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I've been learning differential geometry for a while, and am now reading up on Gauge Theories in Physics. I've come across the notion that curvatures on our Fiber Bundles correspond to forces a few times (heard this colloquially when talking with Grad Students, and seen some particular examples like the electromagnetic field strength tensor being exactly the curvature 2-form associated with a connection on a $U(1)$-bundle, or more generally, Field Strengths in Yang-Mills Theories).

The way I've been thinking about this is by the analogy with GR where rather than a general Fiber Bundle we mainly deal with Tangent, Cotangent, and Tensor Bundles built out of the previous two.

Intuitively, it makes sense to me that the Curvature of Spacetime in GR could be interpreted as giving us a force (eg. curvature of a sphere pushes two balls starting at the equator and following geodesics to the north pole together). This information is then encoded in the Curvature Tensor, or equivalently in the Curvature 2-form.

Similarly, it makes sense to me that the Curvature form on a fiber bundle we're using to model a system should give us information about the dynamics of the system.

My questions are:

  1. Is this analogy valid? Is there better justification for the curvature -> force correspondence?

  2. Are there forces that can't be described as resulting from curvatures? EDIT: A follow up question: Is curvature sufficient for describing all the dynamics in a Gauge Theory?

I'm not super familiar with QFT and Gauge Theory yet, so I apologize if this is a basic question. Any intuitions/examples/suggestions for readings would be appreciated.

EDIT: Found this StackExchange post which gives some reasons for why we associate the Curvature of a gauge field with the field strength in Yang-Mills Theories.

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  • $\begingroup$ Please supply references. $\endgroup$
    – my2cts
    Commented Nov 27 at 11:59
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    $\begingroup$ Regarding GR, search for Palatini action, which expresses the EH action in terms of the spin connection. The starting point for LQG is the formulation of GR as a SU(2) gauge theory. $\endgroup$
    – Simp
    Commented Nov 27 at 14:56

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Question 2: The electro-weak and strong forces cannot be represented by curvatures.

For electromagnetism alone this would require two opposite curvatures, one for each charge.

I leave the fibre bundle stuff to the mathematical physicists here.

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  • $\begingroup$ I think curvature of what needs to be specified. You can represent any field strength tensor $F_{\mu\nu}$ in a gauge theory as a commutator of covariant derivatives, and hence, it can be interpreted as a curvature form. Just not spacetime curvature in the sense of GR $\endgroup$
    – FranDahab
    Commented Nov 27 at 12:02
  • $\begingroup$ @FranDahab I am wondering what kind of aether (space-time if you prefer) might be curved in a way to support the electroweak and strong force, and the charges, colors and spins that come with these. $\endgroup$
    – my2cts
    Commented Nov 27 at 12:26
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    $\begingroup$ The point is that, at least in classical Yang-Mills, much like the Riemann tensor carries information about how a spacetime vector changes as it is parallel transported around a small loop, the Field Strength tensor does the same. Except this vector lives in the gauge space, instead of in tangent spaces of the spacetime manifold. See for example this answer $\endgroup$
    – FranDahab
    Commented Nov 27 at 12:36
  • $\begingroup$ @my2cts I don't get the physics yet, but it seems like the Electroweak and Strong forces are both also described by (non-abelian) Yang-mills theories and are associated with the Curvatures of Fiber bundles with gauge groups $SU(2) \times U(1)$ [2] and $SU(3)$ [3], respectively. References: [1]en.wikipedia.org/wiki/…. [2] en.wikipedia.org/wiki/…. $\endgroup$
    – kdeoskar
    Commented Nov 27 at 12:40
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    $\begingroup$ The ToE is basically quantum gravity with the standard model of particle physics (and beyond for dark matter, for some people). So without quantum gravity, this is just a fancy and complicated way to describe the gauge part of the standard model in curved space... But you are right that Yang-Mills theories could have been expressed in this fancy way. This is, by the way, one approach I used to study Yang-Mills theories in an article I submitted to Communications in Mathematical Physics. $\endgroup$
    – Jeanbaptiste Roux
    Commented Nov 27 at 13:31

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