International Journal of Modern Physics A

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 e^σ, 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.

It is shown that time-independent solutions to the (2+1)-dimensional nonlinear O(3) sigma model may be placed in correspondence with surfaces of constant mean curvature in three-dimensional Euclidean space. The tools required to establish this correspondence are provided by the classical differential geometry of surfaces. A constant-mean-curvature surface induces a solution to the O(3) model through the identification of the Gauss map, or normal vector, of the surface with the field vector of the sigma model. Some explicit solutions, including the solitons and antisolitons discovered by Belavin and Polyakov, and a more general solution due to Purkait and Ray, are considered and the surfaces giving rise to them are found explicitly. It is seen, for example, that the Belavin-Polyakov solutions are induced by the Gauss maps of surfaces which are conformal to their spherical images, i.e. spheres and minimal surfaces, and that the Purkait-Ray solution corresponds to the family of constant-mean-curvature helicoids first studied by do Carmo and Dajczer in 1982. A generalization of this method to include time dependence may shed new light on the role of the Hopf invariant in this model.

In this paper we study the quantum Kantowski-Sachs universe. We use the requirement that physical states be independent of the choice between the representations of canonical coordinate and conjugate momentum to pick out unphysical states of the universe from the mathematically general solutions of quantum Hamiltonian constraint equations.

We discuss the effect of the renormalization prescription in the computation of the unification point for running coupling constants. We explore the effects of threshold crossing on the β functions. We compute the running of the coupling constants of the Standard Model, between mz and Mp, using a mass-dependent subtraction procedure, and then compare the results with and with the θ function approximation. We also do this for the minimal supersymmetric extension of the Standard Model. In the latter, the bounds on SUSY masses that one obtains by requiring perturbative unification are dependent, to some extent, on the procedure.

We obtain high temperature results for the one-loop effective action for composite fields in interaction with an Abelian gauge field and minimally coupled to gravity in a curved background space-time, using the Vilkovisky-DeWitt approach, by making a local expansion in the Riemann tensor and its derivatives. We also give results for the fields minimally coupled to gravity in an arbitrary curved background space-time at zero temperature, and for the case where the fields are nonminimally coupled to gravity in Euclidean de Sitter space. The results obtained are gauge-invariant and gauge-condition-independent on-shell as well as off-shell.

The behavior of the master field in “induced QCD” near the edge of its support is studied. An extended scaling domain, where the shape of the master field is a universal function, is found. This function is determined explicitly for the case of dimensions close to 1, and the D−1 expansion is constructed. The problem of the meson spectrum corresponding to this solution is analyzed. As a by-product of these calculations, a new, explicit equation for the meson spectrum in induced QCD with a general potential is derived.