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Introduction to double field theory 1/2

Guest blog by Olaf Hohm of MIT
The second part will be there

First of all I would like to thank Luboš for giving me the opportunity to write a guest blog on double field theory (previously mentioned here).

This is a subject that in some sense is rather old, almost as old as string theory, but that has seen a remarkable revival over the last five years or so and that, as a consequence, has reached a level of maturity comparable to that of many other sub-disciplines of string theory. In spite of this, double field theory is viewed by some as a somewhat esoteric theory in which unphysical higher-dimensional spacetimes are introduced in an ad-hoc manner for no reasons other than purely aesthetic ones and that, ultimately, does not give any results that might not as well be obtained with good old-fashioned supergravity. It is the purpose of this blog post to introduce double field theory (DFT) and to explain that, on the contrary, even in its most conservative form it allows us to attack problems several decades old that were beyond reach until recently.




Concretely, in the first part I will review work done in collaborations with Warren Siegel and Barton Zwiebach on a formulation of DFT that includes higher-derivative \(\alpha'\) corrections and that describes certain subsectors of string theory in a way that is exact to all orders in \(\alpha'\). This casts the old problem of determining and understanding these corrections into a radically new form that, we believe, provides a significant step forward in understanding the interplay of two of the main players of string theory: \(\alpha'\) and duality symmetries. In the second part, I will explain how an extension of DFT to exceptional groups, now commonly referred to as exceptional field theory, allows us to settle open problems in Kaluza-Klein truncations of supergravity that, although of conventional nature, were impossible to solve with standard techniques.




So let's start by explaining what DFT is. It is framework for the spacetime (target space) description of string theory that makes the T-duality properties manifest. T-duality implies that string theory on the torus \(T^d\) with background metric and B-field looks the same for any background obtained by an \(O(d,d;\ZZ)\) transformation. The discrete nature of the group is due to the torus identifications (periodicity conditions) which the transformations need to respect. In the supergravity approximation to string theory the dimensional reduction on the torus truncates the massive Kaluza-Klein modes (as it should be, since in the effective supergravity we have also truncated massive string modes), and so all memory of the torus is gone. Consequently, the duality symmetry visible in supergravity is actually the continuous \(O(d,d;\RR)\), and in the following I will exclusively consider this group. In contrast to what people sometimes suspect, the continuous symmetry is preserved by \(\alpha'\) corrections, which will be important below.

This implies that gravity in \(D=10\) or \(D=26\) dimensions, extended by the bosonic and fermionic fields predicted by string theory, yields an enhanced global symmetry upon reduction that cannot be understood in terms of the symmetries present in the standard formulation of gravity. Consider the minimal field content of all closed string theories, the metric \(g_{ij}\), the antisymmetric b-field \(b_{ij}\) and a scalar (dilaton) \(\phi\), with effective two-derivative action \[ S = \int d^Dx\,\sqrt{g}e^{-2\phi}\big[R+4(\partial\phi)^2-\tfrac{1}{12}H^2\big] \] This action is invariant under standard diffeomorphisms (general coordinate transformations) \(x^i\rightarrow x^i-\xi^i(x)\) and b-field gauge transformations \(b\rightarrow b+{\rm d}\tilde{\xi}\), with vector gauge parameter \(\xi^i\) and one-form parameter \(\tilde{\xi}_i\). The diffeomorphism symmetry explains the emergence of the \(GL(d,\RR)\) subgroup of \(O(d,d)\), representing global reparametrizations of the torus, while the b-field gauge symmetry permits a residual global shift symmetry \(b\rightarrow b+c\), with antisymmetric constant c. The full symmetry is larger, however: the complete \(O(d,d)\), as predicted by string theory. String theory is trying to teach us a lesson that we fail to understand by writing the spacetime actions as above. DFT is the framework that for the first time made the full symmetries manifest before dimensional reduction.

The idea behind DFT is to introduce a doubled space with coordinates \(X^M=(\tilde{x}_i, x^i)\), \(M=1,…,2D\), on which \(O(D,D)\) acts naturally in the fundamental representation. (Note that here, at least to begin with, we have doubled the number of all spacetime coordinates.) This idea is actually well motivated by string theory on toroidal backgrounds, where these coordinates are dual both to momentum and winding modes. In fact, in closed string field theory on such backgrounds, the doubled coordinates are a necessity and not a luxury. [Perhaps it's time now for some references: the idea of doubling coordinates in connection to T-duality is rather old, going back at least to the early 90's and work by Duff, Tseytlin, Kugo, Zwiebach and others, but the most important paper for the present story is hep-th/9305073 by Warren Siegel. The modern revival of these ideas was initiated in a paper by Chris Hull and Barton Zwiebach, 0904.4664, and then continued with myself in 1003.5027, 1006.4823. There is also a close relation to 'generalized geometry', which I will comment on below. For more references see for instance the review 1309.2977 with Barton Zwiebach and Dieter Lust.]

In DFT we reorganize the fields into \(O(D,D)\) covariant variables as follows \[

\eq{
{\cal H}_{MN} &= \begin{pmatrix} g^{ij} & -g^{ik}b_{kj}\\[0.5ex] b_{ik}g^{kj} & g_{ij}-b_{ik}g^{kl}b_{lj}\end{pmatrix}, \\
e^{-2d} &= \sqrt{g}e^{-2\phi}
}

\] where the 'generalized metric' \({\cal H}_{MN}\) transforms as a symmetric 2-tensor and \(e^{-2d}\) is taken to be an \(O(D,D)\) singlet. Moreover, \({\cal H}_{MN}\) can be thought of as an \(O(D,D)\) group element in the following way: Defining \[ {\cal H}^{MN} \equiv \eta^{MK}\eta^{NL}{\cal H}_{KL}\;, \qquad \eta_{MN} = \begin{pmatrix} 0 & 1\\[0.5ex] 1 & 0 \end{pmatrix} \] where \(\eta_{MN}\) is the metric left invariant by \(O(D,D)\), it satisfies \[ {\cal H}^{MK}{\cal H}_{KN} = \delta^{M}{}_{N}\;, \qquad \eta^{MN}{\cal H}_{MN} = 0 \] Conversely, the parametrization above is the most general solution of these constraints. Thus, we may forget about \(g\) and \(b\) and simply view \({\cal H}\) as the fundamental gravitational field, which is a constrained field.

If we forget about \(g\) and \(b\), how do we write an action for \({\cal H}\)? We can write an action in the Einstein-Hilbert-like form \[ S_{\rm DFT} = \int d^{2D}X\,e^{-2d}\,{\cal R}({\cal H},d) \] where the scalar \({\cal R}\), depending both on \({\cal H}\) and \(d\), denotes a generalization of the Ricci scalar in standard differential geometry. But how is it constructed? There is a beautiful story here, closely analogous to conventional Riemannian geometry with its notions of Levi-Civita connections, invariant curvatures, etc., but also with subtle differences. Most importantly, there is a notion of generalized diffeomorphisms, infinitesimally given by generalized Lie derivatives \({\cal L}_{\xi}\) parametrized by an \(O(D,D)\) vector \(\xi^{M}=(\tilde{\xi}_i,\xi^i)\) that unifies the diffeomorphism vector parameter with the one-form gauge parameter. These Lie derivatives form an interesting algebra, \([{\cal L}_{\xi_1},{\cal L}_{\xi_2}]={\cal L}_{[\xi_1,\xi_2]_C}\), defining the 'C-bracket' \[

\eq{
\!\big[\,\xi_1\,,\;\xi_2\,\big]_{C}^M &= \xi_1^N\partial_N\xi_{2}^M \!-\xi_2^N\partial_N\xi_{1}^M\!-\\
&-\frac{1}{2}\xi_{1N}\partial^M \xi_{2}^N \!+ \frac{1}{2}\xi_{2N}\partial^M\xi_{1}^N
}

\] where indices are raised and lowered with the \(O(D,D)\) invariant metric. The first two terms look like the standard Lie bracket between vector fields, but the remaining two terms are new. Incidentally, this shows that these transformations are not diffeomorphisms on the doubled space, for these would close according to the Lie bracket, not the C-bracket. Due to lack of space I cannot review the geometry further, but suffice it to say that the generalized diffeomorphisms uniquely determine the Ricci scalar, and since I haven't introduced connections, etc., let me just give the explicit and manifestly \(O(D,D)\) invariant expression, written in terms of the derivatives \(\partial_M\) dual to the doubled coordinates, \[

\begin{split} {\cal R} \ \equiv &~~~4\,{\cal H}^{MN}\partial_{M}\partial_{N}d -\partial_{M}\partial_{N}{\cal H}^{MN}- \\[1.2ex]
~&-4\,{\cal H}^{MN}\partial_{M}d\,\partial_{N}d + 4 \partial_M {\cal H}^{MN} \,\partial_Nd+ \\[1.0ex] ~&+\frac{1}{8}\,{\cal H}^{MN}\partial_{M}{\cal H}^{KL}\, \partial_{N}{\cal H}_{KL}-\\
~&-\frac{1}{2}{\cal H}^{MN}\partial_{M}{\cal H}^{KL}\, \partial_{K}{\cal H}_{NL} \end{split}

\] With this form of the generalized Ricci scalar, the above DFT action reduces to the standard low-energy action upon truncating the extra coordinates by setting \(\tilde{\partial}=0\).

So far I have remained silent about the nature of the extended coordinates. Surely, we don't mean to imply that the theory is defined in 20 dimensions, right? Indeed, the gauge invariance of the theory actually requires a constraint, the 'strong constraint' or 'section constraint', \[ \eta^{MN}\partial_M\partial_N = 2\tilde{\partial}^{i}\partial_{i} = 0 \] This is supposed to mean that \(\partial^M\partial_MA=0\) for any field or gauge parameter A but also \(\partial^M\partial_M(AB)=0\) for any products of fields, which then requires \(\partial^MA\partial_MB=0\). One may convince oneself that the only way to satisfy these constraints is to set \(\tilde{\partial}=0\) (or \(\partial_i=0\) or any combination obtained thereof by an \(O(D,D)\) transformation). This constraint is therefore an \(O(D,D)\) invariant way of saying that the theory is only defined on any of the half-dimensional subspaces, on each of which it is equivalent to the original spacetime action. (This will change, however, once we go to type II theories or M-theory, in which case different theories emerge on different subspaces, but more about this later.)

Is it possible to relax this constraint? Indeed, some of the initial excitement about DFT was due to the prospect of having a framework to describe so-called non-geometric fluxes, which are relevant for gauged supergravities in lower dimensions that apparently cannot be embedded into higher-dimensional supergravity/string theory in any conventional 'geometric' way. Those non-geometric compactifications most likely require a genuine dependence on both types of coordinates, and various proposals have been put forward of how to relax the above constraint. (I should also mention that in the full closed string field theory the constraint is relaxed, where one requires only the level-matching constraint, allowing for certain dependences on both \(x\) and \(\tilde{x}\).) While these preliminary results are intriguing and, in my opinion, capture part of the truth, I think it is fair to say that we still do not have a sufficiently rigorous framework. I will therefore assume the strong form of the constraint, and so if you don't like this constraint you may simply imagine at any step that \(\partial_M\) is really only a short-hand notation for \[ \partial_M = (0,\partial_i) \] In this way of thinking the extended coordinates play a purely auxiliary role. They are necessary in order to make certain symmetries manifest and thus analogous, for instance, to the fermionic coordinates in superspace (although technically they appear to be rather different). We then have a strict reformulation of supergravity; in particular, we are not tied to the torus. I want to emphasize that it is not only possible to use this theory for general curved backgrounds (spheres, for instance), but it is actually highly beneficial to do so, as will be explained in a follow-up post.

[Let me also point to a close connection with a beautiful field in pure mathematics called 'generalized geometry', going back to work by Courant, Severa, Weinstein, Hitchin, Gualtieri and others, in which the generalized metric is also a central object. Subsequently, this was picked up by string theorists, suggesting that one should forget about \(g\) and \(b\) and view the generalized metric as the fundamental object. Curiously, however, before the advent of DFT, no physicist seemed to bother (or to be capable) to take the obvious next step and to formulate the spacetime theory in terms of this object, although this is technically straightforward once phrased in the right language. What is the reason for this omission? Of course I can only speculate, but the reason must be that the (auxiliary) additional coordinates, which are absent in generalized geometry, are really needed to get any idea of what kind of terms one could write to construct an action.]

Finally we are now ready to turn to higher-derivative \(\alpha'\) corrections. We constructed a particular subset of these corrections in the paper 1306.2970 with Warren Siegel and Barton Zwiebach by using a certain chiral CFT in the doubled space with a novel propagator and simplified OPEs, earlier introduced by Warren. This is a beautiful story, but slightly too technical to be properly explained here. Therefore, let me give the idea in an alternative but more pedagogical way. First, recall that the generalized metric satisfies a constraint, which we can simply write as \({\cal H}^2=1\) (leaving implicit the metric \(\eta\)). Is there a way to define the theory for an unconstrained metric? The trouble is that when checking gauge invariance of the action we use this constraint, and so simply replacing the constrained \({\cal H}\) by an unconstrained field, which in the paper we called the 'double metric' \({\cal M}\), violates gauge invariance. However, by construction, the failure of gauge invariance must be proportional to \({\cal M}^2-1\), and therefore we can restore gauge invariance, at least to first order, by adding the following term to the action: \[

\eq{
S &= \int e^{-2d}\big[\tfrac{1}{2}{\eta}^{MN}({\cal M}-\tfrac{1}{3}{\cal M}^3)_{MN}+\\
&\qquad+{\cal R}'({\cal M},d) +\dots \big]
}

\] The variation of the first term is proportional to \({\cal M}^2-1\) and so whatever the 'anomalous' transformation of the generalized Ricci scalar is, we can cancel it by assigning a suitable extra gauge variation to \({\cal M}\). (The scalar \({\cal R}'\) carries a prime here, because one actually has to augment the original Ricci scalar by terms that vanish when the metric is constrained.) Since \({\cal R}\) contains already two derivatives this means we have a higher-derivative (order \(\alpha'\)) deformation of the gauge transformations.

Now the problem is that we also have to use these \({\cal O}(\alpha')\) gauge transformations in the variation of \({\cal R}'\), which requires extra higher-derivative terms in the action, in turn necessitating yet higher order terms in the gauge transformations. One would think that this leads to an iterative procedure that never stops, thus giving at best a gauge and T-duality invariant action to some finite order in \(\alpha'\). Remarkably, however, we found an exact deformation, with gauge transformations carrying a finite number of higher derivatives. Moreover, the chiral CFT construction allowed as to define an explicit action with up to six derivatives, which is of the same structural form as above. Intriguingly, the action in terms of \({\cal M}\) is cubic. Since the \(O(D,D)\) metric \(\eta\) is used to raise and lower indices, we never need to use the inverse of \({\cal M}\) (in fact, the action it completely well-defined for singular \({\cal M}\)) and so we have a truly polynomial action for gravity.

How can this be, given the standard folklore that gravity must be non-polynomial? The resolution is actually quite simple: since \({\cal M}\) is now unconstrained it encodes more fields than the expected metric and \(b\)-field, and the extra field components act as auxiliary fields. Integrating them out leads to the non-polynomial form of gravity, but including infinitely many higher-derivative corrections. Actually, the fact that gravity can be made polynomial by introducing auxiliary fields is well known, but in all cases I am aware of this is achieved by using connection variables; the novelty here is that components of the metric itself (of the doubled metric, however) serve as auxiliary fields, in a way that does not simply reproduce Einstein gravity but also infinitetly many higher-derivative corrections!

What are these higher-derivative corrections in conventional language? This is actually a quite non-trivial question, since beyond zeroth order in \(\alpha'\) the conventional metric and b-field are not encoded in \({\cal M}\) in any simple manner, as to be expected, given the rather dramatic reorganization of the spacetime theory. Barton and I managed to show last year that to first-order in \(\alpha'\) the theory encodes in particular the deformation due to the Green-Schwarz mechanism in heterotic string theory. In this, the spacetime gauge symmetries (local Lorentz transformations or diffeomorphisms) are deformed in order to cancel anomalies, which in turn requires higher-derivative terms in the action in the form of Chern-Simons modifications of the three-form curvature. We are currently trying to figure out what exactly the higher derivative modifications are to yet higher order in \(\alpha'\). These are only a subsector of all possible \(\alpha'\) corrections. For instance, this theory does not describe the Riemann-square correction present in both bosonic and heterotic string theory. This is not an inconsistency, because T-duality is not supposed to completely constrain the \(\alpha'\) corrections; after all, there are different closed string theories with different corrections. This theory describes one particular \(O(d,d)\) invariant, and we are currently trying to extend this construction to other invariants.

I hope my brief description conveys some of the reasons why we are so excited about double field theory. In a follow-up blog post I will explain how the extension of double field theory to exceptional groups, exceptional field theory, allows us to solve problems that, although strictly in the realm of the two-derivative supergravity, were simply intractable before. So stay tuned.

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