Invariants and Structures

Thursdays at 2pm

Šarūnas Kaubrys, Kavli IPMU
10/15, SIMIS room 1110

Deformed Drinfeld coproducts via Joyce vertex coalgebras
Cohomological Hall algebras of quivers with potentials give a geometric way to realize quantum groups. In this talk, I will define a certain vertex coproduct called a Joyce vertex coalgebra on the Cohomological Hall algebras of a quiver with potential. This structure is compatible with the multiplication and upgrades the cohomological Hall algebra into a vertex bialgebra. I will then explain how to compare this structure to deformed Drinfeld coproducts on ADE Yangians. This is joint work with Shivang Jindal and Alexei Latyntsev.

Noah Arbesfeld, Utrecht University

10/27, SIMIS room TBA

Vafa-Witten invariants from framed sheaves
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.

Past talks

Ionut Ciocan-Fontanine, Academia Sinica
05/27, SIMIS room 710

Integrality of coefficients of mirror maps (after Jockers and Mayr)
In a 2018 Physics paper, Jockers and Mayr argued that the “3-d, N=2 abelian Gauged Linear Sigma Model” leads, via the renormalization flow, to Givental’s permutation-equivariant quantum K-theory of a complete intersection in a toric variety. Among the consequences they explored was a new derivation of the integrality of the Taylor coefficients of the mirror map for Calabi-Yau complete intersections, which they worked out for the quintic threefold, reproving the result of Lian and Yau. They did this by leveraging the intrinsic integrality of the 3-d GLSM partition function. In this talk, I will explain how the theory of quasimaps to GIT quotients can be used to justify rigorously their argument (and potentially extend its applicability to the non-abelian case as well). This is joint work with Jorin Schug.

Satoshi Nawata, Fudan University
05/20, SIMIS room 710

Quantum Toroidal Algebra of gl(1): Representations and Enumerative Geometry
In this talk, I give a self-contained review of the quantum toroidal algebra of gl(1) and its connections to representation theory and enumerative geometry. We begin with the algebraic structure itself, discussing its relationships to deformed W-algebras, DAHA, and affine Yangians. We then turn to its representation theory, covering the vertical modules — including the vector, Fock, and MacMahon representations — as well as the horizontal modules. In the final part, we explain how these representations are connected to qq-characters and to equivariant DT/PT invariants, illustrating the power of the algebraic approach to enumerative geometry.

Mauricio Romo, SIMIS
05/13, SIMIS room 710

Window categories and GLSM’s: old results and new speculations
I will briefly review the appearance of window categories on gauged linear sigma models and then move on to their uses in relation to local systems associated to Calabi-Yau manifolds and beyond.

Yu Tung Yau, Kavli IPMU
05/06 SIMIS room 710

Brane quantization and SYZ mirror symmetry
Coisotropic A-branes were introduced by Kapustin and Orlov to enlarge the Fukaya category of a symplectic manifold in a way that aligns with predictions from homological mirror symmetry. From a mathematical perspective, however, the categorical framework governing such branes remains largely undeveloped. In this talk, I focus on the semi-flat SYZ mirror symmetry setting and present a mathematical realization of the self-Hom spaces of semi-affine, space-filling coisotropic A-branes (based on joint work in progress with K. Chan, N. C. Leung, Q. Li, and Y.-H. Suen). Our construction yields a non-formal holomorphic deformation quantization of the underlying A-model manifold.

This approach not only reflects ideas from the Gukov–Witten brane quantization proposal and the family Floer theory framework, but also leads to a mirror correspondence identifying these A-model morphism spaces with the corresponding B-model morphism spaces on the mirror manifold. Moreover, the resulting actions on Lagrangian A-branes supported on torus fibers, and on their mirror B-branes, are shown to be functorial.

Zhe Wang, RIKEN
04/29 SIMIS room 710

Universal identities and Dubrovin-Zhang hierarchies
Universal identities are certain relations satisfied by the correlation functions of a cohomological field theory, whose form is independent of the details of the underlying cohomological field theory. Such identities are powerful tools for studying topology of moduli space of stable curves. In this talk, we will discuss the relation between universal identities and integrable hierarchies of topological type, which are also known as Dubrovin-Zhang hierarchies. We will show that it is possible to construct universal identities purely from the loop equations of the Dubrovin-Zhang free energy. In particular, this indicates that the existence of universal identities is a consequence of the existence of an integrable hierarchy of topological type, and hence universal identities can be studied from the view point of integrable hierarchies. This talk is based on joint work with S. Shadrin.

Taro Kimura, IMB, Université Bourgogne Europe, CNRS
04/15 SIMIS room 1010

qq-characters and sheaf counting
The qq-character introduced by Nekrasov in the context of supersymmetric gauge theory is yet another deformation of Frenkel-Reshetikhin’s q-character of a finite-dimensional module of quantum affine algebras. As in the case of q-character, the qq-character allows a geometric realization based on Nakajima’s quiver variety. More recently, we have extended the notion of qq-character to more generalized situations based on the formalism of quiver W-algebras, whose geometric description relies on equivariant cohomology/K-theory of moduli spaces of coherent sheaves. I’d like to address a geometric formulation of these types of qq-characters with emphasis on its relation to enumerative invariants, quiver descriptions, and wall-crossing structures. Joint work with G. Noshita (Pekin U).

Kwok Wai Chan, CUHK
Friday 4 pm 04/03, SIMIS room 1110

Coulomb branch action on quantum cohomology
In his 2014 ICM address, Teleman conjectured that there should be a nonabelian generalization of shift operators. In this talk, I will explain such a construction, which gives rise to an action of Coulomb branches on equivariant quantum cohomology of smooth semi-projective varieties that satisfy suitable conditions. This leads to several applications including a geometric characterization of the BFN Coulomb branch, an action of the affine nil-Hecke algebra on equivariant quantum cohomology of a flag variety which recovers the Peterson-Lam-Shimozono isomorphism, and a proof of a conjecture of Baverman-Finkelberg-Nakajima. This talk is based on recent joint works arXiv:2505.23340 and arXiv:2601.19691 with Ki Fung Chan, Chi Hong Chow and Chin Hang Eddie Lam.

Konstantin Aleshkin, Kavli IPMU
03/25, SIMIS room 1010

Critical cohomology of semiprojective varieties
Let f : U -> C be a regular function on a quasiprojective variety or an orbifold. There exist several related constructions of cohomology groups (called critical cohomology) associated to such a pair. These cohomology groups appear naturally in enumerative geometry of Landau-Ginzburg models such as mathematical theory of GLSM. In general, these cohomology groups are difficult to compute and different constructions are not canonically isomorphic. The situation simplifies when f is weighted homogeneous with respect to a torus action. In the talk I will explain relations between various constructions of critical cohomology and the decomposition theorem in the weighted homogeneous case. I will comment on the orbifold generalizations and provide an explicit description of the critical cohomology if U is a toric DM stack.

Ryo Ohkawa, RIMS, Kyoto University
03/11, SIMIS room 1010

$K$-theoretic wall-crossing formulas and multiple basic hypergeometric series
We study $K$-theoretic integrals over famed quiver moduli via wall-crossing phenomena. In particular, we focus on integrals over the handsaw quiver varieties of type $A_{1}$, and get functional equations for generating series of the integrals. These equations give geometric interpretation of transformation formulas for multiple basic hypergeometric series including the Kajihara transformation formula, and the one studied by Langer-Schlosser-Warnaar and Hallnäs-Langman-Noumi-Rosengren. We also study the chainsaw quiver varieties, and give conjectures for generating series of the integrals. This is based on joint work with Shiraishi.