Related papers: Polyakov relation for the sphere and higher genus …

We propose the form of the Liouville action satisfying Polyakov conjecture on the accessory parameters for the hyperbolic singularities on the Riemann sphere.

Liouville field theory on hyperelliptic surface is considered. The partition function of the Liouville field theory on the hyperelliptic surface are expressed as a correlation function of the Liouville vertex operators on a sphere and the…

A consistent method for obtaining a well-defined Polyakov action on the supertorus is presented. This method uses the covariantization of derivative operators and enables us to construct a Polyakov action which is globally defined.

Initiated by Polyakov in his 1981 seminal work, the study of two-dimensional Liouville Conformal Field Theory has drawn considerable attention over the past decades. Recent progress in the understanding of conformal geometry in dimension…

We work out the constraints imposed by SL(2C) invariance for sphere topology and modular invariance for torus topology, on the discretized form of Liouville action in Polyakov's non local covariant form. These are sufficient to completely…

We give a short and rigorous proof of the existence and uniqueness of the solution of Liouville equation with sources, both elliptic and parabolic, on the sphere and on all higher genus compact Riemann surfaces.

We characterize all possible relative positions between a hyperboloid of one sheet and a sphere through the roots of a characteristic polynomial associated to these quadrics. The classification is also suitable for a hyperboloid and a…

We discuss links in thickened surfaces. We define the Khovanov-Lipshitz-Sarkar stable homotopy type and the Steenrod square for the homotopical Khovanov homology of links in thickened surfaces with genus$>1$. A surface means a closed…

We construct the Fukaya category of a surface with genus greater than one and compute its Grothendieck group. We consider here a topological variant, in which we disregard the area form and use instead an admissibility condition borrowed…

The purpose of this paper is to survey the structure of closed and transitive transformation groups acting on a closed surface. In particular, we prove a number of relations between groups acting on the sphere that contain the rotation…

A proof is given of Polyakov conjecture about the auxiliary parameters of the SU(1,1) Riemann-Hilbert problem for general elliptic singularities. Such a result is related to the uniformization of the the sphere punctured by n conical…

For a non-isotrivial family of surfaces of general type over a complex projective curve, we give upper bounds for the degree of the direct images of powers of the relative dualizing sheaf. They imply that, fixing the curve and the possible…

A simple property of the integrals over the hyperelliptic surfaces of arbitrary genus is observed. Namely, the derivatives of these integrals with respect to the branching points are given by the linear combination of the same integrals. We…

Two-dimensional conformal field theory is a powerful tool to understand the geometry of surfaces. Here, we study Liouville conformal field theory in the classical (large central charge) limit, where it encodes the geometry of the moduli…

We show that the structure of the Liouville action on a two dimensional Regge surface of the topology of the sphere and of the torus is determined by the invariance under the transformations induced by the conformal Killing vector fields…

Assume that two algebraic varieties of finite type over the complex numbers are related by a morphism whose fibers are precisely the orbits for the action of a unipotent group. We show that the two varieties have the same topological Euler…

We obtain an explicit expression for the number of ramified coverings of the sphere by the torus with given ramification type for a small number of ramification points, and conjecture this to be true for an arbitrary number of ramification…

The paper describes relations between Liouville type theorems for solutions of a periodic elliptic equation (or a system) on an abelian cover of a compact Riemannian manifold and the structure of the dispersion relation for this equation at…

The present paper is a sequel to our work on hybrid geometry of curves and their moduli spaces. We introduce a notion of hybrid Laplacian, formulate a hybrid Poisson equation, and give a mathematical meaning to the convergence both of the…

We give upper bounds of the numbers of holomorphic sections of Veech holomorphic families of Riemann surfaces. The numbers depend only on the topological types of base Riemann surfaces and fibers. We also show a relation between types of…