How are curvature and field strength exactly the same?

[victorvmotti]

I am watching these lecture series by Fredric Schuller.

[Curvature and torsion on principal bundles - Lec 24 - Frederic Schuller][1] @minute 34:00

In this part he discusses the Lie algebra valued one and two forms on the principal bundle that are pulled back to the base manifold.

He shows the relationship between general relativity and electromagnetism in the classical theory.

He emphasize that for instance that the curvature of space-time is exactly the same as the non-abelian Yang-Mills field strength.

It is not clear to me in what sense they are "exactly" the same.

Isn't the physical electromagnetism filed strength tensor a real valued object and abelian?

So is it right to say that in the case of electromagnetism the Lie group U(1) is different from the case of the general relativity Lie group which is GL? Otherwise, curvature and field strength are exactly the same?

[1]:

 

[haushofer]

See e.g.

https://www.physicsforums.com/insights/revival-Newton-cartan-theory/

:)

 

[haushofer]

Sorry, wrong link. This is the right one:

https://www.physicsforums.com/insights/general-relativity-gauge-theory/

 

[haushofer]

So, the field strength in this (GR) case is the field strength of local Lorentz transformations. The general coordinate transformations show up differently, as the insight above emphasizes. The field strength of the local translations is just the torsion, which in GR vanishes.