Noncommutative coordinates on Riemann surfaces

We show that the definition of a projective coordinate frame within a Laguerre–Forsyth scheme, leads to the extension of the factorized diffeomorphism algebra. The quantum improvement of this symmetry can be performed only if these coordinates switch, at the quantum level, into a noncommutative regime.     

Reduction of 4-dim self-dual super Yang-Mills equations onto super Riemann surfaces

Recently, a self-dual super Yang-Mills equation over a super Reimann surface was obtained as the zero set of a moment map on the space of superconnections to the dual of the super Lie algebra of gauge transformations. We present a new formulation of the 4-dim Euclidean self-dual super Yang-Mills equations in terms of constraints on the supercurvature. By dimensional reduction, we obtain the same set of superconformal field equations which define self-dual connections on a super Rieman surface.     

Integrability of coupled KdV equations

The integrability of coupled KdV equations is examined. The simplified form of Hirota’s bilinear method is used to achieve this goal. Multiple-soliton solutions and multiple singular soliton solutions are formally derived for each coupled KdV equation. The resonance phenomenon of each model will be examined.     

Nonlinear holomorphic supersymmetry on Riemann surfaces

We investigate the nonlinear holomorphic supersymmetry for quantum-mechanical systems on Riemann surfaces subjected to an external magnetic field. The realization is shown to be possible only for Riemann surfaces with constant curvature metrics. The cases of the sphere and Lobachevski plane are elaborated in detail. The partial algebraization of the spectrum of the corresponding Hamiltonians is proved by the reduction to one-dimensional quasi-exactly solvable families. It is found that these families possess the “duality” transformations, which form a discrete group of symmetries of the corresponding 1D potentials and partially relate the spectra of different 2D systems. The algebraic structure of the systems on the sphere and hyperbolic plane is explored in the context of the Onsager algebra associated with the nonlinear holomorphic supersymmetry. Inspired by this analysis, a general algebraic method for obtaining the covariant form of integrals of motion of the quantum systems in external fields is proposed.     

A Polyakov action on Riemann surfaces

We present a calculation of the effective action for induced conformal gravity on higher genus Rieman surfaces. Our expression, generalizing Polyakov's formula, depends holomorphically on the Beltrami differential and integrates the diffeomorphism anomaly.     

Global vertex operators on Riemann surfaces

We develop an approach towards construction of conformal field theory starting from the basic axioms of vertex operator algebras.     

Line bundles on super Riemann surfaces

We give the elements of a theory of line bundles, their classification, and their connections on super Riemann surfaces. There are several salient departures from the classical case. For example, the dimension of the Picard group is not constant, and there is no natural hermitian form on Pic. Furthermore, the bundles with vanishing Chern number aren't necessarily flat, nor can every such bundle be represented by an antiholomorphic connection on the trivial bundle. Nevertheless the latter representation is still useful in investigating questions of holomorphic factorization. We also define a subclass of all connections, those which are compatible with the superconformal structure. The compatibility conditions turn out to be constraints on the curvature 2-form.     

Global Laurent expansions on Riemann surfaces

We discuss global Laurent expansions for meromorphich-forms on a compact Riemann surface of genusg2. Our approach is motivated by Krichever and Novikov's work on string theory.Supported in part by the Department of Energy under Grant DE-FG02-88ER25065     

Krichever-Novikov-like bases on punctured Riemann surfaces

Bases of holomorphic -differentials on N-punctured Riemann surfaces of arbitrary genus are constructed. The resulting extension of the Virasoro algebra on N-punctured spheres is displayed explicitly.     

On determinants of Laplacians on Riemann surfaces

Determinants of Laplacians on tensors and spinors of arbitrary weights on compact hyperbolic Riemann surfaces are computed in terms of values of Selberg zeta functions at half integer points.Research supported in part by the U.S. Department of EnergyResearch supported in part by the National Science Foundation under Grant DMS-84-02710     

Determinants of Laplace-like operators on Riemann surfaces

We calculate determinants of second order partial differential operators defined on Riemann surfaces of genus greater than one using a relation between Selberg's zeta function and functional determinants. In addition, we perform a calculation of these determinants directly using Selberg's trace formula, and compare our results with previous computations which followed the latter route.     

Moduli of super Riemann surfaces

The basic properties of super Riemann surfaces are presented, and their supermoduli spaces are constructed, in a manner suitable for the application of algebro-geometric techniques to string theory.Supported in part by NSF Grant No. DMS-8704401Supported in part by NSF Grants No. DMS-8501783 and No. DMS-86107301(1)     

Analytic fields on Riemann surfaces. II

The properties of analytic fields on a Riemann surface represented by a branch covering of ¹ are investigated in detail. Branch points are shown to correspond to the vertex operators with simple conformal properties. As applications we compute determinants of operators forZ _n -symmetric surfaces and obtain various representations for the two-loop measure in the bosonic string theory together with various identities for theta-functions of hyperelliptic surfaces. We also present an integral representation for the quantum part of the twist field correlation functions, which describe propagation of the string on the orbifold background. We also calculate the quantum part of the structure constants of the twist-field operator algebra, generalizing the results of Dixon, Friedan, Martinec, and Shenker.     

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.     

Geodesic length spectrum on compact Riemann surfaces

In this paper we use techniques linking combinatorial structures (symbolic dynamics) and algebraic–geometric structures to study the variation of the geodesic length spectrum, with the Fenchel–Nielsen coordinates, which parametrize the surface of genus τ=2$\tau =2$. We explicitly compute length spectra, for all closed orientable hyperbolic genus two surfaces, identifying the exponential growth rate and the first terms of a growth series.