A Portal Special Presentation- Geometric Unity: A First Look: Difference between revisions

Let's think about unified content. We know that we want a space of connections \(A\) for our field theory. But we know, because we have a Levi-Civita connection, that this is going to be equal on the nose to ad-valued 1-forms as a vector space. The gauge group represents on ad-valued 1-forms.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ A = \Omega^1(ad) $$</div>

So just to fix bundle notation, we let \(H\) be the structure group of a bundle \(P_H\) over a base space \(B\). We use \(\pi\) for the projection map. We've reserved the variation in the \(\pi\) orthography for the field content, and we try to use right principal actions. I'm terrible with left and right, but we do our best. We use \(H\) here, not \(G\), because we want to reserve \(G\) for the inhomogeneous extension of \(H\) once we move to function spaces.

So with function spaces, we can take the bundle of groups using the adjoint action of \(H\) on itself and form the associated bundle, and then move to \(C^\infty\) sections to get the so-called gauge group of automorphisms. We have a space of connections, typically denoted by \(A\), and we're going to promote \(\Omega^1(B,\text{ ad}(P_H))\) to a notation of script \(\mathcal{N}\) as the affine group, which acts directly on the space of connections.