Why the particular number of dimensions or particular gauge group is chosen is still not known.

This is the condition that fails to hold for non-Abelian gauge groups.

In this case instantons exist even when the gauge group is U(1).

In the simplest case the gauge group is U(1).

G is the gauge group, and it acts on each fiber of the bundle separately.

They differ by the gauge group in 10 dimensions.

We assume the real representation where the gauge group is .

The gauge group here is U(1), just the phase angle of the field, with a constant θ.

These transformations are together described by a mathematical object known as a gauge group.

The factor in front is the volume of the gauge group, and it contributes a constant, which can be discarded.