Mini-Course on Lattice Gauge Field Theory and Prismatic Sets

Lecture 1: 17 September (Wed): 14h30-15h30 (Room 1.02 ) Bundle Theory: Firstly I will try to give the idea on the importance of gauge theory. The purpose of this talk is to give the theory of principal G-bundles, where G is a Lie group. This talk contains the theory of connection and curvature in a principal G-bundle. I will mention trivial bundle and give some examples. Lecture 2: 18 September (Thu): 14h30-15h30 (Room 1.02 ) Prismatic Sets, Prismatic Triangulation and The Classifying Space: I will talk about simplicial sets and classifying space. I will define prismatic sets and study with their various geometric realizations. I will introduce the prismatic triangulation of a simplicial map and in particular of a simplicial set. If time permits, I will give some topological properties of these triangulations and some related maps. The important thing is to see a bundle over a simplicial set and construct the transition functions for a simplex which are generalized lattice gauge fields. ( This part will be given in the last talk with details) So I hope that I can prepare you with aid of the first two talks for the last one. Lecture 3: 19 September (Fri): 14h30-15h30 (Room 1.02 ) Lattice Gauge Field Theory and Prismatic Sets (joint work with Johan L. Dupont): We study prismatics sets analogously to simplicial sets except that realization involves prims. Particular examples are the prismatic subdivision of a simplicial set S and the prismatic star of S. Both have the same homotopy type as S and in particular the latter we use to study lattice gauge theory in the sense of Phillips and Stone. Thus for a Lie group G and a set of parallel transport functions defining the transition over faces of the simplices, we define a classifying map from the prismatic star to a prismatic version of the classifying space of G. In turn this defines a G-bundle over the prismatic star.