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Announcement | Course on Gromov-Witten, Donaldson-Thomas and Vafa-Witten Theory
发布时间:2020-06-19 14:42:58

【开课说明】

开课日期:20200706

举办地点:线上课程(zoom会议)

授课时间:7月6日(含)起 每周一、周四上午 10:00-11:15 课程持续十周

周一课程会议ID:815 4998 2426  周四课程会议ID:899 0981 1480  密码:123456

主讲人:蒋云峰 (University of Kansas)  导师简介见本页面底部附件

主办单位:国家天元数学中部中心、三峡数学研究中心

 点击此处下载zoom会议客户端 


【报名流程及联系人】

1. 通过Email提交报名申请,附件报名表电子版见本页面底部

(文件命名格式为20200706蒋云峰短期课程报名表+学员姓名+单位名);

2. 报名邮件发至:dxmeisx@126.com,邮件主题为20200706蒋云峰短期课程报名申请

3. 报名截止日期为 20200703,以电子邮件收信时间为准。

    联系人:邓老师  电话:13469861099

    Email:  dxmeisx@126.com


ABSTRCT

This course is an introduction to the theories of counting invariants in modern enumerative geometry, which includes Gromov-Witten, Donaldson-Thomas, and more recent Vafa-Witten invariants.  We first review some foundational work of prefect obstruction theory following Jun Li and Gang Tian, Behrend-Fatechi. We will talk about the basic notion of normal cones, intrinsic normal cones and virtual fundamental classes. We then apply the construction to define Gromov-Witten like invariants and Donaldson-Thomas invariants.  

A special but very important case is the ``Symmetric obstruction theory”, which was defined by Behrend and used to prove that Donaldson-Thomas invariants are motivic invariants.  We will state the basic idea of the proof.  Inspired by the notion of p-feild by Huai-Liang Chang and Jun Li in Gromov-Witten theory and cotangent theory in physics of Cosetllo, we will also talk about the work of signed Euler characteristics of Jiang-Thomas, and its applications to define the Vafa-WItten invariants for algebraic surfaces for both the gauge group SU(r) and its Langlands dual gauge group SU(r)/Z_r. Applications of the Vafa-Witten theory will be given to prove the S-duality conjecture of Vafa-Witten inspired by N=4 supersymmetric Yang-Mills theory in physics.


OUTLINE

1. Introduction: outline of the course

2. Basic deformation and obstruction theory

3. Normal cone and intrinsic normal cone, Virtual classes

4. Deformation and obstruction of Gromov-Witten theory: known results and open problems

5. Symmetric obstruction theory

6. Behrend function and Donaldson-Thomas invariants

7. Virtual signed Euler characteristic

8. Vafa-Witten invariants

9. S-duality conjecture

10. S-duality conjecture II



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