The Number of Ramified Covering of a Riemann Surface by Riemann Surface

Interpreting the number of ramified covering of a Riemann surface by Riemann surfaces as the relative Gromov–Witten invariants and applying a gluing formula, we derive a recursive formula for the number of ramified covering of a Riemann surface by Riemann surface with elementary branch points and prescribed ramification type over a special point. Received: 10 June 1999 / Accepted: 7 July 2000     

The Szegő Kernel on a Sewn Riemann Surface

We describe the Szegő kernel on a higher genus Riemann surface in terms of Szegő kernel data coming from lower genus surfaces via two explicit sewing procedures where either two Riemann surfaces are sewn together or a handle is sewn to a Riemann surface. We consider in detail the examples of the Szegő kernel on a genus two Riemann surface formed by either sewing together two punctured tori or by sewing a twice-punctured torus to itself. We also consider the modular properties of the Szegő kernel in these cases.     

Integrability of Liouville System on High Genus Riemann Surface.Ⅰ. Classical Case

By using the theory of uniformization of Riemann surfaces,we study properties of the Liouville equation and its general solution on a Riemann surface of genus g>1.After obtaining Hamiltonian formalism in terms of free fields and calculating classical exchange matrices,we prove the classical integrability of Liouville system on high genus Riemann surface.     

Matrix models as CFT: Genus expansion

We show how the formulation of the matrix models as conformal field theories on a Riemann surfaces can be used to compute the genus expansion of the observables. Here we consider the simplest example of the Hermitian matrix model, where the classical solution is described by a hyperelliptic Riemann surface. To each branch point of the Riemann surface we associate an operator which represents a twist field dressed by the modes of the twisted boson. The partition function of the matrix model is computed as a correlation function of such dressed twist fields. The perturbative construction of the dressing operators yields a set of Feynman rules for the genus expansion, which involve vertices, propagators and tadpoles. The vertices are universal, the propagators and the tadpoles depend on the Riemann surface. As a demonstration we evaluate the genus-two free energy using the Feynman rules.     

AN APPROXIMATION OF THE KINETIC ENERGY OF A SUPERFLUID FILM ON A RIEMANN SURFACE

The flow of a superfluid film adsorbed on a porous medium can be modeled by a meromorphic differential on a Riemann surface of high genus. In this paper, we define the mixed Hodge metric of meromorphic differentials on a Riemann surface and justify using this metric to approximate the kinetic energy of a superfluid film flowing on a porous surface.     

Riemann surfaces and partial wave models

Within the context of simple partial wave models for elastic scattering the problem of uniformizing the partial wave amplitude and classifying its Riemann surface is studied. Starting with the analytic continuation of the amplitude an analysis of the Riemann surface is made through its group of covering transformations relative to a simpler base surface. A model based on the Yukawa potential is studied in this manner and the Riemann surface of interest is found to be the universal covering surface of the thrice punctured sphere. The uniformization of the amplitude can be done explicitly in this case by use of the elliptic modular function. In terms of the uniformizing variable, the original discontinuity relations for the amplitude then reduce to functional equations involving elements of the modular group.     

Hurwitz number of triple Ramified covers

Using symplectic cut-and-gluing formulae of the relative Gromov–Witten invariants, we get a recursive formula for the Hurwitz number of triple ramified coverings of a Riemann surface by a Riemann surface.     

The inverse problem for the Schrödinger equation and the Riemann boundary problem

A new class of reflection finite-gap potentials for the one-dimensional Schrödinger equation is investigated. The inverse problem for this class is reduced to the 2×2-matrix Riemann boundary problem on a hyperelliptic Riemann surface.     

Generating Functional in CFT and Effective Action for Two-Dimensional Quantum Gravity on Higher Genus Riemann Surfaces

We formulate and solve the analog of the universal Conformal Ward Identity for the stress-energy tensor on a compact Riemann surface of genus g > 1, and present a rigorous invariant formulation of the chiral sector in the induced two-dimensional gravity on higher genus Riemann surfaces. Our construction of the action functional uses various double complexes naturally associated with a Riemann surface, with computations that are quite similar to descent calculations in BRST cohomology theory. We also provide an interpretation of the action functional in terms of the geometry of different fiber spaces over the Teichmüller space of compact Riemann surfaces of genus g > 1. Received: 12 September 1996 / Accepted: 6 January 1997     

Laplace–Beltrami Operator on a Riemann Surface¶and Equidistribution of Measures

We consider the Laplace–Beltrami operator on a compact Riemann surface of a constant negative curvature. For any eigenvalue of the Laplace–Beltrami operator there is an associated sequence of measures on the Riemann surface. These measures naturally appear in Quantum Chaos type questions in the theory of electro-magnetic flow on a Riemann surface. The main result of the paper is the claim that this sequence of measures has the Liouville measure as the (weak^*) limit. We prove a quantitative version of this equidistribution claim. Received: 12 March 2001 / Accepted: 23 April 2001     

The Resultant on Compact Riemann Surfaces

We introduce a notion of the resultant of two meromorphic functions on a compact Riemann surface and demonstrate its usefulness in several respects. For example, we exhibit several integral formulas for the resultant, relate it to potential theory and give explicit formulas for the algebraic dependence between two meromorphic functions on a compact Riemann surface. As a particular application, the exponential transform of a quadrature domain in the complex plane is expressed in terms of the resultant of two meromorphic functions on the Schottky double of the domain.     

Some Remarks on the Cohomology of Krichever–Novikov Algebras

We calculate the continuous cohomology of the Lie algebra of meromorphic vector fields on a compact Riemann surface from the cohomology of the holomorphic vector fields on the open Riemann surface pointed in the poles. This cohomology has been given by Kawazumi. Our result shows the Feigin–Novikov conjecture.     

The Green’s functions of the boundaries at infinity of the hyperbolic 3-manifolds

The work is motivated by a result of Manin in [1], which relates the Arakelov Green’s function on a compact Riemann surface to configurations of geodesics in a 3-dimensional hyperbolic handlebody with Schottky uniformization, having the Riemann surface as a conformal boundary at infinity. A natural question is to what extent the result of Manin can be generalized to cases where, instead of dealing with a single Riemann surface, one has several Riemann surfaces whose union is the boundary of a hyperbolic 3-manifold, uniformized no longer by a Schottky group, but by a Fuchsian, quasi-Fuchsian, or more general Kleinian group. We have considered this question in this work and obtained several partial results that contribute towards constructing an analog of Manin’s result in this more general context.     

Discrete Riemann Surfaces and the Ising Model

We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward discretisation of the Cauchy–Riemann equation. A lot of classical results in Riemann theory have a discrete counterpart, Hodge star, harmonicity, Hodge theorem, Weyl's lemma, Cauchy integral formula, existence of holomorphic forms with prescribed holonomies. Giving a geometrical meaning to the construction on a Riemann surface, we define a notion of criticality on which we prove a continuous limit theorem. We investigate its connection with criticality in the Ising model. We set up a Dirac equation on a discrete universal spin structure and we prove that the existence of a Dirac spinor is equivalent to criticality. Received: 23 May 2000/ Accepted: 21 November 2000     

Hyper-Kähler induction and self-duality on Riemann surfaces

Universal hyper-Kähler spaces are constructed from Lie groups acting on flat Kähler manifolds. These spaces are used to describe the moduli space of solutions of Hitchin's equation — self-duality equations on a Riemann surface — as the contangent bundle of the moduli space of flat connections on a Riemann surface.     

On the Schwinger model on Riemann surfaces

In this paper the Schwinger model or two-dimensional quantum electrodynamics is exactly solved on a Riemann surface providing the explicit expression of the partition function and of the generating functional of the amplitudes between the fermionic currents. This offers one of the few examples in which it is possible to integrate in an explicit way a gauge field theory interacting with matter on a Riemann surface.     

Zeta functions that hear the shape of a Riemann surface

To a compact hyperbolic Riemann surface, we associate a finitely summable spectral triple whose underlying topological space is the limit set of a corresponding Schottky group, and whose “Riemannian” aspect (Hilbert space and Dirac operator) encode the boundary action through its Patterson–Sullivan measure. We prove that the ergodic rigidity theorem for this boundary action implies that the zeta functions of the spectral triple suffice to characterize the (anti-)complex isomorphism class of the corresponding Riemann surface. Thus, you can hear the complex analytic shape of a Riemann surface, by listening to a suitable spectral triple.     

SYZ duality for parabolic Higgs moduli spaces

We prove the SYZ (Strominger–Yau–Zaslow) duality for the moduli space of full flag parabolic Higgs bundles over a compact Riemann surface. In Hausel and Thaddeus (2003) [12], the SYZ duality was proved for moduli spaces of Higgs vector bundles over a compact Riemann surface.     

Computing the Abel map

We present the next step in an ongoing research program to allow for the black-box computation of the so-called finite-genus solutions of integrable differential equations. This next step consists of the black-box computation of the Abel map from a Riemann surface to its Jacobian. Using a plane algebraic curve representation of the Riemann surface, we provide an algorithm for the numerical computation of this Abel map. Since our plane algebraic curves are of arbitrary degree and may have arbitrary singularities, the Abel map of any connected compact Riemann surface may be obtained in this way. This generality is necessary in order for these algorithms to be relevant for the computation of the finite-genus solutions of any integrable equation.     

Magnetic Monopoles over Topologically Nontrivial Riemann Surfaces

An explicit canonical construction of monopole connections on nontrivial U(1) bundles over Riemann surfaces of any genus is given. The class of monopole solutions depends on the conformal class of the given Riemann surface and a set of integer weights. The reduction of Seiberg--Witten 4-monopole equations to Riemann surfaces is performed. It is then shown that the monopole connections constructed are solutions to these equations.