Implementing Category Theory in General Relativity

Question

I was thinking if it may be possible to implement category theory in general relativity. I don't mean writing simply in terms of categories, but actual fundamental ideas (i.e. physics of the theory itself). For example, Prof. John Baez has written a pretty neat paper on the categorification of Yang-Mills theory. Also, is there a way or any recent development(s) to get this IR limit of general relativity from the least input (symmetries and propagating degrees of freedom) via category theory? If not, this is worth thinking about, IMO.

This post imported from StackExchange Physics at 2016-03-31 15:51 (UTC), posted by SE-user Janus Boffin

Comments

I am somewhat intrigued by the use of Category theory for the representation of theory we use (like GR in the case of the OP). For example, the Metric and Cartan formalism.

This post imported from StackExchange Physics at 2016-03-31 15:51 (UTC), posted by SE-user Janus Boffin

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I remember reading a paper a long time ago which proposed the UV limit in case of QG, however, I am unable to locate the paper for now. I will link it as soon as I find it (and also if your objection still holds).

This post imported from StackExchange Physics at 2016-03-31 15:51 (UTC), posted by SE-user Janus Boffin

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Well, one problem is how do you even model GR category-theoretically? The category of smooth manifolds is not nice (hence the fascination with diffeological spaces --- because they do form a "good" category). Or do we model it as a gauge theory with beins as the field? Or are the connections the field? How do you categorify either one? These are nontrivial problems...

This post imported from StackExchange Physics at 2016-03-31 15:51 (UTC), posted by SE-user Alex Nelson

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@AlexNelson: Indeed. An immediate example which came to my mind was that of the metric vs. the Cartan formalism, because the representations are different in both the scenarios and so are the implementations. In the metric setting, we'd need to prove that for a timeline geodesic we can always construct a Fermi normal coordinate system where the first derivatives of the metric are 0, but in the Cartan case, this reduces to finding sections of the bundle wherein you can have the first structure equation being trivial.

This post imported from StackExchange Physics at 2016-03-31 15:51 (UTC), posted by SE-user Janus Boffin

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Although, starting off with Cartan formalism would be beneficial (not as a priori but as differential geometry is well structured in terms of category theory). IIRC, the problem of finding sections of a bundle (which is just a differential form, as other kinds of sections are not worth investigating, IMO), we look for a trivial Maurer-Cartan form, and that is neatly done via de Rham. Moreover, we can have a sheaf of sections over a bundle comprising of some classes of differential forms. But I am not sure if this forms a category. If not, it is worth investigating.

This post imported from StackExchange Physics at 2016-03-31 15:51 (UTC), posted by SE-user Janus Boffin

Answer 1

I don't really understand your question, but since you link to my paper on higher Yang-Mills theory (which I never tried publish because it has problems, even though everything stated in it is true to the best of my knowledge), it sounds like maybe you're interested in approaches that treat gravity using ideas from higher gauge theory. For this, I urge you to read the work of Urs Schreiber. He has lots of papers on the arXiv, but a less strenous place to start is his series of articles on Physics Forums.

This post imported from StackExchange Physics at 2016-03-31 15:51 (UTC), posted by SE-user John Baez

Comments

Isn't it true that if we look at GR as a gauge theory invariant under diffeomorphisms, then a generalised formalism that will allow us to treat gauge theories in general should allow us to do the same with GR? For example, this can be seen at Metric affine gravity and BF -> Holst via breaking topological invariance. Thank you for the reference, I will check them up ASAP! :)

This post imported from StackExchange Physics at 2016-03-31 15:51 (UTC), posted by SE-user Janus Boffin

Answer 2

Here is how higher category theory (homotopy theory) arises in gravity:

First of all, the precise version of the statement that "gravity is a gauge theory" is that gravity in first order formulation is "Cartan geometry" for Minkowski spacetime regarded as the quotient of the Poincare group by the Lorentz spin-group. This statement generalizes to super-gravity, with the Poincare group replaced by the super-Poincare group ("the spacetime supersymmetry group"), see here:

https://ncatlab.org/nlab/show/super-Cartan+geometry

The corresponding Cartan-connections encode the vielbein field and the "spin connection" that are the mathematical incarnation of the field of (super-)gravity, whose quanta are the graviton and the gravitino.

But now something special happens: higher dimensional supergravity by necessity contains not just the graviton and the gravitino, but also higher degree form fields. It is the higher degree of these form fields which is the entry point of higher category theory/homotopy theory in gravity.

Namely these tensor multiplets are no longer encoded by a Cartan-connection with values in an ordinary group (the Poincare group), but they are encoded by higher Cartan connections with values in higher categorical groups!

This is a long and fascinating story, which does not fit into this comment box here. To get started you might try these lecture notes here

https://ncatlab.org/schreiber/show/Structure+Theory+for+Higher+WZW+Terms

or some of the links provided there.

This post imported from StackExchange Physics at 2016-03-31 15:51 (UTC), posted by SE-user Urs Schreiber