# Gauge group acts on affine space of connections

## Contents

## Statement

Suppose is a differential manifold and is a vector bundle over . We define the gauge group of as the group of all smooth maps that sends the fiber over any to itself, and is a linear automorphism for every such fiber.

The gauge group acts on the affine space of connections of . Here, we describe this action in three ways, using the three alternative descriptions of a connection.

### Action with respect to the usual definition of connection

In the usual definition, a connection is defined *globally*, as a map , satisfying certain conditions. Suppose and is a connection on . We define the connection as follows:

The inverse sign comes to preserve the left action condition.

With this definition, it is fairly easy to see that the new map is again a connection, and moreover, the action of any is a linear map, and in particular, an affine map.

### Action with respect to the view of a connection as a module structure

`Further information: Connection is module structure over connection algebra`

A connection on a vector bundle over can be viewed as equipping with the structure of a module over the connection algebra of . Let us understand how an element acts on .

A connection is viewed in terms of its action map:

The action map of is given by:

In other words, a given element now acts on the way it would originally have acted on .

### Action with respect to the view of a connection as a splitting

*Fill this in later*

## Related facts

### The difference tensor

Given any , we can define the map:

This difference is no longer a connection: in fact, since the connections form an affine space, the difference is a -bilinear map, or is a 2-tensor. This turns out to have important applications.

### The notion of a gauge

All connections that are in the same orbit under the action of the gauge group are essentially equivalent, so choosing a specific representative connection, is sometimes termed choosing a *gauge*, and is treated analogous to choosing a basis.

### The particular case of the tangent bundle

`Further information: Gauge group of manifold acts on affine space of linear connections`