Wednesday, July 8, 2015 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere

This is the second part of a guest blog on double field theory (thanks again to Lubos for giving me this opportunity). I will introduce the extension of double field theory to `exceptional field theory', a subject developed in collaboration with Henning Samtleben, and explain how it allowed us to resolve open problems in basic Kaluza-Klein theory that could not be solved by standard techniques.

Exceptional field theory is the completion (in a sense I shall make precise below) of a research program that goes back to the early 80s and attempts to understand why maximal supergravity knows about exceptional groups, such as \(E_6\), \(E_7\) and \(E_8\). These groups emerge, miraculously, as global (continuous) symmetries upon compactifying maximal supergravity on tori. This looks like a miracle because exceptional groups had no role to play in the original construction of, say, 11-dimensional supergravity. Although these symmetries are now understood as the supergravity manifestations of the (discrete) U-dualities of string-/M-theory, they remained deeply mysterious from the point of view of conventional geometry. Exceptional field theory (EFT) makes these symmetries manifest prior to dimensional reduction, in the same sense that double field theory (DFT) makes the T-duality group \(O(d,d)\) manifest.

It should be emphasized that U-dualities are tied to toroidal backgrounds. Similarly, the continuous exceptional symmetries of supergravity only emerge for compactification on tori. For compactifcations on curved backgrounds, such as spheres, there is no exceptional symmetry. Understandably, this fact led various researchers to conclude that DFT and EFT are consistently defined only on toroidal backgrounds. This is not correct, however, despite the continuing claims by some people. In its most conservative interpretation, EFT (like DFT) is simply a reformulation of (maximal) supergravity that makes its duality properties manifest. In particular, it is background-independent, and so one may describe any desired compactification. The real question therefore is whether this formalism is useful for compactifications other than toroidal ones.

Since on curved backgrounds none of the exceptional symmetries are preserved, it is reasonable to expect that EFT is more awkward than useful for such compactifications. Remarkably, it turns out that, on the contrary, EFT allows one to describe such compactifications very efficiently as generalized Scherk-Schwarz compactifications, governed by `twist matrices' taking values in the duality group. For instance, the compactification of type IIB on \(AdS_5\times S^5\) can be described by a matrix valued in \(E_6\) (the U-duality group in \(D=5\)). Moreover, in this formulation one can solve problems that could not be addressed otherwise. Thus, although physically there is no \(E_6\) symmetry in any conventional sense, this group somehow still governs these spaces `behind the scenes'.

Before I explain this and EFT in more detail, let me first discuss what exactly the issues in conventional Kaluza-Klein theory are that we resolved recently. They are related to the `consistency of Kaluza-Klein truncations', a subject that unfortunately is not appreciated even by many experts. In Kaluza-Klein theory we start with some higher-dimensional theory and decompose fields and coordinates in a way that is appropriate for a lower-dimensional theory. For instance, the metric \(G\) is written as \[ G = \begin{pmatrix} g_{\mu\nu}+A_{\mu}{}^m A_{\nu}{}^n g_{mn} & A_{\mu}{}^{k}g_{kn}\\[0.5ex] A_{\nu}{}^{k} g_{km} & g_{mn} \end{pmatrix} \] with `external' indices \(\mu,\nu\) and `internal' indices \(m,n\). The resulting fields will eventually be interpreted as lower-dimensional metric, vectors and scalars. The question is how the fields depend on the internal coordinates \(y^m\), in other words, what the `Kaluza-Klein ansatz' is.

In one extreme we may declare the fields to be completely independent of the internal coordinates, which means we are effectively truncating to the massless modes of a torus compactification. In another extreme, we may keep the full \(y\)-dependence but expand the fields in a complete basis of harmonics (such a Fourier modes on a torus or spherical harmonics on a sphere), which means keeping the full tower of Kaluza-Klein modes. In both cases there is no danger of inconsistency. The interesting question is whether there is anything in between, i.e., a non-trivial truncation that is nevertheless consistent.

The standard lore is that for a compactification on a manifold with metric \(g_{mn}\) and isometry group \(G\) the appropriate ansatz is written in terms of the Killing vectors \(K_{\alpha}=K_{\alpha}{}^{m}\partial_m\) as \[ G_{\mu n}(x,y) = A_{\mu}{}^{\alpha}(x) K_{\alpha}{}^m(y) g_{mn}(y) \] and similarly for the other metric components. Working out the infinitesimal general coordinate transformations of the metric (using standard formulas from differential geometry that give these transformations in terms of Lie derivatives), one finds that the \(A_{\mu}{}^{\alpha}\) transform like Yang-Mills gauge fields with the gauge group given by the isometry group \(G\) of the internal manifold. Concretely, in order to verify this we have to use that the Killing vectors satisfy the following algebra \[ \big[K_{\alpha}, K_{\beta}\big] \ = \ f_{\alpha\beta}{}^{\gamma} K_{\gamma} \] where \(f_{\alpha\beta}{}^{\gamma}\) are the structure constants of \(G\). Moreover, if we take the gravity action and integrate over the internal manifold, we obtain a lower-dimensional Einstein-Yang-Mills theory. This is the famous `Kaluza-Klein miracle' in which an internal gauge symmetry (the Yang-Mills gauge group) is `geometrized' in terms of a higher-dimensional manifold and its spacetime (diffeomorphism) symmetry.

The trouble with this ansatz is that in general it is inconsistent! A Kaluza-Klein truncation is consistent if and only if any solution of the (truncated) lower-dimensional theory can be embedded into a solution of the (original) higher-dimensional theory. One way to see that Kaluza-Klein truncations on curved manifolds in general are inconsistent is to insert the Kaluza-Klein ansatz discussed above into the Einstein equations and to observe that the \(y\)-dependence does not factor out consistently: one may obtain equations in which the left-hand side depends only on \(x\), but the right-hand side depends on \(x\) and \(y\). (A nice discussion of this can be found in a classic 1984 PLB paper by Duff, Nilsson, Pope and Warner.) Consequently, a solution of the Einstein-Maxwell equations, following from the action obtained by simply integrating over the internal manifold, in general does not give rise to a solution of the original field equations. Consistency only holds for very specific theories and very special internal geometries and requires a suitable non-linear extension of the Kaluza-Klein ansatz.

The known consistent truncations include 11-dimensional supergravity on \(AdS_4\times S^7\), as established by de Wit and Nicolai in 1986, and \(AdS_7\times S^4\), shown to be consistent by Nastase, Vaman and van Nieuwenhuizen in hep-th/9911238. In contrast, until our recent paper, for the celebrated case of type IIB on \(AdS_5\times S^5\) there was no proof of consistency, except for certain truncations and sub-sectors. At this point let me stress that the size of the sphere is of the same order as the AdS scale. There is no low-energy sense which would justify to keep only the `massless' modes, and hence it is especially important to actually prove that the truncation is consistent.

What was known already since 1984 is 1) the complete Kaluza-Klein spectrum of type IIB on \(AdS_5\times S^5\), determined by Gunaydin and Marcus, which requires only the linearized theory, and 2) the complete \(SO(6)\) gauged supergravity in five dimensions, constructed directly in \(D=5\) by Gunaydin, Romans, and Warner, which was believed (and is now proven) to be a consistent truncation of type IIB. What was missing since 1984 is a way to uplift the \(D=5\) gauged supergravity to type IIB. This means that we didn't even know in principle how to obtain the \(D=5\) theory from the type IIB theory in \(D=10\), because we simply didn't have the Kaluza-Klein ansatz that needs to be plugged into the higher-dimensional action and field equations.

After this digression into the consistency issues of Kaluza-Klein theory, let me return to exceptional field theory (EFT) and explain how the above problems are resolved in a strikingly simple way. As in DFT, EFT makes the duality groups manifest by introducing extended/generalized spacetimes and organizing the fields into covariant tensors under these groups. In contrast to DFT, the coordinates are not simply doubled (or otherwise multiplied). Rather, the coordinates are split into `external' and `internal' coordinates as in Kaluza-Klein, but without any truncation, and the internal coordinates are extended to live in the fundamental representation.

EFT has been constructed for \(E_6\), \(E_7\) and \(E_8\) in the series of papers 1308.1673, 1312.0614, 1312.4542, 1406.3348, but for the present discussion I will focus on the \(E_6\) case. So let's first recall some basic facts about this group, which more precisely is here given by \(E_{6(6)}\). The extra 6 in parenthesis means that we are dealing with a non-compact version of \(E_6\), in which the number of non-compact and compact generators differs by 6. \(E_{6(6)}\) has two fundamental representations of dimension 27, denoted by \({\bf 27}\) and \(\bar{\bf 27}\), with corresponding lower and upper indices \(M,N=1,\ldots, 27\). There is no invariant metric to raise and lower indices, and so these two representations are inequivalent. \(E_{6(6)}\) admits two cubic fully symmetric invariant tensors \(d^{MNK}\) and \(d_{MNK}\).

The generalized spacetime of the \(E_{6(6)}\) EFT is given by `external' coordinates \(x^{\mu}\), \(\mu=0,\ldots,4\), and (extended) `internal' coordinates \(Y^M\) in the 27-dimensional fundamental representation. As for DFT, this does not mean that the theory is physically 32-dimensional. Rather, all functions on this extended space are subject to a section constraint, which is similar to the analogous constraint in DFT. In the present case it takes the manifestly \(E_{6(6)}\) covariant form \[ d^{MNK}\partial_N\partial_K A = 0 \qquad d^{MNK}\partial_NA\,\partial_K B = 0 \label{section0} \] with \(A,B\) denoting any fields or gauge parameters. Interestingly, this constraint allows for at least two inequivalent solutions: one preserves \(GL(6)\) and leaves six physical coordinates; the other preserves \(GL(5)\times SL(2)\) and leaves five physical coordinates. The first solution leads to a theory in \(5+6\) dimensions and turns out to be equivalent to 11-dimensional supergravity; the second solution leads to a theory in \(5+5\) dimensions and turns out to be equivalent to type IIB supergravity.

The field content of the theory comprises again a generalized metric, here denoted by \({\cal M}_{MN}\), which takes values in \(E_{6(6)}\) in the fundamental representation. Due to the splitting of coordinates, however, more fields are needed. The bosonic field content is given by \[ g_{\mu\nu}\;, \quad {\cal M}_{MN}\;, \quad {\cal A}_{\mu}{}^{M}\;, \quad {\cal B}_{\mu\nu M}\;. \] Here \(g_{\mu\nu}\) is the external, five-dimensional metric, while \({\cal A}_{\mu}{}^{M}\) and \({\cal B}_{\mu\nu M}\) are higher-form potentials needed for consistency. The fields depend on all \(5+27\) coordinates, subject to the above constraint.

The theory is uniquely determined by its invariance under the bosonic gauge symmetries, including internal and external generalized diffeomorphisms. Again, there is not enough space to explain this properly, but in order to give the reader at least a sense of the extended underlying geometry, let me display the generalized Lie derivative, which satisfies an algebra governed by the analogue of the `C-bracket' in DFT (which we call the `E-bracket'), and which encodes the internal generalized diffeomorphisms. Specifically, w.r.t. to vectors \(V^M\) and \(W^M\) in the fundamental representation it reads \[ \big(\mathbb{L}_{V}W\big)^M \ \equiv \ V^N\partial_NW^M-W^N\partial_N V^M+10\,d^{MNP}\,d_{KLP}\,\partial_NV^K\,W^L \] As for the C-bracket, the first two terms coincide with the Lie bracket between vector fields, but the new term, which explicitly requires the \(E_{6(6)}\) structure, shows that the full symmetry cannot be viewed as conventional diffeomorphisms on an extended space.

There is one more fascinating aspect of the symmetries of EFT that I can't resist mentioning. The vector fields \({\cal A}_{\mu}{}^{M}\) act as Yang-Mills-like gauge potentials for the internal diffeomorphisms. The novelty here is that the underlying algebraic structure is not a Lie algebra, because the E-bracket does not satisfy the Jacobi identity. The failure of the E-bracket to satisfy the Jacobi identity is, however, of a certain exact form. As a consequence, one can construct covariant objects like field strengths by introducing higher-form potentials, in this case the two-forms \({\cal B}_{\mu\nu M}\), and assigning suitable gauge transformations to them. This is referred to as the tensor hierarchy. One can then write a gauge invariant action, which structurally looks like five-dimensional gauged supergravity, except that it encodes, through its non-abelian gauge structure, the full dependence on the internal coordinates, as it should be in order to encode either 11-dimensional or type IIB supergravity. EFT explains the emergence of exceptional symmetries upon reduction, because the formulation is already fully covariant before reduction. (For more details see the recent review 1506.01065 with Henning Samtleben and Arnaud Baguet.)

[At this point let me stress that, as always in science, exceptional field theory did not originate out of thin air, but rather is the culmination of efforts by many researchers starting with the seminal work by Cremmer and Julia. The most important work for the present story is by de Wit and Nicolai in 1986, which made some symmetries, normally only visible upon reduction, manifest in the full \(D=11\) supergravity. It did not, however, make the exceptional symmetries manifest. These were discussed in more recent work dealing with the truncation to the internal sector governed by \({\cal M}_{MN}\). Notable work is due to West, Hillmann, Berman, Perry and many others.]

We are now ready to address the issue of consistent Kaluza-Klein truncations in EFT, following the two papers 1410.8145, 1506.01385. The Kaluza-Klein ansatz takes the form of a generalized Scherk-Schwarz reduction, governed by `twist matrices' \(U\in E_{6(6)}\). For technical reasons that I can't explain here we also need to introduce a scale factor \(\rho(Y)\). The ansatz for the bosonic fields collected above then reads \[ \begin{split} g_{\mu\nu}(x,Y) &= \rho^{-2}(Y)\,{g}_{\mu\nu}(x)\nonumber\\ {\cal M}_{MN}(x,Y) &= U_{M}{}^{{K}}(Y)\,U_{N}{}^{{L}}(Y)\,M_{{K}{L}}(x) \nonumber\\ {\cal A}_{\mu}{}^{M}(x,Y) &= \rho^{-1}(Y) A_{\mu}{}^{{N}}(x)(U^{-1})_{{N}}{}^{M}(Y) \nonumber\\ {\cal B}_{\mu\nu\,M}(x,Y) &= \,\rho^{-2}(Y) U_M{}^{{N}}(Y)\,B_{\mu\nu\,{N}}(x) \end{split} \] The \(x\)-dependent fields on the right-hand side are the fields of five-dimensional gauged supergravity.

We have to verify that the \(Y\)-dependence factors out consistently both in the action and equations of motion. This is the case provided some consistency conditions are satisfied, which have a very natural geometric interpretation within the extended geometry of EFT. To state these it is convenient to introduce the combination \[ {\cal E}_{{M}}{}^{N} \ \equiv \ \rho^{-1}(U^{-1})_{{M}}{}^{N} \] [Here I am deviating from the notation in the paper in order to simplify the presentation.] The consistency condition can then be written in terms of the \(E_{6(6)}\) generalized Lie derivative discussed above. It takes the form \[ \mathbb{L}_{\,{\cal E}_{{M}}}\,{\cal E}_{{N}} \ = \ -X_{{M}{N}}{}^{{K}}\, {\cal E}_{{K}} \] where the \(X_{{M}{N}}{}^{{K}}\) are the `structure constants' of gauged supergravity that encode the gauge group. This relation is the extended geometry version of the Lie bracket algebra of Killing vector fields given above. Thus, we can view the \({\cal E}_{M}\) as generalized Killing vectors on the extended space of EFT. An important and intriguing difference is that the \(X_{MN}{}^{K}\) in general are not the structure constants of a Lie group. In general they are not even antisymmetric in their lower two indices. They do, however, satisfy a quadratic Jacobi-type identity, leading to a structure that in the mathematics literature is referred to as a `Leibniz algebra'.

Due to these novel algebraic structures, there is no general procedure of how to solve the above consistency equations, i.e., of how, for given structure constants \(X_{MN}{}^{K}\), to find a twist matrix \(U\) satisfying the above equation. I think it is a mathematically fascinating open problem to understand systematically how to integrate the above Leibniz algebra to the corresponding `Leibniz group' (whatever that could mean). What we did in the paper instead is to solve the equations `by hand' for a few interesting cases, in particular spheres and their non-compact counterparts (inhomogeneous hyperboloidal spaces \(H^{p,q}\)).

These twist matrices take a surprisingly simple universal form, which then allows us to cover in one stroke the sphere compactifications of \(D=11\) supergravity (\(AdS_4\times S^7\) and \(AdS_7\times S^4\)) and of type IIB (\(AdS_5\times S^5\)). This, finally, settles the issue of consistency of the corresponding Kaluza-Klein truncation and also gives the explicit uplift formulas: they are given by the above generalized Scherk-Schwarz ansatz. Thus, for any solution of five-dimensional gauged supergravity, given by \(g_{\mu\nu}(x)\), \(M_{MN}(x)\), etc., we can directly read off the corresponding solution of EFT and, via the embedding discussed above, of type IIB. In particular, every stationary point and every holographic RG flow of the scalar potential directly lifts to a solution of type IIB.

This concludes my summary of exceptional field theory and its applications to Kaluza-Klein compactifications. Far from being impossible to describe in exceptional field theory, spheres and other curved spaces actually fit intriguingly well into these extended geometries, which allows us to resolve open problems. This is one example of a phenomenon I have seen again and again in the last couple of years: the application of this geometry to areas where duality symmetries are not present in any standard sense still leads to quite dramatic simplifications. I believe this points to a deeper significance of these extended geometries for our understanding of string theory more generally, but of course it remains to be seen which radically new geometry (if one may still call it that) we will eventually have to get used to.

Thursday, July 2, 2015 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere

First of all I would like to thank Luboš for giving me the opportunity to write a guest blog on double field theory. 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 \[

{\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' \[

\!\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: \[

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.