Quotients of the double loop group

• Andrea Maffei (Università di Pisa)

Room: Department of Mathematics For a smooth affine algebraic group $G$ over an algebraically closed field, we consider some two-variables generalizations of the affine Grassmannian $G((t))/G[[t]]$, given by quotients of the double loop group $G((x))((y))$. We prove that they are representable by ind-schemes if $G$ is solvable. Given a smooth surface $X$ and a flag of subschemes of $X$, we provide a geometric interpretation of the two-variables Grassmannians, in terms of bundles and trivialisation data defined on appropriate loci in $X$, which depend on the flag. This is a joint work with Valerio Melani and Gabriele Vezzosi.

Line Operators, cohomological Hall algebras, and affine Grassmannian

• Yaping Yang (University of Melbourne)

Room: Department of Mathematics Motivated by the study of categories of line operators in 4d N=2 gauge theories, we construct a triangulated monoidal category with a weak braiding starting from a symmetric 3 Calabi-Yau category equipped with additional structures. When the 3 Calabi–Yau category arises from the root datum of an algebraic group, we compare the resulting category with the equivariant derived category of coherent sheaves on the affine Grassmannian. This talk is based on ongoing joint work with Fujita, Soibelman, and Zhao.

Tensor products of quiver bundles

• Juan Sebastian Numpaque Roa (Universidade do Porto)

Room: Department of Mathematics In this work we introduce a notion of tensor product of (twisted) quiver representations with relations in the category of $\mathcal{O}_X$-modules. As a first application of our notion, we see that tensor products of polystable quiver bundles are polystable and later we use this to both deduce a quiver version of the Segre embedding and to identify distinguished closed subschemes of $\mathsf{GL}(n,\mathbb{C})$-character varieties of free abelian groups.

Cohomology of singular moduli spaces in the non-abelian Hodge theory of a curve

• Tommaso Scognamiglio (Università di Bologna)

Room: Department of Mathematics The geometry of the moduli spaces appearing in the non-abelian Hodge theory of a Riemann surface (i.e. moduli spaces of Higgs bundles and character stacks) is well understood in the smooth case. In the singular case, the situation is more complicated. In this article, we give a conjectural formula for the mixed Poincaré series of character stacks for Riemann surfaces. Furthermore, we verify it under Euler's specialization (i.e., we calculate the E-series). Such a formula was previously known only in the smooth case, thanks to the work of Hausel, Letellier, and Rodriguez-Villegas. Our formula expresses the singular case as a sort of "symmetric power" of the smooth ones. Similar results have appeared in multiple works in related areas. The results of the article provide important evidence in support of the chi-independence property of the coomology of these stacks.

Quantum toroidal algebras, affine Yangians and abelian qKZ equations

• Andrea Appel (Università degli Studi di Parma)

Room: Department of Mathematics In this talk, I will present the construction of abelian meromorphic R-matrices for quantum toroidal algebras, providing a meromorphic braiding on category O representations with respect to the (rational) tensor product induced by Drinfeld coproduct. I will then explain how, under the equivalence of Gautam and Toledano Laredo, these R-matrices compute the monodromy of the abelian qKZ equations for the corresponding affine Yangians. This is based on joint work with S. Gautam.

Quantum K-Theory of Critical Loci

• Gufang Zhao (University of Melbourne)

Room: Department of Mathematics In this talk, I will define a pullback map from the Grothendieck group of coherent matrix factorizations to that of coherent sheaves on a (-1)-shifted Lagrangian inside the critical locus of a function. This map satisfies natural functoriality under composition of Lagrangian correspondences, along with expected properties such as bivariance and base change. I will explain how this construction arises naturally in the study of quantum K-theory for critical loci, with examples drawn from moduli spaces associated to quivers with potentials. This is based on joint work with Y. Cao and Y. Toda.