Invariants and Structures

Coorganized with Yang Zhou

Wednesdays at 2pm

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.

Zhe Wang, RIKEN
04/29 SIMIS room 710

Borislav Mladenov, Academia Sinica
05/20, SIMIS room 710

Differential graded categories of D-branes, virtual de Rham complexes and deformation quantisation
Given a holomorphic symplectic manifold X, I will associate to X a virtual de Rham dg category and a dg category of canonical D-branes of type B wrapped on spin complex Lagrangians in X along with its deformation quantisation. For any suitable collection of complex Lagrangians, I will upgrade the deformation quantisation, with supports in the collection, to a formal deformation whose central fibre is the category of D-branes and whose generic fibre is the deformation quantisation. I will then show that the latter is quasi-isomorphic to the (base-change of) virtual de Rham category and explain the formality of the de Rham category, thus making the formal deformation “generically formal”. Time permitting, I will introduce the Kaledin class obstructing formality of a dg category and explain how the proper Calabi-Yau structure on the formal deformation leads to “generic formality => formality”, thus showing the formality of the dg category of D-branes. This story can be thought of as a B-side analogue of Ivan Smith’s conjecture on the formality of the Solomon-Verbitsky Fukaya category under Kapustin’s duality between type A and type B D-branes on X.

Past talks

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.