Noah Arbesfeld:
Crossed instantons in algebraic geometry
In a series of papers, Nekrasov introduces moduli spaces of “crossed instantons,” spaces of quiver representations modeling instantons on unions of 2-planes in 4-space. He uses the geometry of these spaces to produce relations between tautological bundles on quiver varieties and construct ``qq-characters,’’ deformations of q-characters of quantum affine algebras.
I will explain a project in progress with Martijn Kool and Woonam Lim in which we study Nekrasov’s moduli spaces from the perspective of algebraic geometry. I’ll explain how moduli spaces of crossed instantons can be equipped with a virtual class using a construction from Donaldson-Thomas theory. Time permitting, I will also state a speculative description of these spaces as moduli spaces of framed sheaves on 4-folds.
Lucien Hennecart:
Langlands duality for character varieties via BPS cohomology
In this talk, I will introduce and explain a proof of the Langlands duality conjecture for local systems on the three-dimensional torus. The argument takes advantage of the local structure of the moduli stack of local systems, expressed in terms of stacks of commuting elements in a Lie algebra, together with the cohomological integrality isomorphism, to compute the BPS sheaf associated with the commuting variety. The computation has a Lie-theoretic flavor, involving the classification of distinguished nilpotent orbits in the Lie algebra (due to Premet). It also involves the classification of (quasi-)isolated elements in reductive groups (due to Bonnafé, Digne--Michel). Our study of the BPS sheaf also has an application to the topological mirror symmetry conjecture for G-Higgs bundles on elliptic curves. These conjectures were introduced in Tasuki Kinjo's talk, and this talk is based on joint work with Tasuki Kinjo.
Victoria Hoskins
Projectivity of good moduli spaces of quiver representations and vector bundles on stacky curves
There are many parallels between moduli of quiver representations and vector bundles on smooth projective (stacky) curves, where moduli spaces have been constructed using GIT. Alternatively, by applying the existence criteria of Alper, Halpern-Leistner and Heinloth to the stack of semistable objects, one can construct good moduli spaces, which (if the quiver is acyclic) are proper algebraic spaces. The focus of this talk will be to describe how to produce ample line bundles on these moduli spaces to prove their projectivity (if the quiver is acyclic). Our approach involves characterising semistability via Hom-vanishing conditions in order to produce enough sections of a determinantal line bundle, and moreover yield effective (semi)ampleness bounds. I will highlight the similarities and differences between the case of quiver representations and vector bundles on stacky curves. This is based on joint papers with Pieter Belmans, Chiara Damiolini, Hans Franzen, Svetlana Makarova, Lisanne Taams and Tuomas Tajakka.
Konstantin Jakob
Counting absolutely indecomposable G-bundles
About 10 years ago, Schiffmann proved that the number of absolutely indecomposable vector bundles on a curve over a finite field (with degree coprime to the rank) is equal to the number of stable Higgs bundles of the same rank and degree (up to a power of q). Dobrovolska, Ginzburg and Travkin gave another proof of this result in a slightly more general formulation, but neither proof generalizes in an obvious way to G-bundles for other reductive groups G.
In joint work with Zhiwei Yun, we generalize the above results to G-bundles. Namely, we express the number of absolutely indecomposable G-bundles on a curve X over a finite field in terms of the cohomology of the moduli stack of stable parabolic G-Higgs bundles on X.
Tasuki Kinjo
BPS cohomology for Hitchin systems and 3-manifolds
Mirror symmetry for the G-Hitchin system has long been of interest due to its connection with geometric Langlands duality. In this talk, we propose a formulation of the Hodge-theoretic mirror symmetry conjecture for the Hitchin system using BPS cohomology — a newly defined cohomology constructed using shifted symplectic geometry — for the loop stack of the stacky Hitchin system. This conjecture generalizes the conjecture of Hausel—Thaddeus for type A groups. We also propose a formulation of geometric Langlands duality for real 3-manifolds and explain its relation to the mirror symmetry for the Hitchin system. Computational evidence for our conjecture will be explained in the subsequent talk by Lucien Hennecart. This talk is based on joint work with Chenjing Bu, Ben Davison, Andrés Ibáñez Núñez, and Tudor Pădurariu.
Michael McBreen
Symplectic duality and the Tutte polynomial
The Tutte polynomial was introduced in the 1940s as a two-variable generalisation of the chromatic polynomial of a graph. It is the universal matroid invariant satisfying a deletion-contraction relation, and is the subject of much recent work.
I will describe a geometric realisation of the Tutte polynomial via the cohomology of a symplectic dual pair of hypertoric varieties. The same construction associates an interesting two-variable polynomial to any pair of symplectically dual spaces, whose one-variable specialisations recover the respective Poincare polynomials. Joint work with Ben Davison.
Tudor Pădurariu
On Hall algebras of quivers and curves
I will discuss joint work with Yukinobu Toda (in progress).
Hall algebras may be used to construct quantum groups associated to quivers or curves. The Hall algebra of a curve is expected to have an SL(2,Z)-symmetry. Such a symmetry should have many manifestations, for example the \chi-independence phenomenon due to Kinjo-Koseki.
I will discuss a comparison between part of the Hall algebra of a curve and the K-theoretic Hall algebra associated to a loop quiver, following previous work of Schiffmann-Vasserot. I then plan to mention conjectures about the above mentioned SL(2,Z)-symmetry for both K-theoretic and categorical Hall algebras. In the latter case, the symmetry is related to a Dolbeault version of the Langlands equivalence.
James Pascaleff
Toward Higher Fukaya Categories
This talk explores a possible theory of higher categories associated to shifted symplectic stacks. These categories generalize the Fukaya category of a 0-shifted real symplectic manifold. At all levels they are defined by some kind Lagrangian intersection theory: either derived intersection theory, or Floer theory. The examples of the 1-shifted cotangent bundle and the coadjoint quotient stack will be studied in detail. This work is joint with Nicolò Sibilla.
Manish Patnaik
Loop Groups, Bordifications, and their Arithmetic Quotients
Although infinite-dimensional, symmetric spaces attached to loop groups behave in many ways like their finite-dimensional counterparts. We will describe the natural symmetric and locally symmetric spaces attached to loop groups. In the function field setting, this is related to certain moduli spaces of bundles on some simple surfaces. Following this, we explain a process of compactifying them using an extension of Harder-Narasimhan’s theory of stability.
Joint with Punya Satpathy
Claudia Scheimbauer
Dualizability for topological field theories
Thanks to the famous Cobordism Hypothesis, duals, or rather (highly) dualizable objects, play an important role when looking for topological field theories and relative versions thereof. I will explain why and how one may go about finding such highly dualizable objects. For instance, in higher categories of correspondences and versions thereof, for instance decorated by (shifted) symplectic forms, high dualizability is built in [Calaque—Haugseng—S]. I will explain how to prove a conjecture by Lurie from 2009 about highly dualizable objects in higher Morita categories, a relative version thereof and an invertibility conjecture by Brochier—Jordan—Safronov—Snyder. We obtain new TFTs attached to rigid E_n-monoidal V-categories; e.g. Shv_Vectk (X) for a locally compact space and Rep(G) for G a compact Lie group. This is joint work with Pelle Steffens and Will Stewart.
Nicolò Sibilla
What is (equivariant) elliptic cohomology?
In this talk I will give an introduction to equivariant elliptic cohomology surveying the role it plays within chromatic homotopy theory and the seminal work of Grojnowski in the equivariant setting. Then I will explain recent work joint with Scherotzke and Tomasini which gives an algebro-geometric approach to (equivariant) elliptic cohomology, and shows that this theory can be understood in a very similar way to well known constructions in algebraic geometry, such as Hochschild homology. I will highlight relations between our work and recent contributions by Moulinos-Robalo-Toen and Bouaziz-Khan.
Tanguy Vernet
I will report on joint work with Victoria Hoskins and Joshua Jackson, where we build moduli spaces of quiver representations with multiplicities, using recent techniques in non-reductive geometric invariant theory. These new moduli spaces include analogues of Nakajima quiver varieties for quivers with multiplicities. In good cases, we also show that their cohomology carries a pure Hodge structure, as is the case for classical Nakajima quiver varieties.
The 3d-index introduced by Dimofte-Giaotto-Gukov gives q-series invariants of cusped three-manifolds. It was conjectured by Garoufalidis-Gu-Mariño that these invariants are given by entries of Stokes matrices of perturbative Chern-Simons theory. I will explain how this can be proved using intersections in a certain homology constructed from the quantum dilogarithm. This is based on joint work with Andersen-Fantini-Kontsevich.
Quantum K-Theory of Critical Loci
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.
Instanton on blow ups and free fermions
The semi orthogonal decomposition of the cohomological theory of grassmannian of two term complexes is studied by a series paper of Jiang. In this talk, we will reinterpret it as a representation of the Clifford algebra.
As an application, we will explain a relation between the basic representation of the affine Lie algebra and the moduli space of the instanton spaces on the blow up of a point in a surface. It verifies predictions of Li-Qin and Feigin-Gukov. Based on joint work with Qingyuan Jiang and Wei-ping Li.
Paul Ziegler
chi-independence for K3 surfaces
BPS invariants naturally appear in the enumerative geometry of sheaves with one-dimensional support on a Calabi-Yau threefold. Toda conjectured that these invariants are independent of the appearing Euler characteristic $\chi$. I will talk about work in progress with M. Groechenig and D. Wyss proving this conjecture in the K3 case. We argue by relating BPS cohomology to p-adic integration on moduli stacks of sheaves, for which $\chi$-independence was shown by Carocci-Orecchia-Wyss. For this, we use a local description of these moduli stacks of sheaves in terms of moduli stacks of quiver representations.
