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The Euler characteristic of a compact complex manifold M is a classical cohomological invariant. Depending on the viewpoint, it is most natural to interpret it as an index of an elliptic differential operator on M, or as a supersymmetric index in superconformal field theories “on M”. Refining the Euler characteristic but keeping with both index theoretic interpretations, one arrives at the notion of complex elliptic genera. We argue that superconformal field theory motivates further refinements of these elliptic genera which result in a choice of several new invariants, all of which have lost their interpretation in terms of index theory. However, at least if M is a K3 surface, then superconformal field theory and higher algebra select the same new invariant as a natural refinement of the complex elliptic genus.

Topological field theories in the sense of Atiyah–Segal are symmetric monoidal functors from a bordism category to the category of complex (super) vector spaces. A field theory E of dimension d associates vector spaces to closed (d-1)-manifolds and linear maps to manifolds of dimension d. It turns out that if E is invertible, i.e., if the vector spaces associated to (d-1)-manifolds have dimension one, then the complex number E(M) that E associates to a closed d-manifold M, is an SKK manifold invariant. Here these letters stand for schneiden=cut, kleben=glue and kontrolliert=controlled, meaning that E(M) does not change when modifying the manifold by cutting and gluing along hypersurfaces in a controlled way. The main result of this joint work with Matthias Kreck and Peter Teichner is that the map described above gives a bijection between topological field theories and SKK manifold invariants.

We describe joint work with L. Hollands on the geometry of the moduli space of flat connections over a Riemann surface. On the one hand, we generalize and compute certain "complexified Fenchel-Nielsen" coordinates for SL(2)-connections to higher rank using the spectral network "abelianization" approach of Gaiotto-Moore-Neitzke. We then use these coordinates to compute superpotentials, following a conjecture of Nekrasov-Rosly-Shatashvili which roughly states the following: a certain low energy effective twisted superpotential arising from compactifying a theory of class S is equal to the generating function (in the sense of symplectic geometry), in some special coordinates, of the Lagrangian submanifold of opers in the associated moduli space of flat connections.

In this talk I would like to briefly sketch how one can use the tools of derived symplectic geometry and holomorphically twisted gauge theories to derive a relationship between symplectic duality and local Langlands. Our starting point will be an observation due to Gaiotto-Witten that a 3d N=4 theory with a G-flavor symmetry is a boundary condition for 4d N=4 SYM with gauge group G. By examining the relationship between boundary observables and bulk lines we will be able to derive constructions originally due to Braverman, Finkelberg, Nakajima. By examine the relationship between boundary lines and bulk surface operators one can derive new connections to local geometric Langlands. This is based on joint work with Philsang Yoo, Tudor Dimofte, and Davide Gaiotto