ConnectionsOverVectorBundles

A connection provides the derivative of a flow along a vector field on a vector bundle over a manifold. A vector bundle is a topological space that is locally a product of a manifold (in this case $\mathbb{R}^3$) and a vector space, and globally might have more complicated structure.

To understand why a connection can be seen as a kind of function, it's useful to understand what a connection does. Given a vector field X on the manifold and a section s of the vector bundle (which assigns a vector in the fiber over each point of the manifold), a connection provides a way to differentiate the section along the vector field, producing another section of the bundle. The connection is said to be $\mathfrak{k}$-valued if the coefficients of the connection, when expressed in a local basis, are elements of the Lie algebra $\mathfrak{k}$.

In your text, the space of gauge equivalence classes of connections on $\mathbb{R}^3$$ is being considered. Two connections are said to be gauge equivalent if they can be transformed into each other by a gauge transformation, which is a kind of "change of basis" in the fibers of the vector bundle that varies from point to point in a smooth way.

This space of gauge equivalence classes of connections can be seen as a kind of function space, because each connection can be described by a certain set of functions (the coefficients of the connection in a local basis) on the manifold. However, not every set of functions gives a valid connection, and two connections might be considered "the same" (i.e., gauge equivalent) if their coefficients can be transformed into each other by a certain kind of transformation (a gauge transformation).

A well-defined probability measure on this space of gauge equivalence classes of connections would complete the proof of non-pertubative quantization of Yang-Mills theory.

Reiterated

• A connection on a vector bundle provides a way to differentiate sections of the bundle along a vector field on the base manifold. This essentially gives a rule for how to "transport" vectors in the bundle along curves in the manifold.

• In the context of gauge theory (which includes Yang-Mills theory), connections can be thought of as functions because they can be described locally by a set of functions (the components of the connection form in a local basis).

• The gauge group acts on this space of connections, and two connections are considered equivalent if they are related by a gauge transformation. This leads to the concept of the space of gauge equivalence classes of connections.

• A key challenge in the non-perturbative quantization of Yang-Mills theory is to define a suitable probability measure on this space of gauge equivalence classes of connections. This measure would play the role of the quantum state of the Yang-Mills field, providing probabilities for different configurations of the field.