Gauge group acts on affine space of connections
- 1 Statement
- 2 Related facts
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
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