Beltrami Differentials Research Articles

Light cone Wn geometry and its symmetries and projective field theory

It is shown that the generalized Beltrami differentials and projective connections which appear naturally in induced light cone Wn gravity are geometrical fields parametrizing in one-to-one fashion generalized projective structures on a fixed base Riemann surface. It is also shown that Wn symmetries are nothing but gauge transformations of the flat SL(n,c) vector bundles canonically associated with the generalized projective structures. This provides an original formulation of classical light cone Wn geometry. From the knowledge of the symmetries, the full BRS algebra is derived. Inspired by the results of literature, it is argued that quantum Wn gravity may be formulated as an induced gauge theory of generalized projective connections. This leads to projective field theory. The possible anomalies arising at the quantum level are analysed by solving Wess-Zumino consistency conditions. The implications for induced covariant Wn gravity are briefly discussed. The results presented, valid for arbitrary n, reproduce those obtained for n=2,3 by different methods.

Holomorphic mappings into Teichmüller spaces

Let T ( F ) T(F) be the Teichmüller space of a finitely generated Fuchsian group F F of the first kind. We shall show that there exists no nontrivial family of holomorphic proper mappings of the unit disc into T ( F ) T(F) with a complex analytic parameter. Using this result, we shall investigate a family of quasi-conformal mappings with a complex analytic parameter whose Beltrami differentials vanish on a fixed set of positive area.

Holomorphic families of geodesic discs in infinite dimensional Teichmüller spaces

The theory of quasiconformal mappings plays an important role in Teichmüller theory. The Teichmüller spaces of Riemann surfaces are defined as quotient spaces of the spaces of Beltrami differentials, and the Teichmüller distances are defined to measure quasiconformal deformations between the Riemann surfaces representing points in the Teichmüller spaces. The Teichmüller spaces are complex Banach manifolds equipped with natural complex structures such that the canonical projections are holomorphic. It is known (see Gardinar [4]) that the Teichmüller distance, defined independently of the complex structures, coincides with the Kobayashi distance.In spite of the naturality of the definition of a Teichmüller space as a quotient of Beltrami differentials, for given two Beltrami differentials it is hard to determine whether they are equivalent or not. For this reason, it is not trivial to describe geodesic lines with respect to the Teichmüller-Kobayashi metric.

Super Beltrami differentials

Superconformal structures on a given 2/2-dim supermanifold give rise to the notion of super Riemann surfaces (SRS's). We investigate super Beltrami coefficients which parametrize (almost) superconformal structures on the supermanifold. The integrability condition of the structure reduces to a simple relation among the coefficients. Taking this into account, we can write the super Beltrami equations in a transparent form. Then an analysis of these equations enables us to prove the possibility of the special gauge choice of the Wess-Zumino type for the super Beltrami differentials. This gauge choice simplifies the description of the deformations of SRS's considerably and its existence will afford a better understanding of the structure of the super Teichmuller space.

On holomorphic factorization for free conformal fields

For free spin j conformal fields on a compact Riemann surface without boundary, we prove, locally over the space of complex structures - parametrized by Beltrami differentials - the existence of a class of partition functions which are holomorphically factorized, and whose diffeomorphism anomalies split holomorphically. For a collection of such models with vanishing total central charge, one recovers the Belavin-Knizhnik theorem.

Nonuniqueness of geodesics in infinite dimensional teichmuller spaces

Two extremal Beltrami differentials μ0 and μ1 with the following properties are constructed: μ0 and μ1 represent a same point P in the universal Teichmuller space, while the paths tμ0 and tμ1. 0 ≤ t ≤ 1, represent different geodesics joining the original point to the point P. This answers an open problem on the uniqueness of geodesics in infinite dimensional Teichmiiller spaces proposed by E P. Gardiner. Moreover, it is shown that one can get infinitely many geodesis joining two points in such a way.

Construction of topological W 3 gravity

Topological W 3 gravity is constructed in detail from its definition as topological Yang-Mills theory in two dimensions with the gauge group being a contraction of SL(3, R ). The gauge transformation of the Yang-Mills gauge field can be rewritten to resemble that of the spin-2 and -3 gauge fields describing W 3 gravity. This fact provides a local parametrization of the moduli space of the flat connections in terms of spin-2 and -3 Beltrami differentials on Riemann surfaces. The variations of the action with respect to the moduli give the stress tensor and the spin-3 W current, and the BRST charge can be expressed in terms of these currents in a way similar to that of W 3 gravity. The physical content of the model is briefly discussed.

Beltrami differentials, conformal models and their supersymmetric generalizations

Conformally invariant couplings of two-dimensional field theories to gravity can be formulated either in a Riemannian surface or manifold (metric) framework. After a detailed presentation of the Riemannian surface approach (and comparison with the metric formalism), the authors develop its supersymmetric generalization. Super Beltrami differentials are introduced without any reference to metric (vielbein) structures and superconformal models are studied in this framework. The main goal of the work consists in the construction of local (and supersymmetric) field theories defined on arbitrary (super-)Riemann surfaces and exhibiting the (super-)holomorphic factorization in a manifest way.

Beltrami differentials and 2d (super)gravity in the chiral gauge

We consider the Beltrami parametrization of a 2-dimensional metric and work out the basic transformation laws of the constituting fields. These results are then applied to investigate the geometric structure of 2-dimensional gravity in Polyakov's light-cone gauge. Subsequently, we elaborate on the supersymmetric generalization of this theory. The general relation to the superconformai gauge is presented in detail and the corresponding change of coordinates is shown to satisfy simple super Beltrami equations.

Unidexterous supergravity, Beltrami parametrization and BRST quantization

For 2-dimensional (1, 0) supergravity we work out the explicit solution of the constraints on the torsion and show that the corresponding prepotentials represent a superspace description of Beltrami differentials. Since these variables do not satisfy any constraints they allow for a gauge fixing and BRST quantization in superspace. By projection to space-time we obtain component field expressions which can be related to previously found results. Complete coincidence with this formulation is found in Polyakov's light-cone gauge for which our superspace theory takes a particularly simple form.

Super Beltrami differentials via 2D supergravity

Super Beltrami equations which describe deformations of super Riemann surfaces are given. Two kinds of super Beltrami differentials of weight (− 1 2 , 0) and (−1, 1 2 ) respectively are introduced, which exactly agrees with 2D supergravity theory. Solutions are given for the linearized Beltrami equations. Mappings between the super Beltrami differentials and super 3 2 -differentials are given properly.

Super Riemann surfaces: Uniformization and Teichm�ller theory

Teichmuller theory for super Riemann surfaces is rigorously developed using the supermanifold theory of Rogers. In the case of trivial topology in the soul directions, relevant for superstring applications, the following results are proven. The super Teichmuller space is a complex super-orbifold whose body is the ordinary Teichmuller space of the associated Riemann surfaces with spin structure. For genusg>1 it has 3g-3 complex even and 2g-2 complex odd dimensions. The super modular group which reduces super Teichmuller space to super moduli space is the ordinary modular group; there are no new discrete modular transformations in the odd directions. The boundary of super Teichmuller space contains not only super Riemann surfaces with pinched bodies, but Rogers supermanifolds having nontrivial topology in the odd dimensions as well. We also prove the uniformization theorem for super Riemann surfaces and discuss their representation by discrete supergroups of Fuchsian and Schottky type and by Beltrami differentials. Finally we present partial results for the more difficult problem of classifying super Riemann surfaces of arbitrary topology.

Perturbative BRST current algebra anomalies for the free bosonic string

A gauged version of the BRST symmetry for a general class of gauge-fixed actions for the bosonic string is used to study the possible anomalies of the corresponding BRST current algebra. One finds that they are rigidly linked to the trace anomaly of the initial theory. A similar analysis is performed for the BRST current algebra associated with the “conformal-factorized theory” obtained by introducing Beltrami differentials. One finds that the factorization is maintained in the anomalous current algebra theory; again, all the current-algebra anomalies are rigidly linked to the “holomorphy” anomaly. None of these current algebras is anomalous in the critical dimension.

Beltrami parametrization and string theory

By introducing Beltrami differentials to describe the conformal structure of the worldsheet, we prove the factorization of two-dimensional diffeomorphisms and of the corresponding BRS symmetry which is relevant for the string theory. Weyl invariance appears as a fundamental symmetry of the problem. Factorization of the connected tree diagrams is proven.

Teichmüller deformations of super Riemann surfaces

Exact solutions are obtained for the linearized super Beltrami equations which describe infinitesimal deformations of super Reimann surfaces. The tangent and cotangent spaces to super Teichmüller space, as well as the super Weil-Petersson metric, are described in terms of Beltrami differentials and superconformal tensors of weight 3/2. Martinec's approach to super Teichmüller deformations is thereby derived from that of Crane and Rabin.

N = 2 super-Riemann surfaces

Super-Riemann surfaces are generalized to N ⩾ 2. Basis differentials and tensors are defined. For even N, a distinction between general O( N) surfaces and SO( N) (untwisted) surfaces arises. This is treated in detail for N = 2, and some examples are given. There is a short discussion of the N = 2 extension of SL 2 and Beltrami differentials.

On partially Teichmüller Beltrami differentials.

On partially Teichmüller Beltrami differentials.