# Vortices, Painlevé integrability and projective geometry

Thesis

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.