# FeaturedA Double field theory: Where is the extra space?

1. Jun 8, 2017

### Demystifier

Double field theory [1] is an attempt to realize T-duality of string theory at the level of field theory. For instance, if a field in ordinary field theory lives in 4 non-compact spacetime dimensions, then a field in double field theory lives in 8 non-compact spacetime dimensions.

I don't understand it from the physical point of view. If the field lives in 8 non-compact dimensions, then why do we see only 4 dimensions? Or if fields of double field theory are not supposed to be observable, then what is the point of introducing them in the first place? What do I miss?

[1] For a review see e.g. https://arxiv.org/abs/1305.1907

Last edited: Jun 8, 2017
2. Jun 8, 2017

### mitchell porter

The superfluous dimensions are eliminated by applying a constraint. See part 3.3 of that review, especially the last paragraph.

3. Jun 9, 2017

### Demystifier

Thanks for the hint, but I still don't get it. If the superfluous dimensions are eliminated, then what's the point of introducing them in the first place? What does 8-dimensional DFT tell us of the 4-dimensional world what we didn't already know without DFT? Perhaps the starting 8-dimensional theory looks simpler then the resulting 4-dimensional one?

Last edited: Jun 9, 2017
4. Jun 9, 2017

### mitchell porter

This is arguably about having a single framework for string theory rather than a patchwork of string theories (heterotic, Type II,..) which have some overlaps via dualities. In T-duality, momentum modes (strings moving along the compact dimensions) map to winding modes (strings that are wrapped around the compact dimensions) and vice versa, so if you want a single framework in which T-duality is just a field transformation, from the start you might work with extra coordinates for the winding modes. Double geometry is the maximum generalization of this, in which the entire space-time starts out doubled, though it's only in the compact dimensions where the "double vision" is useful.

One really practical use is to describe "non-geometric" compactifications where a particular string theory only applies locally, and there are T-duality transition functions joining together the different coordinate patches, similar to ordinary differential geometry. You can see a picture at the top of page 4 here: A string propagates around that conch-like torus, and transforms from momentum mode to winding mode as it passes through the transition region. Apparently double geometry gives you a way to describe this. But that would be double geometry for strings, perhaps for the worldsheet. Double field theory is just the field-theoretic limit of that.

5. Jun 9, 2017

### Demystifier

How about this? Suppose that one starts from string field theory, working with the string field $\Phi[X^{\mu}(\sigma)]$, where $\sigma$ is a continuous parameter in a finite range, say $\sigma\in[0,1]$. And suppose that one studies open strings, so that $X^{\mu}(0)\neq X^{\mu}(1)$. And suppose that, in some limit, only the initial and final point on the string are relevant, so that
$$\Phi[X^{\mu}(\sigma)] \approx \Phi(X^{\mu}(0),X^{\mu}(1))$$
Would that be some kind of a natural emergence of DFT?

6. Jun 9, 2017

### haushofer

Because it makes T-duality explicit.

I regard Double Field Theory as some sort of "Stückelberg trick", in which one introduces gauge degrees of freedom to make a symmetry explicit, in the same way as gauge invariance makes the symmetries of special relativity explicit (why introduce four components for a photon via gauge symmetries when it only has 2 polarisations? Because then you can write down special-relativistic covariant field theories).

So the hope is that making string theory explicitly covariant under T-duality, new insights are given.

7. Jun 9, 2017

### Demystifier

That was very illuminating. Do you know a reference (preferably a review) which presents such a view more explicitly?

8. Jun 9, 2017

### haushofer

I have to check; I learned some DFT 7 years ago in a course given by Zwiebach, but never really read papers about it. If I find something, I'll post it. :)

9. Jun 27, 2017

### davidge

What does non-compact means here? Is it about compactness of a topological space?

10. Jun 27, 2017

### haushofer

I've checked several papers on DFT, but I'm not seeing any of the motivation I'm giving here, which surprises me.

String theory contains many symmetries, T, U and S, which are only manifest after a lot of work. I guess DFT is part of the philosophy that currently we have a silly way of writing string theory down. I'd say that one would like to reformulate string theory in a p-brane democratic ways which has T, U and S symmetries manifest (T-duality for open strings works only if one includes p-branes). Maybe this is what people try to achieve doing string field theory, but that's beyond my knowledge.

11. Jun 27, 2017

### haushofer

Yes. Usually, we consider the spacetime of string theory to be the product of a four-dimensional spacetime and a six-dimensional Calabi-Yau. The Calabi-Yau is a compact (complex) space.

The problem with this is that one could wonder why there are 4 non-compact directions, and how to reconcile this with inflation. These questions are part of what we call "moduli stabilization", in which one uses the fluxes of the different p-forms along cycles of the Calabi-Yau to stabilize their volumes.

12. Jul 12, 2017

### Urs Schreiber

Yes, the "doubled" spacetime is a tool to make T-duality symmetry manifest, that's why some people speak of "duality covariant formalism", such as Chris Hull in his early articles on the topic (Hull 06).

In the mathematical literature, the double of the torus fiber coordinates is called the "correspondence space", because it serves to make the two sides of the T-duality transformation "correspond" via a Fourier-Mukai-type transformation, see the literature on what is called "topological T-dualiry" here.

A derivation of this doubling from first principles is in our arXiv:1611.06536

13. Jul 23, 2017 at 2:00 AM

### suprised

This is a classic example which shows that importing too much supergravity philosophy into string theory can do more harm than good. Supergravity people usually consider tori and at most spheres as backgrounds. And indeed, this doubled field theory works essentially only for tori but not for general backgrounds, so it can hardly elucidate any deeper structures of string theory. Only for tori the dimension of the manifold matches the homology.

On the other hand, the T-duality group for a Calabi-Yau can be almost arbitrarly complicated (some unnamed subgroup of say, Sp(200,Z)), and strongly depends on the concrete CY chosen - so you get like 6 space-time dimension plus like 200 homology dimensions (which are here on a different footing). Good luck with trying to linearize the action of T- or U-duality. For good reason, this topic has been avoided in the doubled geometry literature.

Last edited: Jul 23, 2017 at 2:15 AM
14. Jul 23, 2017 at 2:53 AM

### Urs Schreiber

I cannot speak for the "double field theory" community, but it is a mathematical fact that there is an aspect of T-duality, including the Buscher rules for RR-fields, which works on the level of just the super-tangent spaces of type IIA/B spacetime, and which is made manifest by a doubling. This is not a matter of SuGra philosophy, but of analysis of the relevant cocycles (section 6 of arXiv:1611.06536).

15. Jul 23, 2017 at 9:40 AM

### suprised

What is a matter for Sugra philosophy, is the fixation on tori. Which is fine for some applications, but one should not expect to draw more general conclusions from this geometry. Try to make T-duality for CY's manifest and you see what I mean.....

16. Jul 23, 2017 at 11:05 AM

### Urs Schreiber

Yes, thanks, I did understand what you are saying. What I was saying is that there is a sector of what is going on in T-duality, that is seen super-tangent space wise and is hence independent of the global topology. This is the sector that Bouwknegt-Mathai-Evslin had called "topological T-duality" which disregards the metric and focuses on the RR-charges in twisted K-theory. Briefly, as T-duality takes the (probe-) IIA-branes to the IIB-branes, it needs to take the GS-WZW terms of these branes into each other, but these are super-tangentspace wise fixed to be $\propto \exp(F) \underset{p}{\sum} c_p \overline{\Psi} \wedge \Gamma_{a_1 \cdots a_p} \Psi \wedge E^{a_1} \wedge \cdots \wedge E^{a_p}$, where $(E^a, \Psi^\alpha)$ is the super-vielbein field. The T-duality rule which takes these expressions into each other (even $\leftrightarrow$ odd $p$) is the T-duality Buscher rules for the RR-fields and it is made manifest by a super-tangentspace-wise doubling, independently of the global topology. There is extra data involved in globalizing this super-tangentspace "topological" T-duality to a global T-duality, which is what you are highlighting, but the spacetime doubling does take care of making this local aspect manifest. And this is something supergravity does not see, instead it's all in the GS-WZW terms of the (probe-) super D-branes.