The many facets of exceptional field theory
aa r X i v : . [ h e p - t h ] J un The many facets of exceptional field theory
Olaf Hohm a and Henning Samtleben b a Institute for Physics, Humboldt University Berlin, Zum Großen Windkanal 6, D-12489 Berlin,Germany b Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon,FranceE-mail: [email protected] , [email protected] Exceptional field theories are the manifestly duality covariant formulations of the target spacetheories of string/M-theory in the low-energy limit (supergravity) or for certain truncations. Thesetheories feature a rich system of sub-theories, corresponding to different consistent truncations.We review the structure of exceptional field theory and elaborate on various potential applications.In particular, we discuss how exceptional field theories capture some of the the magic supergravitytriangles, Hull’s ∗ -theories related to timelike dualities, as well as the generalized supergravitiesrelated to certain integrable deformations of AdS/CFT. Corfu Summer Institute 2018 "School and Workshops on Elementary Particle Physics and Gravity"(CORFU2018)31 August - 28 September, 2018Corfu, Greece c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ xceptional field theory
1. Introduction
There is mounting evidence for a unifying theory encompassing all 10-dimensional string theo-ries and 11-dimensional supergravity, often referred to as ‘M-theory’. The defining features of thisconjectured theory include the so-called U-dualities, which are motivated by hidden symmetriesthat emerge upon toroidal compactification of maximal supergravity. Specifically, dimensionallyreducing 11-dimensional supergravity on a d -dimensional torus leads to global symmetries in theexceptional series E d ( d ) ( R ) , d = , . . . ,
8, (of non-compact split signature), leading in particular tothe largest finite-dimensional exceptional group E ( ) ( R ) in three dimensions. M-theory is conjec-tured to be invariant under the discrete subgroups E d ( d ) ( Z ) . The natural question arises of how todefine a theory of (quantum) gravity that possesses such exceptional symmetries. Exceptional fieldtheory, which is an extension of double field theory and defined on enlarged spaces with extendedcoordinates, provides, conservatively, a reformulation of supergravity that is manifestly U-dualitycovariant prior to any compactification. More audaciously, exceptional field theory can be viewedas a first step towards a theory in between supergravity and the full string/M-theory that includesmassive string or M-theory modes in duality-complete multiplets and that is a consistent truncationof the full M-theory.Exceptional field theory (ExFT) as currently understood is defined in a Kaluza-Klein inspired‘split formulation’, with fields depending on ‘external’ coordinates x µ and generalized ‘internal’coordinates Y M in the fundamental representation of the U-duality group E d ( d ) under consideration,for instance the adjoint representation for E ( ) , where M , N = , . . . , M MN parametrizing the coset space E d ( d ) / K ( E d ( d ) ) ,Kaluza-Klein-type vectors A µ M , and an external metric g µν . All fields depend on external andinternal coordinates, with the dependence on the latter coordinates restricted by a duality-covariant‘section constraint’. This constraint can be solved in various ways. Most simply, one may takethe fields to be independent of the Y M coordinates, in which case one recovers directly the di-mensionally reduced theory with enhanced global symmetries that originally motivated the notionof U-duality. More non-trivially, there are solutions of the section constraint that complete theexternal coordinates to the ten or eleven coordinates of type IIB or 11-dimensional supergravity.These theories are thereby reconstructed in a split formulation that decomposes all coordinates andtensor components as in Kaluza-Klein but without truncating the coordinate dependence. In par-ticular, the components of the generalized metric M MN then encode the components of the internalRiemannian metric g mn and the internal components of various other tensor fields present in su-pergravity. In this way, ExFT provides a theory encompassing the complete, untruncated 10- or11-dimensional supergravities, thereby rendering manifest the emergence of exceptional symme-tries in lower dimensions.While one can argue, under certain assumptions, that there are only two different ‘U-dualityorbits’ of solutions of the section constraint, leading to M-theory/type IIA or type IIB, respectively,we will use the opportunity to illustrate how ExFT provides a rich set of consistent truncations and‘alternative parametrizations’ that give rise to novel applications. For instance, we will give ExFTin a formulation that is appropriate for timelike dualities. While in the conventional formulationthe internal sector is taken to be Euclidean (in the sense that the internal metric g mn is Euclidean),one may also take the internal sector to be Lorentzian and the external sector to be Euclidean.1 xceptional field theory The U-duality transformations, acting in particular on the generalized metric, then include timelikedualities. It should be emphasized that such dualities, while somewhat unconventional, are ofphysical interest since they apply whenever one has timelike Killing vectors (or BPS solutions).In addition, theories with a ‘Lorentzian generalized metric’ are intriguing in that the latter maygo through a phase in which the conventional metric components g mn become singular while thegeneralized metric is still perfectly regular. In this fashion, ExFT encodes truly ‘non-geometric’configurations.Another application we will discuss here is the relation to the ‘magic tables’ that arose in theliterature since the early days of supergravity. Specifically, there is an abundance of interesting,often exceptional groups arising in lower dimensions, with various intriguing interrelations andsymmetries between them. Here we consider the ‘magic triangle’ discovered by Cremmer et. al. incompactifications to three dimensions [15]. We will show that the E ( ) ExFT unifies all theoriescorresponding to the different entries of that table into a single Lagrangian that, moreover, realisesthe symmetry of the triangle following a simple group-theory argument. Finally, we review howthe generalized type IIB supergravity theory that appeared recently in integrable deformations ofAdS/CFT is naturally embedded in ExFT.The rest of this article is organized as follows. In sec. 2 we review ExFT with a focus on the E ( ) and E ( ) theories that will be employed later. In sec. 3 we introduce the magic triangle ofCremmer et. al. and explain how it is accommodated within the E ( ) ExFT. We then turn to timelikedualities and Hull’s M ∗ -theories in sec. 4. In sec. 5 we discuss the embedding of generalized IIBsupergravity theory, while we close with a brief outlook in sec. 6.
2. Brief Review of Exceptional Field Theory
In this section we give a review of exceptional field theory (ExFT), originally constructedin [1, 2, 3, 4]. We focus on the special theories corresponding to U-duality groups E ( ) (firstsubsection) and E ( ) (second subsection). For an extensive general review of ExFT we refer to theprevious Corfu proceedings [5]. For the lower-rank exceptional groups, the associated ExFTs havebeen constructed in [6, 7, 8, 9]. The first example of an ExFT based on an infinite-dimensionalduality group ( E ( ) ) is under construction [10]. ( ) We begin by reviewing some relevant facts about the non-compact Lie group E ( ) . It is 78-dimensional, with generators that we denote by t α with the adjoint index α = , . . . ,
78. Apartfrom the adjoint representation, E ( ) carries two inequivalent fundamental representations of di-mension 27, denoted by and ¯ , which we label by lower indices M , N = , . . . ,
27 for andupper indices for ¯ . The (rescaled) Cartan-Killing form can then be expressed in terms of gener-ators ( t α ) MN in a fundamental representation as κ αβ ≡ ( t α ) MN ( t β ) N M . Moreover, the fundamentalrepresentations carry two completely symmetric invariant tensors of rank 3, the d -symbols d MNK and d MNK , which are normalized as d MPQ d NPQ = δ NM . Finally, the tensor product ⊗ ¯ containsthe adjoint representation, and the corresponding projector onto the adjoint representation can be2 xceptional field theory expressed explicitly as P MN KL ≡ ( t α ) N M ( t α ) LK = δ MN δ KL + δ KN δ ML − d NLR d MKR , (2.1)satisfying P MN NM = Y M in the ¯27 of E ( ) . All fields and gauge parameters appearing in the following willbe functions of these coordinates, subject to a ‘section constraint’ to be defined shortly. There is anotion of (infinitesimal) generalized diffeomorphisms, Y M → Y M − Λ M ( Y ) , acting on tensor fieldsaccording to generalized Lie derivatives. For instance, on a vector in the ¯27 it reads δ V M = L Λ V M ≡ Λ K ∂ K V M − P MN KL ∂ K Λ L V N + λ∂ P Λ P V M , (2.2)where λ is arbitrary (intrinsic) density weight. The action of L Λ on tensor fields in other represen-tations of E ( ) follows similarly, such that all group-invariant operations are also invariant undergeneralized Lie derivatives. For instance, the d -symbols are gauge invariant, L Λ d MNK =
0, and theaction on the E ( ) -valued generalized metric M MN preserves its group property. The generalizedLie derivatives form a closed algebra, (cid:2) L Λ , L Λ (cid:3) = L [ Λ , Λ ] E , (2.3)governed by the so-called ‘E-bracket’ (cid:2) Λ , Λ (cid:3) M E = Λ K [ ∂ K Λ M ] − d MNP d KLP Λ K [ ∂ N Λ L ] , (2.4)provided the functions defining the vector fields Λ M satisfy the section constraint d MNK ∂ N ⊗ ∂ K = . (2.5)This somewhat symbolic presentation of the constraint is to be interpreted in the sense that for anyfunctions f , g on the extended spacetime d MNK ∂ N ∂ K f = , d MNK ∂ N f ∂ K g = . (2.6)These constraints are the M-theory version of the (stronger version of the) level-matching constraintin the string-inspired double field theory.Having defined the gauge structure of generalized Lie derivatives, our next goal is to define agauge theory based on it. Specifically, we introduce fields on an external five-dimensional space-time with coordinates x µ , taking values in the internal generalized space with coordinates Y M ,subject to generalized diffeomorphisms. Thus, we consider fields and gauge parameters dependingon coordinates ( x µ , Y M ) . As a consequence, partial derivatives like ∂ µ will no longer be covariantunder generalized Lie derivatives w.r.t. parameters Λ M ( x , Y ) , requiring the introduction of gaugecovariant derivatives. We thus introduce gauge fields A µ M (of intrinsic density weight ) and co-variant derivatives D µ ≡ ∂ µ − L A µ . (2.7)3 xceptional field theory These derivatives transform covariantly under generalized diffeomorphisms, provided the gaugeconnections transform as δ A µ M ≡ D µ Λ M . (2.8)Following the usual textbook treatment of gauge theories one would next define gauge covariantfield strengths for A µ M , but this turns out to be more subtle since the E-bracket does not actuallydefine a Lie algebra. However, its failure to define a Lie algebra, and consequently the failure ofthe naive field strength to be gauge covariant, is of a controlled form, so that covariance can berepaired by introducing 2-forms B µν M in the (of intrinsic density weight ) and defining F µν M ≡ ∂ [ µ A ν ] M − (cid:2) A µ , A ν (cid:3) M E + d MNK ∂ K B µν N . (2.9)This curvature is fully gauge covariant, provided we assign appropriate gauge transformations tothe 2-forms, a construction known as tensor hierarchy. Next, one can define a gauge covariant3-form curvature for the 2-form field, which is implicitly determined by the hierarchical Bianchiidentity 3 D [ µ F νρ ] M = d MNK ∂ K H µνρ N . (2.10)(See [11] for an extensive review of this ‘higher gauge theory’ aspect of ExFT, [12] for a review of L ∞ algebras, and [13] for a general theory of tensor hierarchies.)We have now all structures in place in order to define the E ( ) ExFT [2]. Its field content isgiven by (cid:8) e µ a , M MN , A µ M , B µν M (cid:9) , (2.11)with A µ and B µν the tensor fields introduced above, and where e µ a is the frame field of the five-dimensional external space (“fünfbein"), and M MN is the generalized metric of the 27-dimensionalinternal generalized space. More precisely, M MN parameterizes the coset space E ( ) / USp ( ) .The fünfbein is a scalar density under internal generalized diffeomorphisms, with intrinsic densityweight . The generalized metric M MN is an E ( ) valued symmetric tensor under generalizeddiffeomorphisms (of density weight zero). The complete bosonic action is given by S ExFT = Z dx dY e (cid:16) b R + g µν D µ M MN D ν M MN − M MN F µν M F µν N + e − L top − V ( g , M ) (cid:17) , (2.12)written in terms of the gauge covariant derivatives and field strengths defined above. Moreover, thevarious terms are defined as follows. The generalized Ricci scalar b R building the Einstein-Hilbertterm is defined by taking the familiar definition of the Ricci scalar and replacing every ∂ µ by D µ ,acting on g µν as a scalar density of weight . The topological term can be defined concisely interms of the curvatures (2.9), (2.10) as a boundary integral in one higher dimension, S top = κ Z d Y Z M (cid:0) d MNK F M ∧ F N ∧ F K − d MNK H M ∧ ∂ N H K (cid:1) , (2.13)where M is a six-manifold whose boundary is the (external) five-dimensional spacetime. The pre-factor is determined by (external) diffeomorphism invariance to be κ = p /
32. Finally, the last4 xceptional field theory term in (2.12) — sometimes referred to as ‘the potential’, because it reduces upon compactificationto the scalar potential of supergravity — is defined by V = − M MN ∂ M M KL ∂ N M KL + M MN ∂ M M KL ∂ L M NK − g − ∂ M g ∂ N M MN − M MN g − ∂ M g g − ∂ N g − M MN ∂ M g µν ∂ N g µν . (2.14)Its relative coefficients are uniquely determined by (internal) generalized diffeomorphism invari-ance. This term can be written more geometrically, in terms of a generalized Ricci scalar R ( M , g ) of the internal geometry that, however, also depends on the external metric.Being defined in terms of gauge covariant derivatives and field strengths, the above action ismanifestly invariant under internal generalized diffeomorphisms with gauge parameters Λ M ( x , Y ) .In particular, every term in (2.12) is separately invariant. On the contrary, all relative coefficientsin (2.12) are fixed by external diffeomorphisms with parameters ξ µ . They act on the fields as δ e µ a = ξ ν D ν e µ a + D µ ξ ν e ν a , δ M MN = ξ µ D µ M MN , δ A µ M = ξ ν F νµ M + M MN g µν ∂ N ξ ν , ∆ B µν M = κ ξ ρ e ε µνρστ F στ N M MN , (2.15)where we used the ‘covariant variations’ of the 2-forms defined by ∆ B µν M ≡ δ B µν M + d MNK A [ µ N δ A ν ] K . (2.16)Note that this invariance is manifest for gauge parameters ξ µ depending only on x µ coordinates,since all terms in the action are tensorial in the sense of the usual tensor calculus. However, in-variance under parameters ξ µ ( x , Y ) is no longer manifest, due to the presence of ∂ M derivatives.The complete invariance is verified by a quite tedious computation, some aspects of which will bereviewed below, which employs in particular the following identity between the generalized metricand the d -tensor: M KL M MN M PQ d LNQ = d KMP . (2.17)In the remainder of this subsection we discuss how this theory relates to conventional super-gravity, in particular how the section constraints are solved. In order to recover 11-dimensionalsupergravity in a 5 + ( ) down to its subgroupSL ( ) × GL ( ) ⊂ SL ( ) × SL ( ) ⊂ E ( ) , (2.18)with the fundamental representation of E ( ) breaking as −→ + + ′ + − , (2.19)with the subscripts referring to the GL ( ) charges. For the coordinates Y M in this representationthis corresponds to the decomposition (cid:8) Y M (cid:9) −→ (cid:8) y m , y mn , y ¯ m (cid:9) . (2.20)5 xceptional field theory An explicit solution to the section condition (2.6) is given by restricting the Y M dependence of allfields to the six coordinates y m , as can be seen by working out the d -tensor in this decomposition,see [2] for details.The type IIB supergravity theory in a 5 + ( ) down to its subgroup SL ( ) × SL ( ) × GL ( ) IIB ⊂ E ( ) , (2.21)such that −→ ( , ) + + ( ′ , ) + + ( , ) − + ( , ) − , (2.22)and thus for the coordinates (cid:8) Y M (cid:9) −→ { y a , ˜ y a α , ˜ y ab , ˜ y α } , a = , . . . , , α = ± . (2.23)The section constraint is then solved by restricting the internal coordinate dependence of all fieldsto the coordinates y a . This solution is inequivalent to (2.20) in the sense that no E ( ) rotation canmap the y a coordinates from (2.23) into (a subset of) the y m coordinates from (2.20).We close with a few remarks on the dictionary between the ExFT fields and the conventionalsupergravity fields. To this end one must pick a solution of the section constraint and a groupdecomposition as above. For instance, under the decomposition (2.20) appropriate for D = M MN decomposes into theblock form M KM = M km M kmn M k ¯ m M kl m M kl , mn M kl ¯ m M ¯ km M ¯ kmn M ¯ k ¯ m . (2.24)These components should now be expressed in terms of supergravity fields. For instance, its lastline reads M ¯ mn = g − / g mk ε klpqrs c nlp c qrs − g − / g mn ϕ , M ¯ mkl = − √ g − / g mn ε nklpqr c pqr , M ¯ m ¯ n = g − / g mn , (2.25)parametrized by ϕ , c mnk and the six-dimensional internal Euclidean metric g mn with determinant g ≡ det g mn . The internal metric originates directly from the Kaluza-Klein-type decompositionof the 11-dimensional metric. The antisymmetric tensor c mnk encodes the internal components ofthe 11-dimensional 3-form field, while the scalar ϕ is the dualization (in five dimensions) of thepurely external components of the 11-dimensional 3-form field. The dictionary (2.25) is straightfor-wardly established by comparing the action of generalized diffeomorphisms (2.2) onto the variousblocks of (2.24) to the internal diffeomorphism and p -form gauge transformations of the higher-dimensional supergravity fields. The remaining blocks of (2.24) can be expressed in compact formvia the matrix ˜ M MN ≡ M MN − M M ¯ m ( M ¯ m ¯ n ) − M ¯ nN , (2.26)6 xceptional field theory which take the form ˜ M mn = g / g mn + g / c mkp c nlq g kl g pq , ˜ M mkl = − √ g / c mpq g pk g ql , ˜ M kl , mn = g / g m [ k g l ] n . (2.27)The type IIB dictionary takes an analogous form based on the block decomposition of the symmet-ric matrix M MN based on the decomposition (2.23). Details are spelled out in [5]. ( ) The exceptional field theory based on the Lie group E ( ) has been constructed in [4]. Itis built starting from the reduction of eleven-dimensional supergravity down to three dimensionswhich exhibits a global E ( ) symmetry. Like all exceptional field theories with only three externaldimensions, the construction of this theory brings about some particular features, related to theEhlers type symmetry enhancement [14] which necessitates to describe within the scalar sectorsome of the dual graviton degrees of freedom.Vector fields in the E ( ) ExFT transform in the adjoint representation of E ( ) . Accord-ingly, generalized diffeomorphisms take the form analogous to (2.2) δ Λ V M = L Λ V M ≡ Λ K ∂ K V M − P MN KL ∂ K Λ L V N + λ∂ P Λ P V M , (2.28)with indices M , N , . . . now labelling the adjoint representation of E ( ) , raised and lowered by theCartan-Killing form η MN , and the projector P MNKL = f MNP f PKL onto the adjoint representation.In contrast to the lower rank exceptional field theories, the transformations (2.28) no longer closeinto an algebra among themselves but give rise to an enhanced symmetry algebra which also carriescovariantly constrained local E ( ) rotations as δ Σ V M = − Σ K f KMN V N , (2.29)with the structure constants f MN K . The full algebra (2.28), (2.29) now closes according to (cid:2) δ ( Λ , Σ ) , δ ( Λ , Σ ) (cid:3) = δ [( Λ , Σ ) , ( Λ , Σ )] E , [( Λ , Σ ) , ( Λ , Σ )] E ≡ ( Λ , Σ ) , (2.30)with the effective parameters Λ M ≡ Λ N [ ∂ N Λ M ] − ( P ) MK NL Λ N [ ∂ K Λ L ] − η MK η NL Λ N [ ∂ K Λ L ] + f MN P ∂ N ( f PKL Λ K Λ L ) , Σ M ≡ − Σ [ M ∂ N Λ N ] + Λ N [ ∂ N Σ ] M − Σ N [ ∂ M Λ ] N + f NKL Λ K [ ∂ M ∂ N Λ L ] , (2.31)with ( P ) MK NL denoting the projector of the symmetric product ⊗ onto the repre-sentation. The closure (2.30) requires the E ( ) section contraints η MN ∂ M ⊗ ∂ N = , f MNK ∂ N ⊗ ∂ K = , (cid:0) P (cid:1) MN KL ∂ K ⊗ ∂ L = , (2.32)7 xceptional field theory in analogy with (2.5) for the internal coordinate dependence, accompanied by the algebraic con-straints (cid:0) P + + (cid:1) MN KL Σ K ⊗ ∂ L = = (cid:0) P + + (cid:1) MN KL Σ K ⊗ Σ L , (2.33)for the symmetry parameter of (2.29).The field content of the E ( ) ExFT comprises scalar fields, parametrizing the coset spaceE ( ) / SO ( ) in terms of the symmetric positive matrix M MN , together with gauge fields A µ M , B µ M , associated to the internal diffeomorphisms (2.28), (2.29) with minimal couplings via covari-ant derivatives D µ ≡ ∂ µ − L ( A µ , B µ ) , (2.34)to the scalar fields. The full action reads S ExFT = Z dx dY e (cid:16) b R + g µν D µ M MN D ν M MN + e − L CS − V ( g , M ) (cid:17) . (2.35)with Einstein-Hilbert term and scalar kinetic term defined as in (2.12) above. The non-abelianChern-Simons term is most conveniently defined as the boundary contribution of a manifestlygauge invariant exact form in four dimensions. S CS ∝ Z Σ d x Z d Y (cid:16) F M ∧ G M − f MN K F M ∧ ∂ K F N (cid:17) , (2.36)in terms of the non-abelian field strengths F µν M , G µν M associated to the gauge fields A µ M , B µ M .We refer to [4] for their explicit definitions and to [11] for an interpretation in terms of Leibnizgauge theories. The potential term V ( M , g ) is finally given by V ( M , g ) = − M MN ∂ M M KL ∂ N M KL + M MN ∂ M M KL ∂ L M NK (2.37) + f NQP f MSR M PK ∂ M M QK M RL ∂ N M SL − g − ∂ M g ∂ N M MN − M MN g − ∂ M g g − ∂ N g − M MN ∂ M g µν ∂ N g µν , generalizing the potentials of the lower-rank exceptional fields theories such as (2.14) by a new con-tribution explicitly carrying the E ( ) structure constants. Again, every term in the action (2.35) isseparately invariant under generalized internal diffeomorphisms, while the relative coefficients areuniquely fixed by imposing invariance under external diffeomorphisms. Upon solving the sectionconstraints and implementing the proper dictionary, the field equations of (2.35) again reproducethose of the full D =
11 and IIB supergravity, respectively.
3. The magic triangle
In this section we consider an instance of the ‘magic tables’ that appeared in the supergravityliterature since the discovery of maximal supergravity in eleven dimensions. As briefly recalledin the introduction, 11-dimensional supergravity has the intriguing property of giving rise to theexceptional symmetry groups E d ( d ) in compactifications, on a d -torus, to D = − d dimensions.In particular, in three dimensions ( D =
3) one encounters the largest of the finite-dimensional8 xceptional field theory exceptional groups, E ( ) , which is the global symmetry of (ungauged) maximal three-dimensionalsupergravity whose propagating bosonic degrees of freedom are entirely encoded in the scalar fieldsof a non-linear sigma model based on E ( ) / SO ( ) .Irrespective of their origin in maximal supergravity one may consider such G / H coset modelscoupled to (topological) three-dimensional gravity in their own right. The action reads S = Z d x √ g (cid:16) R − P µ A P µ A (cid:17) , (3.1)with the Einstein-Hilbert term for the three-dimensional metric g µν and a scalar kinetic term for thecoset currents P µ A ≡ [ V − ∂ µ V ] A , where V is a G -valued field and [ · ] denotes the projection ontothe non-compact part (the complement of H ), labelled by indices A = , . . . , dim ( G ) − dim ( H ) . Wemay consider the coset models of the exceptional series in D =
3, as done by Cremmer et. al. [15],who also gave a systematic analysis of the higher dimensional origin of these models. (See also theearlier work in [16] and the subsequent completions in [17, 18].) The various higher-dimensionaltheories that these three-dimensional models can be uplifted to are indicated in the table below. Thecolumns are labelled by the rank of the group in D =
3, and the rows are labelled by the highestdimension to which the three-dimensional models can be uplifted to. × R A × × R R × A A × R A A A × R A × R R × A × A A × R R A R A A A × R A × R A D A A × A A × R A × R A R × E E D A A × A A × R A R × D r
Let us now discuss some features of this triangle. The first row ( D =
3) displays the ‘initialdata’, the exceptional series of global symmetries under consideration. The first column ( r = D = D =
11. By construction, the groupsof the first row equal the groups of the first column, for the groups in the first row were chosen inthis way. However, remarkably this symmetry extends to the entire triangle (justifying its ‘magic’)in that under the exchange of rank and dimension according to ( r , D ) ↔ ( − D , − r ) (3.2)the groups stay the same. This is intriguing, for a priori the corresponding theories have little todo with each other, typically being defined in completely different dimensions. This raises thequestion whether there is an overarching framework that not only explains this ‘duality’ but alsoprovides a theory from which the models corresponding to the different entries of the triangle canbe derived by suitable truncations.In the following we will explain that precisely the E ( ) ExFT provides this framework. Inthis we will use a group-theoretical argument anticipated by Keurentjes some time ago [17] that9 xceptional field theory unfolds its full force in the context of ExFT. This group-theoretical argument relies on the fact thatE ( ) can be decomposed according toE ( ) −→ SL ( D − ) × SL ( − r ) × U D , r , (3.3)where U D , r is the U-duality group labelled by ( D , r ) in the table. In this sense, U D , r can be obtainedfrom E ( ) by singling out two SL ( n ) factors. There is, of course, no intrinsic difference betweenthese two factors, and since they are interchanged under the duality (3.2) we infer the symmetry U D , r = U − r , − D [17], thereby explaining this equality of groups, see also [19].More intriguingly, the E ( ) ExFT provides a unifying theory from which all entries of the tablecan be obtained through truncation, following the breaking of E ( ) displayed in (3.3). Specifically,one then decomposes the internal coordinates in the adjoint representation according to (cid:8) Y M (cid:9) → n Y i j , Y ab , . . . o , (3.4)where i , j = , . . . , D − ( D − ) indices and a , b = , . . . , − r label SL ( − r ) indices. Wedisplayed the coordinates in the adjoint of SL ( D − ) and SL ( − r ) , while the ellipsis indicatesthe remaining coordinates transforming non-trivially under U D , r . In the next step one decomposes i = ( i , ) , where i = , . . . , D − ( D − ) index, and identifies the D − y i ≡ Y i , (3.5)which solves the E ( ) section constraint. Together with the three external coordinates x µ , thesedescribe the D -dimensional space-time on which the theory with duality group U D , r is defined.In order to obtain this theory one, finally, truncates the E ( ) ExFT to singlets under SL ( − r ) .Since it is always consistent to truncate to the singlets of a symmetry group, this yields a consistenttruncation. It would be interesting to see if these consistent truncations can be extended to gaugedsupergravities, using the techniques of generalized Scherk-Schwarz compactifications [20, 21, 22]and to see to which extent the magic of the triangle is inherited by its gauged deformations. Itwould also be interesting to explore to which extent other magic triangles [23] and pyramides [24]can be accommodated in ExFT. These typically feature non-split forms of the exceptional groups.Intriguingly, the ExFT construction is largely independent of the particular real form used. Thelatter is only visible via the solution space of the section constraints and via the parametrization ofthe scalar matrix M MN .
4. Timelike dualities
As reviewed above, the E d ( d ) ExFT Lagrangian is modelled after the theory obtained bytoroidal reduction of 11-dimensional supergravity on a spacelike torus T d in such a way that it stillincludes the full 11-dimensional supergravity. In particular, is is based on the scalar coset spaceE d ( d ) / K d with K d the compact subgroup of E d ( d ) . It is then natural to expect an equivalent ExFTformulation of the higher-dimensional theories based on their toroidal reduction on a torus includ-ing a timelike circle. These reductions to Euclidean theories have been studied in [25, 26, 27]. Theyare closely related to spatial reductions, however, based on coset spaces E d ( d ) / ˜K d with ˜K d now rep-resenting a different real form of K d . This accounts for the fact that timelike dimensional reduction10 xceptional field theory in particular results in certain sign flips in the lower-dimensional kinetic terms. Extending thesetheories to their full ExFT form thus gives rise to yet an alternative formulation of 11-dimensionalsupergravity based on a Euclidean external space and a Lorentzian exceptional geometry.Moreover, by construction these theories should (and do) also accommodate Hull’s ∗ -theories[28, 29], defined by T-duality along timelike isometries, as well as the underlying 11-dimensionalsupergravities of exotic signature. More precisely, T-duality along a timelike circle maps IIA andIIB supergravity into the so-called IIB ∗ and IIA ∗ supergravity, respectively, which differ from theirunstarred counterparts by certain sign flips in the kinetic terms. Under timelike dimensional re-duction, all these theories give rise to the same lower-dimensional Euclidean supergravities basedon the coset spaces from the E d ( d ) / ˜K d series. Then it does not come as a surprise that the dual-ity covariant ExFT formulations based on these coset spaces do indeed accommodate in a singleframework IIA/IIB supergravity together with their IIA ∗ /IIB ∗ counterparts.In this section, we illustrate these structures for the Euclidean version of E ( ) ExFT. Hull’s ∗ -theories appeared earlier in type II double field theory [30, 31]. For the internal sector of SL ( ) ExFT these scenarios have also been discussed in [32], then completed in [33], see also [34] forearlier work. ( ) ExFT with Euclidean external space
Reduction of 11-dimensional supergravity on a 6-torus T , including a timelike circle leadsto a Euclidean five-dimensional theory based on the coset space E ( ) / USp ( , ) of indefinite sig-nature [25, 26, 27]. More precisely, under USp ( ) × USp ( ) the 42 physical scalars parametrizingthis coset space decompose according to −→ ( , ) + ( , ) + ( , ) , (4.1)with the ( , ) corresponding to 16 compact directions within E ( ) , thus carrying an opposite signin the kinetic sigma-model term. A quick counting shows that 16 = + + T , yields a coset space GL ( ) / SO ( , ) with 5 compact directions,the 11D three-form adds 10 scalars C mn of opposite kinetic term, as well as a three-form C µνρ ,which in 5 dimensions is dualized to a scalar of opposite kinetic sign (since the metric g µν is nowEuclidean).It is straightforward to adapt the ExFT construction to this setting. First, we note that thestructure of generalized diffeomorphisms including the full tensor hierarchy depends only on thealgebra of E ( ) and therefore remains identical to the previous construction. The alternative E ( ) ExFT Lagrangian can then be constructed from first principles and takes a form which is formallyidentical to (2.12)˜ S EFT = Z dx dY e (cid:16) b R + g µν D µ M MN D ν M MN − M MN F µν M F µν N + e − L top − V ( M , g ) (cid:17) , (4.2)and only differs in the signatures of external and internal metric. Specifically, in contrast to (2.12),the external five dimensions now come with a Euclidean metric g µν while the internal metric11 xceptional field theory M MN parametrizing the coset space E ( ) / USp ( , ) , is no longer positive definite, but of signature ( , ) . In particular, it satisfies the relation M KL M MN M PQ d LNQ = − d KMP , (4.3)which differs in sign from the relation (2.17) satisfied on the non-compact coset space.As in the construction of (2.12), each term in (4.2) is uniquely fixed by invariance undergeneralized diffeomorphisms (which are blind to issues of space-time signature), while the relativefactors between the five terms are fixed by invariance of the action under external diffeomorphisms.The latter are given by the same transformation rules (2.15) with a single change of sign in thetransformation of two-forms, ∆ B µν M = − κ ξ ρ e ε µνρστ F στ N M MN , (4.4)required such as to achieve the standard on-shell transformation ∆ B µν M ∼ ξ ρ H ρµν M , (4.5)upon applying the first order duality equation in now Euclidean external space. As for invarianceof (4.2) under external diffeomorphisms, going through the original calculation [2], the new signin (4.4) precisely compensates the additional sign arising from (4.3) and the effect of a Euclideanexternal metric implying ε µνρστ ε µνρστ = + ∗ -theory, and M’-theory With the Lagrangian (4.2) uniquely fixed by internal and external generalized diffeomorphismstogether with the signatures of external and generalized internal space-time, the embedding ofthe higher-dimensional theories is based on solving the section constraints and establishing thedictionary between the ExFT fields and the components of the higher-dimensional fields in analogyto (2.24)–(2.27).The symmetric matrix M MN in (4.2) parametrizes the coset space E ( ) / USp ( , ) . The iden-tification of the fields from eleven-dimensional supergravity is accomplished by solving the sectionconstraints according to (2.20) and choosing an explicit associated parametrization (2.24) with thelast line now given by M ¯ mn = | g | − / g mk ε klpqrs c nlp c qrs − | g | − / g mn ϕ , M ¯ mkl = − √ | g | − / g mn ε nklpqr c pqr , M ¯ m ¯ n = − | g | − / g mn , (4.6)differing in the sign of the last component from (2.24). The internal metric g mn is now of signature ( , ) with negative determinant. The remaining blocks of (2.24) are expressed in compact formthrough the matrix (2.26) via˜ M mn = | g | / g mn + | g | / c mkp c nlq g kl g pq , ˜ M mkl = − √ | g | / c mpq g pk g ql , ˜ M kl , mn = | g | / g m [ k g l ] n . (4.7)12 xceptional field theory Upon combining this parametrization with the proper dictionary for the ExFT p -forms A µ M and B µν M , the Lagrangian (4.2) reproduces full eleven-dimensional supergravity. We refrain fromspelling out all the details and give a simple illustration from the leading kinetic term of the vectorfields A µ M . Upon setting the scalars c kmn and ϕ to zero, the Yang-Mills term from (4.2) reduces to L vec = − e M MN F µν M F µν N = − p | g D | (cid:16) g mn F µν m F µν n + g kl g mn F µν km F µν ln − | g | − g mn F µν ¯ m F µν ¯ n (cid:17) , (4.8)where we have split vectors A µ M according to (2.20). The first term descends from the 11 D Einstein-Hilbert term, the next two terms descend from the kinetic term of the 11 D four-form fieldstrength. The opposite sign of the last term is due to the sign appearing in the last component of(4.6) and reflects the fact that it descends from dualization of two-forms B µν ¯ m on Euclidean space.With the internal metric g mn of signature ( , ) , it is straightforward to confirm that the signatureof the matrix M MN splits over the three terms in (4.8) as ( , ) + ( , ) + ( , ) = ( , ) , (4.9)precisely as consistent with the coset E ( ) / USp ( , ) .A crucial difference with the standard ExFT reviewed in section 2 above is due to the fact thescalar matrix M MN is no longer positive definite. As a result, the parametrization (4.6) does notguarantee that the internal metric g mn extracted from the submatrix M ¯ m ¯ n is non-degenerate. Thedictionary with D =
11 supergravity might thus break down even though the ExFT configurationremains perfectly regular. This is comparable to the situation in [30, 31, 35, 36] in the O ( d , d ) case.Passing through such a degenerate point, one may reach a configuration in which the blocks (2.24)of the matrix M MN are parametrized as M ¯ mn = | g | − / g mk ε klpqrs c nlp c qrs − | g | − / g mn ϕ , M ¯ mkl = − √ | g | − / g mn ε nklpqr c pqr , M ¯ m ¯ n = | g | − / g mn , (4.10)for the last line, and˜ M mn = | g | / g mn − | g | / c mkp c nlq g kl g pq , ˜ M mkl = √ | g | / c mpq g pk g ql , ˜ M kl , mn = − | g | / g m [ k g l ] n , (4.11)for the remaining blocks via the matrix (2.26). In this parametrization, the internal metric g mn is of signature ( , ) . As an illustration of its higher-dimensional origin, we again evaluate theYang-Mills term from (4.2) at c kmn = = ϕ to find L vec = − e M MN F µν M F µν N = − p | g D | (cid:16) g mn F µν m F µν n − g kl g mn F µν km F µν ln + | g | − g mn F µν ¯ m F µν ¯ n (cid:17) (4.12)13 xceptional field theory W.r.t. (4.8), the last two terms have switched sign. With this parametrization of M MN , keeping thesame solution (2.20) of the section constraint, the theory (4.2) thus describes a D =
11 theory ofsignature ( + , ) = ( , ) in which the kinetic term of the three-form, responsible for the last twoterms in (4.12), has switched its sign. This is precisely the field theory limit of Hull’s M ∗ theory[29]. It is defined such that reduction on a timelike circle yields the IIA ∗ supergravity, which wehave thus also embedded into (4.2). Similar to (4.9), the signatures of the terms in (4.12) still addup to the same result ( , ) + ( , ) + ( , ) = ( , ) , (4.13)fixed by the coset space E ( ) / USp ( , ) .Let us finally mention that the remaining eleven-dimensional supergravity proposed by Hullin [29], the so-called M ′ theory, is of signature ( , ) and thus simply embeds into (4.2) by meansof the dictionary (4.6)–(4.7) with the flip g mn → − g mn , in order to account for an internal metric ofsignature ( , ) . ∗ Here, we briefly describe how IIB and IIB ∗ supergravity are embedded in (4.2) in a similar way.Both are based on the IIB solution (2.23) of the section constraint and amount to parametrizationsof the scalar matrix M MN that again differ from the E ( ) / USp ( ) parametrization worked out in[5] by a number of sign flips in order to account for a coset space of indefinite signature. As anillustration, we simply state the Yang-Mills term from (4.2), evaluated for vanishing forms b mn α and c klmn .In the IIB parametrization, this term reads L vec = − e M MN F µν M F µν N = − p | g D | (cid:16) g mn F µν m F µν n + g mn m αβ F µν m α F µν n β − | g | − g mk g ln F µν mn F µν kl − | g | − m αβ F µνα F µνβ (cid:17) , (4.14)with g mn and m αβ of signature ( , ) and ( , ) , respectively. The first two terms directly descendfrom the D =
10 Einstein-Hilbert and three-form field strength terms. The last two terms areobtained from dualization of the D = D =
10 four-form and two-forms, respectively. Their minus signs account for the fact that this dualization has been performedw.r.t. the Euclidean external metric g µν . A quick count confirms that the signature of the matrix M MN splits over the four terms in (4.8) as ( , ) + ( , ) + ( , ) + ( , ) = ( , ) , (4.15)as determined by the coset E ( ) / USp ( , ) .In turn, the IIB ∗ parametrization of M MN gives rise to a Yang-Mills term that takes the form L vec = − e M MN F µν M F µν N = − p | g D | (cid:16) g mn F µν m F µν n + g mn m αβ F µν m α F µν n β + | g | − g mk g ln F µν mn F µν kl − | g | − m αβ F µνα F µνβ (cid:17) , (4.16)14 xceptional field theory with g mn and m αβ of signature ( , ) and ( , ) , respectively. W.r.t. (4.14), the sign of the third termhas changed, precisely capturing the sign flip of RR fields in the IIB ∗ theory not accounted for bythe signature change of the axion-dilaton matrix m αβ .The ExFT (4.2) with its different solutions to the section constraint and different parametriza-tions of the scalar matrix M MN thus captures the full IIA/IIB and IIA ∗ /IIB ∗ supergravities togetherwith the D =
11 supergravities of exotic signature. It would be very exciting to further explore thepossibility of regular ExFT solutions that interpolate between different higher-dimensional theo-ries, along the lines of [37, 38, 33].
5. Generalized IIB supergravity
We finish by sketching yet another application of exceptional field theory: the embeddingof the so-called generalized type II supergravities. Following the recent discovery of integrabledeformations of the string sigma model [39, 40, 41], it has subsequently been understood that(unlike for λ -deformations [42]) the action of (most of) the η -deformations do not give rise tosolutions of the standard supergravity equations, but to a generalization thereof worked out in[44, 45]. Solutions of these generalized type II equations are obtained by applying a generalizedT-duality to a solution of the original supergravities however (in contrast to standard T-duality)with non-isometric linear dilaton. This suggests that the generalized supergravities should have anatural place in the manifestly duality covariant formulation of exceptional field theory, which isindeed the case, as we shall briefly review following [46]. This provides yet a different scenario forcombining solutions of the section constraint (2.20), (2.23), with particular parametrizations of theExFT fields.Let us start from the IIB split of cordinates (2.23) and in addition impose the existence ofa Killing vector field in the IIB theory, such that the coordinates further split as { y a } = { y i , y ∗ } , ( i = , . . . , ) , with ∂ ∗ Φ = { y i , ˜ y } , i = , . . . , , (5.1)including a specific additional coordinate ˜ y ≡ ˜ y ∗ + from among the { ˜ y a α } . The section condition(2.6) is still satisfied for the section defined by (5.1). As a final step, one imposes a particularadditional ˜ y -dependence on the ExFT fields, according to a generalized Scherk-Schwarz ansatz ofthe form [21] M MN = U MK ( ˜ y ) U NL ( ˜ y ) M KL ( x , y i ) , g µν = ρ − ( ˜ y ) g µν ( x , y i ) , A µ M = ρ − ( ˜ y )( U − ) N M ( ˜ y ) A µ N ( x , y i ) , B µν M = ρ − ( ˜ y ) U MN ( ˜ y ) B µν N ( x , y i ) , (5.2) See, however, [43]. xceptional field theory Here, the matrix U MN lives in the ( SL ( ) × SL ( )) subgroup of E ( ) and is defined as a product ofthe two diagonal matrices ( U SL ( ) ) αβ = U ++ U −− ! = ρ ( ˜ y ) ρ − ( ˜ y ) ! , ( U SL ( ) ) ˆ a ˆ b = U i j U ∗∗
00 0 U = δ i j ρ ( ˜ y )
00 0 ρ − ( ˜ y ) , (5.3)in the basis associated to the splitting (2.23), (5.1), and with the scale factor ρ given by a linearfunction ρ ( ˜ y ) = ˜ y + c of the extra coordinate.Upon expanding the ExFT equations derived from (2.12) in the IIB parametrization of thefields, the coordinate dependence on ˜ y entirely factors out, while the linear dependence of ρ ( ˜ y ) induces a deformation of the IIB equations parametrized by the Killing vector field K [46]. Thisprecisely reproduces the results from [44, 45]. It is important to note that the choice of coordinates(5.1) is equivalent (after rotation of the 27 coordinates) to selecting five coordinates among the y i in (2.20). Applying the same rotation to the IIA parametrization of ExFT fields such as (2.25),(2.27) would simply recover the IIA theory. This is a manifestation of the fact that the generalizedIIB supergravity equations can be obtained via T-duality from a sector of IIA supergravity. Sincethe framework of exceptional field theory is manifestly duality covariant, the above sketched con-struction via (5.2), (5.3) corresponds to absorbing the effect of this duality into a rotation of thecoordinates on the extended space.
6. Conclusions
We have reviewed the construction of exceptional field theory as the manifestly duality co-variant formulation of supergravity theories. Its dynamics is entirely determined by external andinternal diffeomorphism invariance. Upon solving the section constraints and choosing the properparametrization of the ExFT fields, these equations of motion reproduce the full higher-dimensional D =
11 and IIB supergravity equations. What we have further sketched in these proceedings is thepotential of this formulation to provide a framework for embedding the magic triangles of super-gravity, the IIA ∗ /IIB ∗ supergravities completing the orbit of supergravities under timelike dualities,as well as the generalized type II supergravities emerging from integrable deformations.Looking ahead, we may envision that ExFT provides solutions that transcend supergravity,despite the section constraints being everywhere satisfied. One may have a generalized metric orgeneralized frame that evolves through a phase that is singular from the viewpoint of ordinarygeometry but perfectly regular from the viewpoint of generalized exceptional geometry. This ap-proach towards ‘non-geometry’, which is in the spirit of [35, 36] and for which new examples havebeen worked out in [33, 38], is quite different from the idea of relaxing the section constraints,which at least in the full string/M-theory must be viable too. It remains an exciting endeavor toexplore the ultimate scope of exceptional field theory.16 xceptional field theory Acknowledgements
We would like to thank the organizers of the Corfu Summer Institute 2018. The work ofO.H. is supported by the ERC Consolidator Grant “Symmetries & Cosmology".
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