Beltrami Differentials Research Articles

Variation of geodesic length functions in families of Kähler-Einstein manifolds and applications to Teichmüller space

In the study of Teichmuller spaces the second variation of the logarithm of the geodesic length function plays a central role. So far, it was accessible only in a rather indirect way. We treat the problem directly in the more general framework of the deformation theory of Kahler–Einstein manifolds. For the first variation we arrive at a surprisingly simple formula, which only depends on harmonic Kodaira–Spencer forms. We also compute the second variation in the general case and then apply the result to families of Riemann surfaces. Again we obtain a simple formula depending only on the harmonic Beltrami differentials. As a consequence a new proof for the plurisubharmonicity of the geodesic length function on Teichmuller space and its logarithm together with upper estimates follow. The results also apply to the previously not known cases of Teichmuller spaces of weighted punctured Riemann surfaces, where the methods of Kleinian groups are not available. We use our methods from [A-S], where the result was announced.

First order string theory and the Kodaira-Spencer equations. II

The first-order bosonic string theory, perturbed by primary operator, corresponding to the deformation of target-space complex structure is considered. We compute the correlation functions in this theory and study their divergencies. It is found, that consistency of these correlation functions with the world-sheet conformal invariance requires the Kodaira-Spencer equations to be satisfied by target-space Beltrami differentials. This statement is checked explicitly for the three-point and four-point correlators, containing one probe operator. We discuss the origin of these divergences and their relation with beta-functions or effective action and polyvertex structures in BRST approach.

First order string theory and the Kodaira-Spencer equations. I

We consider first-order bosonic string theory, perturbed by the primary operator, corresponding to deformation of the target-space complex structure. We compute the effective action in this theory and find that its consistency with the world-sheet conformal invariance requires necessarily the Kodaira-Spencer equations to be satisfied by target-space Beltrami differentials. We discuss the symmetries of the theory and its reformulation in terms of the vielbein background fields.

ON INFINITESIMAL TEICHMÜLLER SPACE

AbstractMuch research has been done on the geometry of Teichmüller space and Hamilton sequences of extremal Beltrami differentials. This paper discusses some problems concerning infinitesimal Teichmüller geodesic discs and Hamilton sequences of extremal Beltrami differentials in the tangent space of an infinite-dimensional Teichmüller space.

Some characterizations of unique extremality

In this paper, it is shown that some necessary characteristic conditions for unique extremality obtained by Zhu and Chen are also sufficient and some sufficient ones by them actually imply that the uniquely extremal Beltrami differentials have a constant modulus. In addition, some local properties of uniquely extremal Beltrami differentials are given.

Holomorphic extensions of Laplacians and their determinants

The Teichmüller space Teich ( S ) of a surface S in genus g > 1 is a totally real submanifold of the quasifuchsian space QF ( S ) . We show that the determinant of the Laplacian det ′ ( Δ ) on Teich ( S ) has a unique holomorphic extension to QF ( S ) . To realize this holomorphic extension as the determinant of differential operators on S, we introduce a holomorphic family { Δ μ , ν } of elliptic second order differential operators on S whose parameter space is the space of pairs of Beltrami differentials on S and which naturally extends the Laplace operators of hyperbolic metrics on S. We study the determinant of this family { Δ μ , ν } and show how this family realizes the holomorphic extension of det ′ ( Δ ) as its determinant.

On the unique extremality of quasiconformal mappings with dilatation bounds

Concerning the problem of extremality of quasiconformal mappings with dilatation bounds, we discuss the unique extremality of the problem and prove the if part of a conjecture on the unique extremality. To this end, we need to investigate a new extremal problem in the infinitesimal setting. In particular, we give a complete description of the unique infinitesimal extremality of partially zero Beltrami differentials.

Strebel differentials and Hamilton sequences

Let T ( S ) be a Teichmuller space of a hyperbolic Riemann surface S , viewed as a set of Teichmuller equivalence classes of Beltrami differentials on S . It is shown in this paper that for any extremal Beltrami differential μ 0 at a given point τ of T ( S ), there is a Hamilton sequence for μ 0 formed by Strebel differentials in a natural way. Especially, such a kind of Hamilton sequence possesses some special properties. As applications, some results on point shift differentials are given.

The role of complex structures in w-symmetry

In a symplectic framework, the infinitesimal action of symplectomorphisms together with suitable reparametrizations of the two-dimensional complex base space generate some type of w-algebras. It turns out that complex structures parametrized by Beltrami differentials play an important role in this context. The construction parallels very closely two-dimensional Lagrangian conformal models where Beltrami differentials are fundamental.

On unique extremality *

Let M(R) denote the unit ball of the space L∞(R) of all essentially bounded Beltrami differentials on a Riemann surface R. It is well known that μ∈M(R) is infinitesimally extremal if and only if μ/(1−|μ|2)∈L∞(R) is infinitesimally extremal. In this paper, it is proved that the analogous result holds for the unique extremality, that is, μ∈M(R) is uniquely infmitesimally extremal if and only if μ/(1−|μ|2)∈L∞(R) is uniquely infinitesimally extremal. An application is also given.

On non-separating simple closed curves in a compact surface

We introduce a semi-algebraic structure on the set  of all isotopy classes of non-separating simple closed curves in any compact oriented surface and show that the structure is finitely generated. As a consequence, we produce a natural finite dimensional linear representation of the mapping class group of the surface. Applications to the Teichmüller space, Thurston's measured lamination space, the harmonic Beltrami differentials, and the first cohomology groups of the surface are discussed.

Chiral Lagrangians, anomalies, supersymmetry, and holomorphy

We investigate higher-dimensional analogues of the bc systems of 2D RCFT. When coupled to gauge fields and Beltrami differentials defining integrable holomorphic structures, the bc partition functions can be explicitly evaluated using anomalies and holomorphy. The resulting induced actions generalize the chiral algebras of 2D RCFT to 2 n dimensions. Moreover, bc systems in four and six dimensions are closely related to supersymmetric matter. In particular, we show that d = 4, N = 2 hypermultiplets induce a theory of self-dual Yang-Mills fields coupled to self-dual gravity. In this way the bc systems fermionize both the algebraic sector of the WZW 4 theory, as defined by Losev et al., and the classical open N WS = 2 string.

Covariantising the Beltrami equation in W-gravity

Recently, certain higher dimensional complex manifolds were obtained in [hep-th/9412078] by associating a higher dimensional uniformisation to the generalised Teichm\"uller spaces of Hitchin. The extra dimensions are provided by the ``times'' of the generalised KdV hierarchy. In this paper, we complete the proof that these manifolds provide the analog of superspace for W-gravity and that W-symmetry linearises on these spaces. This is done by explicitly constructing the relationship between the Beltrami differentials which naturally occur in the higher dimensional manifolds and the Beltrami differentials which occur in W-gravity. This also resolves an old puzzle regarding the relationship between KdV flows and W-diffeomorphisms.

UNIFORMIZATION THEORY AND 2D GRAVITY I: LIOUVILLE ACTION AND INTERSECTION NUMBERS

This is the first part of an investigation concerning the formulation of 2D gravity in the framework of the uniformization theory of Riemann surfaces. As a first step in this direction we show that the classical Liouville action appears in the expression for the correlators of topological gravity. Next we derive an inequality involving the cutoff of 2D gravity and the background geometry. Another result, still related to uniformization theory, concerns a relation between the higher genus normal ordering and the Liouville action. We introduce operators covariantized by means of the inverse map of uniformization. These operators have interesting properties, including holomorphicity. In particular, they are crucial for showing that the chirally split anomaly of CFT is equivalent to the Krichever-Novikov cocycle and vanishes for deformation of the complex structure induced by the harmonic Beltrami differentials. By means of the inverse map we propose a realization of the Virasoro algebra on arbitrary Riemann surfaces and find the eigenfunctions for the holomorphic covariant operators defining higher order cocycles and anomalies which are related to W algebras. Finally we face the problem of considering the positivity of eσ, with σ the Liouville field, by proposing an explicit construction for the Fourier modes on compact Riemann surfaces. These functions, whose underlying number-theoretic structure seems related to Fuchsian groups and to the eigenvalues of the Laplacian, are quite basic and may provide the building blocks for properly investigating the long-standing uniformization problem posed by Klein, Koebe and Poincaré.

On non-separating simple closed curves in a compact surface

Abstract We introduce a semi-algebraic structure on the set  of all isotopy classes of non-separating simple closed curves in any compact oriented surface and show that the structure is finitely generated. As a consequence, we produce a natural finite dimensional linear representation of the mapping class group of the surface. Applications to the Teichmuller space, Thurston's measured lamination space, the harmonic Beltrami differentials, and the first cohomology groups of the surface are discussed.

Deformation theory of holomorphic vector bundles, extended conformal symmetry and extensions of 2D gravity

Developing the ideas of Stora and coworkers, a formulation of two-dimensional field theory endowed with extended conformal symmetry is given, which is based on deformation theory of holomorphic and Hermitian spaces. The geometric background consists of a vector bundle E over a closed surface endowed with a holomorphic structure and a Hermitian structure subordinated to it. The symmetry group is the semidirect product of the automorphism group Aut(E) of E and the extended Weyl group Weyl(E) of E and acts on the holomorphic and Hermitian structures. The extended Weyl anomaly can be shifted into a chirally split automorphism anomaly by adding to the action a local counterterm, as in ordinary conformal field theory. The dependence on the scale of the metric on the fibre of E is encoded in the Donaldson action, a vector bundle generalization of the Liouville action. The Weyl and automorphism anomaly split into two contributions corresponding, respectively, to the determinant and projectivization of E. The determinant part induces an effective ordinary Weyl or diffeomorphism anomaly and the induced central charge can be computed. As an application, it is shown that to any embedding t of into a simple Lie algebra and any faithful representation R of one can naturally associate a flat vector bundle DS(t,R) on . It is further shown that there is a deformation of the holomorphic structure of such a bundle whose parameter fields are generalized Beltrami differentials of the type appearing in light-cone W geometry, and that the projective part of the automorphism anomaly reduces to the standard W anomaly in the large central charge limit. A connection between the Donaldson action and Toda field theory is also observed.

New linear systems for 2d Poincaré supergravities

A new linear system is constructed for Poincaré supergravities in two dimensions. In contrast to previous results, which were based on the conformal gauge, this linear system involves the topological world sheet degrees of freedom (the Beltrami and super-Beltramidifferentials). The associated spectral parameter likewise depends on these and is itself subject to a pair of differential equations, whose integrability condition yields one on the equations of motion. These results suggest the existence of an extension of the Geroch group mixing propagating and topological degrees of freedom on the world sheet. We also develop a chiral tensor formalism for arbitrary Beltrami differentials, in which the factorization of 2d diffeomorphisms is always manifest.

SUPERCONFORMAL STRUCTURES AND HOLOMORPHIC 1/2-SUPERDIFFERENTIALS ON N=1 SUPER RIEMANN SURFACES

Using the super Riemann-Roch theorem we give a local expression for a holomorphic ½-superdifferential in a superconformal structure parametrized by special isothermal coordinates on an N=1 super Riemann surface. The holomorphy of these coordinates with respect to super Beltrami differentials is proved. The monodromy of these differentials is discussed.

Beltrami differentials and ghost correlators in the Schottky parametrization

We derive explicit expressions in the Schottky parametrization for the periods of the harmonic Beltrami differentials on an arbitrary Riemann surface. Together with the N-point g-loop vertex for the reparametrization ghost system this allows a direct calculation of any ghost correllator encountered in the picture changing operator approach to superstring theory.

(2,0) Super-Beltrami parametrization and superconformal structures

The superconformal structure of the (2,0) supergeometry of two-dimensional superspace is characterized by the Beltrami differentials. The superdiffeomorphism transformations of the Beltrami variables are given and the superfields with arbitrary conformal weights are studied.