New Index Formulas as a Meromorphic Generalization of the Chern–Gauss–Bonnet Theorem

Laplace operators perturbed by meromorphic potential on the Riemann and separated-type Klein surfaces are constructed and their indices are calculated in two different ways. The topological expressions for the indices are obtained from the study of the spectral properties of the operators. Analytical expressions are provided by the heat kernel approach in terms of functional integrals. As a result, two formulae connecting characteristics of meromorphic (real meromorphic) functions and topological properties of Riemann (separated-type Klein) surfaces are derived.     

Meromorphic Continuations of Finite Gap Herglotz Functions and Periodic Jacobi Matrices

We find a necessary and sufficient condition for a Herglotz function m to be the Borel transform of the spectral measure of an exponential decaying perturbation of a periodic Jacobi matrix. The condition is in terms of meromorphic continuation of m to a natural Riemann surface and the structure of its zeros and poles. The analogous result is also established for the Borel transform of the spectral measure of eventually periodic Jacobi matrices. This paper generalizes the corresponding result from the author’s (Constr Approx 36(2):267–309, 2012) for exponential perturbations of the free Jacobi matrix.     

Riemann面上多极点亚纯向量场的代数及其在亚纯λ-微分的实现(Ⅱ)

本文构造了高亏格紧Riemann面上多极点亚纯λ-微分基的一般表达式,并给出了一般方格上亚纯向量场的代数关系。 关键词：     

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.     

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.     

Quasiperiodic Solutions of the Heisenberg Ferromagnet Hierarchy

We present quasi-periodic solutions in terms of Riemann theta functions of the Heisenberg ferromagnet hierarchy by using algebro-geometric method. Our main tools include algebraic curve and Riemann surface, polynomial recursive formulation and a special meromorphic function.     

Meromorphic zeta functions for analytic flows

We extend to hyperbolic flows in all dimensions Rugh's results on the meromorphic continuation of dynamical zeta functions. In particular we show that the Ruelle zeta function of a negatively curved real analytic manifold extends to a meromorphic function on the complex plane.Dedicated to Steve SmaleThis work was supported in part by I.H.E.S. and the National Science Foundation.     

Riemann面上多极点亚纯向量场的代数及其在亚纯λ-微分的实现(Ⅰ)

本文给出球面上亚纯向量场代数的明显构造,并给出这一代数的中心扩充。研究了作为这一代数的实现球面上亚纯λ-微分的性质。     

Representations of the Heisenberg algebra on a Riemann surface

We formulate reflection positivity for meromorphic functions and for 1-forms on a Riemann surface. This construction yields representations of the Heisenberg algebra on a Riemann surface.Supported in part by the Department of Energy under Grant DE-FG02-88ER25065     

Meromorphic continuation approach to noncommutative geometry

Following an idea of Nigel Higson, we develop a method for proving the existence of a meromorphic continuation for some spectral zeta functions. The method is based on algebras of generalized differential operators. The main theorem states, under some conditions, the existence of a meromorphic continuation, a localization of the poles in supports of arithmetic sequences and an upper bound of their order. We give an application in relation to a class of nilpotent Lie algebras.     

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.     

N=2 supersymmetric quantum mechanics on Riemann surfaces with meromorphic superpotentials

We constructN=2 supersymmetric quantum Hamiltonians with meromorphic superpotentials on compact Riemann surfaces and investigate the topological properties of these Hamiltonians.L ₂-cohomology groups for supercharge (a deformed operator) are considered and the Witten index for the supersymmetric Hamiltonian with meromorphic superpotential is calculated in terms of Euler characteristic of the Riemann surface and the degree of a divisor of poles for the differential of the superpotential.This work was supported, in part, by a Soros Foundation Grant awarded by the American Physical Society     

Spectral Determinants on Mandelstam Diagrams

We study the regularized determinant of the Laplacian as a functional on the space of Mandelstam diagrams (noncompact translation surfaces glued from finite and semi-infinite cylinders). A Mandelstam diagram can be considered as a compact Riemann surface equipped with a conformal flat singular metric \({|\omega|^2}\), where \({\omega}\) is a meromorphic one-form with simple poles such that all its periods are pure imaginary and all its residues are real. The main result is an explicit formula for the determinant of the Laplacian in terms of the basic objects on the underlying Riemann surface (the prime form, theta-functions, the canonical meromorphic bidifferential) and the divisor of the meromorphic form \({\omega}\). As an important intermediate result we prove a decomposition formula of the type of Burghelea–Friedlander–Kappeler for the determinant of the Laplacian for flat surfaces with cylindrical ends and conical singularities.     

Riemann surfaces,Clifford algebras and infinite dimensional groups

We introduce a class of Riemann surfaces which possess a fixed point free involution and line bundles over these surfaces with which we can associate an infinite dimensional Clifford algebra. Acting by automorphisms of this algebra is a gauge group of meromorphic functions on the Riemann surface. There is a natural Fock representation of the Clifford algebra and an associated projective representation of this group of meromorphic functions in close analogy with the construction of the basic representation of Kac-Moody algebras via a Fock representation of the Fermion algebra. In the genus one case we find a form of vertex operator construction which allows us to prove a version of the Boson-Fermion correspondence. These results are motivated by the analysis of soliton solutions of the Landau-Lifshitz equation and are rather distinct from recent developments in quantum field theory on Riemann surfaces.     

Pseudodifferential Symbols on Riemann Surfaces and Krichever–Novikov Algebras

We define the Krichever-Novikov-type Lie algebras of differential operators and pseudodifferential symbols on Riemann surfaces, along with their outer derivations and central extensions. We show that the corresponding algebras of meromorphic operators and symbols have many invariant traces and central extensions, given by the logarithms of meromorphic vector fields. Very few of these extensions survive after passing to the algebras of operators and symbols holomorphic away from several fixed points. We also describe the associated Manin triples and KdV-type hierarchies, emphasizing the similarities and differences with the case of smooth symbols on the circle.     

A hierarchy of long wave-short wave type equations: quasi-periodic behavior of solutions and their representation

Based on the Lenard recursion relation and the zero-curvature equation, we derive a hierarchy of long wave-short wave type equations associated with the 3 × 3 matrix spectral problem with three potentials. Resorting to the characteristic polynomial of the Lax matrix, a trigonal curve is defined, on which the Baker-Akhiezer function and two meromorphic functions are introduced. Analyzing some properties of the meromorphic functions, including asymptotic expansions at infinite points, we obtain the essential singularities and divisor of the Baker-Akhiezer function. Utilizing the theory of algebraic curves, quasi-periodic solutions for the entire hierarchy are finally derived in terms of the Riemann theta function.     

The Multi-Pole Neveu-Schwarz Algebra on Riemann Supersphere

According to the Riemann-Roch theorem, we construct bases H_-n⁽ⁱ⁾ and N_-m^(f), for the meromorphic λ = -1 and λ = -1/2 differentials on the Riemann sphere S². The dual bases, A_-n⁽ⁱ⁾, and D_-m^(j), of these meromorphic λ differentials on C_r curves are defined. Expanding the component fields TB(z) and TF(z) of the stress-energy tensor T(z) in the superconformal field theory by the dual bases A_-n⁽ⁱ⁾, and D_-m^(j), respectively, we obtain a series of expanding coefficients. The commutation relations among these coefficients are given explicitly, which,is just the multi-pole Neveu-Schwarz algebra with central extensions on the Riemann supersphere S. Physical implics tions of the algebra are also discussed.     

Existence of a Meromorphic Extension of Spectral Zeta Functions on Fractals

We investigate the existence of the meromorphic extension of the spectral zeta function of a Laplacian on self-similar fractals using the results of Kigami and Lapidus (based on renewal theory) and the newer results by Hambly and Kajino based on heat kernel estimates and other probabilistic techniques. We also formulate conjectures which hold true for the examples that have been analyzed in the existing literature.     

Further Riccati Differential Equations with Elliptic Coefficients and Meromorphic Solutions

We exhibit some families of Riccati differential equations in the complex domain having elliptic coefficients and study the problem of understanding the cases where there are no multivalued solutions. We give criteria ensuring that all the solutions to these equations are meromorphic functions defined in the whole complex plane, and highlight some cases where all solutions are, furthermore, doubly periodic.     

Krichever-Novikov algebras for more than two points

Krichever-Novikov algebras of meromorphic vector fields with more than two poles on higher genus Riemann surfaces are introduced. The structure of these algebras and their induced modules of forms of weight is studied.