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.     

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.     

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     

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.     

Generating Functional in CFT on Riemann Surfaces II: Homological Aspects

We revisit and generalize our previous algebraic construction of the chiral effective action for Conformal Field Theory on higher genus Riemann surfaces. We show that the action functional can be obtained by evaluating a certain Deligne cohomology class over the fundamental class of the underlying topological surface. This Deligne class is constructed by applying a descent procedure with respect to a Čech resolution of any covering map of a Riemann surface. Detailed calculations are presented in the two cases of an ordinary Čech cover, and of the universal covering map, which was used in our previous approach. We also establish a dictionary that allows to use the same formalism for different covering morphisms. The Deligne cohomology class we obtain depends on a point in the Earle–Eells fibration over the Teichmüller space, and on a smooth coboundary for the Schwarzian cocycle associated to the base-point Riemann surface. From it, we obtain a variational characterization of Hubbard's universal family of projective structures, showing that the locus of critical points for the chiral action under fiberwise variation along the Earle–Eells fibration is naturally identified with the universal projective structure. Received: 29 June 2000 / Accepted: 16 January 2002     

Selberg supertrace formula for super Riemann surfaces

This paper is the third in a sequel to develop a super-analogue of the classical Selberg trace formula, the Selberg supertrace formula. It deals with bordered super Riemann surfaces. The theory of bordered super Riemann surfaces is outlined, and the corresponding Selberg supertrace formula is developed. The analytic properties of the Selberg super zeta-functions on bordered super Riemann surfaces are discussed, and super-determinants of Dirac-Laplace operators on bordered super Riemann surfaces are calculated in terms of Selberg super zeta-functions.Address from August 1993: II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany     

A geometrical model for the double octahedral group based on a genus two Riemann surface

Geometrical models of double point groups are provided by double-sheeted Riemann surfaces of nonzero genus. This avoids the geometrically confusing picture of the doubling operation R as a rotation by 2π (i.e. 360°), which should instead be the identity operation E. In this manner a Riemann surface of genus 2 doubly covering a sphere and platonically tessellated b 16 equilateral triangles provides a geometrical model for the double octahedral group ²O_h. Stretching this Riemann surface along one axis to convert the 16 equilateral triangles to isosceles triangles and the underlying sphere to a prolate ellipsoid provides a model for the ²D_4h double group arising from the Jahn-Teller elongation of the regular octahedron into a prolate tetragonal bipyramid.     

The Modular Duality of Specific Heat on Genus g Riemann Surface

The free energy on genus g Riemann surface is derived by introducing two-dimensional scalar field and using Vafa's operator approach.The modular duality relation of specific heat is also obtained.     

The Riemann Surface of the Chiral Potts Model Free Energy Function

In a recent paper we derived the free energy or partition function of the N-state chiral Potts model by using the infinite lattice inversion relation method, together with a non-obvious extra symmetry. This gave us three recursion relations for the partition function per site T _pq of the infinite lattice. Here we use these recursion relations to obtain the full Riemann surface of T _pq . In terms of the t _p ,t _q variables, it consists of an infinite number of Riemann sheets, each sheet corresponding to a point on a (2N–1)-dimensional lattice (for N>2). The function T _pq is meromorphic on this surface: we obtain the orders of all the zeros and poles. For N odd, we show that these orders are determined by the usual inversion and rotation relations (without the extra symmetry), together with a simple linearity ansatz. For N even, this method does not give the orders uniquely, but leaves only [(N+4)/4] parameters to be determined.     

Construction of constant curvature punctured Riemann surfaces with particle-scattering interpretation

A class of punctured constant curvature Riemann surfaces, with boundary conditions similar to those of the Poincaré half plane, is constructed. It is shown to describe the scattering of particle-like objects in two Euclidian dimensions. The associated time delays and classical phase shifts are introduced and connected to the behaviour of the surfaces at their punctures. For each such surface, we conjecture that the time delays are partial derivatives of the phase shift. This type of relationship, already known to be correct in other scattering problems, leads to a general integrability condition concerning the behaviour of the metric in the neighbourhood of the punctures. The time delays are explicitly computed for three punctures, and the conjecture is verified. The result, reexpressed as a product of Riemann zeta-functions, exhibits an intringuing number-theoretic structure: a p-adic product formula holds and one of Ramanujan's identities applies. An ansatz is given for the corresponding exact quantum S-matrix. It is such that the integrability condition is replaced by a finite difference relation only involving the exact spectrum already derived, in the associated Liouville field theory, by Gervais and Neveu.     

Local index theorem for orbifold Riemann surfaces

We derive a local index theorem in Quillen’s form for families of Cauchy–Riemann operators on orbifold Riemann surfaces (or Riemann orbisurfaces) that are quotients of the hyperbolic plane by the action of cofinite finitely generated Fuchsian groups. Each conical point (or a conjugacy class of primitive elliptic elements in the Fuchsian group) gives rise to an extra term in the local index theorem that is proportional to the symplectic form of a new Kähler metric on the moduli space of Riemann orbisurfaces. We find a simple formula for a local Kähler potential of the elliptic metric and show that when the order of elliptic element becomes large, the elliptic metric converges to the cuspidal one corresponding to a puncture on the orbisurface (or a conjugacy class of primitive parabolic elements). We also give a simple example of a relation between the elliptic metric and special values of Selberg’s zeta function.

     

Generalized Verlinde Formulae for Some Riemann Surface Automorphisms

The generalized Verlinde formulae expressing traces of mapping classes corresponding to automorphisms of certain Riemann surfaces, and the congruence relations on allowed modular representations following from them are presented. The surfaces considered are families of algebraic curves given by suitably chosen equations, the modular curve X(11), and a factor curve of X(8). The examples of modular curves illustrate how the study of arithmetic properties of suitable modular representations can be used to gain information on automorphic properties of Riemann surfaces.     

Graded Riemann surfaces and Krichever-Novikov algebras

Following the work of Krichever and Novikov, Bonora, Martellini, Rinaldi and Russo defined a superalgebra associated to each compact Riemann surface with spin structure. Noting that this data determines a graded Riemann surface, we find a natural interpretation of the BMRR-algebra in terms of the geometry of graded Riemann surfaces. We also discuss the central extensions of these algebras (correcting the form of the central extension given by Bonoraet al.). It is hoped that this work will be the first step towards defining Krichever-Novikov algebras for (the more general) super-Riemann surfaces; in particular we emphasise the importance ofgraded conformal vectorfields.     

Yang—Mills instantons over Riemann surfaces

Exact solutions to the self-dual Yang—Mills equations over Riemann surfaces of arbitrary genus are constructed. They are characterized by the conformal class of the Riemann surface. They correspond to U(1) instantonic solutions for an Abelian-Higgs system. A functional action of a genus g Riemann surface is constructed, with minimal points being the two-dimensional self-dual connections. The exact solutions may be interpreted as connecting initial and final nontrivial vacuum states of a conformal theory, in the sense of Segal, with a Feynman functor constructed from the functional integral of the action.     

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.     

Wave Fronts for Hamilton-Jacobi Equations:¶The General Theory for Riemann Solutions in

The Hamilton-Jacobi equation describes the dynamics of a hypersurface in . This equation is a nonlinear conservation law and thus has discontinuous solutions. The dependent variable is a surface gradient and the discontinuity is a surface cusp. Here we investigate the intersection of cusp hypersurfaces. These intersections define (n-1)-dimensional Riemann problems for the Hamilton-Jacobi equation. We propose the class of Hamilton-Jacobi equations as a natural higher-dimensional generalization of scalar equations which allow a satisfactory theory of higher-dimensional Riemann problems. The fist main result of this paper is a general framwork for the study of higher-dimensional Riemann problems for Hamilton-Jacobi equations. The purpose of the framwork ist to unterstand the structure of Hamilton-Jacobi wave interactions in an explicit and constructive manner. Specialized to two-dimensional Riemann problems (i.e., the intersection of cusp curves on surfaces embedded in ), this framework provides explicit solutions to a number of cases of interest. We are specifically interested in models of deposition and etching, important processes for the manufacture of semiconductor chips. We also define elementary waves as Riemann solutions which possess a common group velocity. Our second main result, for elementary waves, is a complete characterization in terms of algebraic constraints on the data. When satisfied, these constraints allow a consistently defined closed form expression for the solution. We also give a computable characterization for the admissibility of an elementary wave which is inductive in the codimension of the wave, and which generalizes the classical Oleinik condition for scalar conservation laws in one dimension. Received: 9 September 1996 / Accepted: 22 April 1997     

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