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D 103, 046002 (2021)

Toward exotic 6D supergravities

† ‡ Yannick Bertrand ,1,* Stefan Hohenegger ,1, Olaf Hohm,2, and Henning Samtleben 3,§ 1Univ Lyon, Univ Claude Bernard Lyon 1, CNRS/IN2P3, IP2I Lyon, UMR 5822, F-69622, Villeurbanne, France 2Institute for Physics, Humboldt University Berlin, Zum Großen Windkanal 6, D-12489 Berlin, Germany 3Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon, France

(Received 14 October 2020; accepted 8 December 2020; published 3 February 2021)

We investigate exotic theories in 6D with maximal (4,0) and (3,1) , which were conjectured by Hull to exist and to describe strong coupling limits of N ¼ 8 theories in 5D. These theories involve exotic gauge fields with nonstandard Young tableaux representations, subject to (self-) duality constraints. We give novel actions in a 5 þ 1 split of coordinates whose field equations reproduce those of the free bosonic (4,0) and (3,1) theory, respectively, including the (self-)duality relations. Evidence is presented for a master exceptional field theory formulation with an extended section constraint that, depending on the solution, produces the (4,0), (3,1) or the conventional (2,2) theory. We comment on the possible construction of a fully nonlinear master exceptional field theory.

DOI: 10.1103/PhysRevD.103.046002

I. INTRODUCTION (5D) theories with maximal supersymmetry (32 real super- charges) were classified by Strathdee [6] and further Among the surprising features of /M-theory is clarified by Hull in [7]. The 5D superalgebra reads the possible existence of exotic superconformal field and theories in six dimensions, which display general- a b ab μ ab ab Qα;Q Ω Γ C Pμ Cαβ Z Ω K ; 1:1 izations of electric-magnetic duality. Specifically, the f βg¼ ð Þαβ þ ð þ Þ ð Þ supermultiplets of these theories are such that the corre- where α; β; … ¼ 1; …; 4 are the space-time spinor indices sponding fields must be subject to (self-)duality constraints, … 1 … 8 8 some of which involve exotic Young tableaux representa- and a; b; ¼ ; ; are USpð Þ R-symmetry indices. ab tions. In this paper our focus will be on a conjecture by Hull This superalgebra features 27 central charges Z , satisfy- ab − ba Ωab 0 [1,2] according to which there are strong coupling limits ing Z ¼ Z , Zab ¼ , and a singlet central charge ’ of N ¼ 8 theories in five dimensions that are given by K. The short (or BPS after Bogomol nyi-Prasad- six-dimensional theories with chiral N ¼ð4; 0Þ and N ¼ Sommerfield) multiplets of this superalgebra describe the ð3; 1Þ supersymmetry, respectively. Such theories must be possible Kaluza-Klein towers that any six-dimensional exotic or nongeometric since they feature mixed symmetry (6D) theory with maximal supersymmetry displays when tensors of Young tableaux type and , respectively, compactified on a circle. For the conventional maximal 6D supergravity, which features N ¼ð2; 2Þ supersymmetry, instead of a conventional , hence suggesting the the massive Kaluza-Klein states do not carry the singlet need for a generalized notion of spacetime and diffeo- central charge K. Instead, they carry a particular central morphism invariance. They are set to play a distinguished ab – charge Z transforming as a singlet under the six-dimen- role among the maximally supersymmetric theories [3 5]. 4 4 A possible window into these somewhat mysterious sional R-symmetry group USpð Þ ×USpð Þ. In contrast, structures is offered by a Kaluza-Klein perspective from the massive multiplets of the exotic theories carry non- five dimensions. The supermultiplets of five-dimensional vanishing singlet charge K [together with nonvanishing Zab, singlet under USpð4Þ ×USpð4Þ in the case of the N ¼ ð3; 1Þ multiplets] [7]. This points to a unifying framework *[email protected] † in the spirit of exceptional field theories [8–11] which we [email protected] ‡ [email protected] will elaborate on in this paper. §[email protected] Exceptional field theory (ExFT) provides in particular a formulation of 11-dimensional (11D) and type IIB super- Published by the American Physical Society under the terms of gravity in a form that is covariant under the global the Creative Commons Attribution 4.0 International license. symmetry group 6 of 5D maximal supergravity, thanks Further distribution of this work must maintain attribution to ð Þ the author(s) and the published article’s title, journal citation, to extended coordinates in the 27 representation of this and DOI. Funded by SCOAP3. group, which are added to the five coordinates of 5D

2470-0010=2021=103(4)=046002(22) 046002-1 Published by the American Physical Society BERTRAND, HOHENEGGER, HOHM, and SAMTLEBEN PHYS. REV. D 103, 046002 (2021) supergravity. The resulting theory is thus based on a section constraint that makes sense for the nonlinear theory (5 þ 27)-dimensional spacetime split, in which the 27 and that reduces to (1.2) in the appropriate limit. “ coordinates are subject to an E6ð6Þ covariant section As a technical result, we present novel actions for the constraint” restricting them to a suitable physical subspace, bosonic sectors of the N ¼ð3; 1Þ and the N ¼ð4; 0Þ from which the complete (untruncated) 11D supergravity model that are based on a 5 þ 1 split of the six-dimensional can be reconstructed, albeit in a Kaluza-Klein type formu- space-time, sacrificing manifest 6D Poincar´e invariance. In lation with a 5 þ 6 split of coordinates. (Equivalently, one the spirit of ExFT, these are two-derivative actions which may think of this as coupling the infinite towers of massive upon dimensional reduction to five dimensions reduce to Kaluza-Klein multiplets to 5D supergravity, which recon- the same action of linearized maximal 5D supergravity. All structs the complete 11D supergravity.) A different physical dual fields, in particular the entire dual graviton sector, only section reproduces the IIB theory [10,12]. From a higher- appear under derivative along the sixth dimension. The full dimensional perspective, the extra coordinates can be field equations obtained by variation combine the second- thought of as accounting for the possible windings, order Fierz-Pauli equations with first-order duality equa- which in turn are related to the 27 central charges of the tions defining the dual graviton sector. Actions for self-dual supersymmetry algebra (1.1). The structure of the exotic fields based on a 5 þ 1 split of spacetime date back to [13] supermultiplets then suggests an inclusion of their cou- with the description of self-dual 6D tensor fields. More plings within a suitable extension of this framework. recently, actions for the N ¼ð3; 1Þ and N ¼ð4; 0Þ models In this paper we will present actions for (the bosonic have been constructed in [14–16], based on the prepotential sectors of) the free exotic 6D theories that generalize the formalism developed in [17] in the context of linearized gravity. Introduction of prepotentials for the gauge fields (linearized) E6ð6Þ exceptional field theory [10,11] by adding one more “exotic” coordinate to the 27, as suggested by the adapted to their self-duality properties allows for the singlet central charge in (1.1). As one of the most enticing construction of an action of fourth order in spatial deriv- outcomes of our investigation we find evidence for a master atives. Our construction is closer in spirit to the original exceptional field theory formulation in which the conven- construction of [13], albeit dual in a sense discussed in tional N ¼ð2; 2Þ theory as well as the N ¼ð4; 0Þ and more detail in the Appendix A. It provides a novel N ¼ð3; 1Þ theory are obtained through different solutions mechanism for describing self-dual exotic tensor fields. of an extended section constraint of the form The rest of this paper is organized as follows. In Sec. II we review the bosonic sector of the N ¼ð2; 2Þ, N ¼ð3; 1Þ and 1 N ¼ð4; 0Þ theories at the level of the equations motion, MNK MN d ∂ ⊗ ∂ − pffiffiffiffiffi Δ ð∂ ⊗ ∂• þ ∂• ⊗ ∂ Þ¼0; N K 10 N N which are manifestly 6D Lorentz invariant. In order to find actions for these theories we abandon manifest Lorentz ð1:2Þ invariance by performing a 5 þ 1 split of coordinates in Sec. III for each of these models. In Sec. IV we then present, 1 … 27 where M; N ¼ ; ; , are fundamental E6ð6Þ indices, as one of our main technical results, actions whose second- MNK d denotes the E6ð6Þ invariant fully symmetric tensor, order Euler-Lagrange equations can be integrated in order to and ∂• is the derivative dual to the exotic coordinate. reproduce the correct dynamics of the three theories. In Moreover, ΔMN denotes the (constant) background part of Sec. V we present the master formulation as currently the generalized metric MMN encoding all scalar fields. The understood, highlight its successes, which strike us as first term in Eq. (1.2) defines the section constraint of E6ð6Þ significant, but also discuss the structural problems that exceptional field theory, whose solutions restrict to the remain. We close with a brief outlook. standard D ¼ 11 and IIB sections. The second term encodes the extension of the constraint allowing for II. REVIEW OF 6D MODELS N 4 0 two more solutions corresponding to ¼ð ; Þ and We study six-dimensional field theories in Minkowski N ¼ð3; 1Þ, respectively. More precisely, we recover the space with flat metric N ¼ð4; 0Þ exotic theory by dropping all dependence on the 27 standard coordinates, and keeping only the depend- ημˆ νˆ ¼ diagf−1; 1; 1; 1; 1; 1g; μˆ; νˆ ¼ 0; …; 5: ð2:1Þ ence on the exotic coordinate. The N ¼ð3; 1Þ model in turn is recovered by superposing this coordinate with the The Poincar´e algebra in six dimensions admits chiral 27 → 26 1 N N N F4ð4Þ singlet under þ among the 27 coordinates ð þ; −Þ supersymmetric extensions where count of ExFT. While we give several independent pieces of the cumber of right- and left-handed supercharges, res- evidence for the existence of this master formulation (some pectively [6]. In particular, maximal supersymmetry of which entail highly nontrivial numerical agreement), we (32 real supercharges) allows for the three possibilities N N 2 2 N N 3 1 N N also point out some gaps of the master formulation as ð þ; −Þ¼ð ; Þ, ð þ; −Þ¼ð ; Þ, and ð þ; −Þ¼ understood so far. This implies that the complete nonlinear ð4; 0Þ. The corresponding lowest-dimensional massless theory requires new ingredients, not the least of which is a supermultiplets have the field content [6]

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ð2; 2Þ∶ ð3; 3; 1; 1Þþð2; 2; 4; 4Þþð1; 3; 5; 1Þþð3; 1; 1; 5Þ SOð5Þ ×SOð5Þ, as expected for the free theory. Finally, the p two-forms Bμˆ νˆ couple with a standard kinetic term þð1; 1; 5; 5Þþð2; 3; 4; 1Þþð3; 2; 1; 4Þ 2 1; 4 5 1 2; 5 4 1 þð ; ; Þþð ; ; Þ; ð2:2aÞ L − q μˆ νˆ ρˆ q 1 … 5 B ¼ 6 Hμˆ νˆ ρˆ H ;q¼ ; ; ; ð2:8Þ ð3;1Þ∶ ð4;2;1;1Þþð2;2;14;1Þþð3;1;6;2Þþð1;1;140;2Þ q q for Hμˆ νˆ ρˆ ¼ 3∂ μˆ Bνˆ ρˆ . For the following it will be con- þð4;1;1;2Þþð3;2;6;1Þþð2;1;14;2Þ ½ venient to combine these fields together with their magnetic 0 a þð1;2;14 ;1Þ; ð2:2bÞ duals into a set of 10 two-forms Bμˆ νˆ , satisfying first-order (anti-)self-duality field equations ð4; 0Þ∶ ð5; 1; 1Þþð3; 1; 27Þþð1; 1; 42Þþð4; 1; 8Þ 1 þð2; 1; 48Þ; ð2:2cÞ δ b ε η σˆ κˆ λˆ b 1 … 10 abHμˆ νˆ ρˆ ¼ 6 μˆ νˆ ρˆ σˆ κˆ λˆ abH ;a¼ ; ; ; ð2:9Þ organized into representations of the little group with the SOð5; 5Þ invariant constant tensor η . 2 2 2N 2N ab G0 ¼ SUð Þ ×SUð Þ ×USpð þÞ ×USpð −Þ: ð2:3Þ Equations (2.9) amount to a description of these degrees of freedom in terms of 5 self-dual and 5 anti-self-dual In this section, we briefly review the six-dimensional free two-forms. theories associated to these multiplets.

A. The N = ð2;2Þ model B. The N = ð3;1Þ model Let us start from the N ¼ð2; 2Þ multiplet corresponding Let us now turn to the free field equations associated to maximal supergravity in six dimensions. Its bosonic field with the N ¼ð3; 1Þ multiplet (2.2b). This multiplet does content comprises a metric, 25 scalar fields, 16 vectors, and not carry a standard graviton field, but an exotic three-index 5 two-forms. The full nonlinear theory has been con- tensor field of mixed-symmetry type [19] structed in [18] with the scalar fields parametrizing an SOð5; 5Þ=ðSOð5Þ ×SOð5ÞÞ coset space. For the purpose of ð2:10Þ this paper, we will only consider the linearized (free) theory with no couplings among the different types of . Its field equation is given by a self-duality equation [1] The linearized -2 sector carries the symmetric Pauli-Fierz field hμˆ νˆ . With the linearized Riemann tensor 1 ε ηˆ κˆ λˆ given by Sμˆ νˆ ρˆ;σˆ τˆ ¼ 6 μˆ νˆ ρˆ ηˆ κˆ λˆS σˆ τˆ; ð2:11Þ −∂ ∂ ∂ ∂ Rμˆ νˆ;ρˆ σˆ ¼ μˆ ½ρˆ hσˆνˆ þ νˆ ½ρˆ hσˆμˆ ; ð2:4Þ in terms of its second-order curvature linearization of the Einstein-Hilbert Lagrangian gives rise 3∂ ∂ − 3∂ ∂ to the massless Fierz-Pauli Lagrangian Sμˆ νˆ ρˆ;σˆ τˆ ¼ σˆ ½μˆ Cνˆ ρˆ;τˆ τˆ ½μˆ Cνˆ ρˆ;σˆ : ð2:12Þ 1 1 1 μˆ νˆ ρˆ ρˆ σˆ μˆ ρˆ σˆ μˆ Counting reveals that the field equation (2.11) captures the L ¼ − ∂μˆ h ∂νˆ hρˆ þ ∂μˆ h ∂ρˆ hσˆ − ∂μˆ h ∂ hρˆ σˆ h 2 2 4 8 degrees of freedom as counted in the multiplet (2.2b). 1 νˆ μˆ ρˆ Moreover, curvature and field equation are invariant under þ ∂μˆ hνˆ ∂ hρˆ 4 the gauge symmetries

1 μˆ νˆ ρˆ 1 μˆ νˆ ρˆ μˆ ¼ − Ω Ωμˆ νˆ ρˆ þ Ω Ωνˆ ρˆ μˆ þ Ω Ωμˆ ; ð2:5Þ δ 2∂ α ∂ β − ∂ β 4 2 Cμˆ νˆ;ρˆ ¼ ½μˆ νˆρˆ þ ρˆ μˆ νˆ ½ρˆ μˆ νˆ; ð2:13Þ Ω ≡∂ i with μˆ νˆ ρˆ ½μˆ hνˆρˆ . The vector fields Aμˆ couple with a α α β β with parameters μˆ νˆ ¼ ðμˆ νˆÞ and μˆ νˆ ¼ ½μˆ νˆ. An action standard Maxwell term principle for the field equations (2.11) has been constructed 1 in [16] based on the prepotential formalism introduced in L − i μˆ νˆ i 1 … 16 A ¼ 4 Fμˆ νˆ F ;i¼ ; ; ; ð2:6Þ [17] in the context of linearized gravity. In addition to the exotic tensor field, the bosonic field i 2∂ i for Fμˆ νˆ ¼ ½μˆ Aνˆ , while scalar couplings take the form content of the N ¼ð3; 1Þ multiplet (2.2b) contains 14 1 vectors, 12 self-dual two-forms and 28 scalar fields. The L − ∂ ϕα∂μˆ ϕα α 1 … 25 dynamics of vector and scalar fields can be captured by ϕ ¼ 2 μˆ ; ¼ ; ; : ð2:7Þ standard Lagrangians (2.6) and (2.7) (with different range a The couplings (2.6) and (2.7) break the global SOð5; 5Þ of internal indices). The self-dual two-forms Bμˆ νˆ obey a symmetry of the nonlinear theory down to its compact part self-duality equation similar to (2.9)

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1 N 4 0 a σˆ κˆ λˆ a The free ¼ð ; Þ theory is invariant under the Hμˆ νˆ ρˆ ¼ ε ˆH ;a¼ 1; …; 12; ð2:14Þ 6 μˆ νˆ ρˆ σˆ κˆ λ R-symmetry group USpð8Þ. The (yet elusive) interacting theory is conjectured to exhibit a global E6 6 symmetry η ð Þ contrary to (2.9), no indefinite tensor ab appears in this with in particular the 42 scalars parametrizing the coset equation, and all forms are self-dual.1 As a consequence space E6 6 =USpð8Þ [1]. there is no standard action principle for these field ð Þ equations; they can however be derived from an action III. 5 + 1 SPLIT with nonmanifest Lorentz invariance [13] or upon coupling to an auxiliary scalar field [20].2 Upon dimensional reduction to D ¼ 5 dimensions, the The free N ¼ð3; 1Þ theory is invariant under the three models discussed in the previous section all reduce to R-symmetry group USpð6Þ ×USpð2Þ. The (yet elusive) the same theory: the free limit of maximal D ¼ 5 super- interacting theory is conjectured to exhibit a global F4ð4Þ gravity [1,2]. The bosonic sector of this theory carries a symmetry with in particular the 28 scalars parametrizing spin-2 field and 27 vector fields together with 42 scalar 6 2 fields. In particular, the exotic tensor fields of the N ¼ the coset space F4ð4Þ=ðUSpð Þ ×USpð ÞÞ [1]. ð3; 1Þ and the N ¼ð4; 0Þ model after dimensional reduc- tion carry the D ¼ 5 dual graviton and double dual N C. The = ð4;0Þ model graviton, respectively. Within the free theory, these fields The N ¼ð4; 0Þ multiplet carries an exotic four-index can be dualized into the standard Pauli-Fierz field tensor field with the symmetries of the Riemann tensor [1,23,24], and do not represent independent degrees of freedom. In order to make the equivalence explicit, the fields of D ¼ 5 supergravity (together with their on-shell duals) have to be properly identified among the various ð2:15Þ components of the D ¼ 6 fields. In this section, we discuss for every of the three models Its field equation is given by a self-duality equation [1] the reorganization of the D ¼ 6 fields which allows their identification after reduction to five dimensions. However, 1 αˆ βˆ γˆ G ˆ ε ˆ ˆ G ρˆ σˆ τˆ; : throughout this section (and this paper) we keep the full μˆ νˆ λ;ρˆ σˆ τˆ ¼ 6 μˆ νˆ λ αˆ β γˆ ð2 16Þ dependence of all fields on six space-time coordinates. More precisely, we break six-dimensional Poincar´e invari- in terms of its second-order curvature ance down to 5 þ 1 and perform a standard Kaluza-Klein decomposition on the six-dimensional fields without drop- G ˆ ¼ 3∂ρˆ ∂ μˆ T ˆ þ 3∂σˆ ∂ μˆ T ˆ þ 3∂τˆ∂ μˆ T ˆ : μˆ νˆ λ;ρˆ σˆ τˆ ½ νˆ λ;σˆ τˆ ½ νˆ λ;τˆ ρˆ ½ νˆ λ;ρˆ σˆ ping the dependence on the sixth coordinate. We then ð2:17Þ rearrange the equations such that they take the form of the five-dimensional (free) supergravity equations how- Counting confirms that this field equation describes the ever sourced by derivatives of matter fields along the 5 degrees of freedom as counted in the multiplet (2.2c). sixth direction. The resulting reformulation of the six- Moreover, curvature and field equation are invariant under dimensional models casts their dynamics into a common the gauge symmetries framework—which ultimately allows us to construct uni- form actions for the three models. δ ∂ λ ∂ λ 5 1 Tμˆ νˆ;ρˆ σˆ ¼ ½μˆ νˆ;ρˆ σˆ þ ½ρˆ σˆ;μˆ νˆ ; ð2:18Þ For the purpose of this paper, we choose the þ coordinate split with the (2,1) gauge parameter λμˆ ρˆ σˆ ¼ λμˆ ρˆ σˆ , λ μˆ ρˆ σˆ ¼ 0. ; ;½ ½ ; μˆ μ An action principle for (2.16) has been constructed in fx g → fx ;yg; μ ¼ 0; …; 4; ð3:1Þ [14,15] based on the prepotential formalism of [17]. The bosonic part of the N ¼ð4; 0Þ multiplet (2.2c) by singling out one of the spatial coordinates. Of course, an analogous construction can be performed with a split along combines the exotic tensor field Tμˆ νˆ;ρˆ σˆ with 42 scalars and 27 self-dual two-forms. Their dynamics is described by the timelike coordinate which may be of interest for a free Lagrangian (2.7) and self-duality equations (2.14), example in a Hamiltonian context. respectively. A. The N = ð2;2Þ model

1For uniformity, we use the same indices a, b, to label two- With the coordinate split (3.1), we parametrize the forms in all three models, despite the fact that the range of these graviton of the N ¼ð2; 2Þ theory as indices differs among the different models according to the  1  number of two-form fields. This should not be a source of hμν − 3 ημνϕ Aμ confusion. hμˆ νˆ ¼ ; ð3:2Þ 2For more recent constructions, see also [21,22]. Aν ϕ

046002-4 TOWARD EXOTIC 6D SUPERGRAVITIES PHYS. REV. D 103, 046002 (2021) which is the linearized form of the standard Kaluza-Klein reduction ansatz. Recall that all fields still depend on six coordinates. Working out the Lagrangian (2.5) in this parametrization gives rise to its expression

ð3:3Þ

up to total derivatives. As an illustration of the above discussion let us note the explicit form of the equations for the five- dimensional spin-2 field 1 1 2 G − ∂ ∂ η ∂ ∂ ρ − η ∂ ∂ ϕ μν ¼ 2 y yhμν þ 2 μν y yhρ 3 μν y y ; ð3:4Þ in terms of the linearized Einstein tensor 1 1 1 1 G −∂ρ ∂ ρ ∂ ρ η ∂ρ σ − η ∂ ρ σ μν ¼ DðμhνÞρ þ 2 ρD hμν þ 2 ðμDνÞhρ þ 2 μν D hρσ 2 μν ρD hσ ; 2 ≡∂ − ∂ η with covariant derivatives Dμhνρ μhνρ 3 yAμ νρ: ð3:5Þ

The form of (3.4) shows that upon dimensional reduction higher-dimensional supergravities as exceptional field the- to D ¼ 5 dimensions, these equations reproduce the ories. Indeed, Eq. (3.4) can be equivalently obtained upon (linearized) five-dimensional Einstein field equations. In linearizing the corresponding E6ð6Þ ExFT [10,11] upon contrast, the coordinate dependence along the sixth coor- proper identification of the coordinate y among the 27 dinate induces a nontrivial gauge structure via covariant internal coordinates on which this ExFT is based. derivatives (3.5) and nonvanishing source terms in (3.4). Let us also note that the Lagrangian (3.3) can be put to This is very much in the spirit of the reformulation of the more compact form

ð3:6Þ

2 with the linearized (and covariantized) anholonomity δhμν ¼ 2∂ μξν þ ημν∂ λ; objects ð Þ 3 y

δAμ ¼ ∂μλ þ ∂yξμ;

2 δϕ ¼ 2∂yλ; ð3:8Þ Ω ≡∂ − ∂ η Ω ≡ Ω ν μνρ ½μhνρ 3 yA½μ νρ; μ μν : ð3:7Þ upon decomposition of the six-dimensional gauge param- eter as fξμˆ g¼fξμ; λg. The remaining part of the six-dimensional degrees In a similar way, the six-dimensional Maxwell and of freedom described by (3.6) are captured by a (modified) Klein-Gordon Lagrangians (2.6) and (2.7) take the form five-dimensional Maxwell and Klein-Gordon equation 1 1 for Aμ and ϕ, respectively, obtained by varying (3.6). μνi i μ i μi i i LA ¼ − F Fμν − ð∂ ϕ − ∂yA Þð∂μϕ − ∂yAμ Þ; It is useful to note the symmetries of the Lagrangian 4 2 1 1 (3.6) descending from six-dimensional spin-2 gauge μ α α α α Lϕ ¼ − ∂ ϕ ∂μϕ − ∂ ϕ ∂ ϕ ; ð3:9Þ transformations 2 2 y y

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i i i 1 respectively, after splitting fAμˆ g¼fAμ ; ϕ g, and with b κλb κλb η Hμνρ ¼ εμνρκλδ ðF þ ∂ B Þ; ð3:10Þ i 2∂ i ab 2 ab y Abelian Fμν ¼ ½μAν , giving rise to modified Maxwell and Klein-Gordon equations for their components. The rewriting of the tensor field sector is slightly less straightfor- where we use conventions εμνρκλ5 ¼ εμνρκλ,andAbelian a 3∂ a a 2∂ a ward: rather than evaluating the Lagrangian (2.8),wechoose field strengths Hμνρ ¼ ½μBνρ ,andFμν ¼ ½μAν , to evaluate the first-order field equations (2.9) after splitting respectively. These equations can be integrated to a a a a a the 6D tensor fields into fBμˆ νˆ g¼fBμν ;Bμ5 ≡ Aμ g Lagrangian

ð3:11Þ

Again, this Lagrangian can be deduced from the linearized standard action principle. We thus perform the Kaluza- version of exceptional field theory. We discuss this mecha- Klein reorganization of the model on the level of the field nism in more detail in Appendix A2. As we will see in equations. To this end, we again split coordinates as (3.1) the following, this form of the Lagrangian allows for the and parametrize the mixed-symmetry tensor as most uniform treatment of the different six-dimensional 5 −2 η ; ; 2 models. After dimensional reduction to D ¼ dimensions, fCμˆ νˆ;ρˆ g¼fCμν;ρ A½μ νρ Cμ5;ν ¼ hμν þBμν Cμ5;5 ¼ Aμg; it simply reduces to a collection of Maxwell terms, such : that all degrees of freedom of (2.8) are described as ð3 14Þ massless vector fields in five dimensions. In the presence hμν hνμ Bμν −Bνμ of the sixth dimension, the Lagrangian (3.11) gives rise to with symmetric ¼ , antisymmetric ¼ , and modified Maxwell equations while variation with respect a (2,1) tensor Cμν;ρ. After dimensional reduction to five a dimensions, the fields hμν and Aμ satisfy the linearized to the tensor fields Bμν induces Eqs. (3.10) (under ∂y derivative) as duality equations relating vector and tensor Einstein and Maxwell equations while the fields Cμν;ρ and fields. Bμν describe their on-shell duals, together accounting for In summary, the D ¼ 6 N ¼ð2; 2Þ, model can be the 8 degrees of freedom of the six-dimensional tensor equivalently reformulated in terms of a Lagrangian given field. Explicitly, in the parametrization (3.14), the six- by the sum of (3.6), (3.9), and (3.11). Upon dimensional dimensional self-duality equations (2.11) split into two reduction to five dimensions, i.e., setting ∂y → 0, and equations rescaling of the scalar fields, this Lagrangian reduces to   1 λστ ∂ρ Fμν þ εμνλστH 1 ∘ μνρ ∘ 1 ∘ μνρ ∘ ∘ μ ∘ 6 L5 ¼ − Ω Ωμνρ þ Ω Ωνρμ þ Ω Ωμ D 4 2 1 1 ∂ ∂ ε ∂ ∂κ λτ ∂ ∂ − ∂ ∂ η 1 1 ¼ y ½μhνρ þ μνκλτ y C ρ þ y yCμν;ρ y yA½μ νρ − ∂μϕA∂ ϕA − μνM M 4 2 2 μ 4 F Fμν ; 1 στ − ερμνστ∂ F ∂ ∂ μBν ρ − ∂ ∂ρBμν; : M ¼ 1; …; 27;A¼ 1; …; 42; ð3:12Þ 4 y þ y ½ y ð3 15Þ

∘ and with Ωμνρ ≡∂μhν ρ, and where we have combined the ½   various vector and scalar fields into joint objects 1 1 κλ Rμν;ρσ ¼ ∂ρ Hμνσ − εμνσκλF M A 2 2 fAμ g;M ¼ 1; …; 27; fϕ g;A¼ 1; …; 42: ð3:13Þ   1 1 κλ 1 κ λτ − ∂σ Hμνρ − εμνρκλF þ εμνκλτ∂ ∂ ρC σ The Lagrangian (3.12) is the free limit of D ¼ 5 maximal 2 2 2 ½ supergravity [25]. In the interacting theory, the fields (3.13) 1 1 ∂ ∂ − ∂ ∂ − ∂ ∂ η transform in the fundamental and a nonlinear representation þ 2 y ρCμν;σ 2 y σCμν;ρ y ρA½μ νσ of its global symmetry group E6 6 . ð Þ ∂ ∂ η þ y σA½μ νρ; ð3:16Þ N 2∂ 3∂ B. The = ð3;1Þ model with Abelian field strengths Fμν ¼ ½μAν, Hμνρ ¼ ½μBνρ, We now turn to the N ¼ð3; 1Þ model. Its most char- and the linearized Riemann tensor Rμν;ρσ defined as in (2.4) acteristic element is the mixed-symmetry tensor field Cμˆ νˆ;ρˆ however for the field hμν. Contraction of (3.16) gives rise to whose field equation (2.11) cannot be derived from a an equation

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1 ρ 1 ρ 1 ρ σ tensor field into the form of two first-order duality Gμν ¼ − ∂ ∂ Cρ μ ν − ∂ ∂ μCν ρ þ ημν∂ ∂ Cρσ ; 2 y ð ; Þ 2 y ð Þ 2 y equations (3.21) and (3.22), upon parametrizing the six- ð3:17Þ dimensional fields in terms of its components (3.14) and introduction of an additional field uμν. Upon reduction to where Gμν denotes the linearized Einstein tensor defined as five dimensions, these equations constitute the duality in (3.5), however with covariant derivatives now given by equations relating the vector-tensor fields, and the grav- iton-dual graviton fields, respectively. It is instructive to work out the gauge symmetries of Dμhνρ ≡∂μhνρ − ∂yAμηνρ; ð3:18Þ these equations which originate from the D ¼ 6 gauge i.e., with a different value of the coupling constant transformations (2.13). Parametrizing the six-dimensional (which could be absorbed into rescaling the vector field). gauge parameters as Equation (3.17) confirms that upon reduction to five   dimensions (∂y → 0), the field hμν satisfies the linearized 1 αμν − ημνλ 2 ðξμ þ 3ΛμÞ Einstein equations. As in the N ¼ð2; 2Þ model, the αμˆ νˆ ¼ ; 1 ξ 3Λ 2λ coordinate dependence along the sixth coordinate induces 2 ð μ þ μÞ  3  a nontrivial gauge structure (3.18) together with non- βμν 2 ðξμ − ΛμÞ — βμˆ νˆ ¼ ; ð3:23Þ vanishing source terms in (3.17) which differ from those 3 Λ − ξ 0 of (3.4) illustrating the inequivalence of the N ¼ð2; 2Þ and 2 ð μ μÞ the N ¼ð3; 1Þ model before dimensional reduction. The full field equation (3.16) takes the form of a their action on the various components of (3.14) is vanishing curl (in ½ρσ) and can locally be integrated into derived as the first-order equation3   1 1 1 1 δ ∂ λ ∂ ξ − 3Λ ∂ ε ∂κ λτ − ε κλ Aμ ¼ μ þ yð μ μÞ; ½μhνρ þ 4 μνκλτ C ρ þ 2 Hμνρ 2 μνρκλF 2 1 1 δBμν 2∂ μΛν ∂ βμν; ∂ − ∂ η ∂ ¼ ½ þ 3 y þ 2 yCμν;ρ yA½μ νρ ¼ ρuμν; ð3:19Þ δ 2∂ ξ η ∂ λ − ∂ α hμν ¼ ðμ νÞ þ μν y y μν; with an antisymmetric tensor uμν ¼ −uνμ. Combining this δ 2∂ α ∂ β − ∂ β ∂ ξ η − 3Λ η Cμν;ρ ¼ ½μ νρ þ ρ μν ½ρ μν þ yð ½μ νρ ½μ νρÞ: equation with the field equation (3.15) implies that   ð3:24Þ 1 3 ∂ ε κλτ ∂ − ∂ 0 ρ Fμν þ 6 μνκλτH þ 2 yBμν yuμν ¼ ; ð3:20Þ With the field uμν defined by Eq. (3.19), its gauge variation is found by integrating up the variation of (3.19) and takes which can be further integrated into another first-order the form duality equation 1 1 1 κλτ 3 ρ στ ε ∂ − ∂ 0 δuμν ∂ μξν εμνρστ∂ β ∂ βμν: : Fμν þ 6 μνκλτH þ 2 yBμν yuμν ¼ ; ð3:21Þ ¼ ½ þ 6 þ 2 y ð3 25Þ up to a function fμνðyÞ that can be absorbed into uμν. For later use, let us note that contraction of (3.22) with Eventually, we can use (3.21) to bring (3.19) into the form the fully antisymmetric ε-tensor yields 1 ∂ ε ∂κ λτ −∂   ½μhνρ þ μνκλτ C ρ ρuμν 4 1 ρ 1 ρ 1 ρ στ 1 3   ∂ Cμν ρ ∂ μCν ρ εμνρστ∂ u ∂ uμν − Bμν ; 1 3 1 6 ; þ 3 ½ þ 6 ¼ 2 y 2 ε ∂ κλ − κλ − ∂ ∂ η ¼ μνρκλ y u B yCμν;ρ þ yA½μ νρ: ð3:22Þ 4 2 2 ð3:26Þ To sum up, we have cast the original second-order field equations (2.11) of the six-dimensional mixed-symmetry while contraction gives rise to

3 μ μ μ μ Here, and in the following we work locally and ignore ∂μhν − ∂νhμ þ ∂yCμν þ 4∂yAν ¼ 2∂ uμν: ð3:27Þ potential subtleties that may arise from a nontrivial topology. We refer to [26] for a discussion of such issues in the context of chiral p-forms. This gives rise to an equivalent rewriting of (3.22) as

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∂ ∂ ση η ∂ σ ½μhνρ þ σh½μ νρ þ ρ½μ νhσ ε ∂α∂ βγ μναβγ ½ρTστ 1 κ λτ λταβσ − ε ∂ ε η − 3∂ η ¼ −2∂μ∂ ρCστ ν þ 2∂ν∂ ρCστ μ − 2∂ ∂ ρTστ μν; ð3:33Þ ¼ 4 μνκλτ ðC ρ þ uαβ ρσÞ yA½μ νρ ½ ; ½ ; y ½ ;   1 3 1 ε ∂ κλ − κλ − ∂ − ∂ ση with the linearized Riemann tensor Rμν;ρσ defined as in þ μνρκλ y u B yCμν;ρ yCσ½μ νρ: 4 2 2 (2.4) for the field hμν. The second equation (3.33) has the ð3:28Þ form of a curl in ½ρστ and can be integrated up into 1 Let us further note that taking the divergence of (3.21) ε ∂α βγ ∂ − ∂ ∂ 2∂ μναβγ Tστ þ μCστ;ν νCστ;μ þ yTστ;μν ¼ ½σvτ;μν; yields the Maxwell type equation 2 ð3:34Þ 1 1 1 ∂μ ∂ ∂ μ − ∂ ∂ μ ∂ ∂ μ Fμν ¼ 2 y μhν 2 y νhμ þ 2 y yCμν up to a tensor vτ;μν ¼ −vτ;νμ, determined by this equation 3 μ up to the gauge freedom δvτ;μν ¼ ∂τζμν. Combining (3.34) − ∂ ∂ Bμν þ 2∂ ∂ Aν; ð3:29Þ 2 y y y with the first field equation (3.32), we find where we have used (3.27) in order to eliminate the 1 1 1 ∂ ∂ − ∂ ∂ ε ∂ ∂κ λτ divergence of uμν. Rμν;ρσ ¼ 2 y ρCμν;σ 2 y σCμν;ρ þ 2 μνκλτ ½ρ C σ For the remaining fields of the N ¼ð3; 1Þ model, the þ ∂ ∂ ρvσ μν; ð3:35Þ 5 þ 1 Kaluza-Klein split is achieved just as for the N ¼ y ½ ; ð2; 2Þ model discussed above. The six-dimensional field which in turn is a curl in ½ρσ and can be integrated up into equations of the 14 vector fields and 28 scalar fields take the form obtained from variation of Lagrangians of the 1 1 1 ∂ ε ∂λ στ ∂ ∂ ∂ form (3.9), respectively. The field equations of the 12 self- ½μhνρ þ 4 μνλστ C ρ þ 2 yCμν;ρ þ 2 yvρ;μν ¼ ρuμν; dual forms take the form ð3:36Þ 1 a κλa κλa Hμνρ ¼ εμνρκλðF þ ∂ B Þ; ð3:30Þ 2 y up to an antisymmetric field uμν ¼ −uνμ. As for the N ¼ ð3; 1Þ model, we have obtained an equivalent reformulation a after splitting the two-forms according to fBμˆ νˆ g¼ of the dynamics in terms of two first-order equations (3.34) a a a fBμν ;Bμ5 ≡ Aμ g. The equations may be integrated up and (3.36) from which the original second-order field to an action in precise analogy with (3.11), cf. the dis- equations (3.32), (3.33), can be obtained by derivation. cussion in Appendix A2. After reduction to five dimensions, Eqs. (3.34) and (3.36) describe the duality relations between graviton and dual C. The N = ð4;0Þ model graviton and between dual graviton and double dual graviton, respectively. In particular, Eq. (3.36) differs from In this model, the exotic graviton is given by the rank- Eq. (3.22) in the N ¼ð3; 1Þ model only if fields depend on four tensor (2.15) whose dynamics is defined by the self- the sixth coordinate. duality equations (2.16) for its second-order curvature. It is instructive to work out the gauge symmetries of According to the split of coordinates (3.1), we parametrize these equations which originate from the D ¼ 6 gauge the various components of this field as transformations (2.18). Parametrizing the six-dimensional gauge parameters as fTμˆ νˆ ρˆ σˆ g¼fTμν ρσ; Tμν ρ5 ¼ Cμν ρ; Tμ5 ν5 ¼ hμνg: ð3:31Þ ; ; ; ; ;   2 After dimensional reduction to five dimensions, these fields λ λ ; λ 2α − β ; λ 2ξ f ρˆ;μˆ νˆ g¼ ρ;μν μ;ν5 ¼ μν 3 μν 5;μ5 ¼ μ ; describe the graviton, dual graviton and double dual graviton, respectively. Explicitly, in this parametrization ð3:37Þ the six-dimensional field equations (2.16) split into two equations with symmetric αμν, and antisymmetric βμν, their action on 1 1 1 the various components of (3.31) is derived as Rμν;ρσ ¼ ∂y∂μCρσ;ν − ∂y∂νCρσ;μ þ ∂y∂ρCμν;σ 2 2 2 δ 2∂ ξ − 2∂ α hμν ¼ ðμ νÞ y μν; 1 1 κ λτ − ∂ ∂σCμν ρ þ εμνκλτ∂ ρ∂ C σ 1 2 y ; 2 ½ δ 2∂ α ∂ β − ∂ β − ∂ λ Cμν;ρ ¼ ½μ νρ þ ρ μν ½ρ μν 2 y ρ;μν; 1 κ λτ 1 þ εμνκλτ∂ ∂ T ρσ þ ∂ ∂ Tμν ρσ; ð3:32Þ δ ∂ λ ∂ λ 4 y 2 y y ; Tμν;ρσ ¼ ½μ ν;ρσ þ ½ρ σ;μν: ð3:38Þ

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Gauge variations of the two new fields vρ;μν and uμν are For the remaining fields of the N ¼ð4; 0Þ model, the obtained by integrating up the variation of (3.34) and 5 þ 1 Kaluza-Klein split is achieved just as for the previous (3.36), respectively, giving rise to models discussed above. The field equations of the 42 1 1 1 scalar fields are obtained from variation of a Lagrangian of δ ∂ ξ ε ∂λβστ ∂ β ∂ ζ the form Lϕ in (3.9). The field equations of the 27 self-dual uμν ¼ ½μ ν þ 6 μνλστ þ 3 y μν þ 2 y μν; forms take the form of (3.30) above, again after splitting the 1 2 a a a a δ ε ∂κλ λσ 2∂ α ∂ β two-forms according to fBμˆ νˆ g¼fBμν ;Bμ5 ≡ Aμ g. vρ;μν ¼ 4 μνκλσ ρ þ ½μ νρ þ 3 ½μ νρ 1 ∂ ζ ∂ λ IV. ACTIONS FOR (FREE) EXOTIC þ ρ μν þ 2 y ρ;μν; ð3:39Þ GRAVITON FIELDS where the antisymmetric gauge parameter ζμν −ζνμ has ¼ In the above, we have reformulated the dynamics of been introduced after (3.34). the six-dimensional exotic tensor fields in terms of first- Let us finally note that from (3.32) and (3.35),wemay order differential equations upon breaking six-dimensional obtain the modified Einstein equations Poincar´e invariance according to the split (3.1), and intro- ∘ 1 1 1 ducing some additional tensor fields. As a key property of G − ∂ ∂ρ − ∂ ∂ ρ ∂ ∂ ρ μν ¼ 2 y Cρðμ;νÞ 2 y ðμCνÞρ þ 2 y ρvðμ;νÞ the resulting equations, we have put the dynamics of the 1 1 1 different models into a form which reduces to the same − ∂ ∂ ρ η ∂ ∂ρ σ − η ∂ ∂ σρ ∂ → 0 2 y ðμv νÞρ þ 2 μν y Cρσ 2 μν y ρvσ ; equations after dimensional reduction y .Forexample, all three models feature linearized Einstein equations for ð3:40Þ the field hμν, given by (3.4), (3.17),and(3.40), respectively. ∘ The three equations only differ by terms carrying explicit G with the linearized Einstein tensor μν defined as derivatives along the sixth dimension. We will use this as a ∘ 1 1 guiding principle to construct uniform Lagrangians for the G −∂ρ∂ ∂ ∂ρ ∂ ∂ ρ N 3 1 N 4 0 μν ¼ ðμhνÞρ þ 2 ρ hμν þ 2 μ νhρ ¼ð ; Þ and the ¼ð ; Þ model which after setting ∂ → 0 1 1 y both reduce to the Lagrangian (3.12) of linearized η ∂ρ∂σ − η ∂ ∂ρ σ D 5 þ 2 μν hρσ 2 μν ρ hσ ; ð3:41Þ ¼ maximal supergravity. This construction follows the toy model of D ¼ 6 self- which differs from the previous models by the absence of dual tensor fields whose dynamics can be described by a covariant derivatives, cf. (3.5). Lagrangian (3.11)

ð4:1Þ

after a Kaluza-Klein (5 þ 1) decomposition fBμˆ νˆ g¼ six-dimensional self-duality equation. Details are spelled fBμν;Bμ5 ≡ Aμg of the six-dimensional tensor field. After out in Appendix A2. The Lagrangians for exotic dimensional reduction to five dimensions, the 3 degrees of are constructed in analogy to (4.1) with the role of Aμ and Bμν freedom of the self-dual tensor field are described as a now taken by the graviton hμν and its duals, respectively. massless vector with the standard Maxwell Lagrangian to which (4.1) reduces at ∂ → 0. In the presence of the sixth y A. Action for the N = ð3;1Þ model dimension, variation of the Lagrangian (4.1) with respect to the vector field gives rise to modified Maxwell equations The main result of this subsection is the following: the while variation with respect to the tensor field yields the first-order field equations (3.21) and (3.22), which describe duality equation relating Aμ and Bμν, which is of first order in the dynamics of the six-dimensional exotic graviton field N 3 1 the derivatives ∂μ and appears under a global ∂y derivative. Cμˆ νˆ;ρˆ in the ¼ð ; Þ model, can be derived from the Combining these two equations one may infer the full Lagrangian

ð4:2Þ

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b with Maxwell Lagrangian for Aμ; the dual fields Cμν;ρ and Bμν 1 drop out in this limit. In the presence of the sixth Ωb ≡∂ − ∂ η ∂ b dimension, variation of the Lagrangian (4.2) with respect μνρ ½μhνρ yA½μ νρ þ 2 yCμν;ρ; to the dual fields yields the first-order duality equa- b στ Cμν;ρ ≡ Cμν;ρ þ εμνρστu ; tions (3.21) and (3.22), however under an overall deriva- 3 tive ∂y. Together, one recovers the full six-dimensional F ≡ 2∂ ∂ μν ½μAν þ 2 yBμν: ð4:3Þ dynamics. Details of the equivalence are presented in Appendix B1. The Lagrangian (4.2) is invariant under the gauge The bosonic Lagrangian for the full N ¼ð3; 1Þ model is transformations (3.24), (3.25). After reduction to five then given by combining (4.2) with the Lagrangians of the dimensions, i.e., at ∂y → 0, this Lagrangian reduces to type (3.9) and (4.1) for the remaining matter fields of the the Fierz-Pauli Lagrangian for hμν together with a free theory. Putting everything together, we obtain

ð4:4Þ

b with indices ranging along Cμν;ρ. D ¼ 6 Poincar´e invariance is no longer manifest although it can still be realized on the equations of motion. i ¼ 1; …; 14; α ¼ 1; …; 28;a¼ 1; …; 12: ð4:5Þ B. Action for the N = 4;0 model After dimensional reduction to five dimensions (and ð Þ rescaling of the vector field Aμ), this Lagrangian coin- The main result of this subsection is the following: the cides with the Lagrangian (3.12) of linearized maximal first-order field equations (3.34) and (3.36), which describe supergravity. The Lagrangian (4.4) describes the full the dynamics of the six-dimensional exotic graviton field six-dimensional theory, with the field content of five- Tμˆ νˆ;ρˆ σˆ in the N ¼ð4; 0Þ model, can be derived from the dimensional maximal supergravity enhanced by the field Lagrangian

ð4:6Þ

b with the dual fields Cμν;ρ, Cμν;ρ, and Tμν;ρσ drop out in this limit. In the presence of the sixth dimension, variation of the b b 1 Lagrangian (4.6) with respect to the dual fields yields the Ωμνρ ¼ ∂ μhν ρ þ ∂ Cμν ρ − ∂ Cμν ρ; ½ y ; 2 y ; first-order duality equations (3.34) and (3.36), however b στ under an overall derivative ∂y. Together, one recovers the Cμν;ρ ¼ Cμν;ρ þ εμνρστu ; full six-dimensional dynamics. The computation works in C − 3 2ε στ μν;ρ ¼ Cμν;ρ vρ;μν þ v½ρ;μν þ μνρστu : ð4:7Þ close analogy with the derivation for the N ¼ð3; 1Þ model, and is presented in detail in Appendix B2. After reduction to five dimensions, i.e., at ∂y → 0, this Let us spell out the gauge transformations (3.38), (3.39) Lagrangian reduces to the Fierz-Pauli Lagrangian for hμν; in terms of the fields (4.7)

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2 1 δΩb ∂ ∂ ξ − ∂ ∂ β − ∂ ∂ ζ ∂ ∂κλ στε μνρ ¼ ρ ½μ ν 3 y ½μ νρ y ½μ νρ þ 4 y ½μ νκστρ;   1 3 1 δb 2∂ α − 2∂ β ε ∂σξτ ε ∂ βστ ζστ − ∂ λ Cμν;ρ ¼ ½μ νρ ½μ νρ þ μνρστ þ 3 μνρστ y þ 2 2 y ρ;μν;   8 2 3 1 δC 2ε ∂σξτ − ∂ β 2∂ ζ ε ∂ βστ ζστ ε ∂κλ στ − ∂ λ μν;ρ ¼ μνρστ 3 ½μ νρ þ ½μ νρ þ 3 μνρστ y þ 2 þ 2 κστρ½μ ν y ρ;μν; ð4:8Þ which allows us to confirm gauge invariance of the Lagrangian (4.6). The bosonic Lagrangian for the full N ¼ð4; 0Þ model is finally given by combining (4.6) with the Lagrangians of the type (3.9) and (4.1) for the remaining matter fields of the theory. Putting everything together, we obtain

ð4:9Þ

with indices ranging along coordinates of which the latter are formally embedded into a fundamental representation Rv of the global symmetry 1 … 27 1 … 42 M ¼ ; ; ;A¼ ; ; : ð4:10Þ group E11−D;ð11−DÞ of D-dimensional maximal supergrav- ity. Different embeddings of the internal coordinates into After dimensional reduction to five dimensions, this La- Rv then correspond to different higher-dimensional ori- grangian coincides with the Lagrangian (3.12) of linearized gins. Here, we will discuss a similar uniform description of maximal supergravity. The Lagrangian (4.9) describes the the six-dimensional models based on D ¼ 5 external full six-dimensional theory, with the field content of five- dimensions which encompasses the three different models dimensional maximal supergravity enhanced by the fields upon proper identification of the sixth coordinate within the b Cμν;ρ, Cμν;ρ, and Tμν;ρσ. D ¼ 6 Poincar´e invariance is no internal coordinates. As discussed in the Introduction this longer manifest although it can still be realized on the will require an enhancement of the internal coordinates of equations of motion. exceptional field theory by an additional exotic coordinate related to the singlet central charge in the D ¼ 5 super- V. PROGRESS TOWARD AN EXCEPTIONAL symmetry algebra. MASTER ACTION In the previous sections, we have constructed Lagrangians A. Linearized ExFT and embedding N (3.6), (4.4),and(4.9), for the three six-dimensional of the = ð2;2Þ model models which share a number of universal features and The theory relevant for our discussion is E6ð6Þ excep- structures. In particular, after dimensional reduction to tional field theory [10,11]. Its bosonic field content is given M five dimensions they all reduce to the same Lagrangian by a graviton gμν together with 27 vector fields Aμ and (3.12) corresponding to linearized maximal supergravity their dual tensors BμνM, together with 42 scalars para- in five dimensions. The three distinct six-dimensional metrizing the internal metric M ¼ðVVTÞ with V a theories are then described as different extensions of this MN MN representative of the coset space E6 6 =USpð8Þ. Fields Lagrangian by terms carrying derivatives along the sixth ð Þ depend on 5 external and 27 internal coordinates with dimension. In the various matter sectors, these terms 27 ensure covariantization under nontrivial gauge structures the latter transforming in the fundamental of E6ð6Þ and and provide sources to the field equations of five-dimen- with internal coordinate dependence of the fields restricted sional supergravity. by the section constraint [9] This reformulation within a common framework is very KMN∂ ⊗ ∂ 0 much in the spirit of exceptional field theories. In that d M N ¼ ; ð5:1Þ framework, higher-dimensional supergravity theories are reformulated in terms of the field content of a lower- with the two differential operators acting on any couple of dimensional supergravity keeping the dependence on all fields and gauge parameters of the theory. The tensor dKMN coordinates. More precisely, their formulation is based on a denotes the cubic totally symmetric E6ð6Þ invariant tensor, MNP P split of coordinates into D external and n internal which we normalize as d dMNQ ¼ δQ . The section

046002-11 BERTRAND, HOHENEGGER, HOHM, and SAMTLEBEN PHYS. REV. D 103, 046002 (2021) condition (5.1) admits two inequivalent solutions [10] properties of MMN. To quadratic order in the fluctuations, which reduce the internal coordinate dependence of all the ExFT Lagrangian then yields 11 fields to the 6 internal coordinates from D ¼ super- 1 1 gravity, or 5 internal coordinates from IIB supergravity, L − ΩμνρΩ ΩμνρΩ ΩμΩ ExFT;free ¼ 4 μνρ þ 2 νρμ þ μ respectively. For details of the ExFT Lagrangian we refer to 1 [10,11]. Here, we spell out its “free” limit, obtained by μνM N − F F μν ΔMN linearizing the full theory according to 4 5 pffiffiffiffiffi − 10εμνρστ MNK∂ B ∂ B 4 d μ νρM N στK gμν ¼ ημν þ hμν; M ¼ Δ þ ϕ ; ð5:2Þ MN MN MN 1 − ϕMN μϕ L Dμ D MN þ pot; ð5:3Þ around the constant background given by the Minkowski 24 η Δ metric μν and the identity matrix MN. The scalar with indices M, N raised and lowered by ΔMN and its fluctuations ϕMN are further constrained by the coset inverse, and with the various elements of (5.3) given by

2 Ω ∂ − ∂ A Mη Ω ≡ Ω ν μνρ ¼ ½μhνρ 3 M ½μ νρ; μ μν ; F M 2∂ A M 10 MNK∂ B μν ¼ ½μ ν þ d N μνK; 2 ϕMN ∂ ϕMN 2∂ A ðMΔNÞK ∂ A KΔMN − 20∂ A L RKðMΔNÞP Dμ ¼ μ þ K μ þ 3 K μ K μ dPLRd ; 1 1 1 L − ΔMN∂ ϕKL∂ ϕ ΔMN∂ ϕKL∂ ϕ − ∂ ν∂ ϕMN pot ¼ 24 M N KL þ 2 M L NK 2 Mhν N 1 1 ΔMN∂ μ∂ ν − ΔMN∂ μν∂ þ 4 Mhμ Nhν 4 Mh Nhμν: ð5:4Þ

The Lagrangian we have presented above for the six- and keeping only coordinate-dependence along the N 2 2 5 5 dimensional ¼ð ; Þ model naturally fits into this SOð ; Þ singlet. In this split, the E6ð6Þ invariant symmetric framework. This does not come as a surprise since the tensor dMNK has the nonvanishing components six-dimensional model is nothing but linearized maximal 1 1 supergravity known to be described by E6ð6Þ ExFT upon d0ab ¼ pffiffiffiffiffi ηab;daij ¼ pffiffiffi ðΓaÞij; ð5:6Þ proper selection of the sixth coordinate among the internal 10 2 5 ∂ . This choice is uniquely fixed by the requirement that M 5 5 Γ ηab the resulting theory exhibits the global SOð5; 5Þ symmetry in terms of SOð ; Þ matrices and its invariant tensor of group of maximal six-dimensional supergravity, thus signature (5,5), showing that the section constraint (5.1) is ∂ 0 ∂ breaking trivially satisfied as i ¼ ¼ a. Putting this together with the linearized ExFT Lagrangian (5.3), and splitting fields as

→ 5 5 27 → 1 ⊕ 16 ⊕ 10 A M i a E6ð6Þ SOð ; Þ; ; f μ g¼fAμ;Aμ ;Aμ g; etc:; ð5:7Þ ∂ → ∂ ∂ ∂ f Mg f 0; i; ag; ð5:5Þ we arrive at

1 1 1 1 L − ΩμνρΩ ΩμνρΩ ΩμΩ − μν − μνi i ð2;2Þ ¼ 4 μνρ þ 2 νρμ þ μ 4 F Fμν 4 F Fμν 1 1 1 − a ∂ a μνa ∂ μνa − εμνρστη ∂ a b − ∂μϕα∂ ϕα 4 ðFμν þ yBμν ÞðF þ yB Þ 24 ab yBμν Hρστ 2 μ  rffiffiffi  rffiffiffi  1 8 8 1 − ∂μϕ − ∂ μ ∂ ϕ − ∂ − ∂μϕi − ∂ μi ∂ ϕi − ∂ i 2 3 yA μ 3 yAμ 2 ð yA Þð μ yAμ Þ; 1 5 2 1 1 − ∂ ϕα∂ ϕα ∂ ϕ∂ ϕ − ∂ σ∂ ϕ ∂ σ∂ ρ − ∂ μν∂ 2 y y þ 6 y y 3 yhσ y þ 4 yhσ yhρ 4 yh yhμν; ð5:8Þ

046002-12 TOWARD EXOTIC 6D SUPERGRAVITIES PHYS. REV. D 103, 046002 (2021) which precisely produces the sum of Lagrangians (3.6), ð3;1Þ 2 ð3; 1Þ∶ ∂ ¼ pffiffiffi ∂0 ¼ −2∂•; (3.9), (3.11), after proper rescaling of the singlet scalar field y 3 ϕ. The nontrivial checks of this coincidence include all the with the 4 singlet ∂0 ⊂ ∂ ; coefficients in the various connection terms, as well as in the ð Þ M Stückelberg-type couplings betweenvector and tensor fields, ð4;0Þ ð4; 0Þ∶ ∂y ¼ −∂•; ∂M ¼ 0; ð5:12Þ and the coefficients in front of the various ∂yϕ∂yϕ terms in the last line. Again, this is not a surprise but a consequence of corresponding to the two exotic six-dimensional models in the proven equivalence of ExFT with higher-dimensional precise correspondence with the central charges carried by maximal supergravity. Note that although the free theory the corresponding BPS multiplets [7]. While the (4,0) only exhibits a compact USpð4Þ ×USpð4Þ global symmetry, solution trivially solves the constraint (5.10), the N ¼ the couplings exhibited in (5.8) are far more constrained than ð3; 1Þ solution is based on the decomposition allowed by this symmetry and witness the underlying E6ð6Þ 5 5 → 27 → 1 ⊕ 26 structure broken to SOð ; Þ according to (5.5), (5.6). E6ð6Þ F4ð4Þ; ; The ExFT Lagrangian is to a large extent determined by ∂ → ∂ ∂ invariance under generalized internal diffeomorphisms f Mg f 0; Ag; ð5:13Þ acting with a gauge parameter ΛM in the fundamental under which the symmetric d-tensor decomposes into 27. After linearization (5.2) these diffeomorphisms act as 2 1 2 000 − ffiffiffiffiffi 0AB ffiffiffiffiffi ηAB ABC δϕ 2Δ ∂ ΛK ∂ ΛKΔ d ¼ p ;d¼ p ;d; ð5:14Þ MN ¼ KðM NÞ þ 3 K MN 30 30 − 20dPKRd Δ ∂ ΛL; ηAB RLðM NÞP K with the F4ð4Þ invariant symmetric tensor of signature 2 ABC δA M ∂ ΛM δ ∂ ΛMη (14,12), and the symmetric invariant tensor d satisfying μ ¼ μ ; hμν ¼ 3 M μν; ð5:9Þ 14 ABCη 0 ABD δ D and one can show invariance of the linearized Lagrangian d BC ¼ ;dABCd ¼ 15 C : ð5:15Þ (5.8), provided the section constraint (5.1) is satisfied. This shows explicitly how the (3,1) assignment of (5.12) also provides a solution to the extended section con- B. Beyond standard ExFT: Embedding N straint (5.10). of the = ð3;1Þ and (4,0) couplings It is intriguing to study the fate of diffeomorphism As we have discussed in the Introduction, the charges invariance of the ExFT Lagrangian (5.3) if the original carried by the massive BPS multiplets in the reduction of section constraint is relaxed to (5.10). Except for the last the N ¼ð3; 1Þ and the N ¼ð4; 0Þ model, respectively, term in (5.3), the Lagrangian remains manifestly invariant suggest that an inclusion of these models into the frame- without any use of the section constraint. Explicit variation L work of ExFT necessitates an extension of the space of of the potential term pot under linearized diffeomorphisms 27 internal coordinates by an additional exotic coordinate (5.9) on the other hand yields (up to total derivatives) corresponding to the singlet central charge [7]. Denoting derivatives along this coordinate by ∂•, this would amount δ L 5Δ LMN KPQ − 10ΔMNΔKL RPQ Λ pot ¼ð LSd d dLSRd Þ to a relaxation of the standard section constraint (5.1) to a ΛS∂ ∂ ∂ ϕ constraint of the form × P Q M NK μ MK RPQ L − 10hμ Δ dKLRd ∂M∂P∂QΛ ; ð5:16Þ 1 KMN KM d ∂ ⊗ ∂ − pffiffiffiffiffi Δ ð∂ ⊗ ∂• þ ∂• ⊗ ∂ Þ¼0; M N 10 M M which consistently vanishes modulo the standard section constraint (5.1). For the weaker constraint (5.10), this ð5:10Þ variation no longer vanishes and may be recast in the following form: which at the present stage only makes sense in the KM linearized theory where Δ is a constant background KM N ν N δΛL ¼ Δ Λ ∂•∂•∂ ϕ − 4hν ∂•∂•∂ Λ ; ð5:17Þ tensor. Apart from the standard ExFT solutions pot M NK N after repeated use of (5.10) and further manipulation of the KMN∂ ⊗ ∂ 0 ∂ 0 d M N ¼ ; • ¼ ; ð5:11Þ expressions. In order to compensate for this variation let us first note that there is no possible covariant extension of the M of this constraint, which allow the embedding of the N ¼ transformation rules (5.10) by terms carrying ∂•Λ , such ð2; 2Þ model as described above, the extended section that invariance can only be restored by extending the constraint also allows for two exotic solutions potential. A possible such extension is given by

046002-13 BERTRAND, HOHENEGGER, HOHM, and SAMTLEBEN PHYS. REV. D 103, 046002 (2021) 1 3 L L − ∂ ϕ ∂ ϕMN − ∂ σ∂ ρ relaxation (5.10) of the section constraint together with pot;• ¼ pot • MN • •hσ •hρ 24 4 generalized diffeomorphism invariance precisely implies 3 μν the correct scalar couplings in the Lagrangians of the exotic þ ∂•h ∂•hμν; ð5:18Þ 4 models. In addition, the ∂•h∂•h terms in (5.18) cancel the L and it is straightforward to verify that the variation of the corresponding terms in pot (5.4) upon selecting the (3,1) additional terms in (5.18) precisely cancels the contribu- solution of the section constraint (5.12), just as required in N 3 1 tions in (5.17), such that order to reproduce the correct Lagrangian of the ¼ð ; Þ model (4.2).4 δ L 0 Λ pot;• ¼ : ð5:19Þ We may continue the symmetry analysis for the tensor gauge transformations given by a gauge parameter ΛμM in For the exotic solutions of the section constraint, the standard ExFT. For these transformations there is a natural ∂ ϕ ∂ ϕMN • MN • terms in (5.18) give rise to additional contri- extension of the standard ExFT transformation rules in the ∂ ϕ∂ ϕ butions of the type y y in the Lagrangian. Collecting all presence of the exotic coordinate and exotic fields as such terms in (5.18) for the two exotic solutions (5.12) yields pffiffiffiffiffi M MNK MK 1 δΛ Aμ ¼ −10d ∂ Λμ − 10Δ ∂•Λμ ; 3 1 → − ∂ ϕα∂ ϕα α 1 … 28 μ N K K ð ; Þ 2 y y ; ¼ ; ; ; δΛ Bμν ¼ 2∂ μΛν : ð5:21Þ 1 μ M ½ M 4 0 → − ∂ ϕA∂ ϕA 1 … 42 ð ; Þ 2 y y ;A¼ ; ; : ð5:20Þ Computing the action of these transformations on the These are precisely the terms found in our explicit construc- connection featuring in the covariant scalar derivatives MN 5 tion of actions (4.4) and (4.9) above. In other words, the Dμϕ in (5.4), we obtain after some manipulation

  1 MN MN Q QðM NÞ SðM NÞQR PKL δΛ Dμϕ ¼ 10 Δ δ þ Δ δ − 10Δ d d d ∂ ∂ Λμ μ 3 P P RSP K L Q   pffiffiffiffiffi 1 − 2 10 ΔMNδ Q δ ðMΔNÞQ − 10ΔLðM NÞQR ΔPK∂ ∂ Λ 3 P þ P d dRPL K • μQ: ð5:22Þ

The resulting expression precisely vanishes with the modified section constraint (5.10). This shows the necessity of the MN μ ∂•ΛμM terms in (5.21) in order to maintain gauge invariance of the kinetic term Dμϕ D ϕMN in presence of the relaxed section constraint. It is straightforward to verify that these additional terms in the transformation induce a modification of the gauge invariant vector field strengths to pffiffiffiffiffi F M ≡ 2∂ A M 10 MNK∂ B 10ΔMK∂ B μν ½μ ν þ d N μνK þ • μνK; ð5:23Þ as well an extension of the topological term, such that the combined vector-tensor couplings take the form 1 5 pffiffiffiffiffi L − Δ F μνMF N − εμνρστ∂ B 10 MNK∂ B ΔMK∂ B vt;• ¼ 4 MN μν 4 μ νρMð d N στK þ • στKÞ; ð5:24Þ and are invariant under these gauge transformations. Let us work out the effect of these modifications for the exotic solutions of the section constraint. With the kinetic scalar term unchanged, the resulting couplings are directly inferred from evaluating the covariant derivatives (5.4) for the d-symbol (5.14), giving rise to 1 1 3 1 ⟶ − ∂μϕi − ∂ μi ∂ ϕi − ∂ i − ∂μϕα∂ ϕα 1 … 14 α 1 … 28 ð ; Þ 2 ð yA Þð μ yAμ Þ 2 μ ;i¼ ; ; ; ¼ ; ; ; 1 4 0 ⟶ − ∂μϕA∂ ϕA 1 … 42 ð ; Þ 2 μ ;A¼ ; ; : ð5:25Þ

4In contrast, these terms appear in conflict with embedding the spin-2 sector of the N ¼ð4; 0Þ model as they survive under the (4,0) solution in (5.12) but should be absent in the final Lagrangian (4.6). We come back to this in Sec. VC. 5A useful identity for this computation is given by 1 1 1 1 dPLQd dKMR∂ ∂ δKdLQM∂ ∂ δMdQKL∂ ∂ δQdMKL∂ ∂ − dQMRd dPKL∂ ∂ ; PSR K L ¼ 10 S K L þ 20 S K L þ 20 S K L 2 RSP K L generalizing Eqs. (2.12), (2.13) of [11].

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This precisely reproduces the vector-scalar couplings found in the explicit Lagrangians (4.4), (4.9) above. As for the vector- tensor couplings, evaluating the Lagrangian (5.24) with (5.14) for the solutions (5.12) gives rise to the explicit couplings  pffiffiffi  pffiffiffi  1 3 3 3 3 1 3 1 ⟶ − ∂ μν ∂ μν − a ∂ a μνa ∂ μνa ð ; Þ 4 Fμν þ 2 yBμν F þ 2 yB 4 ðFμν þ yBμν ÞðF þ yB Þ 1 3 1 − μνi i − εμνρστ∂ − εμνρστ∂ a a 4 F Fμν 16 yBμνHρστ 24 yBμν Hρστ ; 1 1 4 0 ⟶ − M ∂ M μνM ∂ μνM − εμνρστ∂ M M ð ; Þ 4 ðFμν þ yBμν ÞðF þ yB Þ 24 yBμν Hρστ ; ð5:26Þ with indices in range i ¼ 1; …; 14, a ¼ 1; …; 12, despite the fact that the free theory only exhibits invariance 1 … 27 2N M ¼ ; ; , as above. Again, this precisely reproduces under the compact R-symmetry subgroup USpð þÞ × the couplings found above (after proper rescaling of the USpð2N −Þ which might in principle allow for much more vector field Aμ). general couplings. We take this as evidence for the To summarize, in the scalar, vector, and tensor sector, we conjectured E6ð6Þ and F4ð4Þ invariance of the putative have constructed an extension of the ExFT Lagrangian (at interacting theories [1]. the linearized level), given by 1 C. The spin-2 sector L − ϕMN μϕ L L ¼ Dμ D MN þ vt;• þ pot;•; ð5:27Þ 2 The above findings have revealed a very intriguing which is invariant under the gauge transformations (5.9), common structure of the couplings in the scalar, vector, (5.21) modulo the relaxed section constraint (5.10). The and tensor sectors of the different models which can be weaker section constraint necessitates a number of addi- consistently embedded into an extension of (linearized) tional contributions to the Lagrangian (and transformation exceptional field theory. For the spin-2 sector carrying the rules) which precisely reproduce the explicit couplings Pauli-Fierz field and its duals on the other hand the picture found in the Lagrangians of the exotic models (4.4), (4.9) appears not yet complete. Extrapolation of the Lagrangian constructed above. It is remarkable that this match confirms of the N ¼ð4; 0Þ model (4.6) suggests an extension of the the couplings that have been determined from an under- standard ExFT Lagrangian by couplings carrying ∂• derivatives and the dual graviton fields as lying noncompact E6ð6Þ and F4ð4Þ structure, respectively,

1 1 1 L − Ωb μνρΩb ΩbμνρΩb ΩbμΩb ε ∂μ bνσ ∂ bκλ;ρ ¼ 4 μνρ þ 2 νρμ þ μ þ 8 μνσκλ C ρ •C 1 1 1 − ε ∂μCνσ ∂ Cκλ;ρ ∂ C ∂ μν;στ − ∂ C ∂κ λσ;τ 32 μνσκλ ρ • þ 8 • στ;ν μT 4 • κλ;τ T σ 1 1 1 − ∂ C μ∂ στ;ν ∂ C μ∂σ τν ε ∂α βγ∂ μν;στ 4 ν σμ •T τ þ 8 • σμ Tτν þ 64 μναβγ Tστ •T 1 1 1 − ∂ ∂ μν;στ ∂ μ∂ στ;ν − ∂ μν∂ στ 32 •Tστ;μν •T þ 8 •Tσμ;ν •T τ 32 •Tμν •Tστ 5 εμνρστ KMN∂ B ∂ b þ 4 d K μνM NCρσ;τ; ð5:28Þ with can find its place in this construction. In particular, the appearance of the extra fields Cμν ρ and Tμν ρσ appearing in 2 1 ; ; b M b (5.28), whose couplings remain present upon selecting the Ωμνρ ¼ ∂ μhν ρ − ∂ A μ ην ρ − ∂•Cμν ρ þ ∂•Cμν ρ: ½ 3 M ½ ; 2 ; (3,1) solution (5.12) of the section constraint, poses a ð5:29Þ challenge for recovering the Lagrangian (4.4) of the N ¼ ð3; 1Þ model. The structure of the gauge transformations of By construction, this reproduces the N ¼ð2; 2Þ and the C as extrapolated from (4.8) appears to suggest a gauge ζ λ — N ¼ð4; 0Þ models upon choosing the corresponding fixing of the μν and ρ;μν gauge symmetries absent in the solutions of the section constraint. It remains unclear N ¼ð3; 1Þ model—in order to remove this field. Another however, how the spin-2 sector of the N ¼ð3; 1Þ model apparent problem in the spin-2 sector is the lacking

046002-15 BERTRAND, HOHENEGGER, HOHM, and SAMTLEBEN PHYS. REV. D 103, 046002 (2021) reconciliation between the ∂•h∂•h terms from (5.18) and supported by the ERC Consolidator Grant “Symmetries & the ∂•T∂•T terms of (5.28) which mutually violate the Cosmology.” correct limits to the exotic models. Resolution of this problem may require to implement algebraic relations Note added.—While finalizing the present paper, [29] between the Pauli-Fierz hμν field and the double dual appeared, which also investigates exotic theories in 6D. graviton [2] (see also [27]). APPENDIX A: ACTIONS FOR SELF-DUAL VI. CONCLUSIONS AND OUTLOOK TENSOR FIELDS In this paper we have taken the first step in constructing It is well known that the first-order field equations for action principles for exotic supergravity theories in 6D by D ¼ 6 self-dual tensor fields giving such actions for the free bosonic part. These actions show already intriguing new features such as the simulta- 1 ˆ ε σˆ κˆ λ 3∂ neous appearance of (linearized) diffeomorphisms and Hμˆ νˆ ρˆ ¼ 6 μˆ νˆ ρˆ σˆ κˆ λˆ H ;Hμˆ νˆ ρˆ ¼ ½μˆ Bνˆ ρˆ ðA1Þ dual diffeomorphisms, which are realized on exotic Young tableaux fields as well as more conventional gravity fields. do not integrate to a standard action principle, yet various Our formulation abandons manifest 6D Lorentz invariance, mechanisms with different characteristics have been as expected to be necessary on general grounds, by being devised such as to provide a Lagrangian description of based on a 5 þ 1 split of coordinates. Remarkably, the field these equations [13,20–22]. In this Appendix, we briefly equations implied by our actions can be integrated to review the construction of Henneaux and Teitelboim [13] reconstruct the correct dynamics of these exotic super- which is somewhat closest in spirit to the construction gravites. Moreover, we have seen the first glimpses of an employed in this paper, together with its dual formulation exceptional field theory master formulation, in which the that is naturally embedded within exceptional field theory. conventional N ¼ð2; 2Þ, as well as the exotic N ¼ð3; 1Þ Both formulations are based on a coordinate split (3.1) N 4; 0 and ¼ð Þ models all emerge through different μˆ μ solutions of an extended section constraint, but clearly fx g → fx ;yg; ðA2Þ much more needs to be done. We close with a brief 6 6 discussion of possible future developments. and sacrifice manifest D ¼ Poincar´e invariance. With ≡ First, it remains to exhibit the (maximal) supersymme- the corresponding split fBμˆ νˆ g¼fBμν;Bμ5 Aμg of the tries in these nonstandard formulations, even just at the free six-dimensional tensor field, the self-duality equations (A1) level. We have no doubt that this can be achieved as in take the form exceptional field theory where different 1 (such as type IIB versus type IIA) are realized within a ρστ F μν þ εμνρστH ¼ 0; for F μν ≡ Fμν þ ∂yBμν: ðA3Þ single master formulation. Second, it would be interesting 6 to study possible embeddings into exceptional field theo- In particular, the divergence and curl of this equation give ries of higher rank, such as for U-duality groups 7 and ð Þ rise to E8ð8Þ, which may illuminate some issues and which can μ also be done already at linearized level. Finally, the most ∂ F μν ¼ 0; important outstanding problem is clearly the question εμνλστ∂ − 6∂ λμν 0 whether our formulation can be extended to the nonlinear yHλστ λH ¼ ; ðA4Þ interacting theory. We would like to emphasize that the present formulations seem quite promising in this regard respectively. since they feature not only the exotic fields but also the more conventional gravity fields, which come with an 1. Henneaux-Teitelboim Lagrangian action that allows a natural embedding into the full non- The Lagrangian proposed by Henneaux and Teitelboim linear Einstein-Hilbert action. In turn this suggests that all [13] for the description of the self-dual tensors takes the these fields might become part of a tensor hierarchy that form extends to the gravity sector. If so this could quite naturally lend itself to a formulation of nonlinear dynamics in terms 1 μνρστ 1 μνρ L ε F μνHρστ − HμνρH A5 of a hierarchy of duality relations as in [28]. ¼ 24 12 ð Þ

ACKNOWLEDGMENTS 6The original construction of [13] defines the 5 þ 1 split by singling out the time coordinate, but the method obviously We wish to thank X. Bekaert, M. Henneaux, C. Hull, and applies equally well for the split based on a spatial distinguished V. Lekeu for enlightening discussions. The work of O. H. is dimension.

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1 when applied to Eq. (A3), i.e., evaluated for spacelike split ˜ ρ ˜ στ ðFμν þ ∂ BμνÞþ εμνρστ3∂ B ¼ 0; ðA13Þ and flat background. As a first observation, this Lagrangian y 6 depends on the vector field Aμ only via total derivatives, such that it does not show up in the field equations with the modified two-form ˜ ≡ − 2 ρστ ρ Bμν Bμν bμν: ðA14Þ εμνρστ∂yH ¼ 6∂ Hμνρ; ðA6Þ ˜ In terms of the fields Aμ, Bμν, we thus recover the desired reproducing the second equation of (A4). This equation original field equations (A3). Note finally, that the now serves as an integrability equation in order to locally Lagrangian (A8) precisely comes with a gauge freedom define the vector field Aμ via the equation of the type (A14) which allows one to absorb bμν into Bμν. We thus arrive at two complementary Lagrangians (A5), 1 2∂ −∂ − ε ρστ (A8), which both describe the six-dimensional self-dual ½μAν ¼ yBμν 6 μνρστH : ðA7Þ tensor field upon sacrificing manifest D ¼ 6 Poincar´e invariance. They are dual to each other in the sense that Indeed, the curl of the right-hand side vanishes by virtue of upon dimensional reduction to D ¼ 5 dimensions, i.e., upon (A6). Defining the vector field Aμ by (A7), we precisely setting ∂y → 0, the Lagrangian (A8) describes the 3 degrees recover the equations of motion (A3). of freedom in terms of a free Maxwell field whereas (A5) describes them in terms of the dual massless tensor field Bμν. 2. ExFT type Lagrangian Similarly, the two Lagrangians (A5) and (A8) can be dualized into each other in presence of the sixth dimension. Exceptional field theory typically yields formulations of higher-dimensional supergravity theories based on the field content of lower-dimensional theories. In particular, it APPENDIX B: 6D FIELD EQUATIONS FROM offers actions for theories that do not admit actions in THE NEW LAGRANGIANS terms of their original variables, such as IIB supergravity, In this Appendix, we present in detail how the second- cf. [12]. In the context of (anti-)self-dual tensor fields order field equations obtained by variation of the appearing in six dimensions, an exceptional field theory Lagrangians (4.2) and (4.6) can be integrated to the formulation based on a split (A2) gives rise to an action first-order field equations which in turn imply the original 6D second-order self-duality equations of the N ¼ð3; 1Þ 1 1 N 4 0 L − F F μν − εμνρστ∂ model and the ¼ð ; Þ model, respectively. ¼ 4 μν 24 yBμνHρστ; ðA8Þ 1. The N = ð3;1Þ model carrying the fields of Eq. (A3). The field equations are now given by Here, we show how the second-order field equations obtained by variation of the Lagrangian (4.2) for the N ¼ ν ν ν ð3; 1Þ model can be integrated to the first-order field 0 ∂ F νμ ∂ Fνμ ∂ ∂ Bνμ; ¼ ¼ þ y ðA9Þ equations (3.21) and (3.22) which in turn imply the original   6D second-order self-duality equations (2.11). 1 ρστ 0 ¼ ∂y F μν þ εμνρστH : ðA10Þ 6 a. Field equations Let us first spell out the field equations obtained from In particular, Eq. (A10) implies the original field variation of the Lagrangian (4.2). equations (A3) up to some function that does not depend Variation with respect to Aμ: on y: μ 3 μ ∂ Fμν þ ∂y∂ Bμν 1 ρστ 2 F μν εμνρστH χμν; ∂ χμν 0: A11 þ 6 ¼ y ¼ ð Þ 1 − ∂ ∂μ − ∂ μ − ∂ b μ 4∂ 0 2 yð hμν νhμ yCνμ þ yAνÞ¼ ; ðB1Þ Comparing the divergence of this equation to (A9), we find that locally the field χμν can be integrated to which is exactly (3.29). Variation with respect to Bμν: μ ρ στ   ∂ χμν ¼ 0 ⇒ χμν ¼ εμνρστ∂ b ; ðA12Þ 3 1 1 ∂ ∂ ε ∂ρ στ ∂ ε bρστ 0 y Fμν þ 2 yBμν þ 2 μνρστ B þ 12 y μνρστC ¼ ; in terms of a function bμν, such that the field equations (A11) can be rewritten as ðB2Þ

046002-17 BERTRAND, HOHENEGGER, HOHM, and SAMTLEBEN PHYS. REV. D 103, 046002 (2021) which is the ∂y derivative of Eq. (3.21). with the linearized Einstein tensor as it appears in Variation with respect to hμν: (3.17); this variation thus exactly reproduces the Einstein equation (3.17). 1 b G ∂ ∂ρ b ∂ b ρ − η ∂ρ b σ 0 Variation with respect to Cμν;ρ: μν þ 2 yð Cρðμ;νÞ þ ðμCνÞρ μν Cρσ Þ¼ ; ðB3Þ

 1 1 ∂ ∂ ∂σ η − η ∂ σ ε ∂λ bστ ∂ b − b − b y ½μhνρ þ hσ½μ νρ ρ½ν μhσ þ 4 μνλστ C ρ þ 4 yðCμν;ρ Cνρ;μ Cρμ;νÞ  3 − η ∂ b σ 3∂ η ε στ 0 ρ½ν yCμσ þ yA½μ νρ þ 8 μνρστB ¼ ; ðB4Þ which we can further project onto its (2,1) part and totally antisymmetric part  1 1 ∂ ∂ ∂σ η − η ∂ σ ε ∂λ bστ − ε ∂λ bστ y ½μhνρ þ hσ½μ νρ ρ½ν μhσ þ 4 μνλστ C ρ 4 λστ½μν C ρ  1 ∂ b − b − b b − η ∂ b σ 3∂ η 0 þ 4 yðCμν;ρ Cνρ;μ Cρμ;ν þ C½μν;ρÞ ρ½ν yCμσ þ yA½μ νρ ¼ ; ðB5Þ   1 1 3 ∂ ε ∂λ bστ − ∂ b ∂ ε στ 0 y 4 λστ½μν C ρ 4 yC½μν;ρ þ 8 y μνρστB ¼ : ðB6Þ

We now show how to recover the full 6D system (3.19) b. A-B duality and (3.21) from the equations derived above. Let us first Combining (B7) and (B8) gives rewrite these equations in terms of the original fields of the ∂ 1 (3,1) model and integrate all the equations under y by ∂ ∂μ ∂ ∂μ − ∂ μ − ∂ μ 4∂ − ∂μχ μ μ μ y uμν ¼ 2 yð hμν νhμ yCνμ þ yAνÞ μν; introducing three functions χμνðx Þ, ψ μνρðx Þ and φμν;ρðx Þ which are respectively antisymmetric, antisymmetric and of ðB12Þ (2,1) type, and do not depend on the sixth coordinate: while the trace of (B11) in ðμρÞ gives 3 μ μ 1 1 ∂ Fμν þ ∂y∂ Bμν μ μ μ μ μ 2 ∂ uμν ¼ ð∂ hμν − ∂νhμ − ∂yCνμ þ 4∂yAνÞþ φμν : 1 2 3 − ∂ ∂μ − ∂ μ − ∂ μ 4∂ 0 2 yð hμν νhμ yCνμ þ yAνÞ¼ ; ðB7Þ ðB13Þ

3 1 Together, these two equations imply that locally, we can ∂ − ∂ ε ∂ρ στ χ define a two-form b such that Fμν þ 2 yBμν yuμν þ 2 μνρστ B ¼ μν; ðB8Þ

  1 ρ στ μ 3 χμν ¼ εμνρστ∂ b ðx Þ: ðB14Þ ε ∂λ στ − 4∂ − ∂ ε στ − στ ψ 2 λστ½μν C ρ ½ρuμν y μνρστ u 2 B ¼ μνρ; This two-form can be absorbed in B (following exactly the ðB9Þ same process as in Sec. A2) such that Eq. (B8) repro- duces (3.21). 1 ρ ρ ρ σ Gμν þ ∂ ð∂ Cρ μ ν þ ∂ μCν ρ − ημν∂ Cρσ Þ¼0; ðB10Þ 2 y ð ; Þ ð Þ c. h-C duality Contracting (B11) with ∂μ, we can extract both sym- 1 ∂ ∂σ η − η ∂ σ ε ∂λ στ metric and antisymmetric parts: ½μhνρ þ hσ½μ νρ ρ½ν μhσ þ 4 μνλστ C ρ 1 1 1 − ε ∂λ στ ∂ − η ∂ σ 3∂ η νρ ∶ G ∂ ∂μ − η ∂μ σ ∂ σ 4 λστ½μν C ρ þ 2 yCμν;ρ ρ½ν yCμσ þ yA½μ νρ ð Þ νρ þ 2 yð Cμðν;ρÞ νρ Cμσ þ ðρCνÞσ Þ − ∂ ∂ − 2∂σ η φ ∂μφ ρuμν þ ½ρuμν uσ½μ νρ ¼ μν;ρ: ðB11Þ ¼ μðν;ρÞ; ðB15Þ

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1 νρ ∶ − ε ∂μ∂λ στ ∂ ∂μ ∂ σ − 3∂ 2∂μ∂ 2∂μφ ½ 6 λστνρ C μ þ yð Cμ½ν;ρ þ ½ρCνσ ½ρAνÞþ ½μuνρ ¼ μ½ν;ρ: ðB16Þ

Using (B10) we can conclude that ∂μφ 0 μðν;ρÞ ¼ : ðB17Þ The divergence of (B9) reads   1 1 3 1 − ε ∂ρ∂α αβ 2∂ρ∂ ∂ ε ∂ρ αβ − αβ − ∂ρψ 6 μναβγ C ρ þ ½ρuμν þ 2 y μνραβ u 2 B ¼ 2 μνρ; ðB18Þ and combining it with (B16) and (B8), we eventually get 1 2∂μφ − ∂ρψ μ½ν;ρ ¼ 2 μνρ: ðB19Þ

Together with (B17), one has   1 ∂μ 2φ ∂ρψ 0 μν;ρ þ 2 μνρ ¼ ; ðB20Þ such that locally there exist 5D tensors cμν;ρ and aμνρ, where c is of (2,1) type and a is completely antisymmetric, such that 1 1 2φ ∂ρψ ε ∂α βγ βγ μν;ρ þ 2 μνρ ¼ 2 μναβγ ðc ρ þ a ρÞ: ðB21Þ

Consequently 1 φ μ ε ∂α βγμ μν ¼ 4 μναβγ a ; ðB22Þ 1 1 φ ε ∂α βγ βγ − ε ∂α βγ βγ μν;ρ ¼ 4 μναβγ ðc ρ þ a ρÞ 4 αβγ½μν ðc ρ þ a ρÞ; ðB23Þ

ψ ε ∂α βγ βγ μνρ ¼ αβγ½μν ðc ρ þ a ρÞ: ðB24Þ

Plugging the expression for φ and its trace back into (B11), one has 1 1 2∂ ε ∂λ στ − ε ∂λ στ ∂ − 2∂ η − 2∂ 2∂ ½μhνρ þ 2 μνλστ C ρ 2 λστ½μν C ρ þ yCμν;ρ yA½μ νρ ρuμν þ ½ρuμν 1 1 1 ε ∂α βγ βγ − ε ∂α βγ βγ ε ∂α βγση ¼ 2 μναβγ ðc ρ þ a ρÞ 2 αβγ½μν ðc ρ þ a ρÞþ3 αβγσ½μ a νρ: ðB25Þ

Then, using the following two Schouten identities

ε αβγ∂ 0 −εαβγ ∂ 3∂ εαβγ ½μν αaβγρ ¼ ¼ ½μν ρaαβγ þ α ½μνaρβγ; ðB26Þ ε ∂α βγση 0 2ε ∂α βγση − ∂ ε βγσ 3ε ∂α βγ ½σμαβγ a νρ ¼ ¼ αβγσ½μ a νρ ρ βγσμνa þ αβγμν a ρ; ðB27Þ one obtains   1 1 2∂ ε ∂λ στ − στ − ε αβ ∂ − 2 η ½μhνρ þ 2 μνλστ ðC ρ c ρÞþ Hμνρ 2 μνραβF þ yðCμν;ρ A½μ νρÞ   1 − 2∂ ε αβγ 0 ρ uμν þ 12 μναβγa ¼ : ðB28Þ

We recover the 6D equation (3.19) after the following redefinitions

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1 αβγ N ¼ð4; 0Þ model can be integrated to the first-order field uμν → uμν þ εμναβγa ; 12 equations (3.34) and (3.36) which in turn imply the original 6D second-order self-duality equations (2.16). Bμν → Bμν − bμν;Cμν;ρ → Cμν;ρ − cμν;ρ: ðB29Þ

One can check that these redefinitions are consistent with a. Field equations the expression of ψ in (B9). Finally, the derivatives of Let us first spell out the field equations obtained from (3.19) and (3.21) give rise to the original 6D equations of variation of the Lagrangian (4.6). Apart from the Einstein motion (2.11) as discussed in Sec. III B above. equations (3.40) obtained by variation with respect to hμν, all other field equations appear under total derivative ∂ and N y 2. The = ð4;0Þ model can be integrated into first-order equations up to functions Here, we show how the second-order field equations that only depend on the five coordinates xμ. Explicitly, we obtained by variation of the Lagrangian (4.6) for the may put them to the form

1 1 ε ∂α βγ − ∂ ∂ χ 4 αβγ½μν C ρ ½ρuμν þ 2 yv½ρ;μν ¼ μνρ; ðB30Þ 1 1 1 ε ∂α βγ ε ∂α βγ − ∂ ∂ ψ 8 αβγ½μν C ρ þ 4 αβγ½μν vρ ½ρuμν þ 2 yv½ρ;μν ¼ μνρ; ðB31Þ 1 1 ∂ ∂α η − η ∂ α ε ∂α βγ − ε ∂α βγ − ∂ ∂ ½μhνρ þ hα½μ νρ ρ½ν μhα þ 4 μναβγ C ρ 4 αβγ½μν C ρ ρuμν þ ½ρuμν 1 − 2∂α η ∂ − − ∂ η α α φ uα½μ νρ þ 2 yðCμν;ρ þ vρ;μν v½ρ;μνÞ y ρ½νðCμα þ v μαÞ¼ μν;ρ ðB32Þ 1 1 ∂ ∂α η − η ∂ α ε ∂α βγ − ε ∂α βγ − ∂ ∂ ½μhνρ þ hα½μ νρ ρ½ν μhα þ 8 μναβγ C ρ 8 αβγ½μν C ρ ρuμν þ ½ρuμν 1 1 − 2∂α η ∂ − − ∂ η α α ε ∂α βγ uα½μ νρ þ 2 yðCμν;ρ þ vρ;μν v½ρ;μνÞ y ρ½νðCμα þ v μαÞþ4 μναβγ vρ 1 1 1 1 1 − ε ∂α βγ − ∂α ∂ α η ∂α β − η ∂ αβ θ 4 αβγ½μν vρ 4 Tαρ;μν þ 2 ½μTνα;ρ þ 2 ρ½ν Tμβ;α 4 ρ½ν μTαβ ¼ μν;ρ; ðB33Þ

1 1 ε ∂α βγ ε ∂α βγ − ε ∂α βγ 2∂ − 2 αβγμν T ρσ þ 2 αβγρσ T μν αβγ½μν T ρσ þ ðμðCjρσj;νÞ vνÞ;ρσÞ 2∂ − − 2∂α − η − 2∂α − η þ ðρðCjμνj;σÞ vσÞ;μνÞ ðCαρ;ðμ vðμ;jαρjÞ νÞσ ðCασ;ðμ vðμ;jασjÞ νÞρ − 2∂α − η − 2∂α − η − 2η ∂ α − α ðCαμ;ðρ vðρ;jαμjÞ σÞν ðCαν;ðρ vðρ;jανjÞ σÞμ σðμ νÞðCρα v ραÞ − 2η ∂ α − α − 2η ∂ α − α − 2η ∂ α − α ρðμ νÞðCσα v σαÞ μðρ σÞðCνα v ναÞ νðρ σÞðCμα v μαÞ ∂ 2 − 4η α − 4η α 2η η αβ þ yð Tμν;ρσ σðνTμÞα;ρ ρðνTμÞα;σ þ μðρ σÞνTαβ Þ 4η η ∂α β − β ϕ þ μðρ σÞν ðCαβ v αβÞ¼ μν;ρσ: ðB34Þ

Here, the functions χμνρ and ψ μνρ are totally antisymmetric, This implies that locally, we can define a totally antisym- φ θ ϕ while μν;ρ and μν;ρ are of (2,1) type, and μν;ρσ is (2,2). All metric aμνρ and a cμν;ρ of (2,1) type, such that of these functions do not depend on the sixth coordinate.

1 α βγ βγ φμνρ þ χμνρ ¼ εμναβγ∂ ða ρ þ c ρÞ: ðB36Þ b. h-C duality 4 Using (3.40), one can show that the derivatives of (B30) and (B32) imply Moreover 1 1 ∂μ φ χ ∂ ε χαβγ − ψ αβγ 0 φ μ ε βγμ ð μνρ þ μνρÞ¼2 yð νραβγð ÞÞ ¼ : ðB35Þ μν ¼ 4 μναβγa ; ðB37Þ

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3∂ Ωμν 3∂μ∂ ν − 3∂ν∂ μ − 3∂ ∂ νμ thus ½μ ρσ ¼ ½μCρσ ½μCρσ y ½μTρσ 42 1 1 ðB: Þ3∂μ∂ ν − 3∂ν∂ μ ∂ ε ∂α βγ − ∂ ∂ ¼ ½μCρσ ½μCρσ ½μhνρ þ 4 μναβγ C ρ ρuμν þ 2 yðCμν;ρ þ vμν;ρÞ 1 μκλρσ ν ν ν 2 − ε ∂ ð∂μCκλ þ ∂λv μκ − ∂ vλ μκÞ φ χ φ αη 2 y ; ¼ μνρ þ μνρ þ 3 α½μ νρ ðB:41Þ 1 1 ¼ 0; ðB47Þ ε ∂α βγ ∂ ε αβγ ¼ 4 μναβγ c ρ þ 12 ρ μναβγa : ðB38Þ vanishes. In particular, contraction of this equation shows We thus recover the 6D equation (3.36) after redefining that μν μν μ μ ν 2∂μΩ ρν − ∂ρΩ μν ¼ 0 ⇒ ∂μðΩ ρ − δ ρΩ νÞ¼0: Cμν;ρ → Cμν;ρ − cμν;ρ ðB39Þ ðB48Þ 1 → ε αβγ uμν uμν þ 12 μναβγa : ðB40Þ This can be integrated to 1 Ωμ − δμ Ων ∂ ωλμ ⇔ Ωμ ∂ ωλμ − δμ ∂ ωλκ ρ ρ ν ¼ λ ρ ρ ¼ λ ρ 4 ρ λ κ: c. C-T duality ðB49Þ Having established equation (3.36), we can hit it with another derivative to obtain the (modified) Curtright Next, we observe that (B47) gives rise to the second- equation order differential equation 0 ∂ ∂ Ωμν ¼ μ ½τ ρσ: ðB50Þ ½τ μν ½τ μν 3∂τ∂ C ρ − 3∂ρ∂ C τ 1 Using the parametrization (B44) and the relation (B48), this εμντκλ∂ ∂ ∂ − ∂ ¼ 2 yð τCκλ;ρ þ τvρ;κλ ρvτ;κλÞ: ðB41Þ equation reduces to ˆ μν 0 ¼ ∂μ∂ τΩ ρσ ; ðB51Þ It remains to recover the duality equation (3.34). From ½ (B30)–(B34), we obtain for the traceless part of Ω. Dualizing the first two indices on Ωˆ , defines the object Eμν ρσ ¼ Ωμν ρσðxÞ; ðB42Þ ; ; 1 ˆ ˆ μν Ωαβγ;ρσ ≡ εαβγμνΩ ρσ; ðB52Þ where 2

1 α βγ corresponding to an irreducible Young tableau . Using Eμν;ρσ ≡ εαβγμν∂ T ρσ þ2∂ μC ρσ ;ν −2∂ ρvσ ;μν þ∂yTμν;ρσ; 2 ½ j j ½ the generalized Poincar´e Lemma [30] for this object then ðB43Þ allows to integrate up equation (B51) into 1 4 Ω θ ϕ Ωˆ μν εμναβγ∂ 2∂ μν δ ½μ∂ νκ and μν;ρσ is a combination of μν;ρ and μν;ρσ. For later use, ρσ ¼ 2 αtβγ;ρσ þ ½ρs σ þ 3 ½ρj κs jσ; we parametrize this field as ðB53Þ μν μν 4 μ ν Ω Ωˆ δ½ Ω in terms of two gauge parameters tμν ρσ and sμν ρ. The first ρσ ¼ ρσ þ 3 ½ρ σ; ðB44Þ ; ; one corresponds to an irreducible Young tableau , while ˆ μν with traceless Ω ρσ and the trace part given by the parameter sμν;ρ is antisymmetric in its first two indices ν and traceless sμν ¼ 0,—such that its dual sˆαβγ;ρ ¼ 1 1 μν Ωμν Ωμ δμ Ων 2 εαβγμνs ρ corresponds to an irreducible Young tableau ρν ¼ ρ þ 3 ρ ν: ðB45Þ . The last term in (B53) implements the projection onto Equation (3.34) amounts to showing that the traceless part. Putting together (B53) and (B49), we arrive at Eμν ρσ ¼ 0 ðB46Þ ; 1 4 Ωμν εμναβγ∂ 2∂ μν δ ½μ∂ νκ ρσ ¼ 2 αtβγ;ρσ þ ½ρs σ þ 3 ½ρj κs jσ (after potential re-definition of Tμν;ρσ and vρ;μν). To begin 4 1 with, using the field equations that are already established, ½μ νκ μν λκ − δ ρ ∂κω σ − δ ρσ∂λω κ: ðB54Þ we can show that the particular combination of derivatives 3 ½ j j 3

046002-21 BERTRAND, HOHENEGGER, HOHM, and SAMTLEBEN PHYS. REV. D 103, 046002 (2021)   1 2 We can plug this back into (B47) to arrive at Ωμν εμναβγ∂ 2∂ μν δ ½μξν ρσ ¼ 2 αtβγ;ρσ þ ½ρ s σ þ 3 σ : ðB57Þ 4 4 0 3∂ Ωμν ∂ ∂ μν − ∂ ∂ ωμν ¼ ½μ ρσ ¼ 3 μ ½ρs σ 3 μ ½ρ σ Using this result in (B42), we finally arrive at 1 Eμν ρσ 0; B58 δ ν∂ ∂ ωλκ ; ¼ ð Þ þ ½σ ρ λ κ; ðB55Þ 3 upon shifting 2 which in turn is a curl in ½ρσ and can be integrated into → → η ξ Tμν;ρσ Tμν;ρσ þ tμν;ρσ;vρ;μν vρ;μν þ sμν;ρ þ 3 ρ½μ ν:

μν μν 1 ν λκ ν ðB59Þ ∂μs σ − ∂μω σ þ δσ ∂λω κ ¼ ∂σξ : ðB56Þ 4 We have thus shown that also the Eq. (3.34) follows from the Lagrangian (4.6). Finally, derivatives of (3.34) and This further reduces the result (B54) and allows to put it (3.36) give rise to the original 6D equations of motion into the form (2.16) as discussed in Sec. III C above.

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