Recent Developments in SL(2,C) Gauge Theory

主讲人 / Speaker

Siqi He 何思奇

Morningside Center of Mathematics, Chinese Academy of Sciences

时间 / Time

Thursday & Tuesday, 15:30-17:00

Dec. 11, 16, 23, 2025

地点 / Venue

B541, Shuangqing Complex Building A

Recent Developments

in SL(2,C) Gauge Theory

Course Description

This mini-course gives a gentle introduction to gauge theory with structure group SL(2,C) and some recent developments. We will start with the basic language of connections and curvature on vector bundles, and explain how flat SL(2,C) connections naturally appear in geometry and low-dimensional topology. The course will be divided into four parts:

1. Connections and Curvature on SL(2,C)-Bundles.We review the definitions of connection, curvature, and gauge transformation on complex rank-2 bundles with structure group SL(2,C). Simple examples and basic identities (such as the Bianchi identity) will be discussed.

2. Flat SL(2,C) Connections and Character Varieties. We introduce flat connections, their holonomy, and the associated representation and character varieties of fundamental groups. Examples from surfaces and 3-manifolds will be used to illustrate the geometry of these moduli spaces.

3. Compactness and Z/2 Harmonic 1-Forms. We explain how sequences of flat SL(2,C) connections can degenerate, and how Z/2 harmonic 1-forms arise in the description of the ideal boundary of certain moduli spaces, following work of Taubes.

4. Recent Work and Further Directions. We give a brief overview of some recent progress in SL(2,C) gauge theory and related geometry problems, including measured foliation, special Lagrangian submanifolds, etc.

主讲人简介 / About the Speaker

I am now a faculty member at Academy of Mathematics and Systems Science, Chinese Academy of Sciences.

My research focuses on gauge theory and differential geometry, in particular, the Kapustin-Witten equations, Higgs bundles, special Lagrangian submanifolds, and manifolds with special holonomy.