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Hello,
I noticed DFT page has not been updated in a while and I added a couple of sections: some sketchy introductory material (analogy between Kaluza-Klein and DFT) and a little insight about a more rigorous geometrical formulation of DFT.
It is still quite sketchy but I would be happy to refine it.
PS: this is my first edit, I hope I played by the rules. And thank you all for this wiki
Luigi
Thanks very much for this, Luigi! I don’t know much about the mathematics here, so will leave it to others to comment on that if they wish, but it is really great when new people make substantial additions like this!
If you let me know the syntax error you encountered, I can look into it.
[Removed: double posted.]
Thanks, Luigi. Looks good.
One question I had wanted to raise in one of the discussion sessions at HSiMT18 was:
Did it occur to anyone (else) that the “correspondence space” appearing in topological T-duality is clearly a “doubled geometry” as appearing in DFT? Did anyone try to use this observation to work out a closer relationship between DFT and topological T-duality?
Unfortunately, due to an ambitious schedule of talks, we ended up cancelling all the discussion sessions except for the first one, which I was chairing myself, and so I never got around to sending this question/comment/suggestion to the community in Durham.
Right, but 3) is a special case of 1), the case where the fiber being dualized is the whole spacetime and the base space is the point. And conversely, one may consider DFT with only some of the coordinates doubled. The exceptional analog of this is in fact the default in EFT (exceptional field theory).
To my knowledge this is still an open problem in DFT.
Yes, so that’s where the identification of doubled geometry with the correspondendence space in topological T-duality becomes useful. For topological T-duality there is a good global concept of gluing:
added pointer to today’s impressive
Luigi Alfonsi, Global Double Field Theory is Higher Kaluza-Klein Theory (arXiv:1912.07089)
(relating Kaluza-Klein compactification on principal ∞-bundles to double field theory, T-folds, non-abelian T-duality, type II geometry, exceptional geometry, …)
Thank you! I’m sure there are things to fix, but I hope this point of view can be useful to DFT community
Good to see someone taking up some of your ideas like this
We are intrigued by the possibility that a super non-abelian higher Kaluza-Klein Theory on the total space of the (twisted) M2-M5-brane gerbe over the 11d super-spacetime of [FSS18a] can be something closer to a geometrized M-theory than what previously allowed.
Yes, these lines are a corollary in DFT/ExFT of the ideas discussed by FSS .
Of course, if I see that this is not clear for someone who’s less familiar with the references, I’ll definitely expand these lines to make it very clear in V2!
@Luigi
might be worth updating the reference to Bundle gerbes to be the published version: https://doi.org/10.1112/jlms/54.2.403 (it’s in J. London Math. Soc.)
And if you want to be pedantic with referencing, the original reference for what came to be called “bundle gerbes with connection” is Deligne 71, Section 2.2. :-)
Thank you for correcting my sloppy citations!
I will certainly fix this in a revised version of the paper
Please do!
added pointer to today’s
What is the relationship between double field theory and double copy?
DFT is the best framework for describing the double copy structure (slide 10)
I hope Luigi sees this and can say something. I find the claim surprising, as the concepts being doubled are different in two cases. If true, it would seem to need more justification than the half-sentence on that slide 10.
The claim seems to be elaborated in
which Luigi describes at classical double copy as extending classical double copy to double field theory.
This reference here will be probably interesting: https://arxiv.org/abs/1912.02177
In principle the two concept of doubling are very different, however (by quoting the reference) “the doubled local Lorentz group originates in the left-right mode decomposition of the closed string, and shares the same origin as the KLT relations in string scattering amplitudes, which underlie the double copy.”
As far as I know the supporting argument is that the general double copy of a point charge is a bosonic supergravity background (consisting in a metric, Kalb-Ramond field and dilaton) given by a Kerr-Schild-like solution of Double Field Theory, which relaxes the Kerr-Schild solution.
It is true that strong-constrained DFT is perfectly equivalent to bosonic supergravity just in a different fashion, and thus a DFT solution is nothing more than a supergravity solution. But it’s also nice that the solution is formally quite similar to the Kerr-Schild solution.
In DFT (at least in its doubled torus bundles version) one has a “doubled” gauge field $A^I$ with $I=1,...,2n$, while in this double copy one has 2 gauge fields $A^i, \bar{A}^i$ with $i=1,...,n$. The origin of both these facts should lie in the left-right mode decomposition of the closed string…
Thanks! So that ’$O(D,D)$ constraint’ doesn’t arise “by itself” in DFT but rather is induced by the requirement of left-right mode decomposition?
A priori, the $O(D,D)$-symmetry in double field theory is that of T-duality, mixing winding and momentum modes and not left/right moving modes.
And T-duality is a phenomenon firmly in the closed string sector – it applies even to heterotic string theory where open strings do not even exist!
Hence a claim that DFT secretly knows about open/closed string duality would seem to have some deep non-obvious content and would need a lot of scrutiny.
The relation with open strings has been studied sometimes, for instance see the review https://arxiv.org/abs/1803.08861 around (1.10), even if I think this far from clear
If DFT and Double Copy are originated from the same principle, this is still almost all to be understood
Some other people interested in a possible connection:
The Double-Copy and Double-Field Theory [Hohm, Plefka]. We will investigate whether the double- copy structure of gravity can be made manifest at the Lagrangian level using double field theory, where a doubled Lorentz invariance implies a left-right factorization of space-time indices to all orders in perturbation theory. In year 1 the computation of tree-level amplitudes in double field theory shall be performed in order to see whether it matches the double copy construction ones. Based on this, a more refined formulation of what it means for a Lagrangian to “make double copy manifest”, beyond the simple factorization of indices will be possible (year 2). The long-term goal is to arrive at a first- principle understanding of the double-copy prescription using double field theory and its mathematical structures thereby possibly shedding light on the nature of the kinematical algebra.
This seems to be motivated by
added pointer to today’s
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