The N-Vortex Problem on a Riemann Sphere

Vortex Streets on a Sphere

Invariant Dynamical Systems in the N-vortex Problem on the Sphere and Unstable Motion of the N-ring

We consider the system of the N vortex points with the identical strength on a sphere. As a local effect of the rotation of the sphere, we fix vortex points at the both poles, which are called the pole vortices. When the N vortex points are equally spaced along the line of latitude of the sphere, the configuration is called the “N -ring” or “N -gon”. It is a relative fixed configuration[3,4] an...

The Automorphism Group on the Riemann Sphere

In order to study the geometries of a hyperbolic plane, it is necessary to understand the set of transformations that map from the space to itself. This paper shows that in the Poincaré half-plane model, this group of transformations is the general Möbius group. This is done by showing that the general Möbius group is a group of homeomorphisms that map circles to circles on the extended complex...

More transition amplitudes on the Riemann sphere

We consider a conformal field theory for bosons on the Riemann sphere. Correlation functions are defined as singular limits of functional integrals. The main result is that these amplitudes define transition amplitudes, that is multilinear Hilbert-Schmidt functionals on a fixed Hilbert space.

Liouville Quantum Gravity on the Riemann sphere

In this paper, we rigorously construct 2d Liouville Quantum Field Theory on the Riemann sphere introduced in the 1981 seminal work by Polyakov Quantum Geometry of bosonic strings. We also establish some of its fundamental properties like conformal covariance under PSL2(C)-action, Seiberg bounds, KPZ scaling laws, KPZ formula and the Weyl anomaly (Polyakov-Ray-Singer) formula for Liouville Quant...