| Higher Gauge Theory, Homotopy Theory and n-Categories Lectures at Topics in Homotopy Theory, (2009) | |||||||||||||||
Abstract | |||||||||||||||
| These are rough notes for four lectures on higher gauge theory, aimed at explaining how this theory is related to some classic themes from homotopy theory, such as Eilenberg–Mac Lane spaces. After a brief introduction to connections on principal bundles, with a heavy emphasis on the concept of ‘torsor’, we describe how to build the classifying space BG of a topological group G starting from the topological category of its torsors. In the case of an abelian topological group A, we explain how this construction can be iterated, with points of B n A corresponding to ‘finite collections of A-charged particles on S n ’. Finally, we explain how B n A can be constructed from the n-category of n-torsors of A. In the process, we give a quick introduction to some simple concepts from n-category theory. References provide avenues for further study. 1 A Taste of Gauge Theory Gauge theory describes the forces of nature using the mathematical formalism of connections on principal bundles, which physicists call ‘gauge fields’. We will not explain how this works — our goal is instead to explain how principal bundles and their categorified generalizations relate to some basic themes in homotopy theory — but a taste of the original physics motivation will still be helpful. The easiest example is gravity. A physical object can be used to define a ‘frame ’ in the n-dimensional smooth manifold M representing spacetime: Definition 1. A frame at a point x in some smooth manifold M is a basis of the tangent space TxM. The set of all frames at x is denoted FxM. The set of all frames at all points of M is denoted F M, and called the frame bundle of M. Ignoring the fourth dimension (time), the picture looks like this: 1 The frame bundle of M can be made into a smooth manifold, and the motion of a freely falling nonrotating object traces out a path in the frame bundle: More generally, we may carry an object without rotating it along any smooth path in spacetime — this is called ‘parallel transport’. Parallel transport along a smooth path γ from x ∈ M to y ∈ M gives rise to a map | |||||||||||||||
Publication details | |||||||||||||||
| |||||||||||||||