The fifty-fourth meeting of the North British Mathematical
Physics Seminar was held on **Saturday 17th November
2018** in Durham, in Room CM221 in the Department of Mathematical Sciences.
These were the
participants at the meeting.

11:00-11:30

Tea/Coffee in CM211

11:30-12:30

Christiana PANTELIDOU
(Durham University)

Holography has been proven to be a very powerful tool for studying strongly coupled Condensed Matter systems. In this talk, I will discuss the behaviour of the simplest holographic superconductor in an external magnetic field. As we will see, a lattice of vortices will be spontaneously created at low temperatures. What are the properties of the system in this phase and how universal are they?

12:30-13:00

Maxime Fairon
(University of Leeds)

Ruijsenaars-Schneider systems are a particular family of integrable systems describing the motion of particles on a line. In 1995, Krichever and Zabrodin introduced generalisations of the classical RS systems, where each particle is now endowed with internal degrees of freedom called spins. In 1998, Arutyunov and Frolov derived these spin systems in the complex rational case by Hamiltonian reduction, which allowed them to determine the Hamiltonian structure underlying that case. Moreover, they conjectured the form of some of the Poisson brackets for the complex trigonometric system.

In this talk, I will explain how to derive the phase space in the latter case by quasi-Hamiltonian reduction. This gives a way to compute Poisson brackets and to finally prove the Arutyunov-Frolov conjecture. Time allowing, I will describe some features of the system, such as degenerate integrability, Liouville integrability, or how to understand the geometry of the phase space from a quiver using Van den Bergh's theory of double brackets. This is based on joint work with Oleg Chalykh.

13:00-14:00

Lunch in CM211

14:00-15:00

Benoit VICEDO
(University of York)

I will begin by reviewing how classical integrable sigma models can be recast as classical Gaudin models associated with affine Kac-Moody algebras, or affine Gaudin models for short. One usually thinks of the Gaudin model as a spin chain, since it can be obtained from a certain limit of the XXX spin chain. When viewed in this way, integrable sigma models are essentially described by affine Gaudin models with a single site. It is then very natural, in this formalism, to consider affine Gaudin models with arbitrarily many sites. I will then go on to show that such multi-site classical Gaudin models can be used to construct new relativistic classical integrable field theories that couple together an arbitrary number of integrable sigma models. This talk is based on joint work with F. Delduc, M. Magro and S. Lacroix.

15:00-15:30

Iain Findlay
(Heriot-Watt University)

In this talk I will describe how the Lax/zero-curvature formalism can be used to view integrable systems in terms of their "space-evolution" rather than the usual idea of time-evolution. This leads to defining a dual Hamiltonian structure and a family of commuting quantities that are all conserved under space-evolution. This methodology is applied to the Non-Linear Schroedinger and Isotropic Landau-Lifshitz (the continuous classical analogue of the Heisenberg ferromagnet, or XXX spin chain) models. I will also discuss how generic integrable boundary conditions can be incorporated into this dual picture, and give examples of such time-like boundary conditions for both of the models under consideration.

This work is based off of a recent paper joint with A. Doikou and S. Sklaveniti (arXiv:1810.10937) as well as a forthcoming paper.

15:30-16:00

Tea/Coffee in CM211

16:00-17:00

Mathew BULLIMORE
(Durham University)

The product of local operators in a topological quantum field theory in dimension greater than one is commutative, as is more generally the product of extended operators of codimension greater than one. In theories of cohomological type these commutative products are accompanied by secondary operations, which capture linking or braiding of operators, and behave as (graded) Poisson brackets with respect to the primary product. We describe the mathematical structures involved and illustrate this general phenomenon in a range of physical examples arising from supersymmetric field theories in spacetime dimension two, three, and four.

17:00--

Post-meeting discussions in pub and/or over dinner. All are welcome.

Douglas Smith
Last modified: 19 November 2018