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• CommentRowNumber1.
• CommentAuthorLuigi
• CommentTimeAug 21st 2018

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

• CommentRowNumber2.
• CommentAuthorLuigi
• CommentTimeAug 21st 2018
It looks like there was syntax mistake in the first part so I removed it, at least for now
1. 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.

2. Used new LaTeX syntax for definitions, etc. Main purpose was to make them numbered. Hope I made the breaks at the correct points; just correct if not!

• CommentRowNumber5.
• CommentAuthorRichard Williamson
• CommentTimeAug 21st 2018
• (edited Aug 21st 2018)

[Removed: double posted.]

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeAug 22nd 2018

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.

• CommentRowNumber7.
• CommentAuthorLuigi
• CommentTimeAug 23rd 2018
Thank you, I try to answer to the best of my knowledge.

In literature there is an ambiguity: 3 slightly different (but simply related) things have been called with the same name "doubled manifold":

1) the total space of a bundle where only the fibre is doubled,
2) the fibre of such a bundle,
3) a whole doubled spacetime (with all the coordinates doubled).
Nomenclature 1) was more frequent at the beginning, but 3) should be the most frequently used at the moment. (And let me say also that most of the DFT maths is totally not rigorous).

The underlying manifold of 1) is the correspondence space of T-duality, but it is true that (to my knowledge) this point has never been really investigated. (I recall only Papadopoulos' papers mentioning it explicitly). Global aspects of DFT in general have not been too much investigated. "Doubled geometry" means usually the (patchwise) O(d,d)-covariant geometry (bracket, metric, ...) on a 2d-dimensional spacetime, so maybe this point of view hides the relationship.
• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeAug 24th 2018

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).

• CommentRowNumber9.
• CommentAuthorLuigi
• CommentTimeAug 26th 2018
• (edited Aug 26th 2018)
You are right, I try to answer a little better

It would be very natural to have the correspondence space as doubled manifold and a constraint that let us "select" the two T-dual submanifolds, but the community does not agree about how we should patch DFT.

DFT patches should be glued together in compatibility with strong constraint. This means gluing (x',y') and (x,y) (where x's are physical and y's are the "tilde" coords) by

x' = f(x), y' = y + v(x).

This gives the DFT infinitesimal gauge transformations.
The problem is that this patching condition should give automatically a trivial topology to the fibration of the y's over the physical spacetime. And inside the correspondence space we know that the dual U(1)-bundle over the physical U(1)-bundle has in general non-trivial topology.

To my knowledge this is still an open problem in DFT.

Para-hermitian formalism is surely a step forward, but we are still in proposal stage.
For instance I was mentioning Papadopoulos, who proposed that the dual S^1-coordinates are actually the angular coords defined on overlaps of (physical spacetime) patches that appears in the description of the gerbe of the B-field.

(I hope I have been not too much inaccurate)
• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeAug 26th 2018

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:

• T. Nikolaus, K. Waldorf, Higher geometry for non-geometric T-duals (arXiv:1804.00677)
• CommentRowNumber11.
• CommentAuthorLuigi
• CommentTimeSep 3rd 2018
I just say thank you for indicating me this paper, there are actually chances that I could use something later
3. Added reference for Svoboda

Anonymous

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeDec 17th 2019
• (edited Dec 17th 2019)

added pointer to today’s impressive

• CommentRowNumber14.
• CommentAuthorLuigi
• CommentTimeDec 17th 2019

Thank you! I’m sure there are things to fix, but I hope this point of view can be useful to DFT community

• CommentRowNumber15.
• CommentAuthorDavid_Corfield
• CommentTimeDec 17th 2019

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.

• CommentRowNumber16.
• CommentAuthorLuigi
• CommentTimeDec 17th 2019
• (edited Dec 17th 2019)

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!

• CommentRowNumber17.
• CommentAuthorDavidRoberts
• CommentTimeDec 17th 2019

@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.)

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeDec 17th 2019

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. :-)

• CommentRowNumber19.
• CommentAuthorLuigi
• CommentTimeDec 18th 2019

Thank you for correcting my sloppy citations!

I will certainly fix this in a revised version of the paper

• CommentRowNumber20.
• CommentAuthorLuigi
• CommentTimeMay 6th 2020

I ordered better sections and references and added the link to exotic branes.

I would like to keep expanding/adjusting this and other related page in the near future :)

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeMay 6th 2020

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeJul 10th 2020

added pointer to today’s

• CommentRowNumber23.
• CommentAuthorDavid_Corfield
• CommentTimeJul 15th 2020

What is the relationship between double field theory and double copy?

DFT is the best framework for describing the double copy structure (slide 10)

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeJul 15th 2020

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.

• CommentRowNumber25.
• CommentAuthorDavid_Corfield
• CommentTimeJul 15th 2020

The claim seems to be elaborated in

• Kanghoon Lee, Kerr-Schild Double Field Theory and Classical Double Copy, (arXiv:1807.08443)

which Luigi describes at classical double copy as extending classical double copy to double field theory.

• CommentRowNumber26.
• CommentAuthorLuigi
• CommentTimeJul 15th 2020

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…

• CommentRowNumber27.
• CommentAuthorDavid_Corfield
• CommentTimeJul 15th 2020

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?

• CommentRowNumber28.
• CommentAuthorUrs
• CommentTimeJul 15th 2020

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.

• CommentRowNumber29.
• CommentAuthorLuigi
• CommentTimeJul 15th 2020

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

• CommentRowNumber30.
• CommentAuthorDavid_Corfield
• CommentTimeJul 16th 2020
• (edited Jul 16th 2020)

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

• CommentRowNumber31.
• CommentAuthorLuigi
• CommentTimeNov 12th 2020

Added draft of relation between para-Hermitian geometry and bundle gerbes. To be refined much more later.

• CommentRowNumber32.
• CommentAuthorUrs
• CommentTimeFeb 22nd 2021

added pointer to today’s

• Luigi Alfonsi, Towards an extended/higher correspondence – Generalised geometry, bundle gerbes and global Double Field Theory (arXiv:2102.10970)