Why is A∧A nonzero in Yang-Mills curvature if ω∧ω=0?

[GoldPheonix]

Right, so in Yang-Mills theory, the vector potential is modified from:

$$F = dA$$​

To:

$$F = dA + A\wedge A$$​

However, it is my understanding that the exterior/wedge product is anticommutitive, so that for a given exterior algebra over a vector space, V:

$$\omega \wedge \omega = 0, \forall \omega \epsilon \Lambda(V)$$​

Why then is the second term in the curvature, F, not non-zero? I assume I'm missing something, could someone fill me in?


(Sorry, this probably fits better in Topology & Geometry section, but the question technically is a question about multilinear algebra)

 

[George Jones]

Argh! I was almost done a long latex post, but I accidentally closed the window.

$A$ is a Lie algebra-valued one-form. not a "normal" one-form. Roughy, think of $A$ as a matrix of "normal" one-forms, and $A \wedge A$ as matrix multiplication, where exterior products of matrix elements (normal one-forms) are used instead of multiplication of numbers.

 

[GoldPheonix]

Thank you George Jones for your reply.

Is there a particular name for the topic of Lie-algebra valued one-forms? (I assume it's actually a differentiable form, i.e. a section of the cotangent bundle). If there is, I would like to study it a little more.I'm assuming there's a little more to it than this. For instance, the Lie algebra here, is it the Lie algebra of the Lie group of diffeomorphisms/isometries on the differentiable manifold/Riemannian manifold (this is the only straight foreword way, that I can imagine, that they'd be matrices)? Or are we assuming the manifold for which our A-field is defined over is also has smooth group structure?

 

[George Jones]

Thank you George Jones for your reply.

Is there a particular name for the topic of Lie-algebra valued one-forms? (I assume it's actually a differentiable form, i.e. a section of the cotangent bundle). If there is, I would like to study it a little more.

They are treated in The Geometry of Physics: An Introduction by Theodore Frankel, and in Geometry, Topology, and Physics by M. Nakahara.

I'm assuming there's a little more to it than this. For instance, the Lie algebra here, is it the Lie algebra of the Lie group of diffeomorphisms/isometries on the differentiable manifold/Riemannian manifold (this is the only straight foreword way, that I can imagine, that they'd be matrices)? Or are we assuming the manifold for which our A-field is defined over is also has smooth group structure?

A Yang-Mills theory involves a internal (i.e., not spacetime) symmetry group G, and uses a structure called a principal G bundle. The field strength that you gave in the first post is the curvature (again, internal, not spacetime) of the principal G bundle.

 

[GoldPheonix]

I own M. Nakahara's book. Is it contained in the section on the theory of connections on fiber bundles?

 

[George Jones]

I own M. Nakahara's book. Is it contained in the section on the theory of connections on fiber bundles?

Nakahar covers this stuff briefly (maybe too briefly, I prefer Frankels's treament) in the section on curvature of fiber bundles.