Program
 Geometry and Physics Seminar Student No.： 40 Time： Tue 10:00-11:30 Instructor： Li Si, He Weiqiang, Zong Zhengyu Place： Conference Room 3, Jin Chun Yuan West Bldg. Starting Date： 2018-3-6 Ending Date： 2018-6-26

2018-6-26 Tuesday

Speaker: Xiudi Tang

Time: 10:00-11:30

Title: Focus-focus singular fibers in integrable systems

Abstract: We classify, up to symplectomorphisms, a neighborhood of a singular fiber of an integrable system (which is proper and has connected fibers) containing $k > 1$ non-degenerate focus-focus critical points.
Our result shows that there is a one-to-one correspondence between such neighborhoods and $k$ formal power series, up to a $(\Z_2 \times D_k)$-action, where $D_k$ is the $k$-th dihedral group.
This proves a conjecture of San V{\~u} Ng{\d{o}}c from 2002.

2018-5-22 Tuesday

Speaker: Xinxing Tang

Time: 08:30-10:00

Title: The toroidal tt^* metric in the LG B model

Abstract: We study the genus 1 Kahler metric on the Hodge bundle over the deformation space of a singularity. We recover its Kahler potential in two different cases: (1) consider the universal deformation of a quasi-homogeneous polynomial with isolated critical points, and define the isomodromic tau function for the tt^* geometry structure on the Hodge bundle; (2) consider the marginal deformation of a homogeneous polynomial, we study torsion type invariants via the heat kernel analysis, and show that the 2nd torsion serves as the Kahler potential.

Speaker: Jun Wang

Time: 10:00-11:30

Title: Phase transition of Kahler-Einstein Metrics via moment maps

Abstract: We study the phase transitions of Kahler Ricci-flat metrics on some open Calabi-Yau spaces with the help of the images of moment maps of natural torus actions on these spaces and the Hessian geometry on them.

2018-4-24 Tuesday

Speaker: Yang Liu

Time: 10:00-11:30

Title: Modular curvature and Hypergeometric functions

Abstract: In noncommutative geometry, an essential question is to extend the notion of metric and curvature in Riemannian geometry to noncommutative spaces in a operator theoretical framework. A fundamental feature, in contrast to Riemannian geometry, is the fact that metrics are parametrized by noncommutative coordinates. In the conformal geometry of noncommutative tori, the new structure in the modular analog of the Gaussian  curvature consists of two spectral functions, which compress the ansatz caused by the noncommutativity between the metric coordinate and its derivatives. In the first part of the talk, I will explain the higher dimensional generalization of a fantastic functional equation between them due to Connes and Moscovici. In the second part, I will show that hypergeometric functions are the build blocks of those spectral functions. A surprising discovery, obtained by combining the power of hypergeometric functions and computer algebra systems, is that Connes-Moscovici functional relation can be extended to a continuous family with respect to the dimension parameter.

2018-4-17 Tuesday

Speaker: Bohan Fang

Time: 10:00-11:30

Title: Topological recursion and modularity of GW invariants

Abstract:I will describe how to use topological recursion to study the modularity of Gromov-Witten invariants for toric CY 3-folds with genus one mirror curve. In particular, with the ring structure of the GW potential, one can show the Yamaguchi-Yau holomorphic anomaly equation. This talk is based on the joint work with Yongbin Ruan, Yingchun Zhang and Jie Zhou.

2018-4-10 Tuesday

Geometry and Physic seminar: basic notion

Speaker: Yan Wenbin [Tsinghua University]

Time: 08:30-09:30

Title: Argyres-Douglas theories, wild Hitchin systems, vertex operator algebra and refined Chern Simons, Part I: introduction to AD theories (fin).

Abstract: We continue the lecture series on the relations among AD theories, wild Hitchin systems, VOA and refined CS. In the third lecture, we will finish analyzing the singularity on the moduli space of pure SU(3) SYM at which two intersecting cycles shrinks at the same time, and write down the monodromies and SW curves around this singularity. One can see from the SW curve that the theory around the singularity has one dimension Coulomb branch and the Coulomb branch operator has fractional dimension. I will also give another construction of the same AD theory which hints a more general (G, G’) construction of AD theories.

Geometry and Physic seminar

Speaker: Rak-Kyeong Seong [Tsinghua University]

Time: 10:30-11:30

Title: Calabi-Yau Volumes, Reflexive Polytopes and Applications of Machine Learning

Abstract: I will review our study on various geometrical quantities for Calabi-Yau varieties realized as cones over Gorenstein Fano varieties. These were obtained as toric varieties from reflexive polytopes in various dimensions. We focus on the minimized volume of the Sasaki-Einstein base of the corresponding Calabi-Yau and identify volume bounds that have interpretations in the context of associated field theories via the AdS/CFT correspondence. I will also review applications of machine learning techniques that have reproduced our computations with astonishing accuracy.

2018-3-27 Tuesday

Speaker: Yan Wenbin [Tsinghua University]

Time: 08:20-09:20

Title: Argyres-Douglas theories, wild Hitchin systems and vertex operator algebra, Part I: introduction to AD theories

Abstract: We continue the lecture series on the relations among AD theories, wild Hitchin systems, VOA and refined CS. In the second lecture, we will continue our discussion on the moduli space of pure SU(3) SYM, Seiberg-Witten (SW) geometry, and singular locus. We will see BPS particles corresponds to cycles on the SW curve and there are singular points on the moduli space where multiple intersecting cycles degenerate, hence leads to isolated fixed points which mutually non-local massless BPS states.

Speaker: Wei Li

Time: 10:00-11:30

Title: Higher spin and Yangian

Abstract: The talk has two parts. First, I will explain a triangle connecting W symmetry, affine Yangian, and plane partition.  I will explain:

(1) How affine Yangian and W symmetry are related.

(2) Plane partitions furnish a natural class of representations for affine Yangians.

(3) Plane partition provides a useful new way to study representation of W algebra. (Time permitted, I will illustrate the last point with one or two examples motivated by higher spin gravity and string theory.) The second half is about the N=2 version of this triangle.

2018-3-13 Tuesday

Speaker: Yan Wenbin [Tsinghua University]

Time: 08:30-09:20

Title: Argyres-Douglas theories, wild Hitchin systems and vertex operator algebra, Part I: introduction to AD theories

Abstract: This is supposed to be a pedagogical lecture series to discuss the (generalized) Argyres-Douglas (AD) theories and their relations to wild Hitchin systems and vertex operator algebra (VOA). I will discuss both the original construction of AD theory, namely as an isolated fixed point of pure N=2 SU(3) SYM and the modern approach as an infrared fixed point of a certain deformation of N=1 SQCD.

Time: 10:00-11:30

Title: Classifying Subspaces of Superspaces

Abstract: Superspaces admit a general classification through non abelian sheaf cohomology. In this talk I will begin by describing this classification. Subsequently I will comment on working progress to give an analogous classification of embedded superspaces. This is useful if, for instance, one wants to study complicated superspaces by reference to simpler ones.

2018-3-6 Tuesday

Speaker:  Martin Vogrin [Hamburg]

Time: 10:00-11:30

Title: Differential rings for elliptic $K_3$ surfaces

Abstract: I will present a recent construction of moduli spaces $\mathsf{T}$ of pairs $(X,\{\alpha_i\})$, where $X$ is a lattice polarized $K_3$ surface and $\{\alpha_i\}$ are elements of a basis of its middle-dimensional cohomology. I will specialize to certain two-parameter families, which are mirrors to elliptic $K_3$ surfaces with fibres of type $E_6$, $E_7$ and $E_8$. For these families I will construct coordinates on $\mathsf{T}$ and show that the local rings are isomorphic to the ring of quasi-modular forms in two variables. I will show that a non-trivial combination of coordinate vector fields on $\mathsf{T}$ generates a Lie algebra $\mathrm{sl}_2(\mathbb{C})\oplus \mathrm{sl}_2(\mathbb{C})$ and I will give an alternative construction of this Lie algebra starting from an action of an algebraic group on $\mathsf{T}$.

References:
[1] S. Yamaguchi and S.-T. Yau, \emph{Topological string partition functions as polynomials}, JHEP, 2004.
[2] M. Alim and J. D. Lange, \emph{Polynomial structure of the (open) topological string partition function}, JHEP, 2007.
[3] H. Movasati, \emph{Quasi-modular forms attached to Hodge structures}, Arithmetic and Geometry of K3 Surfaces and Calabi-Yau Threefolds, Springer New York (567 - 587), 2013.
[4] M. Alim, \emph{Algebraic structure of $tt^*$ equations for Calabi-Yau sigma models}, Commun. Math. Phys., 2017.
[5] M. Alim and M. Vogrin, \emph{Differential rings for elliptic K3 surfaces}, in preparation.

History:
2017:
http://ymsc.tsinghua.edu.cn/sjcontent.asp?id=921
Autumn, 2016: http://ymsc.tsinghua.edu.cn/sjcontent.asp?id=860
Spring, 2016: http://ymsc.tsinghua.edu.cn/sjcontent.asp?id=799
Autumn, 2015: http://ymsc.tsinghua.edu.cn/sjcontent.asp?id=731