# Connections

also known as a path lifting rules in the context of fiber bundles and as gauge fields in particle physics

## Intuitive

The phase of a charged particle moving in an electromagnetic field (e.g., a monopole field) is quite like the internal spinning of our ping-pong ball. We have seen that a phase change alters the wavefunction of the charge only by a factor of modulus one and so does not effect the probability of finding the particle at any particular location, i.e., does not effect its motion through space. Nevertheless, when two charges interact (in, for example, the Aharonov-Bohm experiment), phase differences are of crucial significance to the outcome. The gauge field (connection), which mediates phase changes in the charge along various paths through the electromagnetic field, is the analogue of the room’s atmosphere, which is the agency (“force”) responsible for any alteration in the ball’s internal spinning.page 23 in Topology, Geometry and Gauge Fields: Foundations by Naber

## Concrete

The two most important types of connections are

• The Ehresmann Connection, which is appropriate tool to describe parallel transport in gauge theories. The parallel transport here happens on the fiber bundle, i.e. from one fiber to the next. Each fiber is one copy of our gauge group, e.g. $U(1)$. Our gauge fields, like the electromagnetic field is described by an Ehresmann connection.
• The Levi-Civita Connection, which is the appropriate tool to describe parallel transport in General Relativity. The gravitational field is described by a Levi-Civita connection

## Abstract

The wavefunction of the particle takes values in some vector space $V$ (for our purposes, $V$ will be some $\mathbb{C}_k$ ). The particle is coupled to (i.e., experiences the effects of) a gauge field which is represented by a connection on a principal G-bundle. The connection describes (via Theorem 6.1.4) the evolution of the particle’s internal state. The response of the wavefunction at each point to a gauge transformation will be specified by a left action (representation) of $G$ on $V$. $V$ and this left action of $G$ on $V$ determine an “associated vector bundle” obtained by replacing the $G$-fibers of the principal bundle with copies of $V$. The local cross-sections of this bundle then represent local wavefunctions of the particle coupled to the gauge field. Because of the manner in which the local wavefunctions respond to a gauge transformation the corresponding local cross-sections piece together to give a global cross-section of the associated vector bundle and this, we will find, can be identified with a certain type of $V$-valued function on the original principal bundle space. Finally, the connection on the principal bundle representing the gauge field gives rise to a natural gauge invariant differentiation process for such wavefunctions. In terms of this derivative one can then postulate differential equations (field equations) that describe the quantitative response of the particle to the gauge field (selecting these equations is, of course, the business of the physicists).Topology, Geometry and Gauge Fields: Foundations by Naber

## Why is it interesting?

Our interest in connections was originally motivated (in Chapter 0) by the suggestion that such a structure would provide the unique path lifting procedure whereby one might keep track of the evolution of a particle’s internal state (e.g., phase) as it traverses the field established by some other particle (e.g., the electromagnetic field of a magnetic monopole). Topology, Geometry and Gauge Fields: Foundations by Naber

## History

The historical evolution of our definition of the curvature form from more familiar notions of curvature (e.g., for curves and surfaces) is not easily related in a few words. Happily, Volume II of [Sp2] is a leisurely and entertaining account of this very story which we heartily recommend to the reader in search of motivation. Topology, Geometry and Gauge Fields: Foundations by Naber

[Sp2] is Spivak, M., A Comprehensive Introduction to Differential Geometry, Volumes I–V, Publish or Perish, Inc., Boston, 1979.

Contributing authors: 