Titles and abstracts

Nekrasov's 4D ADHM quiver is a generalization of the 2D and 3D ADHM quivers. I will give a mathematical definition of the corresponding "gauge origami" partition function in terms of certain torus localized Oh-Thomas virtual cycles. I will relate the orientation problem of gauge origami to the one of Hilbert schemes of points on affine 4-space, which was previously analyzed in physics by Nekrasov-Piazzalunga ("Magnificent Four") and in algebraic geometry in a joint work with Rennemo. I will provide a sheaf-theoretic interpretation of gauge origami in terms of certain moduli of 2-dimensional framed sheaves on (P^1)^4 (recently proved by Rennemo). Joint work with Arbesfeld-Lim.

As a follow-up to Martijn Kool’s talk, I’ll present a proof of the correspondence conjectured by Arbesfeld, Kool and Lim between representations of Nekrasov’s 4d ADHM quiver and pure 2-dimensional framed sheaves on (P^1)^4.

Tanaka-Thomas’s algebro-geometric approach to Vafa-Witten theory defines invariants of projective surfaces using moduli spaces of stable Higgs pairs. In work with M. Kool and T. Laarakker, we express certain Vafa-Witten invariants in terms of the equivariant geometry of a quiver variety, the moduli space of torsion-free framed sheaves on P^2. I’ll present wall-crossing results for this moduli space of framed sheaves, one due to Kuhn-Leigh-Tanaka and one new, and discuss their implications for Vafa-Witten invariants. One consequence is a formula for so-called "vertical" Vafa-Witten invariants in rank 2.

We establish an integral basis for the B-brane central charges of certain Calabi-Yau fourfold flops and derive exact expressions for their monodromies using the abelian grade restriction rule and window categories of the associated gauged linear sigma models. The monodromy is interpreted as an EZ twist associated with the contraction of an exceptional surface onto a curve of conifold singularities. We further illustrate a decomposition of the EZ twist into spherical twists for families of splitting configurations of the sixtic Calabi-Yau fourfold and a complete intersection in Grassmannian. Based on 2608.06280.

I will discuss work in progress exploring the computation of DT4 invariants of local Calabi-Yau fourfolds using generalised BPS quivers. These are N=2 supersymmetric quantum mechanics (SQM), which have richer sets of ground states than the N=4 SQM BPS quivers which compute DT invariants of local threefolds, and I will contrast the two. [Based on work with Mario De Marco and Dongwook Ghim.]

The quiver W-algebra addresses the quantum algebraic structure of quiver gauge theories on surfaces. We have recently extended this formalism to higher-dimensional settings, where the equivariant DT/PT generating function is obtained from the qq-character associated with the underlying quiver structure. In this talk, I start with basic ideas of the vertex operator formalism in this context, and discuss the DT/PT theory with emphasis on its wall-crossing structure and the descendent insertions. I will also address a geometric perspective of the deformed Cartan matrix, which plays a role of the structure constant in our formalism, extending the Ext pairing construction by Carlsson and Okounkov for the surface case.

Relative invariants and degeneration formula play an important role in the series of works on Gromov-Witten/Donaldson-Thomas correspondence by MNOP, where the fundamental building blocks are the cases of local curves. In this talk, I will introduce the generalization of this technique to local log CY 4-folds. In particular, it leads to a proof of Nekrasov’s formula for degree 0 DT invariants. This is joint work with Yalong Cao and Gufang Zhao.

In this talk, I will explain the construction of Lagrangian classes for perverse sheaves in cohomological Donaldson-Thomas theory, whose existence was conjectured by Joyce. The two key ingredients are a relative version of the DT perverse sheaves and a hyperbolic version of the dimensional reduction theorem. As a special case, we recover Borisov-Joyce/Oh-Thomas virtual classes in DT4 theory. As applications, I will explain how to construct the following structures from the Lagrangian classes: (1) cohomological field theories for gauged linear sigma models, (2) cohomological Hall algebras for 3-Calabi-Yau categories, (3) relative Donaldson-Thomas invariants for Fano 4-folds with anti-canonical divisors, (4) refined surface counting invariants for Calabi-Yau 4-folds.

Ben-Bassat-Brav-Bussi-Joyce have defined for any oriented (-1)-shifted symplectic stack a DT perverse sheaf which categorifies DT invariants. In the case of the moduli of objects of a 3 Calabi-Yau category C, Kinjo-Park-Safronov and Descombes have defined a cohomological Hall algebra structure on the cohomology of this perverse sheaf. This is achieved by using the compatibility of this perverse sheaf with the (-1)-shifted Lagrangian correspondence given by considering the moduli space of short exact sequences of objects in C. However, computing this algebra in most cases is difficult via direct methods since both the DT sheaf and the multiplication are defined via a gluing procedure. In this talk I will give several examples of how one can use factorization algebra techniques to study examples of 3d CoHAs. This is joint work with Latyntsev, Bu,Li and Descombes.

I will explain the gauge-theoretic approach to Donaldson-Thomas theory of fourfolds, with focus on toric varieties. After reviewing the flat space case and the role of observables in factorization, I will discuss how to fix square-roots of vertex functions with general asymptotics and their gluing in global geometries and comment on the relation to quasimaps from curves (usual ones) and surfaces (new ones). I will also address the higher-rank extension of this story, and connections to other recent developments.

Mirror symmetry implies that the higher genus Gromov-Witten (GW), Pandharipande-Thomas (PT) and certain Donaldson-Thomas (DT) invariants on the A-model side are encoded in generating functions which are related to analytic sections of line bundles over the complex moduli space on the B-model side constructed from the periods. This relates the asymptotic behaviour of the invariants as extracted at the point of maximal unipotent monodromy to the behaviour of these sections near the closest singularities, typically a conifold. Here the relevant analytic information is encoded in closed and open modular period integrals over the modular domain of $\Gamma_0(N)$. The latter structures are determined by properties of the degenerate Calabi-Yau motive in the conifold fibre, which have been proven by the fibering out technique with B\"onisch and Golyshev. We comment on the implications of the asymptotic behaviour of the symplectic invariants for the estimates of the microscopic entropy of supersymmetric black holes and black rings, as investigated recently with Alexandrov and Pioline.

Given a pair of a smooth space X and a regular function f on it, critical cohomology provides a way to construct an invariant capturing properties of the critical locus of f. There are several approaches to defining it such as cohomology of vanishing cycle sheaf, relative cohomology, or hypercohomology of twisted de-Rham and Hodge complexes. Critical cohomology is well understood in several important cases, but it is still hard to compute and study it in general. In the talk I will explain recent progress in the case where a smooth DM stack X admits a C^* action scaling the function f.

We study the torsion of 1-dimensional operator d+dF., where F is a circle-valued function on X. We show that variation of torsion may be interpreted as a one-point correlator in a topological quantum mechanics coupled to one-dimensional topological gravity. Detailed study of this theory leads to Mathai-Quillen formula that relates torsion to the zeta-function, counting periodic orbits of the gradient vector field for F and Morse-Novikov torsion on a covering of X.