I will explain my joint work with R. Abedin, in which we construct Yangian algebra for T^*g of a Lie algebra g, and extract solutions of spectral quantum Yang-Baxter equations. The construction is based on the geometry of equivariant affine Grassmannians, especially its monoidal and factorization structure. This geometric object is closely related to Kapustin's twist of 4d N=2 gauge theories, which is also a main motivation for our work.

In the first half of the talk, I will sketch some general features that we expect from mixed holomorphic-topological (HT) field theories. Focusing on 4d HT theories, I will explain the vision of Costello, that a 4d holomorphic topological field theories can give rise to a integrable system by dimensional reduction. I will then review the relation between equivariant affine Grassmannians and Kapustin’s twist (which gives rise to such a 4d HT theory).

In the second half of the talk, I will sketch our construction in relation to the equivariant affine Grassmannians as well as the HT field theory of Kapustin. Time permitting, I will also explain how we dynamify this quantum algebra (namely twisting it into a quantum groupoid) over formal neighbourhoods in the moduli space of G bundles on curves.