Workshop on Geometric Representation Theory

Arkadij Bojko

Title: T-deformations of vertex algebras from wall-crossing

Abstract: There are two natural ways to deform Joyce's construction of vertex algebras. One is to include tautological insertions, and the other is to work equivariantly. I introduce T-deformed vertex algebras, which unify these two refinements. For the purpose of wall-crossing, they must be treated differently because the associated Lie algebras use different expansions. I will explain this behavior explicitly by studying tautological stable pair wall-crossing and the 2-loop quiver. Using the calculus of T-deformed vertex algebras developed in an upcoming joint work with E. Bouaziz, we derive 7 new Virasoro algebras for the affine plane.

Eric Yen-Yo Chen

Title: Skein theory with boundary conditions

Abstract: The skein module of a closed oriented 3-manifold is a vector space invariant that can be computed via cutting and pasting, reflecting the fact that skein theory defines a 3-dimensional extended TQFT. In this talk, we motivate and survey some recent progress towards enriching skein theory with boundary conditions, which, among other applications, produces new skein-theoretic knot invariants. If time permits, we will also discuss connections to the relative Langlands program and Donaldson–Thomas theory from the perspective of boundary TQFTs.

Dragos Fratila

Title: A Bala-Carter type classification of irreducible components of the global nilpotent cone

Abstract: The global nilpotent cone was introduced by Laumon as an analogue of the nilpotent cone in a reductive Lie algebra. It is defined as the moduli stack of nilpotent G-Higgs bundles on a smooth projective curve, where G is a reductive group. It was proved by Laumon and Ginzburg that it is a Lagrangian substack of the stack of G-Higgs bundles. In the case of GL(n), the irreducible components have been parametrized by Tristan Bozec in terms of combinatorial data akin to Young tableaux. For general reductive groups, Beilinson and Drinfeld proposed a non-explicit description of these components. In joint work in progress with Tristan Bozec and Nicolas Seroux, we combine the Bala-Carter classification of usual nilpotent orbits inside a Lie algebra and the Beilinson–Drinfeld approach in order to give a rather explicit description of the irreducible components in terms of dominant cones in a coroot lattice.

Franco Giovenzanna

Title: The LLV Lie Algebra of Hyperkähler Manifolds and Applications to Enriques Manifolds

Abstract: In the classification of compact K-trivial manifolds, Enriques manifolds arise as higher-dimensional analogues of Enriques surfaces: they are smooth projective varieties whose universal cover is an irreducible holomorphic symplectic, or hyperkähler, manifold. Understanding which hyperkähler manifolds admit such finite étale quotients is a natural and largely open problem.

In this talk, I will first introduce hyperkähler and Enriques manifolds and explain the relation between them. I will then explain how the cohomology of a hyperkähler manifold carries a rich representation-theoretic structure through the Looijenga–Lunts–Verbitsky, or LLV, Lie algebra, which extends the classical Lefschetz \(\mathfrak{sl}_2\)-action. We then show how the LLV action on the cohomology of hyperkähler manifolds can be exploited to study finite-order automorphisms. As an application, we prove that no Enriques manifold can arise as a finite étale quotient of a hyperkähler manifold of O'Grady's ten-dimensional, or OG10, deformation type, answering a question of Pacienza and Sarti.

Based on joint work with S. Billi, L. Giovenzana and A. Grossi.

Archi Kaushik

Title: Hyperquot schemes on curves and shifted Yangians

Abstract: Quot schemes parameterising rank 0 quotients of a fixed vector bundle on a smooth curve are interesting moduli spaces, which in particular generalise the notion of symmetric products. Recently, Marian and Neguţ studied these spaces by constructing an action of a certain algebra, called the shifted Yangian of \(\mathfrak{sl}_2\), on their cohomology. In this talk, I will present my recent work, where I generalise Marian and Neguţ's results to the case of hyperquot schemes, that is, the moduli space of flags of rank 0 quotients of a fixed vector bundle on a smooth curve.

Elsa Maneval

Title: Motivic BPS invariants as motivic integrals?

Abstract: I will report on a work in progress, hoping to relate motivic BPS invariants to motivic integrals. I will first introduce motivic BPS invariants on one side, and give basic ideas about motivic and p-adic integration theory on the other side. In particular, I will mention the existing connections between those integrals and BPS sheaves. Then, I will state a formula I proved, relating the motivic BPS invariant to certain motives of cyclotomic inertia stacks, which in turn can be phrased as the motivic integral of a certain gerbe function. It is analogous to a formula obtained by Groechenig, Wyss and Ziegler over finite fields. Then, I will explain what I hope to do next, which is to understand this gerbe function better in order to perform duality arguments along the Hitchin fibration.

Weite Pi

Title: Parabolic sheaves, CoHA of curves, and the P=C conjecture

Abstract: The P=C conjecture for moduli spaces of 1-dimensional sheaves on del Pezzo surfaces asserts that the perverse filtration on its cohomology, induced by the support map, coincides with a more classical filtration defined in terms of tautological classes. This can be viewed as an analogue of the celebrated P=W conjecture in a compact and Fano setting. In particular, it predicts highly non-trivial properties of tautological integrals on the moduli space. I will present work in progress that confirms some of the predictions, namely certain vanishing and χ-independence of the tautological integrals. Combined with earlier results, this implies the P=C conjecture in high cohomological degrees. Two key ingredients are parabolic 1-dimensional sheaves and the cohomological Hall algebra of torsion sheaves on curves, which I will explain in some detail. This is joint work with Woonam Lim and Miguel Moreira.

Shubham Sinha

Title: Twisted quantum cohomology of Grassmannians and intersection theory of Quot schemes

Abstract: In the first part of the talk, I will define a notion of \(l\)-twisted quantum cohomology rings of Grassmannians \(\mathrm{Gr}(r,n)\). We use intersection theory on Quot schemes over \(\mathbb{P}^1\) to describe the structure constants of this twisted ring, which recovers the quantum cohomology of \(\mathrm{Gr}(r,n)\) when \(l = 0\). We then study the virtual intersection theory on Quot schemes of curves of genus \(g\), and prove a twisted Vafa–Intriligator type formula and further combinatorial identities. In the second part of the talk, I will briefly describe a formula computing all possible virtual intersection pairings on Quot schemes of curves, and how they can be used to study the cohomology ring of the moduli space of stable bundles on curves. This is based on ongoing work: the first part with Arvind Ayyer, and the second with Alina Marian.

Kostiantyn Tolmachov

Title: Nearby cycles, character sheaves and Vinberg monoid

Abstract: The functors of nearby and vanishing cycles are important tools in geometric representation theory. In some favorable geometric situations, they behave exceptionally well. I will report on one such instance, with two applications to the theory of character sheaves. In arXiv:2412.07146, joint with Gonin and Ionov, we give a new proof and an extension to the non-unipotent setting of the t-exactness of the composition of Radon and Harish-Chandra transforms for character sheaves. In arXiv:2305.02980, joint with Bezrukavnikov, Ionov and Varshavsky, we show that this can be used to compute the Drinfeld center of the abelian Hecke category attached to the same reductive group. The unifying theme is the study of character sheaves on the Vinberg monoid.

Olga Trapeznikova

Title: Intersection theory on moduli spaces of semistable bundles on curves

Abstract: Moduli spaces of vector bundles on Riemann surfaces appear in a wide range of subjects, from enumerative geometry to non-abelian gauge theory in physics. While their topology is well understood in the smooth, coprime case, much less is known about their enumerative geometry in the singular, degree-zero case.

In this talk, I will show how a classical tool — moduli spaces of parabolic bundles — can be used to compute intersection pairings in the intersection cohomology of these singular moduli spaces, providing a direct geometric interpretation of the resulting formulas and a simpler alternative to the earlier blow-up approach.

This is based on recent joint work with Camilla Felisetti.

Jiangfan Yuan

Title: Eisenstein sheaves and canonical bases associated with a curve

Abstract: In the representation theory of quantum groups, the discovery of canonical bases was a milestone. For the quantum group associated with a quiver, Lusztig constructed canonical bases by considering Hall algebras and the geometry of representation stacks. In my talk, I will present an analogous construction in the setting of curves, following the ideas of Schiffmann. The main contributors of the canonical bases are Eisenstein sheaves, which play an important role in the geometric Langlands program. I will also explain the microlocal behavior of Eisenstein sheaves from the perspective of Donaldson–Thomas theory.