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Title: Vortices, Painlevé integrability and projective geometry
Author: Contatto, Felipe
ISNI:       0000 0004 7228 7426
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 2018
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GaugThe first half of the thesis concerns Abelian vortices and Yang-Mills theory. It is proved that the 5 types of vortices recently proposed by Manton are actually symmetry reductions of (anti-)self-dual Yang-Mills 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 Painlevé test, when applied to its soliton equations, reveals a complete list of its integrable cases. The corresponding solutions are given in terms of third Painlevé 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 systems. It is shown that the projective structures defined by the Painlevé 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 hydrodynamic-type systems of two components.
Supervisor: Dunajski, Maciej Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
Keywords: Vortices ; Yang-Mills ; Painleve´ integrability ; Integrable systems ; Frobenius integrability ; Projective geometry ; Metrisability ; Killing forms ; Killing vectors ; Hydrodynamic-type systems ; Hamiltonian ; Self-duality ; Instantons ; Solitons ; Moduli space ; Symmetry reduction ; Gauge theory