QuantMod · Quantisation of moduli spaces: Hitchin connections and isomonodromic deformations

Quantisation of moduli spaces: Hitchin connections and isomonodromic deformations

This is a project in (pure) Mathematics, intimately tied with theoretical/mathematical Physics. The main goal is to generalize previous constructions in complex-algebraic Poisson geometry, representation theory, and low-dimensional topology; all related with certain mathematica; models in quantum field theory (QFT). Namely, geometric quantization & deformation quantization are two important mathematically-rigorous entry points into QFT, and they both famously lead to linear actions of mapping class/braid groups. The latter have found important usage, and we aim at: (i) extending the definitions of mapping class/braid groups; (ii) defining `quantum' spaces on which they will act; and (iii) computing the actions of the new groups, both before and after quantization. The potential applications include the definition of `quantum' topological invariants, the proof of Kohno--Drinfel'd type theorems, and the mathematical formalization of QFTs with topological/conformal symmetries: notably, the construction of `irregular' conformal blocks in the WZNW model, complementing the (nowadays) much-studied irregular Liouville theory. The new ingredient are moduli spaces of irregular-singular connections, defined on principal bundles over Riemann surfaces, extending the regular-singular case. We put particular emphasis on nongeneric irregular singularities, as well as twisted ones (which cover the general case): this is current frontier of this much-active line of research. Then the main background result is that the moduli spaces fit into flat Poisson/symplectic fibre bundles, over isomonodromic families of `wild' Riemann surfaces, which in turn generalize ordinary pointed Riemann surfaces. Thus: (i) the fundamental groups of the stacks of wild Riemann surfaces generalize mapping class/braid groups; (ii) the quantization of the moduli spaces leads to new flat (projective) bundles of irregular nonabelian theta-functions & conformal blocks; and (iii) their monodromy yields a new `quantum' action of wild mapping class groups, whose semiclassical limit corresponds to braiding the Stokes data of irregular-singular connections.

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