Quantum Algebras and Toric Calabi-Yau 3-folds’’. The goal is to understand some modern developments on quantum algebras associated to toric Calabi-Yau 3-folds.
Mathematically, the quantum algebras have to do with the corner vertex algebra of Gaiotto and Rapcak, cohomological Hall algebras of Kontsevich and Soibelman, and quiver Yangians of Galakhov, Li, and Yamazaki. Physically, such a quantum algebra is supposed to be the BPS algebra of type IIA string theory compactified on a toric CY 3-fold or the corresponding 4d N=2 supersymmetric theory.

Upcoming Talks

Past Talks

Date：June 15th

Speaker: Xiudi Tang (University of Toronto)

Title: Symplectic excision

Abstract: We consider closed subsets of a noncompact symplectic manifold and determine when they can be removed by a symplectomorphism, in which case we say the subsets are symplectically excisable.

In the first hour, we explain the excision of a ray.

In the second hour, we generalize the excision to any embedded epigraph of a lower semi-continuous function and give several examples.

Date：June 1st

Speaker: Si-Qi Liu (THU)

Title: Matrix Integral, Hodge Integral, and Integrable systems

Abstract: Matrix integral is a classical topic in mathematics. It is introduced by physicist E. Wigner, and has many interesting applications in physics, probability theory, mathematical statistics, numerical analysis, and number theory. It is revealed by the celebrated Witten conjecture that matrix integral is also the bridge among two-dimensional quantum gravity, the moduli space of stable curves, and the Korteweg-de Vries (KdV) hierarchy. Hodge integrals are the integrals of certain natural cohomological classes on the moduli space of stable curves, which are very important in modern mathematical physics. In our previous work, we showed that the generating function of certain Hodge integrals is related to the GUE matrix model and the Volterra hierarchy. We also conjecture a generalization of this correspondence. Recently, we prove this generalization.

Date：May 25th

Speaker: Maosong Xiang (Huazhong University of Science and Technology)

Title: Cohomology of regular Courant algebroids

Abstract: Courant algebroids were introduced by Liu, Weistein and Xu in their study of Manin triples. Roytenberg observed that each Courant algebroid correspond to a degree 2 symplectic graded manifold with a degree 3 Hamiltonian function satisfying the classical master equation; thus produce a 3D TFTs via AKSZ's construction. Classical observables of these TFTs arise from cohomology of the corresponding Courant algebroids. In the pre-talk, I will review some basics on dg geometry with a focus on symplectic graded manifolds of degree $1$ and $2$, respectively. In the second hour, I will discuss cohomology of regular Courant algebroids by constructing a Hodge-to-de Rham type spectral sequence. Time permitted, applications to regular Courant algebroids arising from regular foliations and regular Lie algebroids would be touched, which might be helpful to understand corresponding 3D TFTs.

Date：May 18th

Speaker: William Donovan (YMSC)

Title: Simplices in the Calabi-Yau web

Abstract: Calabi-Yau manifolds of a given dimension are connected by an intricate web of birational maps. This web has deep consequences for the derived categories of coherent sheaves on such manifolds, and for the associated string theories. In particular, for 4-folds and beyond, I will highlight certain simplices appearing in the web, and identify corresponding derived category structures.

Date：May 11th

Speaker: Stavros Garoufalidis (Southern University of Science and Technology)

Title: Meromorphic quantum Jacobi forms in complex Chern-Simons theory

Abstract: Two interesting and closely related concepts, quantum modularity and resurgence, have recently emerged in complex Chern–Simons theory. These two concepts combine to define a meromorphic quantum Jacobi form, a function with remarkable properties which is uniquely determined by its asymptotic expansion in two ways: via resurgence and via quantum modularity. In this talk, we will describe this structure for the complex Chern-Simons invariants of the two simplest hyperbolic knots, the 4_1 and 5_2 knots, and explain their properties. This is joint work (in several collaborations) with Gie Gu, Rinat Kashaev, Marcos Marino and Don Zagier.

Date：April 27th

Speaker: Junwu Tu(Shanghai Tech University)

Title: On the categorical enumerative invariants of a point

Abstract: We briefly recall the definition of categorical enumerative invariants (CEI) first introduced by Costello around 2005. Costello's construction relies fundamentally on Sen-Zwiebach’s notion of string vertices V_{g,n}’s which are chains on moduli space of smooth curves M_{g,n}’s. In this talk, we explain the relationship between string vertices and the fundamental classes of the Deligne-Mumford compactification of M_{g,n}. More precisely, we obtain a Feynman sum formula expressing the fundamental classes in terms of string vertices. As an immediate application, we prove a comparison result that the CEI of the field \mathbb{Q} is the same as the Gromov-Witten invariants of a point.

Date：April 20th

Speaker: Di Yang (University of Science and Technology of China)

Title: Geometry and arithmetic of integrable hierarchies of KdV type

Abstract: For each of the simple Lie algebras \g = A_l, D_l or E_6, we show that the all-genera one-point FJRW invariants of g-type, after multiplication by suitable products of Pochhammer symbols, are the coefficients of an algebraic generating function and hence are integral. Moreover, we find that the all-genera invariants themselves coincide with the coefficients of the unique calibration of the Frobenius manifold of g-type evaluated at a special point. For the A_4 (5-spin) case we also find two other normalizations of the sequence that are again integral and of at most exponential growth, and hence conjecturally are the Taylor coefficients of some period functions. The talk is based on a joint work with Boris Dubrovin and Don Zagier. In the pre-talk, I will recall the definitions of Frobenius manifolds, deformed flat connections, \lambda-periods, etc., as well as topological ODEs of \g-type.

Date：April 13th

Speaker: Shanzhong Sun (Capital Normal University)

(pretalk) Title: Introduction to Ecalle’s resurgent theory

Abstract: Ecalle’s resurgent theory was originally developed to prove Dulac conjecture about finiteness of limiting cycles in ODEs, and it immediately found applications in quantum mechanics which is now called exact WKB analysis. The theory develops steadily during last three decades with many beautiful algebraic structures unrivaled. Recently, it exerts many activities in quantum field theory and topological string theory. In this talk, I will review the basics of resurgence theory including alien derivatives, transseries.

(research talk) Title: Resurgent Deformation Quantization

Abstract: We use resurgence theory to address the issue of convergence in formal deformation quantization. For Moyal quantization, although it has little chance to be convergent, it does have interesting structures by imposing some algebraic conditions on the singularity set of Borel transformations.

Date：April 6th

Speaker: Yehao Zhou (Perimeter Institute)

Title: Generalized affine Grassmannian slices

Abstract: Generalized affine Grassmannian slices are introduced by Braverman, Finkelberg and Nakajima, and they show that these varieties are Coulomb branches of 3d N=4 quiver gauge theories. The first hour of my talk will be a pedagogical introduction to affine Grassmannian, G_O orbits and slices. In the second hour, I will define generalized affine Grassmannian slices and explain my recent work on some conjectures made by BFN on the geometry of these varieties. Then I will talk about how generalized affine Grassmannian slices show up in the study of ’t Hooft operators in 4d Chern-Simons theory.

Date：March 30th

Speaker: Satoshi Nawata (Fudan University)

Title: Instanton counting and O-vertex

Abstract: In the pretalk, I will give a pedagogical introduction to Nekrasov instanton partition function and topological vertex for A-type gauge group. (Certainly, experts can skip it.) It has been an open problem to classify pole structures for instanton partition functions for gauge groups other than A-type. In the main talk, I will present a closed-form expression of unrefined SO(N) and Sp(N) instanton partition functions as a sum over tuples of Young diagrams. We also formulate it by topological vertex with an O5-plane. This is on-going work with Rui-Dong Zhu.

Date：March 16th

Speaker: Emanuel Scheidegger (BICMR, PKU)

Title: On the quantum K-theory of the quintic

Abstract: Quantum cohomology is a deformation of the cohomology of a projective variety governed by counts of stable maps from a curve into this variety. Quantum K-theory is in a similar way a deformation of K-theory but also of quantum cohomology, It has recently attracted attention in physics since a realization in a physical theory has been found. Currently, both the structure and examples in quantum K-theory are far less understood than in quantum cohomology.

In the pre-talk we will review the basic properties of quantum cohomology, in particular, Givental's symplectic formalism. In the research talk, we will explain in a parallel way the corresponding properties of quantum K-theory, and we will discuss the examples of projective space and the quintic hypersurface in P^4.

Date：March 9th

Speaker : Peng Shan

Title: Categorification of quantum loop sl_2

Abstract: Categorification of quantum groups has important applications in representation theory and low dimensional topology.

In the first part, Situ Quan will give an introduction to quiver Hecke algebras and how to categorify negative part of quantum groups via their modules categories.

In the second part, I will explain a recent result on an equivalence of categories between a module category of quiver Hecke algebras associated with the Kronecker quiver and a category of equivariant perverse coherent sheaves on the nilpotent cone of type A. This provides a link between two different monoidal categorifications of the open quantum unipotent cell of affine type A_1.

Date：March 2nd

Speaker : Mauricio Romo

Title : A GLSM view on Homological Projective Duality

Abstract : After an introduction to the concept of window categories of B-branes on gauged linear sigma models (GLSM) and of homological projective duality, I will present a rather general construction of homological projective dual pairs using gauged linear sigma models.

I will mostly focus on the abelian case and, if time permits, will present a few remarks for the nonabelian case. This is based on joint work with Z. Chen and J. Guo: 2012.14109