Showing papers on "Monodromy published in 2003"

Asymptotic black hole quasinormal frequencies

Abstract: We give a new derivation of the quasinormal frequencies of Schwarzschild black holes in d greater than or equal to 4 and Reissner-Nordstrom black holes in d = 4, in the limit of infinite damping. For Schwarzschild in d greater than or equal to 4 we find that the asymptotic real part is THawkinglog(3) for scalar perturbations and for some gravitational perturbations; this confirms a result previously obtained by other means in the case d = 4. For Reissner-Nordstrom in d = 4 we find a specific generally aperiodic behavior for the quasinormal frequencies, both for scalar perturbations and for electromagnetic-gravitational perturbations. The formulae are obtained by studying the monodromy of the perturbation analytically continued to the complex plane; the analysis depends essentially on the behavior of the potential in the 'unphysical' region near the black hole singularity.

Asymptotic black hole quasinormal frequencies

Abstract: We give a new derivation of the quasinormal frequencies of Schwarzschild black holes in d>=4 and Reissner-Nordstrom black holes in d=4, in the limit of infinite damping. For Schwarzschild in d>=4 we find that the asymptotic real part is T_Hawking.log(3) for scalar perturbations and for some gravitational perturbations; this confirms a result previously obtained by other means in the case d=4. For Reissner-Nordstrom in d=4 we find a specific generally aperiodic behavior for the quasinormal frequencies, both for scalar perturbations and for electromagnetic-gravitational perturbations. The formulae are obtained by studying the monodromy of the perturbation analytically continued to the complex plane; the analysis depends essentially on the behavior of the potential in the "unphysical" region near the black hole singularity.

Spectral Analysis of Unitary Band Matrices

Abstract: This paper is devoted to the spectral properties of a class of unitary operators with a matrix representation displaying a band structure. Such band matrices appear as monodromy operators in the study of certain quantum dynamical systems. These doubly infinite matrices essentially depend on an infinite sequence of phases which govern their spectral properties. We prove the spectrum is purely singular for random phases and purely absolutely continuous in case they provide the doubly infinite matrix with a periodic structure in the diagonal direction. We also study some properties of the singular spectrum of such matrices considered as infinite in one direction only.

Structure theorem for compact vaisman manifolds

Abstract: A locally conformally Kahler (l.c.K.) manifold is a complexmanifold admitting a Kahler covering ˜ M, with each deck transformation acting by Kahler homotheties. A compact l.c.K. manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties on ˜ M. We prove a structure theorem for compact Vaisman manifolds. Every compact Vaisman manifold M is fibered over a circle, the fibers are Sasakian, the fibration is locally trivial, and M is reconstructed from the Sasakian structure on the fibers and the monodromy automorphism induced by this fibration. This construction is canonical and functorial in both directions.

Hodge Structure on the Fundamental Group and its Application to p-Adic Integration

Abstract: We study the unipotent completion Π un (x0, x1,XK) of the de Rham fundamental groupoid [De] of a smooth algebraic variety over a local non-archimedean field K of characteristic 0. We show that the vector space Π un (x0, x1,XK) carries a certain additional structure. That is a Q p -space Πun(x0, x1,XK) equipped with a σ-semi-linear operator φ, a linear operator N satisfying the relation Nφ = pφN and a weight filtration W• together with a canonical isomorphism Π un (x0, x1,XK) ⊗K K ≃ Πun(x0, x1,XK) ⊗Qur p K. We prove that an analog of the Monodromy Conjecture holds for Πun(x0, x1,XK). As an application, we show that the vector space Π un (x0, x1,XK) possesses a distinguished element. In the other words, given a vector bundle E on XK together with a unipotent integrable connection, we have a canonical isomorphism Ex0 ≃ Ex1 between the fibres. The latter construction is a generalisation of Colmez’s p-adic integration (rkE = 2) and Coleman’s p-adic iterated integrals (XK is a curve with good reduction). In the second part we prove that, if XK0 is a smooth variety over an unramified extension of Qp with good reduction and r ≤ p−1 2 then there is a canonical isomorphism Π r (x0, x1,XK0)⊗ BDR ≃ Π et r (x0, x1,XK0)⊗BDR compatible with the action of Galois group. ( Π DR r (x0, x1, XK0) stands for the level r quotient of Π un (x0, x1,XK)). In particularly, it implies the Crystalline Conjecture for the fundamental group [Shiho] (for r ≤ p−1 2 ) .

On Hodge spectrum and multiplier ideals

Abstract: We describe a relation between two invariants which measure the complexity of a hypersurface singularity. One is the Hodge spectrum which is related to the monodromy and the Hodge filtration on the cohomology of the Milnor fiber. The other is the multiplier ideal, having to do with log resolutions.

Special Functions, Kz Type Equations, and Representation Theory

Abstract: Hypergeometric solutions of KZ equations Cycles of integrals and the monodromy of the KZ equation Selberg integral, determinant formulas, and dynamical equations Critical points of master functions and the Bethe ansatz Elliptic hypergeometric functions q-hypergeometric solutions of qKZ equations Bibliography Index.

On the holomorphicity of genus two Lefschetz fibrations

Abstract: We prove that any genus-2 Lefschetz fibration without reducible fibers and with ``transitive monodromy'' is holomorphic. The latter condition comprises all cases where the number of singular fibers is not congruent to 0 modulo 40. An auxiliary statement of independent interest is the holomorphicity of symplectic surfaces in S^2-bundles over S^2, of relative degree up to 7 over the base, and of symplectic surfaces in CP^2 of degree up to 17.

A point's point of view of stringy geometry

Abstract: The notion of a ``point'' is essential to describe the topology of spacetime. Despite this, a point probably does not play a particularly distinguished role in any intrinsic formulation of string theory. We discuss one way to try to determine the notion of a point from a worldsheet point of view. The derived category description of D-branes is the key tool. The case of a flop is analyzed and Π-stability in this context is tied in to some ideas of Bridgeland. Monodromy associated to the flop is also computed via Π-stability and shown to be consistent with previous conjectures.

Galois groups and fundamental groups

Abstract: Introduction 1. Monodromy groups of coverings of curves Robert Guralnik 2. On the tame fundamental groups of curves over algebraically closed fields of characteristic > 0 Akio Tamagawa 3. On the specialization homomorphism of fundamental groups of curves in positive characteristic Florian Pop and Mohamed Saidi 4. Topics surrounding the anabelian geometry of hyperbolic curves Shinichi Mochizuki 5. Monodromy of elliptic surfaces Fedor Bogomolov and Yuri Tschinkel 6. Tannakian fundamental groups associated to Galois groups Richard Hain and Makoto Matsumoto 7. Special loci in moduli spaces of curves Leila Schneps 8. Cellulation of compactified Hurwitz spaces Michel Imbert 9. Patching and Galois theory David Harbater 10. Constructive differential Galois theory B. Heinrich Matzat and Marius van der Put.

Structure theorem for compact Vaisman manifolds

Abstract: A locally conformally Kaehler (l.c.K.) manifold is a complex manifold admitting a Kaehler covering $\tilde M$, with each deck transformation acting by Kaehler homotheties. A compact l.c.K. manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties on $\tilde M$. We prove a structure theorem for compact Vaisman manifolds. Every compact Vaisman manifold M is fibered over a circle, the fibers are Sasakian, the fibration is locally trivial, and M is reconstructed as a Riemannian suspension from the Sasakian structure on the fibers and the monodromy automorphism induced by this fibration. This construction is canonical and functorial in both directions.

An Area-Preserving Action of the Modular Group on Cubic Surfaces and the Painlevé VI Equation

Abstract: We construct an area-preserving action of the modular group on a general 4-parameter family of affine cubic surfaces. We present a geometrical background behind this construction, that is, a natural symplectic structure on a moduli space of rank two linear monodromy representations over the 2-dimensional sphere with four punctures, and a natural symplectic action upon it of the braid group on three strings. Studying this action as a discrete dynamical system will be important in discussing the monodromy of the Painleve VI equation.

On braid monodromy factorizations

Abstract: We introduce and develop a?language of semigroups over the braid groups to study the braid monodromy factorizations (bmf's) of plane algebraic curves and other related objects. As an application, we give a?new proof of Orevkov's theorem on the realization of bmf's over a?disc by algebraic curves and show that the complexity of such a?realization cannot be bounded in terms of the types of factors of the bmf. We also prove that the type of a?bmf distinguishes Hurwitz curves with singularities of inseparable type up to -isotopy and -holomorphic cuspidal curves in? up to symplectic isotopy.

Derivatives of p-adic L-functions, Heegner cycles and monodromy modules attached to modular forms

Abstract: In this paper we prove a Gross-Zagier type formula for the anticyclotomic p-adic L-function of an elliptic modular form f of higher weight and of multiplicative type at p. For such f we also decribe explicitely the local Galois representation attached to it at p and compare the Fontaine-Mazur and Teitelbaum L-invariants.

A lecture on the octahedron

Abstract: Algebraic solutions of certain Painlev´ eV I equationsare produced by solving linear systems with monodromy contained in the octahedral subgroup of SO(3). The method uses the algebraic geometry of special plane curves, and makes contact with some classical geometrical problems.

Schubert induction

Abstract: We describe a Schubert induction theorem, a tool for analyzing intersections on a Grassmannian over an arbitrary base ring The key ingredient in the proof is the Geometric Littlewood-Richardson rule, described in a companion paper Schubert problems are among the most classical problems in enumerative geometry of continuing interest As an application of Schubert induction, we address several long-standing natural questions related to Schubert problems, including: the "reality" of solutions; effective numerical methods; solutions over algebraically closed fields of positive characteristic; solutions over finite fields; a generic smoothness (Kleiman-Bertini) theorem; and monodromy groups of Schubert problems These methods conjecturally extend to the flag variety

Braid monodromy and topology of plane curves

Abstract: In this paper we prove that braid monodromy of an affine plane curve determines the topology of a related projective plane curve.

Monodromy in the spectrum of a rigid symmetric top molecule in an electric field

Abstract: We show that for rigid symmetric top molecules in electric fields the phenomenon of monodromy arises naturally as a “defect” in the lattice of quantum states in the energy-momentum diagram. This makes it impossible to use either the total angular momentum or a pendular quantum number to label the states globally. The monodromy is created or destroyed by classical Hamiltonian Hopf bifurcations from relative equilibria. These phenomena are robust and should be observable in quasi-symmetric top molecules with field strengths E satisfying μE/b>4.5, where μ is the dipole moment and b the rotational constant perpendicular to the symmetry axis of the molecule.

The displacement map associated to polynomial unfoldings of planar Hamiltonian vector fields

Abstract: We study the displacement map associated to small one-parameter polynomial unfoldings of polynomial Hamiltonian vector fields on the plane. Its leading term, the generating function $M(t)$, has an analytic continuation in the complex plane and the real zeroes of $M(t)$ correspond to the limit cycles bifurcating from the periodic orbits of the Hamiltonian flow. We give a geometric description of the monodromy group of $M(t)$ and use it to formulate sufficient conditions for $M(t)$ to satisfy a differential equation of Fuchs or Picard-Fuchs type. As examples, we consider in more detail the Hamiltonian vector fields $\dot{z}=i\bar{z}-i(z+\bar{z})^3$ and $\dot{z}=iz+\bar{z}^2$, possessing a rotational symmetry of order two and three, respectively. In both cases $M(t)$ satisfies a Fuchs-type equation but in the first example $M(t)$ is always an Abelian integral (that is to say, the corresponding equation is of Picard-Fuchs type) while in the second one this is not necessarily true. We derive an explicit formula of $M(t)$ and estimate the number of its real zeroes.}

Fibred Kähler and quasi-projective groups.

Abstract: We formulate a new theorem giving several necessary and sufficient conditions in order that a surjection of the fundamental group πι^) of a compact Kahler manifold onto the fundamental group Π^ of a compact Riemann surface of genus g ^ 2 be induced by a holomorphic map. For instance, it suffices that the kernel be finitely generated. We derive as a corollary a restriction for a group G, fitting into an exact sequence 1 -» H -* G —> U g —> 1, where H is finitely generated, to be the fundamental group of a compact Kahler manifold. Thanks to the extension by Bauer and Arapura of the Castelnuovo-de Franchis theorem to the quasi-projective case (more generally, to Zariski open sets of compact Kahler manifolds) we first extend the previous result to the non-compact case. We are finally able to give a topo- logical characterization of quasi-projective surfaces which are fibred over a (quasi-projective) curve by a proper holomorphic map of maximal rank, and we extend the previous restriction to the monodromy of any fibration onto a curve.

Iterated Monodromy Groups

Abstract: We associate a group IMG(f) to every covering f of a topological space M by its open subset. It is the quotient of the fundamental group �1(M) by the intersection of the kernels of its monodromy action for the iterates f n . Every iterated monodromy group comes together with a naturally defined action on a rooted tree. We present an effective method to compute this action and show how the dynamics of f is related to the group. In particular, the Julia set of f can be reconstructed from IMG(f) (from its action on the tree), if f is expanding.

Hypergeometric Abelian Varieties

Abstract: In this paper, we construct abelian varieties associated to Gauss’ and Appell–Lauricella hypergeometric series. Abelian varieties of this kind and the algebraic curves we define to construct them were considered by several authors in settings ranging from monodromy groups (Deligne, Mostow), exceptional sets (Cohen, Wolfart, Wustholz), modular embeddings (Cohen, Wolfart) to CM-type (Cohen, Shiga, Wolfart) and modularity (Darmon). Our contribution is to provide a complete, explicit and self-contained geometric construction.

Opers on the projective line, flag manifolds and Bethe Ansatz

Abstract: We consider the problem of diagonalization of the hamiltonians of the Gaudin model, which is a quantum chain model associated to a simple Lie algebra. The hamiltonians of this model act on the tensor product of finite-dimensional representations of this Lie algebra. We show that the eigenvalues of the Gaudin hamiltonians are encoded by the so-called "opers" on the projective line, associated to the Langlands dual Lie algebra. These opers have regular singularities at the marked points with prescribed residues and trivial monodromy representation. The Bethe Ansatz is a procedure to construct explicitly the eigenvectors of the generalized Gaudin hamiltonians. We show that each solution of the Bethe Ansatz equations defines what we call a "Miura oper" on the projective line. Moreover, we show that the space of Miura opers is a union of copies of the flag manifold (of the dual group), one for each oper. This allows us to prove that all solutions of the Bethe Ansatz equations, corresponding to a fixed oper, are in one-to-one correspondence with the points of an open dense subset of the flag manifold. The Bethe Ansatz equations can be written for an arbitrary Kac-Moody algebra, and we prove an analogue of the last result in this more general setting. For the Lie algebras of types A,B,C similar results were obtained by other methods by I.Scherbak and A.Varchenko and by E.Mukhin and A.Varchenko.

Monodromy of Variations of Hodge Structure

Abstract: We present a survey of the properties of the monodromy of local systems on quasi-projective varieties which underlie a variation of Hodge structure In the last section, a less widely known version of a Noether–Lefschetz-type theorem is discussed

Monodromy of Projective Curves

Abstract: The uniform position principle states that, given an irreducible nondegenerate curve C in the projective r-space $P^r$, a general (r-2)-plane L is uniform, that is, projection from L induces a rational map from C to $P^1$ whose monodromy group is the full symmetric group. In this paper we show the locus of non-uniform (r-2)-planes has codimension at least two in the Grassmannian for a curve C with arbitrary singularities. This result is optimal in $P^2$. For a smooth curve C in $P^3$ that is not a rational curve of degree three, four or six, we show any irreducible surface of non-uniform lines is a Schubert cycle of lines through a point $x$, such that projection from $x$ is not a birational map of $C$ onto its image.

Casimir Operators and Monodromy Representations of Generalised Braid Groups

Abstract: Let g be a complex, simple Lie algebra with Cartan subalgebra h and Weyl group W. We construct a one-parameter family of flat connections D on h with values in any finite-dimensional h-module V and simple poles on the root hyperplanes. The corresponding monodromy representation of the braid group B of type g is a deformation of the action of (a finite extension of) W on V. The residues of D are the Casimirs of the sl(2)-subalgebras of g corresponding to its roots. The irreducibility of a subspace U of V under these implies that, for generic values of the parameter, the braid group B acts irreducibly on U. Answering a question of Knutson and Procesi, we show that these Casimirs act irreducibly on the weight spaces of all simple g-modules if g is sl(3) but that this is not the case if g is not isomorphic to sl(2) or sl(3). We use this to disprove a conjecture of Kwon and Lusztig stating the irreducibility of quantum Weyl group actions of Artin's braid group B_n on the zero weight spaces of all simple U_{q}sl(n)-modules for n greater or equal to 4. Finally, we study the irreducibility of the action of the Casimirs on the zero weight spaces of self-dual g-modules and obtain complete classification results for g=sl(n) or g(2).