Surfaces Of Arbitrary Genus Research Articles

Transitive flows on orientable surfaces

We consider smooth vector fields on closed orientable surfaces with a fixed collection of singularities and a finite number of separatrices none of which connects the equilibrium states. We prove that, on an orientable surface of arbitrary genus g ≥ 2, there exists a vector field with an admissible set of singularities (degenerate saddles) whose trajectory is everywhere dense on the surface.

Abel--Jacobi Mapping for Real Hyperelliptic Riemann Surfaces

We study the degrees of the Abel--Jacobi mapping on hyperelliptic Riemann surfaces of arbitrary genus and the restrictions of the corresponding mappings to the symmetric powers of the real locus of the given Riemann surface.

A Remark on the Stability of Constant Mean Curvature Surfaces of Higher Genus and Circular Boundary

We give sufficient conditions, depending on the degree of the divisor of the Hopf differential, for a constant mean curvature surface of arbitrary genus and circular boundary, to be unstable. The result extends a previous result of Alías, Lopez and the author for genus zero surfaces.

Modularity and total ellipticity of some multiple series of hypergeometric type

We collect some new evidence for the validity of the conjecture that every totally elliptic hypergeometric series is modular invariant and briefly discuss a generalization of such series to Riemann surfaces of arbitrary genus.

The Number of Ramified Coverings of the Sphere by the Double Torus, and a General Form for Higher Genera

An explicit expression is obtained for the generating series for the number of ramified coverings of the sphere by the double torus, with elementary branch points and prescribed ramification type over infinity. Thus we are able to determine various linear recurrence equations for the numbers of these coverings with no ramification over infinity; one of these recurrence equations has previously been conjectured by Graber and Pandharipande. The general form of this series is conjectured for the number of these coverings by a surface of arbitrary genus that is at least two.

AdS membranes wrapped on surfaces of arbitrary genus

We present and analyze solutions of D=11 supergravity describing the “near-horizon” (i.e., asymptotically AdS 4× S 7) geometry of M2-branes wrapped on surfaces of arbitrary genus. We study the forces experienced by test M2-branes in such backgrounds, and find evidence that extremal branes on surfaces of genera higher than the torus are unstable. Using the holographic connection between AdS spaces and superconformal field theories in the large N limit, we discuss the phases of the associated 2+1 dimensional theories. Finally, we also study the extension of these solutions to other branes, in particular to D2-branes.

Kähler Surfaces of Positive Scalar Curvature

We construct Kahler metrics of positive scalar curvature on almost all blown-up ruled surfaces of arbitrary genus. The metrics have an explicit form on ruled surfaces blown up at most twice successively from a minimal model. Our surfaces are generic in the sense that they make up a dense set in the deformations of a given ruled surface.

A Formulation for the Genus Series for Regular Maps

Embedding of vertex regular maps on orientable surfaces have been studied extensively, particularly in the case of low genus where several classes have proved to be tractable. However, the question of determining exact information about embeddings on surfaces of arbitrary genus has proved to be less so. In this paper we show that the genus series for vertex regular maps can be formulated elegantly as the constant term in a series in a way which fully captures information about the genus. It is hoped that the cosntant term formulation will lead to further study of the genus series for regular maps.

Broken supersymmetry in the matrix model on a circle

We consider the discretization of aD=2 surface using polygons. We map the surface onto superspace and integrate over surfaces of arbitrary genus, obtaining a discretized version of the Green-Schwarz string inD=1. Taking an unusual critical limit of the supersymmetric matrix model involved, we construct exact solutions, to all perturbative orders, for the discretized superstring in one dimension, both when the target space is a real line and when the theory is represented in terms of matrix variables on a circle of finite radius. We comment on the behavior of the compactfied perturbative expansion under duality transformations.

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.

HIGHER GENUS WAVEFUNCTION OF ANYON SYSTEM

A method is proposed for constructing the wavefunction of anyons on Riemann surfaces of arbitrary genus. This has been carried out in the framework of computing the correlation function of the chiral vertex operators, involving free scalar fields, with appropriate external momenta. Our technique, as a check, reproduces the already known anyonic wavefunction on the sphere as well as that on the torus.

Constructive methods for higher-genus correlation functions of level-one simply-laced WZW models

Constructive methods for calculating the partition and correlation functions of level-one SU( N) Wess-Zumino-Witten models, valid for any compact Riemann surface, are presented. Both hamiltonian and lagrangian formulations of the fermionic description of s^SU( N) 1, which involves constrained Dirac fermions, are considered. Results are confirmed by comparison with a theory of toroidally compactified free bosons. Bose-Fermi equivalence for s^SU( N) 1 thus exhibited on surfaces of arbitrary genus. In the path integral treatment, modular invariance emerges automatically, while the spectrum is more clearly exhibited in the hamiltonian formulation.

KRICHEVER — NOVIKOV BASES FOR THE BOSONIC STRING

The global operator formalism proposed recently by Krichever and Novikov using meromorphic tensor fields on general Riemann surfaces of arbitrary genus is applied to the bosonic string. The well-known Green's function of the scalar boson fields is obtained in a simple manner from the spectral representation. Proof of Wick's theorem is also given.

Irreducible automorphic functions and then-point functions of conformal field theory

Irreducible automorphic functions for a compact Riemann surface of arbitrary genus are used to expand two- and three-point functions of conformal quantum field theory. The divergence of this expansion at coinciding arguments is studied in particular.

Absence of divergences in type II and heterotic string multi-loop amplitudes

A detailed analysis is given of the two main types of degeneration of Riemann surface of arbitrary genus by domain variational theory. Explicit estimates for first and third Abelian functions are given. These estimates are used to analyse the possible divergences of type II or heterotic superstring multiloop amplitudes for the scattering of massless particles. They are all shown to be finite at arbitrary loop order.