Quantisation of isomonodromy systems (Gabriele Rembado, UniGE)

29.10.2019 15:00 – 17:00

We will at first contemplate the quantisation of the following geometric picture.
Consider a vector bundle on the sphere equipped with a logarithmic connection. The connection defines systems of differential equations with meromorphic coefficients (having simple poles) for the local sections of the bundle. Solutions to such systems are multi-valued, and transform by monodromy upon analytic continuation around a pole: we are interested in deformations of the logarithmic connection such that this monodromy is fixed (up to gauge), and call them isomonodromic deformations. It turns out they are controlled by an Hamiltonian system.
Importantly, a natural quantisation of this system yields the Knizhnik--Zamolodchikov connection in conformal field theory. Thus the final output is a (very important) integrable quantum Hamiltonian system constructed out of the quantisation of a classical system that controls isomonodromic deformations (or an isomonodromy system, for short).
In the first part we will try to review this construction and to introduce the relevant formalism of deformation quantisation (and give some motivation in TQFT if time allows).
In the second more technical part we will consider extensions of this story were higher order poles in the meromorphic connections are allowed. This will involve quantising the symplectic (Nakajima) varieties of representations of certain quivers, as well as introducing the formalism of quantum Hamiltonian reduction.

Lieu

Bâtiment: Battelle

Villa Battelle, Séminaire "Groupes de Lie et espaces des modules"

Organisé par

Faculté des sciences
Section de mathématiques

Intervenant-e-s

Gabriele Rembado, UniGE

entrée libre

Classement

Catégorie: Séminaire