A Double field theory: Where is the extra space?

[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

 

[mitchell porter]

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

 

[Demystifier]

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

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?

 

[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.

 

[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?

 

[haushofer]

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?

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.

 

[Demystifier]

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.

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

 

[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. :)

 

[davidge]

a field in ordinary field theory lives in 4 non-compact spacetime dimensions

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

 

[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.

 

[haushofer]

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

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.

 

[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

 

[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.

 

[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).

 

[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...

 

[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.

 

[suprised]

Hi Urs,

I wasn't that sophisticated, rather than just naive. Upon glancing over the paper you cited I found I understand
next to nothing, at least without studying it for hours..

So to clarify the basic setup, we have the following scalars:

a) 3 complex scalars which are the coordinates of the CY

b) up to several hundred RR scalars (say 200) arising from the harmonic forms of the CY.
Roughly half of them can be seen as coordinates of the moduli space.

Now:

1) which of those will be "doubled" ?

2) How does a manifest duality invariant formulation look? The T-Duality group
of essentially a subgroup of Sp(200) and a priori acts on the homology lattice.
That is, on all sorts of massive states that arise from wrapped branes, and can be split
into electric and magnetic duals.

3) T-duality is an intrinsic string feature which involves alpha-prime. At small volumes
alpha-prime corrections become important. For example at the Fermat point of the quintic CY
there is a Z_5 symmetry which mixes up wrapped branes that naively arise from transporting
wrapped 0,2,4,6-branes from the large radius regime to the Fermat point. In other, loose words,
the duality symmetries mix cycles of different dimensions, so they loose their naive geometric meaning,
once you go away from the large radius limit.

As you know, the right language to describe this is in terms of categories and auto-equivalences thereof.

4) Now back to the main question: how does whatever doubling of scalar coordinates provide a linearisation, or manifest realisation,
of these quantum symmetries acting on the category of wrapped branes ?

 

[Urs Schreiber]

Hi surprised,

thanks for your thoughtful replies! It's good to speak with an expert here on PF. I have to run now, am late for the beginning of StringMath17, but here is a quick reply:

As you know, the right language to describe this is in terms of cathe thetegories and auto-equivalences thereof.

Unless we are talking about the topological string, the right language to describe what happens to the RR-fields is isomorphisms in twisted complex K-theory, or more generally, if there is orientifolding in the background, in twisted KR-theory. The D-branes themselves are cycles in twisted K-theory, the RR-fields are cocycles. There is a Chern-character map from twisted K-theory to twisted de Rham cohomology in even or odd degree, respectively, and that produces those RR-flux forms which may then be taken to be harmonic forms. The fact that under T-duality differential forms of different degree may be mixed is a reflection of the fact that (twisted) K-theory does not decompose as a direct sum of ordinary cohomology theories. In fact the Buscher rules for RR-fields (usually attributed to Hori arXiv:hep-th/9902102 equation (1.1)) says how these forms are taken into each other. It's that Buscher rule for RR-fields which is being made "manifest" by the spacetime doubling. That's the content around prop. 6.4 in the article arxiv:1611.06536 which you say you find hard to read (and sorry for that).

which of those will be "doubled" ?

If T-duality is taken to act on $F$-fibers over some base space $B$, taking them to $\tilde F$ fibers, then in global T-duality its these fibers that are being doubled, hence one considers and $F \times \tilde F$ fiber bundle as the "doubled spacetime". The condition on the T-dual NS-fluxes is that they becomes equivalent once pulled back to this fiber product space, with the equivalence exhibited by a Poincare-line bundle, as in Fourier-Mukai duality, and the induced FM-like transformation through this fiber product space takes the twisted K-theory of the IIA RR-charges to that of IIB, and vice versa.

Now this is where you come in with your point that this is established for toroidal fibers only (Bouwknegt-Mathai-Evsli and Bunke-Schick and others). That's fair enough. Moreover, what is actually considered in "doubled geometry" is (though the community might not state it this way) the tangent-space version of this, where we consider "doubled spacetimes" that are locally modeled on just the tangent space of the $F \times \tilde F$ fiber bundle over $B$. That does lose a lot of global information, which is the point you are making forcefully and rightly so. But not to throw out the child with the bathwater, all I am trying to say is that while this is so, it does serve to make the Buscher rules for the RR-fields manifest.

 

[suprised]

Hi Urs,

thanks fot your answer. Still I am confused: the only instance where I could vaguely understand a doubling to linearize the T-Duality G < Sp(200) is to double the number of coordinates of the moduli space (roughly, there is a symplectic polarization so that half of the periods (minus one to be precise) corresponds to the coordinates, let's say 100 of them. Manifestly realizing the discrete symmetry G requires the other half of periods to be present, so this is kind of a doubling.

But this has to do with the moduli space of the particular CY in question, and not with 6 space-time coordinates. So all what one can possibly learn is about intrinsic properties of the CY, and not about intrinsic features of string theory. For T2 it is a coincidence that moduli coordinates more or less match the number of space-time coordinates.

 

[Urs Schreiber]

You keep referring to the cohomology classes of the differential forms, I keep trying to call attention to the actual form representatives and their tangent-space wise behaviour.