This is a connection theory type question. Let ( M) be a Rienannian manifold, ( E\rightarrow M) a vector bundle over ( M), and ( \mathcal{A}) the space of connections on ( E ). If (A) is a local connection 1-form of some connection, and ( F(A) ) is it’s curvature 2-form, how does one define the square of the L2-norm ( ||F(A)||^2\in\mathbb{R})? I was at a math - physics conference recently and the speaker mentioned connections and denoted the connection as ( A) which I assumed to mean a local connection 1-form and denoted the curvature as ( F(A) ) which I took to mean it’s local curvature 2-form. However I could be wrong.
@Diffgeometer1 wedge with Hodge dual to get a top-degree form, then integrate. Maybe throw in an absolute value before integrating? Look at what this does locally, and it is sensible
@Diffgeometer1 Though, of course, this is just for real-valued forms. If you have a Lie algebra valued form, then you need an inner product on your Lie algebra as well. For a classical Lie algebra, up to a normalisation factor, this can usually be taken to be (\langle A ,B\rangle = -tr(AB)). Since here (F(A)) should be in general just (\mathfrak{gl}_n)-valued, this should be ok.
So you a) wedge with the Hodge dual of (F(A)) and then b) post-compose with the inner product on matrices, to get a scalar-valued top-form, and then c) integrate over the manifold. Steps a) and c) will use the Riemannian metric.
@highergeometer Thanks! This sounds quite reasonable. Also since the curvature 2-forms on overlapping frames (say (\alpha ) and ( \beta )-frames) are related by [ F(A^{(\alpha)})=g^{-1}F(A^{(\beta)})g]it follows that
[\mbox{tr}(F(A^{(\alpha)})\wedge \ast F(A^{(\alpha)}))=\mbox{tr}(F(A^{(\beta)})\wedge \ast F(A^{(\beta)}))]So one has a global n-form for integration where n is the dimension of M. If one is integrating over M, then it seems that we also require M to be compact.
@Diffgeometer1 you don't always need compactness. The curvature could decay sufficiently fast heading out to infinity, for instance. This happens with the original BPST instanton on R^4.
@Diffgeometer1 Of course, this is ultimately because the BPST instanton really lives on S^4, but the construction was explicitly thought of as being on Wick-rotated spacetime (hence R^4) and with a finite-energy condition, which is exactly that the L^2 norm of the curvature was finite.
@Diffgeometer1 - If G is a compact simple Lie group, its Lie algebra 𝔤 has an inner product that's invariant under automorphisms of the Lie algebra. Indeed, up to a constant factor, such a Lie algebra has a unique inner product invariant under all Lie algebra automorphisms. It's called "minus the Killing form":
So, if G is a compact simple Lie group, we can use the Killing form to create an n-form from the curvature 2-form of a connection on a principal G-bundle over a n-dimensional oriented Riemannian manifold, in a way that is invariant under gauge transformations. We simply follow @highergeometer's procedure but with minus the Killing form replacing -tr(AB).
These facts about compact simple Lie groups are some of the reasons physicists prefer them to other Lie groups, e.g. for "grand unified theories".
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