Hyperkähler structures in topological field theory

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Decades ago, Rozansky and Witten surprised the mathematics community by introducing a new, partially defined 3D topological quantum field theory for hyperkähler manifolds. Further study revealed the theory was defined for holomorphic symplectic manifolds, and the perturbative invariants led to new characteristic classes for the latter. Recent work by Teleman, based on early examples by Seiberg and Witten, uncovered a duality between the Rozansky-Witten theory of certain Coulomb branches and pure topological gauge theory in 3D. Conjecturally, the duality has a mirror side which relates algebraic ('B-model') gauge theory with an undiscovered 3D topological gauge theory constructed from Higgs branches of the latter, which does appear to involve the hyperkähler and not just the holomorphic structure, in the form of solutions of a twisted Fueter equation. The project proposes to construct this theory using techniques from algebraic and analytic geometry.Zielinski's main background lies in algebraic geometry and homological algebra, notably in the use of derived categories in algebraic geometry. His interest also include representation theory of Lie algebras and algebraic groups and monoidal categories. In combination, these structures lead to new invariants of algebraic varieties, and can show surprising relationships between different kind of objects. For instance, the first summer reading project Thomas undertook as a second year undergraduate lead to understand Beilinson's theorem about the structure of the derived category of projective space, and some of the relations between exceptional collections in the derived category and derived representations of quivers. Later on, he also studied Bridgeland stability conditions and in particular how to construct such on curves, surfaces and conjecturally on threefolds via results of Bayer-Macri-Toda. More specifically, the project focused on understanding how Bogomolov-Gieseker type inequalities and Bridgeland stability conditions are related, and on some constructions of stability conditions on varieties whose derived category admits an exceptional collection. This has provided good training in the various techniques coming from homotopy theory (localisations of categories, model structures), as well as classical algebraic geometry, such as intersection theory and characteristic classes.These fields show a close interplay with theoretical physics, like topological field theories, as can be observed in the homological mirror symmetry programme. This project fits the EPSRC research areas of Geometry&Topology, but is also relevant to Algebra and the very theoretical side of Mathematical Physics.