Gauge transformation of a vector bundle

Definition

Given data

• A differential manifold $M$
• A vector bundle $E$ over $M$

Definition part

A gauge transformation is a differentiable automorphism of $E$ as a vector bundle over $M$. In other words, it is a map from $E$ to $E$ that:

• Is a diffeomorphism of $E$ as a manifold
• Preserves the vector space over any point in $M$ and restricts to a linear automorphism in each such subspace.

The collection of all gauge transformations of a vector bundle is termed the gauge group of the bundle.

What gauge transformations act upon

The space of all sections

By the natural action of the gauge transformation on the vector bundle, it also acts on the space of sections, namely, by acting on the value of the section at each point as per the gauge transformation.

Since the space of sections of a vector bundle forms a vector space in its own right, the gauge transformations sit inside the space of linear automorphisms of this vector bundle.

Gauge transformations also preserve smoothness of sections. Hence, we can think of the gauge group as sitting inside the group of linear automorphisms of the space of smooth sections of the differential manifold.

Tensor powers of the vector bundle

We know that automorphisms of a vector space also act on tensor powers of that vector space. In a similar way, gauge transformations of a vector bundle also act on tensor powers of that vector bundle. In fact, they also act on the dual of the tensor bundle. Hence, they act on all tensor products of tensor powers of that vector bundle and its dual vector bundle.

The set of connections

The gauge transformations also act on the set of all connections. Namely, given any connection, which after all gives maps from sections of the vector bundle to sections of the vector bundle, we can apply the gauge transformation on it by conjugation. It is easy to check that the resultant mapping is again a connection.

All these actions of gauge transformations are mutually compatible. In some sense, we can thus think of the gauge group as acting on the entire colossum of data: the vector bundle, the dual, the sections, the tensor powers, the connections. Roughly, we can apply the gauge transformation to each of these uniformly everywhere.

Invariance under gauge transformations

Given a gauge transformation, we are interested in those vectors, tensors, vector fields, tensor fields, and connections that are invariant under that gauge transformation. A gauge field that is invariant under a particular gauge transformation is termed gauge-invariant for that gauge transformation.