I'm not a geometer, so I beg for indulgence on the below:(adsbygoogle = window.adsbygoogle || []).push({});

In a modern geometrical description of electromagnetism (either in flat or in curved space-time*), I see at least 3 (or 4) (fiber) bundles over the 4D space-time taken to be the base space:

*1the cotangent bundle and the bundle of p-forms over space time (de Rham complex). Differential operator: d Here the EM field appears as: the potential is a 1 form field, the field strength is a 2 form field. F=dA.

*2the (principal?)[itex] SO_{\uparrow}(1,3) [/itex] bundle of tensors over space time. Here if the group is replaced by SL(2,C), we get the spinor bundle over space time, a concept described in the 13th chapter of Wald's book - in the context of a curved space time. Here I imagine A and F as covariant and double covariant tensors. Differential operator = [itex] \partial_{\mu} [/itex] or [itex] \nabla_{\mu} [/itex].

*3the so-called U(1) gauge bundle over space time (principal/associated bundle ?) , with F and A having particular descriptions. Here we have another differential operator d-bar, such as F = d-bar A.

Questions:

a) How are1,2,3exactly related ?

b) Is d-bar at3related to d at1? I suspect yes. How can it be proven ? The operator at2, how's it related to d-bar at3or to d at1?

Thank you!

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# Geometric description of the free EM field

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