Geometry-Physics Workshop

Organizing committee: Yongbin Ruan (Zhejiang University) 

Contact: Ran Ji

     E-mail: ranji02@zju.edu.cn 

Workshop Venue：

Lecture Hall of Institute for Advanced Study in Mathematics, 1st floor of east 7th teaching building, Zhejiang University (Zijingang Campus) 

Title and Abstract

Friday, October 21 

• Mauricio  Romo (Tsinghua University)

Title: Exponential Networks

Abstract: I will present an introduction of the basics of exponential networks associated to CY 3-folds described by conic bundles. I will focus on how one can formally define a generalization of the nonabelianization map of spectral networks and how this leads to counting of Donaldson-Thomas invariants or, from a physical perspective, 5d BPS states. They can be also be related to stable representations of quivers with potentials. This is based on joint work with S. Banerjee and P. Longhi. 

• Qin Li (Southern University of Science and Technology)

Title: Quantization  of Kapranov $L_\infty$ structure and one-loop exact BV quantization

Abstract: In  this talk, I will introduce a quantization of Kapranov $L_\infty$ structure  on K\ahler manifolds, which gives rise to a special class of solutions  of Fedosov equations. These solutions give rise to one-loop exact  quantization using Costello's theory of effective renormalization. 

• Zijun Zhou (IPMU)

Title: 3d N=2 mirror symmetry for toric varieties

Abstract: In this talk, I will introduce a new construction for the K-theoretic mirror symmetry of toric varieties, based on the 3d N=2 mirror symmetry introduced by Dorey-Tong. Given the toric datum, i.e. a

short exact sequence 0 -> Z^k -> Z^n -> Z^{n-k} -> 0, we consider the toric Artin stack of the form [C^n / (C^*)^k]. Its mirror is constructed by taking the Gale dual of the defining short exact sequence. As an analogue of the 3d N=4 case, we show that the K-theoretic I-function, defined by counting parameterized quasimaps from P^1, satisfies two sets of q-difference equations, with respect to the K\”ahler and equivariant parameters respectively. Under mirror symmetry, the q-difference equations associated with a mirror pair are identified with K\”ahler and equivariant parameters exchanged. This is joint work in progress with Yongbin Ruan and Yaoxiong Wen. 

• Weiqiang He (Sun Yat-sen University)

Title: Landau-Ginzburg theory and G-Frobenious algebra

Abstract: Consider a Landau-Ginzburg pair (W, G), where W is a non degenerate quasi-homogenous polynomial with only one singularity on origin, G is an admissible diagonal symmetry group of W. Via different approaches, Fan-Jarvis-Ruan, Polishchuk-Vaintrob, Kiem-Li construct Cohomological field theory on (W, G). They also prove that the CohFT will induce a Frobenius algebra structure. On the other hand, Kaufmann defines a new structure called G-Frobenius algebra structure, as a extension/generalization of Frobenius algebra. On this talk, I will explain how to find a G-Frobenius algebra structure via a modification on PV theory for LG pair (W, G), and state a G-Frobenius algebra version of LG mirror theorem when W is an invertible polynomial. This is based on a joint work with Polishchuk, Vaintrob and Shen. 

Saturday, October 22 

• Huijun Fan （Peking University）

Title: LG/CY correspondence between $tt^*$ geometries

Abstract: The concept of $tt^*$ geometric structure was introduced by physicists (Cecotti-Vafa, BCOV…), and then studied firstly in mathematics by C. Hertling. It is believed that the $tt^*$ geometric structure contains the whole genus $0$ information of the corresponding two dimensional topological field theory. In this talk, a LG/CY correspondence conjecture for $tt^*$ geometry will be given and partial result is given as follows. Let $f\in\mathbb{C}[z_0, \dots, z_{n+2}]$ be a nondegenerate homogeneous polynomial of degree $n+2$, then it defines a Calabi-Yau model represented by a Calabi-Yau hypersurface $X_f$ in $\mathbb{CP}^{n+1}$ or a Landau-Ginzburg model represented by a hypersurface singularity $(\C^{n+2}, f)$. We build the isomorphism of almost all structures in $tt^*$ geometries between these two models except the isomorphism between real structures. This is a joint work with Lan Tian and Yang Zongrui. 

• Bohan Fang (Peking University)

Title: All-genus  crepant transformation conjecture for toric CY 3-folds

Abstract: Ruan's  crepant transformation conjecture relates Gromov-Witten potentials for a  crepant pair by a symplectic transformation and an analytic continuation. For  a crepant pair of toric CY 3-folds, the conjecture for genus 0 open-closed  Gromov-Witten invariants are proved by Coates-Iritani-Jiang and Yu. For  higher genus, such conjecture involves a graph sum expression relating  Gromov-Witten potentials including lower genus terms. I will describe a proof  based on the remodeling theorem (different from the approach by  Coates-Iritani on $K_{P^2}$), which explicitly depends on the change of  polarization in the fundamental bidifferential form. This talk is based on  the joint work with Chiu-Chu Melissa Liu, Song Yu and Zhengy Zong. 

Sunday, October 23 

• Junwu Tu (ShanghaiTech University)

Title: Invariants of Calabi-Yau A-infinity categories

Abstract: Costello  (in 2005) introduced certain Gromov-Witten type invariants of Calabi-Yau  A-infinity categories. In this talk, we explain Costello's definition in more  down-to-earth terms. This leads to an explicit Feynman graph sum formula of  these invariants. Geometrically, this formula comes from writing down the  fundamental class of the Deligne-Mumford moduli space in terms of tubular  neighborhoods of its various boundary strata. 

• Xiaojun Chen (Sichuan University)

Title: Calabi-Yau categories and the shifted noncommutative symplectic structure

Abstract: In this talk, we study the local structure of Calabi-Yau categories, and show that under some mild conditions, they admit a shifted noncommutative symplectic structure. Such a noncommutative symplectic structure induces a shifted symplectic structure, in the sense of Pantev et al, on the moduli stack of the objects of the Calabi-Yau category. This result is also recently obtained by Brav and Dyckerhoff, but our technique is completely different. This talk is based on an ongoing project joint with A. and F. Eshmatov and Zeng.