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.