Abstract:

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.

Language: English

Personal Website：http://www.mcm.ac.cn/people/members/202204/t20220415_1094313.html