Workshop on Quantum K-theory, 2018
January 19-22, 2018, Sun Yat-sen University
Organizers:
HU, Jianxun (Sun Yat-sen University)
LI, Changzheng (Sun Yat-sen University)
Speakers:
Chen, Xiaojun (Sichuan University)
Du, Chengyong (Sichuan Normal University)
Fan, Huijun (Peking University)
Li, Weiping (Hong Kong University of Science and Technology)
Ruan, Yongbin (University of Michigan)
Sun, Shanzhong (Capital Normal University)
Zhang, Ming (University of Michigan)
Zhou, Jian (Tsinghua University)
Program:
Venue: Room 415 (New Math. Building)
Jan 19: Arrival Day
| Jan. 20 (Sat) | Jan. 21 (Sun) | Jan. 22 (Mon) | |
| 9:00-10:00 | Ruan | Fan | Zhou |
| Tea Break | |||
| 10:30-11:30 | Li | Zhang | Du |
| Lunch Break | |||
| 14:30-15:30 | Chen | Sun | Free discussions |
| Tea Break | |||
| 16:00-17:00 | Zhang | Zhang | |
Banquet: 18:00-20:00 (TBA)
Title and Abstracts:
Title: Level Structure in Quantum K-theory (Lecture series by Yongbin Ruan & Ming Zhang)
Abstract: Quantum K-theory is a K-theoretic version of quantum cohomology introduced by Givental-Lee more than ten years ago. The recent interest in the subject comes from its connection to 3-d quantum field theory and q-hypergeometric series. In many way, quantum K-theory is more closely tied with representation theory than quantum cohomology. In this lecture series, we will introduce the level structure into quantum K-theory and discuss some of its surprising connection to other subject of mathematics such as mocked theta function.
Title: Calabi-Yau algebras and noncommutative symplectic geometry (by Xiaojun Chen)
Abstract: In 2012, Pantev, Toen, Vaquie and Vezzosi introduced the notion of shifted symplectic structures for derived stacks. As remarked by the authors, the shifted symplectic structure, if it exists, always comes from the Poincare duality of the corresponding source spaces; they also outlined how to generalize the shifted symplectic structure to non-commutative spaces, such as Calabi-Yau categories. In this talk, we focus on a special class of non-commutative Calabi-Yau spaces, namely Calabi-Yau algebras, and show that for a Koszul Calabi-Yau algebra, there is a noncommutative shifted symplectic structure, which is also called the shifted bi-symplectic structure, on the cobar construction of its Koszul dual coalgebra adjoining a co-unit. Such shifted bi-symplectic structure naturally induces a shifted symplectic structure on the corresponding differential graded (DG) representation schemes, and also induces a shifted Poisson structure on the derived representation schemes of the Calabi-Yau algebra. Joint work with F. Eshmatov.
Title: Orbifold Gromov–Witten theory of weighted blowups (by Chengyong Du)
Abstract: Determine the GW-theory of a blowup manifold $\tilde X$ of a symplectic manifold $X$ along a symplectic submanifold $S$ is a basic question in GW-theory. By the previous works of Maulik-Pandharipande, Hu-Li-Ruan etc. we now know that the GW-theory of $\tilde X$ is determined by the GW-theories of $X$ and $S$, the natural map $H^*(X)\to H^*(S)$, modular the relative GW-theory of $\mathbb P^1$ which is solved. In this talk we explain the orbifold version of this problem. In the orbifold case, we could consider weighted blowups. We show that for a symplectic orbifold $\sf X$, a symplectic sub-orbifold $\sf S$, and the weight $\mathfrak a=(\alpha_1, \ldots, \alpha_n)$ blowup $\sf \tilde X$ with exceptional divisor $\sf D_{\mathfrak a}$, the orbifold GW-theory of $\sf \tilde X$ is determined by the orbifold GW-theories of $\sf X, S$ and $\sf D_{\mathfrak a}$, the natural restriction map $H^*_{CR}(\msf X)\to H^*_{CR}(\sf S)$ and the first Chern class of the normal bundle of $\D_{\mathfrak a}$ in $\sf \tilde X$, modular the relative orbifold GW-theory of $[\mathbb P^1/\mathbb Z_r]$ solved by Tseng-You and some certain relative orbifold GW-invariants of weighted projective spaces computed by Chen-D-Hu. Our results have several applications. For example, we could show that the relative orbifold GW-theory of $(\sf X, Z) $ is determined by the absolute orbifold GW-theories of $\sf X, Z$ and $ H^*_{CR} (\msf X) \to H^*_{CR} (\sf Z) $. This talk is based on a joint work with B. Chen and R. Wang.
Title: Fukaya category of Landau-Ginzburg model (by Huijun Fan)
Abstract: In this talk, I will describe the construction of the Fukaya category of Landau-Ginzburg model based on Witten equation with boundary conditions on Lefschetz thimbles. In particular, we can obtain the compactness theorem for the tame LG system. This is a joint work with Wenfeng Jiang and Dingyu Yang.
Title: Genus one GW invariants of quintic CY via MSP (by Weiping Li)
Abstract: I will talk about the computation of the elliptic Gromov-Witten invariants of quantic Calabi-Yau threefolds X. Zinger and J. Li analyzed the moduli space of stable maps from elliptic curves to the quintic CY three-folds, and Zinger used torus localization and some new packaging methods to calculate the elliptic GW invariants of X. We will use the mixed spin P-fields moduli to calculate the elliptic GW invariants of X. Analysis of the moduli spaces of stable maps to X is replaced by introducing FJRW moduli spaces into the moduli set-up. This is the joint work with S. Guo, H.L. Chang and J. Zhou.
Title: Algebraic Resurgent Deformation Quantization. (by Shanzhong Sun)
Abstract: We report our recent progress on deformation quantization using ideas from resurgence theory to address the convergent issue.
Title: A conjectural formula on equivariant K-theory in genus zero. (by Jian Zhou)
Abstract: We present a conjectural formula that generalizes Yuan-Pin Lee's formula on moduli space of genus zero curves to the equivariant case with respect to the action of symmetric groups. We prove some special cases by some sophisticated combinatorial formulas.
Accommodations and Local information:
Hotels on campus:
SYSU hotel & Conference Center（中山大学学人馆）
Address: North gate of Sun Yat-sen University, Bingjiang Dong Road, Haizhu District, Guangzhou
Tel: 020-89222888 Website: http://www.syskaifeng.com
Contact information:
Ms. YU, Jinyun
Email: yujinyun AT mail.sysu.edu.cn
Prof. LI, Changzheng
Email: lichangzh AT mail.sysu.edu.cn
Phone: +86-20-84111738
School of Mathematical, Sun Yat-sen University, No. 135, Xingang Xi Road, Haizhu, Guangzhou, 510275, China