Quantisation of moduli spaces: Hitchin connections and isomonodromic deformations

Our goal is to construct generalisations of the Hitchin and Wess--Zumino--Witten (WZW) and Knizhnik--Zamolodchikov (KZ) connections, both in geometric and deformation quantisation, and of their associated monodromy representations.The Hitchin connection achieved the quantisation of compact Chern--Simons theory and resulted in the construction of a topological quantum field theory.

A different projectively flat connection provides a viable mathematical definition of correlation functions in the WZW model for conformal field theory.

The resulting projectively flat vector bundles are isomorphic, and their monodromies have far-reaching applications in low-dimensional topology/geometry (quantum invariants of knots/3-manifolds) and representation theory (of mapping class/quantum/braid groups).Our guiding viewpoint is that the connections of Hitchin/WZW can be derived from the quantisation of moduli spaces of connections on Riemann surfaces.

We will extend this further, focusing on meromorphic connections with high-order poles (i.e., wild singularities), generalising the above bundles and their applications.The motivation for this project is twofold.

First, there is now a complete understanding of the Poisson/symplectic nature of isomonodromic deformations of wild singularitites, which are naturally amenable to quantisation.

The quantum theory is much less developed than the classical one, and this naturally motivates us to close the gap using the latter as a guide.Second, recent work related the genus-zero WZW connection---that is, the KZ connection---to a new version of the Hitchin connection, and this was then used for the quantisation of moduli spaces of parabolic bundles.

We want to pursue extensions of this identification; in particular, we will use the new flat connections constructed on the deformation quantisation side as candidates for `wild' Hitchin connections, in the geometric quantisation of wild character varieties: a complete novelty.