Applications of quantum groups to 4-dimensional topological field theory

Lead Research Organisation: University of Edinburgh
Department Name: Sch of Mathematics


Recently, Ben-Zvi, Brochier, and Jordan [BZBJ] have introduced a 4-dimensional topological field theory - dubbed Betti Geometric Langlands (BGL) theory - intimately related to N=4 super Yang-Mills theory, and to the moduli spaces of Betti local systems on surfaces and 3-manifolds. The BGL theory produces among other things, quantizations of the Poisson structures on spaces of $G$-local systems of Riemann surfaces.
It is an interesting an important problem to quantize individually the symplectic leaves of these moduli spaces, which correspond to local systems whose monodromy around each puncture lies in a prescribed conjugacy class, and to leverage the output to produce new topological invariants. Specific projects include:

1) Constructing quantizations of conjugacy classes in reductive groups, as co-ideal sub-algebras, and developing the resulting braided module categories for the quantum group
2) Studying topological field theories on orbifolds, and relating these to quantum symmetric pairs.
3) Studying special features of these and related constructions when the quantum parameter is a root of unity.


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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509644/1 01/10/2016 30/09/2021
1974971 Studentship EP/N509644/1 01/09/2015 31/03/2019 Tim Anne Weelinck
Description We have introduced a framework for equivariant topological quantum field theories, generalising work of Lurie, Ayala-Francis, and Costello-Gwilliam on topological quantum field theories viewed through their observables. We characterise this theory through a universal property it satisfies, and hence can recognise that other classical equivariant homology theories are examples of our framework. We also apply our methods to the subject of quantum symmetric pairs, and as such construct new (categorical) invariants of surfaces. Finally, we connect our results to work of D.Jordan and X.Ma on a duality between double affine Hecke algebras of type C-C_n and D-modules on quantum symmetric spaces. We recover their results in a conceptual manner and also extend them to work in greater generality.
Exploitation Route The framework we designed for constructing invariants is very general, and could be applied by others in various settings of pure mathematics: (homotopy) algebra, topological quantum field theory and representation theory.
Sectors Other