Quantization of Calogero-Painlevé System and Multi-Particle Quantum Painlevé Equations II-VI

We consider the isomonodromic formulation of the Calogero-Painlevé multi-particle systems and proceed to their canonical quantization. We then proceed to the quantum Hamiltonian reduction on a special representation to radial variables, in analogy with the classical case and also with the theory of quantum Calogero equations. This quantized version is compared to the generalization of a result of Nagoya on integral representations of certain solutions of the quantum Painlevé equations. We also provide multi-particle generalizations of these integral representations.

Symmetries of the Simply-Laced Quantum Connections and Quantisation of Quiver Varieties

We will exhibit a group of symmetries of the simply-laced quantum connections, generalising the quantum/Howe duality relating KZ and the Casimir connection. These symmetries arise as a quantisation of the classical symmetries of the simply-laced isomonodromy systems, which in turn generalise the Harnad duality. The quantisation of the classical symmetries involves constructing the quantum Hamiltonian reduction of the representation variety of any simply-laced quiver, both in filtered and in deformation quantisation.

DERIVATIONS OF THE TRIGONOMETRIC BCn SUTHERLAND MODEL BY QUANTUM HAMILTONIAN REDUCTION

The BCn Sutherland Hamiltonian with coupling constants parametrized by three arbitrary integers is derived by reductions of the Laplace operator of the group U(N). The reductions are obtained by applying the Laplace operator on spaces of certain vector valued functions equivariant under suitable symmetric subgroups of U(N) × U(N). Three different reduction schemes are considered, the simplest one being the compact real form of the reduction of the Laplacian of GL(2n, ℂ) to the complex BCn Sutherland Hamiltonian previously studied by Oblomkov.

THE SU(2)0 WZNW MODEL

We analyse the correlation functions and operator content of the SU(2)0 model using both the free field description and the solutions to the Knizhnik-Zamolodchikov equations. We show that there is a very close correspondence between this model and the well studied c=-2 theory. We also demonstrate that the quantum Hamiltonian reduction of SU(2)0 leads very directly to the correlation functions of the c=-2 model.

SU(2)0 AND OSp(2|2)-2 WZNW MODELS: TWO CURRENT ALGEBRAS, ONE LOGARITHMIC CFT

We show that the SU (2)0 Wess–Zumino–Novikov–Witten (WZNW) model has a hidden OSp (2|2)-2 symmetry. Both these theories are known to have logarithms in their correlation functions. We also show that, like OSp(2|2)-2, the logarithmic structure present in the SU (2)0 model is due to an underlying c = - 2 sector. We demonstrate that the quantum Hamiltonian reduction of SU (2)0 leads very directly to the correlation functions of the c=-2 model. We also discuss some of the novel boundary effects which can take place in this model.