Gauge group acts on affine space of connections

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Statement

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

The gauge group acts on the affine space of connections of M. 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 \Gamma(TM) \otimes \Gamma(E) \to \Gamma(E), satisfying certain conditions. Suppose g \in G and \nabla is a connection on M. We define the connection g . \nabla as follows:

(g . \nabla)_X(s) = \nabla_X(g^{-1}(s))

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 g \in G 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 E over M can be viewed as equipping \Gamma(E) with the structure of a module over the connection algebra of M. Let us understand how an element g \in G acts on E.

A connection \nabla is viewed in terms of its action map:

a_\nabla:\mathcal{C}(M) \times \Gamma(E) \to \Gamma(E)

The action map of g.\nabla is given by:

a_{g.\nabla}(\alpha,s) = a_\nabla(\alpha,g^{-1}s)

In other words, a given element \alpha \in \mathcal{C}(M) now acts on s the way it would originally have acted on g^{-1}s.

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

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Related facts

The difference tensor

Given any g \in G, we can define the map:

\nabla \mapsto \nabla - g.\nabla

This difference is no longer a connection: in fact, since the connections form an affine space, the difference is a C6\infty-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