Vortices, Painlev\'e integrability and projective geometry

TL;DR

This thesis explores the integrability and geometric structures of vortices, Painlevé equations, and projective geometry, revealing new links between gauge theories, soliton solutions, and Hamiltonian systems.

Contribution

It establishes that certain vortices are symmetry reductions of (A)SDYM equations, classifies integrable cases of a modified Abelian-Higgs model, and characterizes projective structures and Killing forms in 2D geometry.

Findings

Vortices are symmetry reductions of (A)SDYM equations.

Complete classification of integrable cases via Painlevé test.

Conditions for existence of Killing forms linked to Hamiltonian structures.

Abstract

The first half of the thesis concerns Abelian vortices and Yang-Mills (YM) theory. It is proved that the 5 types of vortices recently proposed by Manton are symmetry reductions of (A)SDYM equations with suitable gauge groups and symmetry groups acting as isometries in a 4-manifold. As a consequence, the twistor integrability results of such vortices can be derived. It is presented a natural definition of their kinetic energy and thus the metric of the moduli space was calculated by the Samols' localisation method. Then, a modified version of the Abelian-Higgs model is proposed in such a way that spontaneous symmetry breaking and the Bogomolny argument still hold. The Painlev\'e test, when applied to its soliton equations, reveals a complete list of its integrable cases. The corresponding solutions are given in terms of third Painlev\'e transcendents and can be interpreted as original vortices on surfaces with conical singularity. The last two chapters present the following results in projective differential geometry and Hamiltonians of hydrodynamic-type (HT) systems. It is shown that the projective structures defined by the Painlev\'e equations are not metrisable unless either the corresponding equations admit first integrals quadratic in first derivatives or they define projectively flat structures. The corresponding first integrals can be derived from Killing vectors associated to the metrics that solve the metrisability problem. Secondly, it is given a complete set of necessary and sufficient conditions for an arbitrary affine connection in 2D to admit, locally, 0, 1, 2 or 3 Killing forms. These conditions are tensorial and simpler than the ones in previous literature. By defining suitable affine connections, it is shown that the problem of existence of Killing forms is equivalent to the conditions of the existence of Hamiltonian structures for HT systems of two components.