Gauge group acts on affine space of connections

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 $\mathrm{\Gamma }(TM)\otimes \mathrm{\Gamma }(E)\to \mathrm{\Gamma }(E)$, satisfying certain conditions. Suppose $g\in G$ and $\mathrm{\nabla }$ is a connection on $M$. We define the connection $g\dot{\mathrm{\nabla }}$ as follows:

$(g\dot{\mathrm{\nabla }}{)}_X(s)={\mathrm{\nabla }}_X(g^{-1}(s))$

The inverse sign comes to preserve the left action condition.

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 $\mathrm{\Gamma }(E)$ 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 $\mathrm{\nabla }$ is viewed in terms of its action map:

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

The action map of $g.\mathrm{\nabla }$ is given by:

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

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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