Arbitrary Riemann Surfaces Research Articles

The partition function of two-dimensional quantum gravity

We propose a candidate partition function of two-dimensional quantum gravity on an arbitrary Riemann surface with boundary. The conjecture is based on the SO(2,1) gauge Chern-Simons theory in three dimensions. We show that when the surface is a disk, the wave function coincides with the one obtained by quantizing the Polyakov induced action of gravity.

TWO-DIMENSIONAL CONFORMAL FIELD THEORY AND THREE-DIMENSIONAL TOPOLOGY

We present a survey of two-dimensional conformal field theory and show how the mathematical structures underlying conformal field theory can be used to construct invariants of links imbedded in a general class of three-dimensional manifolds. After a general introduction, we discuss chiral algebras and their representation theory. Chiral vertices are introduced as analogues of Clebsch-Gordan operators in group theory. Braiding and fusing of chiral vertices is analyzed, and it is sketched how to define conformal field theory on arbitrary Riemann surfaces by a sewing procedure. We then show how to construct link invariants from the data provided by a conformal field theory and sketch connections with quantum group theory.

N-point g-loop vertex for a free bosonic theory with vacuum charge Q

Starting fron the N-point vertex on the sphere and using the sewing procedure we construct the N-point g-loop vertex for a free bosonic theory with vacuum charge Q. We then show that, when this vertex is saturated with N highest weight states, it gives their correlation function on an arbitrary Riemann surface of genus g. We also extend our formalism to the case of a free scalar field compactified on a circle, which is related to the Coulomb gas description of minimal models.

The central extension of Kac-Moody-Malcev algebras

We introduce a class of infinite-dimensional Kac-Moody-Malcev algebras. These algebras are the generalization of Lie algebras of the Kac-Moody type to Malcev algebras. We demonstrate that the central extensions of the Kac-Moody-Malcev algebras are given by the same cocycles as in the case of Lie algebras. Analogues of Kac-Moody-Malcev algebras may be also introduced in the case of an arbitrary Riemann surface.

The N-string amplitude. “Conserved charges” and generalized boundary variations

The method of “conserved charges” allows to construct N-string amplitudes partially. A generalization of this method is proposed, which determines such amplitudes uniquely. It makes essentially use of the fact that generalized boundary variations are generated by the first quantized stress-energy tensor. The N-string amplitude of both the X-coordinates as well as of the b-c coordinates on an arbitrary Riemann surface is constructed explicitly.

Superconformal ghost correlators and picture changing

We compute the correlation functions for the system of superconformal ghosts β, γ (λ = 3 2 ) , including the corresponding spin fields, on arbitrary Riemann surfaces. Using fermionization, defined as a change of variables in the functional integration, we derive and generalize previous results obtained by bosonization. As an application we study the picture changing mechanism in the Ramond sector of the superstring.

Bosonization, τ-functions and operator formalism on Riemann surfaces

We propose an operator bosonization of generalized ghost systems of closed string theory (the bc-system of spin J) on Riemann surfaces of arbitrary genus. The bosonization is carried out at the operator level in terms of Bose conformal field theory operators which are well-defined globally on the Riemann surface and are essentially given by operator-valued Baker-Akhiezer functions. The formalism yields an elementary derivation of higher-loop bc-correlation functions as well as a bosonic operator realization of the algebro-geometric tau function of arbitrary Riemann surfaces. It also enables us to construct explicitly an operator that glues a handle to Riemann surfaces.

BRST QUANTIZATION AND QUASI-FREE FIELD THEORY FOR THE KRICHEVER-NOVIKOV STRING

Based on the BRST quantization, a quasi-free field theory for the Krichever-Novikov string on arbitrary genus Riemann surface has been given in analog to the Siegel approach to free closed string field theory. A non-purtubative formalism has also been suggested.

String ghost number violation in the operator formalism

A systematic procedure of incorporating b− c zero modes into an operator formalism on arbitrary Riemann surfaces is presented. It is demonstrated that transition amplitudes on Riemann surfaces must contain certain fermionic δfunctions to ensure analyticity of the “classical” ghost coordinates. A transition amplitude on a Riemann surface of Euler characteristic χ violates ghost number conservation by −3χ units.

Superstring vertex operators and scattering amplitudes on arbitrary Riemann surfaces

The construction of scattering amplitudes involving arbitrary bosonic mass level states is considered in both the closed superstring and in the heterotic string theories at any order of perturbation. From massless particle scattering on a general Riemann surface, the super-covariant form of the vertex operators is derived via factorization. The super-covariant rules, including the normal ordering prescription, to be used in computing amplitudes, are automatically given by this procedure.

N-string, g-loop vertex for the bosonic string

We construct the N-string, g-loop vertex V N, g for the orbital degrees of freedom of the bosonic string in terms of the first abelian differentials, the period matrix and the prime form. We also build the | g> vacuum recently discussed by many people in the framework of an operator formalism on an arbitrary Riemann surface; our expression also contains the measure that takes into account the ghost contribution.

Simple superconformal field theories in two dimensions

Superconformal field theories with first-order lagrangians, the super BC-systems, are considered on an arbitrary Riemann surface with an arbitrary gravitino field. The corresponding functional integrals are carefully defined, and the Weyl, supersymmetry, and holomorphic anomalies are computed. All the anomalies have a common coefficient. The holomorphic anomaly involves terms quadratic in the gravitino field.

A bosonic representation of the ising model

Abstract A bosonic construction of the critical Ising model is given. The stress-energy tensor has a different form from that of a free scalar field. Equivalence is established on an arbitrary compact Riemann surface. It is argued that this description is realized in a sector of the critical Askin-Teller model.

Non-linear sigma models on arbitrary genus Riemann surfaces

Abstract A Ward-Takahashi type identity is obtained for two insertions of the energy-momentum tensor of the non-linear sigma model on an arbitrary Riemann surface. The identity shows explicitly how the Virasoro algebra is violated by spurious terms generated by the trace anomaly. Requiring these terms to vanish leads to a set of constraints on the graviton and dilaton background fields, which are necessary for the algebra to be restored. Although the modular parameters play an important role in the computation, the background field equations turn out to be genus independent up to order α ′.

The lagrangian formulation of chiral scalars

The lagrangian proposed by Siegel describing left- or right-moving scalars in two dimensions is analysed. The existence of anomalies is pointed out and a modification of the lagrangian leading to the elimination of anomalies is proposed. The generalization for chiral scalars defined on an arbitrary Riemann surface is discussed.

BRST operator on a Riemann surface

We discuss construction of the BRST operator for the bosonic closed string theory on an arbitrary Riemann surface of genus g>1. Using the BRST operator we prove the decoupling of the bosonic spurious physical states in general.

Variational formulas on arbitrary Riemann surfaces under pinching deformation

Variational formulas on arbitrary Riemann surfaces under pinching deformation

Functional determinants on Riemann surfaces

We compute the functional determinant of the laplacian acting on n-differentials ( n is a half-integer) on an arbitrary Riemann surface of genus g > 1. We first compute the propagator of the corresponding fermionic field theory and obtain the energy-momentum tensor. Then we integrate the energy-momentum tensor to obtain the functional determinant.

On the modulus of the free homotopy class of a simple closed curve on an arbitrary Riemann surface

On the modulus of the free homotopy class of a simple closed curve on an arbitrary Riemann surface

Square integrable harmonic differentials on arbitrary Riemann surfaces with a finite number of nodes

Square integrable harmonic differentials on arbitrary Riemann surfaces with a finite number of nodes