Vortices and flat connections

Vortices and Flat Connections

At Bradlow’s limit, the moduli space of Bogomol’nyi vortices on a compact Riemann surface of genus g is determined. The Kähler form, and the volume of the moduli space is then computed. These results are compared with the corresponding results previously obtained for a general vortex moduli space. 1. The Abelian Higgs model in (2+1) dimensions is an interesting arena to study vortices. The coup...

Flat Connections and Comodules

This is the text of a talk given at the Isaac Newton Institute in Cambridge on 4th August, 2006. This talk is motivated by a recent paper [A Kaygun and M Khalkhali, Hopf modules and noncommutative differential geometry, Lett. Math. Phys. 76 (2006), 77–91] in which Hopf modules appearing as coefficients in Hopf-cyclic cohomology are interpreted as modules with flat connections. We start by descr...

Flat Projective Connections

Constructing Complete Projectively Flat Connections

Theorem 1. Let T 2 be the two dimensional torus. Then for any positive integer m there is a complete torsion free projectively flat connection, ∇, on T 2 such that for any point p ∈ T 2 there is a point q ∈ T 2 with the property that any broken ∇-geodesic between p and q has at least m breaks. Moreover if T 2 is viewed as a Lie group in the usual manner, this connection is invariant under trans...

Lectures on meromorphic flat connections

These notes form an extended version of a minicourse delivered in Université de Montréal (June 2002) within the framework of a NATO workshop “Normal Forms, Bifurcations and Finiteness Problems in Differential Equations”. The focus is on Poincaré–Dulac theory of “Fuchsian” (logarithmic) singularities of integrable systems, with applications to problems on zeros of Abelian integrals in view.