Enumerative seminar

Coorganized with Yang Zhou

Wednesdays at 2pm

Satoshi Nawata, Fudan University
11/26, SIMIS room 1710

DT/PT Correspondence from Combinatorics
In this talk, I will present explicit methods for computing the generating functions of equivariant Donaldson–Thomas (DT) and Pandharipande–Thomas (PT) invariants on C^3. By using the combinatorial structure of 3d Young diagrams, I will explain quiver descriptions of the corresponding moduli spaces, and how their generating functions can be computed via Jeffrey–Kirwan residues using localization on these moduli spaces. By varying the contour cycles, it is expected to naturally recover the DT/PT correspondence. While the approach provides strong computational evidence, it currently lacks mathematical rigor, and it is desirable to develop a precise mathematical formulation and proof. This talk is a continuation of my previous presentation.

Past talks

Leonardo Santilli, SIMIS
11/19, SCMS room 102

Framed BPS invariants from M-theory with M5-brane defects
I will introduce a new class of enumerative invariants and give a gauge theory computational framework to obtain them. I will study M-theory on a local Calabi-Yau threefold X decorated with M5-branes on non-compact divisors. I will show two derivations of the corresponding partition functions, which, conjecturally, provide the generating functions of virtual counts of framed torsion sheaves on X.
The first derivation is valid only for Calabi-Yau threefolds without compact divisors, but yields a closed-form expression.
The second derivation is valid in general and stems from a 5d gauge theory computation of the partition function, leading to a new class of BPS counting invariants.
Based on work in progress with Mauricio Romo.

Yuuji Tannaka, BIMSA
11/12, SIMIS room 610

Construction of Spin(7)-instantons
After a brief introduction to Gauge theory in higher dimensions, as proposed by Donaldson and Thomas in the late ’90s, we will introduce the notion of Spin(7)-instantons on Riemannian 8-manifolds with holonomy contained in Spin(7); and illustrate the constructions of them on Joyce’s examples of compact Spin(7)-manifolds. This talk is partly based on joint work with Mateo Galdeano, Daniel Platt, and Luya Wang.

Michel van Garrel, University of Birmingham
Friday 10/24, SIMIS room 1610

Intrinsic mirrors to log K3 surfaces and modular forms
Intrinsic Mirror Symmetry (Gross, Siebert) associates to a log Calabi-Yau variety (Y,D) a geometric generating function with support the tropicalisation of (Y,D), and with invariants the corresponding punctured Gromov-Witten invariants. This construction defines a ring and a mirror family. The enumerative mirror conjecture then states that various period integrals on the mirror family compute various Gromov-Witten invariants of (Y,D), not necessarily punctured. I will describe some log K3 surfaces, where this is realized, and where one obtains some new relations for Gromov-Witten invariants, including via modular forms. Joint work with Siebert and Ruddat.

Noah Arbesfeld, University of Vienna
10/01, SIMIS room 1210

Crossed instantons in algebraic geometry
In a series of papers, Nekrasov introduces moduli spaces of “crossed instantons,” certain quiver representations modelling instantons on unions of 2-planes in 4-space. He uses the geometry of these spaces to extract information about bundles on quiver varieties and construct deformations of 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 invariants of moduli spaces of crossed instantons can be defined using a construction from sheaf-counting theories for 4-folds. I will also state a speculative description of these spaces as moduli spaces of framed sheaves on a projective 4-fold.

Todor Milanov, IPMU
09/10, SIMIS room 1310

Reflection vectors in quantum cohomology
Smooth projective varieties with semi-simple quantum cohomology is a very interesting class of varieties from the point of view of mirror symmetry and integrable systems. The goal in the first part of my talk is twofold. I would like to explain an approach to integrability in Gromov-Witten theory based on vertex algebras and more generally Borcherd’s products. This is based on an old joint work with Bojko Bakalov. As a byproduct of our construction, we will see that there is a certain system of vectors, called reflection vectors, that plays a key role in our project. The second goal of my talk is to explain the relation between reflection vectors and the refined Dubrovin conjecture. This part is based on a recent joint work with John Alexander Cruz Morales.

Hyeonjun Park, KIAS
Thursday 06/12, SIMIS room TBA

Lagrangian classes, Donaldson-Thomas theory, and gauged linear sigma models
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 Hall algebras for 3-Calabi-Yau categories, (2) relative Donaldson-Thomas invariants for Fano 4-folds with anti-canonical divisors, (3) refined surface counting invariants for Calabi-Yau 4-folds, (4) cohomological field theories for gauged linear sigma models.

This is joint work in progress with Adeel Khan, Tasuki Kinjo, and Pavel Safronov.

Yehao Zhou, SIMIS & Fudan University
04/16, SCMS room 102, Fudan University

Stable envelope for critical loci
In this talk, we will introduce a generalization of Maulik-Okounkov’s stable envelopes to equivariant critical cohomology. In the case of a tripled quiver variety with standard cubic potential, this reduces to MO’s stable envelope for the Nakajima variety of the corresponding doubled quiver along the dimensional reduction. We define non-abelian stable envelopes for quivers with potentials following a similar construction of Aganagic-Okounkov, and use them to relate critical COHAs to the abelian stable envelopes. Explicit computations are given in three examples: 1) Verma modules and higher spin representations of the Yangian of sl(2); 2) oscillator representations of the shifted Yangian of sl(2); 3) fundamental representation of the Yangian of sl(2|1). This talk is based on joint work in progress with Yalong Cao, Andrei Okounkov, and Zijun Zhou.

Will Donovan, Tsinghua University
On Monday as an exception! 2pm 03/31, SIMIS room 1310

Exceptional surfaces in 3-folds and derived symmetries Video
Crepant resolutions of 3-fold singularities may contain elaborate configurations of exceptional surfaces. Using toric cases as a guide, I review some known contributions of these configurations to the derived autoequivalence group of the resolution, particularly from the work of Seidel-Thomas, and discuss work in progress with Luyu Zheng.

Yingchun Zhang, Zhejiang University
03/26, SCMS room 102, Fudan University

Quantum cohomology/quantum K rings and cluster algebras
I will introduce a relation between the quantum cohomology ring/quantum K ring of a quiver variety and the cluster algebra. More explicitly, given a quiver with potential, there is an injective ring homomorphism from the cluster algebra to quantum cohomology/quantum K ring of the corresponding quiver variety. This relation has been proved for A and D type quivers.