Lattice Gauge Field Theory and Prismatic Sets

TL;DR

This paper introduces prismatic sets, a generalization of simplicial sets involving prisms, and applies them to lattice gauge theory to construct G-bundles via classifying maps.

Contribution

It develops the theory of prismatic sets and demonstrates their application in modeling lattice gauge theories through classifying maps and G-bundles.

Findings

Prismatic sets have the same homotopy type as simplicial sets.

A classifying map from the prismatic star to a prismatic classifying space is constructed.

G-bundles are realized over prismatic stars using parallel transport functions.

Abstract

We study prismatics sets analogously to simplical sets except that realization involves prisms, i.e., products of simplices rather than just simplices. 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.