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The Moore-Tachikawa conjecture posits the existence of certain 2-dimensional topological quantum field theories (TQFTs) valued in a category of complex Hamiltonian varieties. Previous work by Ginzburg-Kazhdan and Braverman-Nakajima-Finkelberg has made significant progress toward proving this conjecture. In this talk, I will introduce a new approach to constructing these TQFTs using the framework of shifted symplectic geometry. This higher version of symplectic geometry, initially developed in derived algebraic geometry, also admits a concrete differential-geometric interpretation via Lie groupoids and differential forms, which plays a central role in our results. It provides an algebraic explanation for the existence of these TQFTs, showing that their structure comes naturally from three ingredients: Morita equivalence, as well as multiplication and identity bisections in abelian symplectic groupoids. It also allows us to generalize the Moore-Tachikawa TQFTs in various directions, raising interesting questions in Lie theory and Poisson geometry. This is joint work with Peter Crooks.

We will discuss the functor of geometric Whittaker coefficients in the context of quantum geometric Langlands. We will prove that tempered twisted D-modules on the stack of G-bundles on a smooth projective curve have non-vanishing Whittaker coefficients. Roughly, this means that a certain natural subcategory of twisted D-modules on the stack of G-bundles can be controlled by the category of twisted D-modules on the Beilinson-Drinfeld affine Grassmannian. The proof will combine generalizations of representation-theoretic and microlocal methods from the preceding works of Faergeman-Raskin and Nadler-Taylor respectively.

In quantum field theory (QFT) the spin-statistics theorem says that in a unitary QFT, a particle has half-integer spin if and only if it is a fermion. I show how to phrase this statement in the language of functorial field theories. More precisely, I explain when a functorial field theory "has fermions" and "has spinors" and when they are "related". I will then restrict to topological field theories (TFTs) and define unitary TFTs. There are counterexamples of the spin-statistics theorem for non-unitary TFTs. I will prove that every unitary TFT satisfies the spin-statistics theorem.

BPS line defects in 4d N=2 supersymmetric QFT are described by a monoidal category with a list of desired properties. For example, the Grothendieck group of this category is supposed to coincide with quantization of functions on Coulomb branch of the theory compactified on a circle. Based on an observation, that at a given vacuum the spectrum of PBS particles can be quipped with an algebra structure – cohomological Hall algebra of the corresponding BPS quiver – we propose a category generated by certain bimodules over this algebra that possesses expected properties of the category of lines. Based on a joint work with Davide Gaiotto and Wei Li. 

The Green-Schwarz anomaly cancellation condition says that the target space of heterotic string theory must come with a string structure for the theory to be consistent. In this talk we discuss a new tangential structure called string^h, first introduced by Devalapurkar, as a spin^c analogue of string. We will show that the spectrum of string^h has the notable property that it orients tmf_1(n), just like how the spectrum of string orients tmf, by the work of Ando-Hopkins-Rezk. Finally we will show that the anomaly of the partition function of type IIA, studied by Diaconescu-Moore-Witten induces a string^h structure on the target space of type IIA, in parallel to the Green-Schwarz anomaly for heterotic string theory, and discuss applications for anomaly cancellation. This is joint work in progress with Arun Debray.