

Abstract: The aim of this work is to analyze Kahler moduli space monodromies of string compactifications. This is achieved by investigating the monodromy action upon D-brane probes, which we model in the Landau-Ginzburg phase in terms of matrix factorizations. The two-dimensional cubic torus and the quintic Calabi-Yau hypersurface serve as our two prime examples.

Differential equations associated with nonarithmetic Fuchsian groups

Abstract: We describe globally nilpotent differential operators of rank 2 defined over a number field whose monodromy group is a nonarithmetic Fuchsian group. We show that these differential operators have an S-integral solution. These differential operators are naturally associated with Teichmueller curves in genus 2. They are counterexamples to conjectures by Chudnovsky--Chudnovsky and Dwork. We also determine the field of moduli of primitive Teichmueller curves in genus 2, and an explicit equation in some cases.

Symmetric and alternating groups as monodromy groups of Riemann surfaces I : generic covers and covers with many branch points

Abstract: Introduction and statement of main results Notation and basic lemmas Examples Proving the main results on five or more branch points--Theorems 1.1.1 and 1.1.2 Actions on $2$-sets--the proof of Theorem 4.0.30 Actions on $3$-sets--the proof of Theorem 4.0.31 Nine or more branch points--the proof of Theorem 4.0.34 Actions on cosets of some $2$-homogeneous and $3$-homogeneous groups Actions on $3$-sets compared to actions on larger sets A transposition and an $n$-cycle Asymptotic behavior of $g_k(E)$ An $n$-cycle--the proof of Theorem 1.2.1 Galois groups of trinomials--the proofs of Propositions 1.4.1 and 1.4.2 and Theorem 1.4.3 Appendix A. Finding small genus examples by computer search--by R. Guralnick and R. Stafford Appendix. Bibliography.

NOTES ON THE KAZHDAN-LUSZTIG THEOREM ON EQUIVALENCE OF THE DRINFELD CATEGORY AND THE CATEGORY OF Uqg-MODULES

Abstract: We discuss the proof of Kazhdan and Lusztig of the equivalence of the Drinfeld cate- goryD(g;~) of g-modules and the category of nite dimensional Uqg-modules,q = e i ~ , for ~2 CnQ . Aiming at operator algebraists the result is formulated as the existence for each ~2 iR of a normal- ized unitary 2-cochainF on the dual ^ G of a compact simple Lie group G such that the convolution algebra of G with the coproduct twisted by F is -isomorphic to the convolution algebra of the q-deformation Gq of G, while the coboundary ofF 1 coincides with Drinfeld's KZ-associator dened via monodromy of the Knizhnik-Zamolodchikov equations.

A weight two phenomenon for the moduli of rank one local systems on open varieties

Abstract: The twistor space of representations on an open variety maps to a weight two space of local monodromy transformations around a divisor com- ponent at infinty. The space of �-invariant sections of this slope-two bundle over the twistor line is a real 3 dimensional space whose parameters correspond to the complex residue of the Higgs field, and the real parabolic weight of a harmonic bundle.

On monodromies of a degeneration of irreducible symplectic Kähler manifolds

Abstract: We study the monodromy operators on the Betti cohomologies associated to a good degeneration of irreducible symplectic manifold and we show that the unipotency of the monodromy operator on the middle cohomology is at least the half of the dimension. This implies that the “mildest” singular fiber of a good degeneration with non-trivial monodromy of irreducible symplectic manifolds is quite different from the generic degeneration of abelian varieties or Calabi–Yau manifolds.

Classification of perturbations of the hydrogen atom by small static electric and magnetic fields

Abstract: We consider perturbations of the hydrogen atom by sufficiently small homogeneous static electric and magnetic fields of all possible mutual orientations. Normalizing with regard to the Keplerian sy...