E$_{8(8)}$ Exceptional Field Theory: Geometry, Fermions and Supersymmetry

We present the supersymmetric extension of the recently constructed E$_{8(8)}$ exceptional field theory -- the manifestly U-duality covariant formulation of the untruncated ten- and eleven-dimensional supergravities. This theory is formulated on a (3+248) dimensional spacetime (modulo section constraint) in which the extended coordinates transform in the adjoint representation of E$_{8(8)}$. All bosonic fields are E$_{8(8)}$ tensors and transform under internal generalized diffeomorphisms. The fermions are tensors under the generalized Lorentz group SO(1,2)$\times$SO(16), where SO(16) is the maximal compact subgroup of E$_{8(8)}$. Vanishing generalized torsion determines the corresponding spin connections to the extent they are required to formulate the field equations and supersymmetry transformation laws. We determine the supersymmetry transformations for all bosonic and fermionic fields such that they consistently close into generalized diffeomorphisms. In particular, the covariantly constrained gauge vectors of E$_{8(8)}$ exceptional field theory combine with the standard supergravity fields into a single supermultiplet. We give the complete extended Lagrangian and show its invariance under supersymmetry. Upon solution of the section constraint the theory reduces to full D=11 or type IIB supergravity.


Introduction
Exceptional field theories provide an E npnq covariant formulation of supergravity which unites eleven-dimensional supergravity and type IIB supergravity in a common framework [1][2][3][4]. This formulation gives a natural explanation for the exceptional symmetries known to appear upon dimensional reduction of supergravity theories [5][6][7]. It is based on the underlying symmetry algebra of E npnq generalized diffeomorphisms on an internal spacetime whose coordinates are embedded into representations of the underlying exceptional groups (together with the associated section constraints) [8,9]. Action functionals invariant under these generalized diffeomorphisms consistently reproduce subsectors of eleven-dimensional supergravity [8,10,11]. Coupling of external tensor fields and further imposing invariance under diffeomorphisms on the external part of spacetime determines a unique action functional -the exceptional field theory -which depending on the solution of the section constraint reproduces the full eleven-dimensional supergravity and full type IIB supergravity, respectively. For the lower rank groups E 5p5q " D 5p5q , E 4p4q " A 4p4q , E 3p3q " A 2p2q b A 1p1q , and E 2p2q " A 1p1q b R`, these theories have been constructed in [12][13][14][15]. Remarkably, this construction uniquely reproduces the full bosonic sectors of the higher-dimensional supergravities without any reference to the fermionic field content and supersymmetry. Nevertheless these actions can be supersymmetrized with fermions transforming under the maximal compact subgroups KpE npnq q. For E 7p7q and E 6p6q the supersymmetric completions have been worked out in [16,17].
The exceptional field theory based on the group E 8p8q appears to differ in some respects from its lower-rank cousins. In the physical context the special role of E 8p8q is often assigned to the 'dual graviton barrier'. Upon dimensional reduction to three dimensions, the Kaluza-Klein vectors are dualized into scalar fields in order to exhibit the full duality group. As a result, the scalar sector of the dimensionally reduced theory carries degrees of freedom descending from the higher dimensional 'dual graviton' [18][19][20][21]. While this dualization goes through straightforwardly for the free vector fields of the dimensionally reduced theory, their non-abelian gauge structure in presence of the higher Kaluza-Klein modes requires a modification of the standard dualization procedure. On the formal side, this is reflected by the fact that the E 8p8q generalized diffeomorphisms do not close into an algebra [8,9] which has obstructed a straightforward extension of the constructions for the lower-rank groups. In E 8p8q exceptional field theory this is taken care of by the appearance of an extra gauge symmetry in the commutator of two generalized diffeomorphisms, such that the combined algebra closes and allows for the construction of a gauge invariant action functional which in turn reproduces the full higher-dimensional supergravities [4]. In particular, the realization of this extra gauge symmetry takes the form of constrained E 8p8q rotations and requires the introduction of an additional (constrained) gauge connection B µ M which is invisible in the dimensionally reduced theory. This extra gauge symmetry is in fact a generic feature of exceptional field theories but for the lower rank groups it only kicks in at the higher-rank p-forms. More specifically, the theories associated to the E npnq groups exhibit such extra constrained gauge symmetry among the p9´nq-forms. Only for E 8p8q this symmetry comes down to the vector fields and becomes an integral part of the algebra of generalized diffeomorphisms.
In this paper, we construct the supersymmetric completion of the E 8p8q exceptional field theory. The theory is defined on a (3+248)-dimensional generalized spacetime. In addition to the usual dependency in spacetime ('external') coordinates x µ , µ " 0, 1, 2, all fields and gauge parameters formally depend also on extended coordinates Y M , M " 1, . . . , 248, transforming in the adjoint representation of E 8p8q . As usual, not all of these internal coordinates are physical. This is taken care of by the E 8p8q covariant section constraints, where η MN and f MN K are respectively the Cartan-Killing form and the structure constants of E 8p8q (see appendix A for more details on the conventions used throughout this paper), and P 3875 is the projector onto the irreducible representation 3875 in the tensor product of two adjoint representation 248 b 248 " 1 ' 248 ' 3875 ' 27000 ' 30380 , (1.2) explicitly given by The bosonic sector of the theory combines an external three-dimensional metric g µν (or dreibein e µ a ), an internal frame field V M K , parametrizing the coset space E 8p8q {SOp16q, and gauge con- in the external and internal coordinates, and for the two factors of the generalized Lorentz group, respectively. In the external sector, the SOp1, 2q connection ω µ ab is defined by the usual vanishing of external torsion according to DrA, ωs rµ e νs a " 0 ðñ Γ rµνs ρ " 0 , (1.5) where (in contrast to standard geometry) derivatives are also covariantized w.r.t. internal generalized diffeomorphisms under which the dreibein e µ a transforms as a weighted scalar. For the internal sector on the other hand, the generalized Christoffel connection is e 8p8q valued (1.6) and the proper condition of vanishing torsion amounts to the projection condition [22] r Γ M,N s 1 ' 3875 " 0 , within the tensor product (1.2). As for the lower-rank exceptional groups [8,23] The spin connections (1.4) are the central object for the description of fermionic couplings and supersymmetry transformation rules in E 8p8qˆS Op16q-covariant form. In this paper, we construct the unique supersymmetric completion of the bosonic E 8p8q exceptional field theory from [4]. Upon explicit solution of the section condition (1.1), the resulting Lagrangian reduces to full D " 11 supergravity and the IIB theory, respectively, for appropriate reformulations of these theories, as pioneered in [24][25][26].
The supersymmetric completion in particular underlines the role of the extra constrained gauge connection B µ M which joins the other fields in an irreducible supermultiplet and whose variation contributes to the supersymmetry invariance of the resulting action. Although its supersymmetry variation is given by some non-covariant expression, remarkably it turns out that the following combination of variations with supersymmetry parameter ǫ I . The structure of the r.h.s. is such that it exhibits the full internal spin connection Q M IJ , however its undetermined parts precisely cancel against the corresponding contributions from the Christoffel connection Γ M,N on the l.h.s., such that the net variation δ ǫ B µ M is uniquely determined and compatible with the constraints this connection is subject to.
The paper is organized as follows. After a brief review of the symmetry structure of bosonic E 8p8q exceptional field theory, we construct the necessary tools such as an internal fully covariant derivative and the SOp1, 2q b SOp16q spin connections needed to describe the coupling to fermions and supersymmetry transformation rules. We discuss their generalized curvatures whose components provide the building blocks for the bosonic field equations. In section 3, we then analyze the algebra of supersymmetry transformations and show that its closure into the bosonic symmetries of the theory entirely determines all the supersymmetry transformations. Finally, in section 4 we present the fermionic completion of the bosonic Lagrangian and prove its invariance under supersymmetry.
2 E 8p8qˆS Op16q exceptional geometry 2.1 E 8p8q generalized diffeomorphisms Let us start by a brief review of the field content and symmetry structures of E 8p8q bosonic exceptional field theory. For details, we refer to [4]. The field content is given by into the adjoint and the spinor of SO (16), i.e.
in terms of which the bosonic theory can be formulated. The inverse 248-bein then is given by Throughout, we raise and lower adjoint indices M, N , . . . , with the Cartan-Killing form η MN . Finally, the 248-bein is an E 8p8q group-valued matrix, which results in the standard decomposition of the Cartan form where X IJ and Y A denote the compact and non-compact generators of E 8p8q , respectively. With the explicit expressions for the structure constants in the SOp16q basis from (A.1), one finds the internal currents which will be our building blocks for the internal spin connection and later the Ricci scalar. This sums up the basic properties of the generalized vielbein.
The local symmetries of this exceptional field theory are generalized internal diffeomorphisms, constrained E 8p8q rotations, and external diffeomorphisms with respective parameters Λ M , Σ M , and ξ µ . Let us first review the generalized internal diffeomorphisms. The generalized Lie derivative acting on a vector W M of weight λ W is defined by Here P M N K L projects onto the adjoint representation 248 and guarantees compatibility with the E 8p8q structure, c.f. the explicit expression (A.3). The weight λ W of the various fields in the theory coincides with the three-dimensional Weyl weight of the fields, i.e. weight 2 and 0 for the external and internal metrics g µν and M MN , respectively, and weights 1 and 0 for the gauge connections A µ M and B µ M , respectively. Fermions come with half-integer weight. This is summarized for all fields in Table 1.
1 See appendix A for more details on the relevant group and algebra conventions.  Unlike the lower-rank E npnq cases with n ď 7, the generalized Lie derivative (2.9) depends on two parameters, Λ M and Σ M , with the latter being subject to the section condition (1.1), i.e.
This is needed together with the section constraints (1.1) in order to ensure closure of the full symmetry algebra. Schematically, we have an algebra rδ pΛ 1 ,Σ 1 q , δ pΛ 2 ,Σ 2 q s " δ pΛ 12 ,Σ 12 q , (2.11) with notably the gauge parameter Σ 12 given by confirming that the Λ transformations do not close among themselves.
Before we describe the associated gauge connections and curvatures, let us make a small digression to discuss connections and torsion compatible with the generalized diffeomorphisms (2.9). For an algebra-valued connection the fact that pure Λ-transformations do not close into an algebra implies that the naive definition of torsion asT 14) does no longer define a tensorial object. Here, L ∇ refers to generalized Lie derivatives (2.9) with partial derivatives replaces by covariant ones ∇ " B´Γ. Following [22], this suggests to rather define torsion as the part of the Christoffel connection that transforms covariantly under the generalized diffeomorphisms. With the transformation of (2.13) under (2.9) given by With the r.h.s. of (2.17) manifestly covariant, this shows that the combinationΣ M behaves as a tensorial object under generalized diffeomorphisms. In this sense it may appear more natural to parametrize generalized diffeomorphisms in terms of the parameters pΛ,Σq. The disadvantage of usingΣ w.r.t. the original formulation is the fact that the constraint (2.10) which Σ M has to satisfy, takes a much less transparent form when expressed in terms ofΣ since the connection Γ M,N in general will not be constrained in its first index and will not even be fully determined by covariant constraints. For the description of generalized diffeomorphisms we thus have the choice between a description with covariant parameters pΛ,Σq and a description in terms of parameters pΛ, Σq in terms of which the constraints (2.10) are well defined and easily expressed. We will in general stick with the latter but observe that the existence of the covariant combinationΣ M gives rise to some compact reformulations of the resulting formulas. 2 The various terms of the bosonic action are constructed as invariants under the generalized internal Lie derivatives (2.9). In the full theory, the gauge parameters Λ M and Σ M depend not only on the internal Y M but also on the external x µ coordinates. From the three-dimensional perspective, these symmetries are implemented as (infinite-dimensional) gauge symmetries, such that external derivatives are covariantized with gauge connections In accordance with (2.10), the connection B µ M is constrained to obey the same constraints as the gauge parameter Σ M . The commutator of the covariant derivatives (2.18) closes into the field strengths The ellipsis denote additional two-form terms required for the proper transformation behavior of the field strengths, c.f. (2.23) below. As required for consistency, the section constraints (1.1) ensure that all these terms drop from the commutators of covariant derivatives where the field strengths are contracted with particular differential operators according to (2.19). Moreover, all the two-form terms drop out from the bosonic Lagrangian.
Under gauge transformations The existence of the covariant combinationΣM may suggest to imposeΣ " 0 in order to reduce the number of independent gauge parameters [22] while preserving closure of the algebra. In view of the constraints (2.10), this is only possible in case the connection ΓM,N is identified with the Weitzenböck connection BMVL P pV´1qP K fN K L which itself is constrained in the first index. We will in the following keep both gauge parameters Λ M and ΣM independent which seems important for the supersymmetric extension.
(where just as the associated gauge connections, the parameters Λ M and Σ M carry weight 1 and 0 under (2.9), respectively, c.f. Table 1), the full field strengths (2.20) transform according to i.e. not the G µν M but only the combinationG µν M " G µν M´ΓM,N F µν N behaves as a tensor under (2.9). This reflects the tensorial structure (2.17) of generalized diffeomorphisms. Pushing this structure further ahead, we are led to introduce the general 'covariant' variation of the connection B µ M as  (2.17). This will turn out to be very useful in the following.

Section constraints
Since the section constraints (1.1) play a central role in the construction of the exceptional field theory, for the coupling of fermions it will be useful to spell out the decomposition of these constraints under the subgroup SOp16q according to (2.2). With the e 8p8q representations of (1.1) decomposing as the section constraints take the explicit form which we will use in the following. Following the above discussion, the same algebraic constraints hold for derivatives B M replaced by the gauge connection B µ M or its gauge parameter Σ M .
Let us recall from [4] that these section constraints allow for (at least) two inequivalent solutions which break E 8p8q to GLp8q or GLp7qˆSLp2q, and in which all fields depend on only eight or seven among the 248 internal coordinates Y M , respectively. The resulting theory then coincides with the bosonic sector of D " 11 and type IIB supergravity, respectively.

Spin connections
Fermions enter the theory as spinors under the SOp1, 2qˆSOp16q generalized Lorentz group and transform as weighted scalars under generalized diffeomorphisms. Their couplings thus require four different blocks of the spin connection that ensure covariance of both external and internal derivatives under SO(1,2) and SO (16), respectively. Via the generalized vielbein postulates for the external and internal frame fields, the spin connections relate to the external and internal Christoffel connections Starting with the external sector, the SO(1,2) connection ω µ ab is defined by the vanishing torsion condition of the external Christoffel connection This leads to the standard expression for the spin connection in terms of the objects of anholonomity Ω abc " 2 e ra µ e bs ν D µ e νc , where however derivatives are covariantized according to (2.18) with the dreibein transforming as a scalar of weight 1 under (2.9). The external SO (16) connection on the other hand is defined by imposing that the external current lives in the orthogonal complement of sop16q within e 8p8q : In analogy to (2.8) this yields the explicit expressions with covariant derivatives from (2.18). According to their definition, the currents P µ and Q µ satisfy Maurer-Cartan integrability conditions W.r.t. the integrability relations of D " 3 supergravity [27], these relations represent a deformation with additional terms in field strengths due to the introduction of the gauge fields A µ M and B µ M . We will see in the next section how these terms take a manifestly covariant form.
In the fermionic sector, the full external covariant derivatives acting on the SOp1, 2qˆSOp16q spinors of the theory are given by In order to define the internal SO(16) connection, we recall that the proper condition of vanishing torsion in the internal sector is given by setting to zero the tensorial part (2.16) of the Christoffel connection Γ MN K . Via (2.29) this condition determines a large part of the SO (16) connection. More precisely, the counting goes as follows [22]: decomposition of (2.16) into SO(16) irreducible representations To explicitly solve these equations (2.43), we use (2.29), to express the internal Christoffel connection in terms of derivatives of the vielbein or, more explicitly To summarize, the full internal covariant derivative act on an E 8p8qˆS Op16q tensor X M I of weight λ X as with the connections defined by (2.47) and (2.45), respectively. This covariant derivative transforms as a generalized tensor of weight λ " λ X´1 under generalized diffeomorphisms. In particular, for the spinor fields of the theory, the covariant internal derivatives take the form We conclude this section with a collection of the different covariant derivatives we have used and will use throughout this paper:

Curvatures
Having defined the various components of the spin connection (2.28), we can now discuss their curvatures which will be the building blocks for the bosonic Lagrangian and field equations. Moreover, we will require a number of identities for the commutators of covariant derivatives in order to prove the invariance of the full Lagrangian under supersymmetry.
Let us start with the commutator of two external covariant derivatives on an SOp1, 2qŜ Op16q spinor ǫ I which is obtained straightforwardly from (2.37) with the tensorial combination of field strengths r G µν M from (2.23). Similarly, one may rewrite the second integrability relation (2.35) into the manifestly covariant form We now turn to the mixed curvature, arising from the commutators of one external and one internal covariant derivatives. We will only be interested in those projections of this commutator, in which the undetermined part of the SO(16) connection drops out. Fortunately, they are the projections relevant to prove the invariance of the Lagrangian under supersymmetry. Evaluating different projections of such a commutator on an SOp1, 2qˆSOp16q spinor ǫ I , we obtain the relations A r∇ M , D µ s ǫ I " up to a boundary currents J M A , I µ A of respective weights λ J A "´1, λ I A "´2 , that do not contribute under the integral.
Finally, for the internal curvature, we are again interested in specific projections of two internal covariant derivative in which the undetermined part of the connection drops out. The pertinent projection for the definition of an internal curvature scalar R in the E 8p8qˆS Op16q exceptional geometry is given bỳ (2.66) By construction it transforms as a scalar (of weight λ R "´2) under generalized diffeomorphisms (2.9). Its dependence on the external metric is such that δpeRq " pδeq R`total derivatives . The other relevant projection of two internal derivatives on a spinor is given bý where again all double derivatives on the l.h.s. vanish due to the section constraints. The generalized curvature R A on the r.h.s. plays the analogue of a Ricci tensor in this geometry and is most conveniently defined by variation of the curvature scalar R w.r.t. to a non-compact local e 8p8q transformation of the internal frame field, i.e.
up to a boundary current J M Σ of weight λ J Σ "´1 . It can be explicitly given in terms of the Cartan forms (2.8) as (2.70) This expression above is given in compact form, after simplification by various Fierz-like identities, some of which are collected in appendix B.

Supersymmetry algebra
In this section we establish the supersymmetry transformation of the various fields and verify that the supersymmetry algebra closes. Before discussing supersymmetry, we briefly review the bosonic symmetries of E 8p8q exceptional field theory, since these are the transformations we are going to recover in the commutator of two supersymmetry transformations.

Bosonic symmetries of E 8p8q exceptional field theory
In section 2.1 we have extensively discussed the structure of internal generalized Lie derivatives which depend on two parameters Λ M and Σ M with associated gauge connections A µ M and B µ M . A closer analysis [4] shows that these gauge connections come with additional shift symmetries which take the form Apart from the internal gauge symmetries, the full set of bosonic symmetries also includes a covariantized version of the (2+1)-external diffeomorphism with the parameter ξ µ depending on both set of coordinates tx µ , Y M u. On the bosonic fields these act as 3

Closure of the supersymmetry algebra
Let us now move on to the fermionic fields and the supersymmetry algebra. In addition to the bosonic fields introduced in section 2, the supersymmetric completion of the E 8p8q exceptional field theory contains the following spinor fields: sixteen gravitinos ψ µ I as well as 128 matter fermions χ 9 A , transforming in the vector and spinor representation of SO(16), respectively. With respect to generalized diffeomorphisms, they transform as scalar densities with half-integer weights given in Table 1. We are working in the Majorana representation and mostly minus signature, i.e. spinors are taken to be real and SO(1,2) gamma matrices γ µ purely imaginary, c.f. [28] for our spinor conventions. In particular, we use γ µνρ "´ieε µνρ .
3 W.r.t. the form of these transformations given in [4], we have expressed the current bosonic current j ρM by the coset current P ρM , see (4.4) below, and furthermore changed the vector transformations by a shift transformation (3.1) with parameter Ξµ M "´gµν BMξ ν , in order to obtain a more compact presentation of the external diffeomorphisms. Also some signs differ from the formulas in [4] due to the fact that in this paper we use mostly minus signature p`´´q for the external metric.
In this section, we present the supersymmetry transformation rules δ ǫ e µ a " iǭ I γ a ψ I µ , As a first test, we use this ansatz to calculate the commutator of two supersymmetry transformations on the dreibein e µ a to obtain The first term reproduces the action of covariantized external diffeomorphisms, the second and third term describe the action of internal generalized diffeomorphisms on the dreibein, and the last term is an SO(1,2) Lorentz transformation, with the respective parameters given by Similarly, one can show closure of the supersymmetry algebra on the 248-bein. Using (3.3), we find the commutator with the adjoint projector from (A.3). We recognize the first term as the action of external diffeomorphisms on the 248-bein. The second term reproduces the action (2.9) of a generalized internal diffeomorphism with parameter Λ L when parametrized covariantly as in (2.17) (note that the transport term Λ N ∇ N V M KL vanishes due to the vielbein postulate (2.29)). The last term thus describes the covariantized E 8p8q rotation from which we read off the parameterΣ Ñ (3.7) As a consistency check, it is straightforward to verify that although the expression for the parameter (3.7) carries the full internal SO(16) spin connection Q N IJ (including its undetermined parts), its form is such that the constrained parameter Σ N "Σ N`ΓN ,M Λ M which actually appears in the rotation term of (2.9) is uniquely determined (with the undetermined part from Q N IJ cancelling the undetermined part from Γ N ,M ) and moreover satisfies the required constraints (2.10).
Also on the gauge field A µ M we obtain closure of the supersymmetry algebra by a standard calculation which gives the explicit result with the parameters Λ M and ξ µ from (3.5) and the shift parameters Ξ µ of the last line defined as corresponding to the shift symmetries (3.1) discussed above. The fact that Ξ µ 3875 pMN q lives in 3875 representations is an immediate consequence of its specific form with a parameter ξ µ IJ in the 135 of SO (16), combined with the fact that the tensor product of two adjoint representations (1.2) contains only a single representation 135 of SO(16) which lives within the 3875 representation of E 8p8q . Moreover, the last term in (3.9) carrying the Christoffel connection ensures that the parameter Ξ µ N K does not carry any of the undetermined parts of the SO(16) connection Q N IJ and furthermore is constrained in its first index, as required by the shift symmetries (3.1).
We have at this point fully determined the supersymmetry algebra with parameters given in (3.5), (3.7), (3.9). As a consistency check of the construction it remains to verify that the algebra closes in the same form on the constrained connection B µ M . This computation is greatly facilitated by the notation of the covariant variation (2.24) in terms of which its supersymmetry variation takes the covariant form (3.3). To lowest order in fermions, the supersymmetry algebra on B µ M is given by For the second term we may use the closure of the algebra on the vector fields A µ M established above. The first term after some calculation yields with the parameters given in (3.5), (3.7), (3.9) and the covariant combinatioñ (3.14) The first line of (3.13) reproduce the covariant variation of B µ M under generalized internal and external diffeomorphisms. For the supersymmetry algebra to close, the second and third line of (3.13) must reproduce the shift symmetries where we have obtained the last equality with the use of the following identitỳ This is reminiscent of standard Riemannian geometry, where the curvature of the Christoffel symbols is the curvature of the spin connection R µν ρσ rΓs " R µν ab rωs e a ρ e b σ , (3.17) albeit here, in a projected fashion.
This proves the closure of the supersymmetry algebra on B µ M and concludes the discussion on the consistency of the supersymmetry algebra (3.11).

Action
Having introduced fermion fields and supersymmetry transformation laws, we now have at our disposal all the necessary tools to construct the fermionic completion of the E 8p8q bosonic Lagrangian constructed in [4]. We start by giving a brief review of the bosonic Lagrangian in the form most suited for the coupling of fermions before presenting its supersymmetric completion.

The bosonic theory
Let us start by a brief review of the action of bosonic E 8p8q exceptional field theory following [4] 4 however adapted to the further coupling of fermions, in particular using the internal frame field (the 248-bein) from (2.2). The bosonic field content has been given and discussed in (2.1). The action of bosonic E 8p8q exceptional field theory is given by 5 where each term is separately invariant under generalized internal diffeomorphisms (2.9). The Einstein-Hilbert Lagrangian is given by the Ricci scalar (2.56) obtained from contraction of the improved Riemann tensor where e denotes the determinant of the dreibein e µ a . The scalar kinetic term in (4.1) is given by where we have used the expression of the scalar currents (with covariant derivatives from (2.18)) in terms of the E 8p8q structure constants f MK L and the coset currents (2.34). The topological term in (4.1) carries the nonabelian Chern-Simons couplings of the gauge connections according to (4.5) Its covariance becomes manifest upon spelling out its variation as with the covariant field strengths from (2.20), (2.23) and the general covariant variation introduced in (2.24). As anticipated above, we note that the two-form contributions to the field strengths F and G (whose explicit form has been suppressed in (2.20)) drop out from this expression due to the section constraint. Moreover, the contributions to the Christoffel connection in ∇ K that are left undetermined by the vanishing torsion condition cancel in this expression against the corresponding contributions in ∆B ρ M .
Finally, the last term in (4.1) carries only derivatives in the internal coordinates and is explicitly given by with the 'potential' V given by in terms of the internal current j M N defined as where we have used (2.45) in the last equality. The scalar potential (4.8) then takes the manifestly covariant form with the general covariant variation of (2.24), the Chern-Simons couplings together with the minimal couplings in the covariant derivatives give rise to duality equations relating the field strengths to matter currents according to (4.12) with the covariant field strengths from (2.20), (2.23) and the current p J ρ M from (2.62). Let us stress that the equations of motion do not imply the full vanishing E pAq M µν " 0 due to the fact that the corresponding variation (4.11) is a variation w.r.t. a constrained gauge connection subject to the section constraint (2.10). In other words, the equations of motion only imply the weaker projected equation where O µν M vanishes when contracted with a field satisfying the section constraints (2.10).

Supersymmetric Lagrangian
We can now present the supersymmetric completion of the bosonic action (4.1). The fermionic field content comprises the gravitinos ψ µ I and spin 1/2 fermions χ 9 A transforming in the fundamental vector 16 and spinor 128 c representations of SO(16), respectively. The full E 8p8q Lagrangian is given by (4.14) The first line is the bosonic Lagrangian (4.1). The terms in the second line are obtained via a direct uplift (and proper covariantization) from D " 3 maximal supergravity [27,28]: a Rarita-Schwinger term for the gravitinos ψ I µ , a kinetic term for the 128 matter fermions χ 9 A , and the Noether coupling between the coset current P µ A and the fermions. The three last terms of (4.14) carrying internal covariant derivative ∇ M have been added to ensure invariance of the Lagrangian under supersymmetry transformations. After proper Scherk-Schwarz reduction of the Lagrangian [29], these terms provide the Yukawa couplings of the gauged three-dimensional supergravity. Finally, L quartic denotes the quartic fermion terms. We expect these to coincide with the corresponding terms of the three-dimensional theory [27,28] e´1 L quartic "´1 2´χ γ ρ Γ IJ χ`ψ I µ γ µνρ ψ J ν´ψ I µ γ ρ ψ µJ˘`χ χ ψ I µ γ ν γ µ ψ I ν1 2´p χχqpχχq´1 12 χγ µ Γ IJ χ χγ µ Γ IJ χ¯, (4.15) but as far as this paper is concerned we will only deal with fermions at quadratic order.
For the proof of invariance of (4.14) under supersymmetry (3.3), we first note that all terms that do not carry internal derivatives cancel precisely as in the three-dimensional theory. Terms carrying internal derivatives arise in the bosonic sector from variation of the potential V and the topological term L top . In the fermionic sector, such terms arise from the corresponding terms in the supersymmetry transformations (3.3), from variation of the last three terms in (4.14), as well as from the modified integrability relations (2.57), (2.58).
We organise these terms according to their structurē and show that they cancel against the contributions from the bosonic Lagrangian. In the rest of this section, we will only focus on the last two types of terms in (4.16), which carry two internal derivatives and thus exhibit an interesting geometric structure of the internal space. The cancellation of the remaining terms is described in detail in appendix C.
Let us start by collecting the terms inψ ∇ M ∇ N ǫ in the variation of the fermionic Lagrangian Upon use of the section constraints (2.27) and together with the identity (2.64), one can show that all the quadratic and linear terms in derivatives of ǫ vanish. Then, the remaining terms cancel the first two lines of the variation of the scalar potential (4.10) under a supersymmetry transformation (up to total derivatives) where for the cancellation we have used the following identity The last line in (4.18) then cancels against the corresponding terms from the variation of the fermionic Lagrangian Using the identity (2.68) and the section constraints (2.27) one finds that all quadratic and linear terms in ǫ vanish while the remaining terms precisely cancel the last line of (4.18). For this, the following relations are useful We have thus sketched the vanishing of all terms carrying two internal derivatives in the supersymmetry variation of (4.14). The cancellation of the remaining terms is described in detail in appendix C. To summarize the result, we have shown invariance of the action (4.14) up to quartic fermion terms.

Conclusions
In this paper we have constructed the supersymmetric completion of the bosonic E 8p8q exceptional field theory. The final result is given by the action (4.14) and the supersymmetry transformation laws (3.3). In particular, we have established the supersymmetry algebra which consistently closes into the generalized internal and external diffeomorphisms together with the tensor gauge transformations of the theory. The geometry of the internal space is constrained by the section condition (1.1) which admits (at least) two inequivalent solutions for which the action (4.14) reproduces the full D " 11 supergravity and full type IIB supergravity, respectively. The fermions of exceptional field theory can consistently accommodate the fermions of the type IIA and type IIB theory, since the E 8p8q -covariant formulation (4.14) does not preserve the original D " 10 Lorentz invariance. The resulting D " 10 fermion chirality thus depends on the solution of the section constraint.
In contrast to the standard formulation of supergravities, in exceptional field theory the bosonic symmetries already uniquely determine the bosonic Lagrangian without any reference to fermions and supersymmetry. Nevertheless, it is important to establish that the resulting bosonic Lagrangian allows for a supersymmetric completion upon coupling of the proper fermionic field content as we have done in this paper. In particular, in the context of generalized Scherk-Schwarz reductions [29] this construction provides the consistent reduction formulas for the embedding of the fermionic sector of lower-dimensional supergravities into higher dimensions.
A particular attribute of E 8p8q exceptional field theory is the appearance of an additional constrained gauge connection B µ M related to an additional gauge symmetry which ensures closure of the algebra of generalized diffeomorphisms. Unlike all other fields of E 8p8q exceptional field theory, this gauge connection is invisible in three-dimensional supergravity. More precisely, upon a consistent truncation of exceptional field theory down to three dimensions by means of a generalized Scherk-Schwarz reduction M MN px, Y q " U M K pY q U N L pY q M KL pxq , g µν px, Y q " ρ´2pY q g µν pxq , with the Y -dependence carried by an E 8p8q matrix U and a scaling factor ρ (satisfying their system of consistency equations), the constrained gauge connection B µ M reduces according to such that its fluctuations are expressed in terms of the same three-dimensional vector fields A µ N pxq that parametrize the fluctuations of the A µ M px, Y q . It is thus tempting to wonder if already in exceptional field theory, and before reduction, the constrained gauge connection can be considered as a function of the remaining fields such as [22] B µ M c.f. (2.17). However, as seen above, coupling to fermions requires a connection Γ M,N other than the Weitzenböck connection, such that (5.3) would obstruct compatibility with the constraints (2.10). Moreover, supersymmetry of the Lagrangian requires a non-trivial transformation law (3.3) for the constrained connection B µ M . It is remarkable that as we have shown above this additional constrained connection consistently joins the remaining bosonic and fermionic fields into a single supermultiplet without the need of additional fermionic matter.
The fact that all transformation laws of B µ M are most compactly expressed in terms of the general covariant variation (2.24) is remnant of structures that appear in a general tensor hierarchy of non-abelian p-forms [30]. This may hint at a yet larger algebraic structure which in particular unifies the topological term and the generalized three-dimensional Einstein-Hilbert term of (4.1) into a single non-abelian Chern-Simons form on an enlarged algebra. If the present construction should allow for a generalization to the infinite-dimensional cases of E 9 [31][32][33], E 10 [34,35], (and maybe E 11 [20,36]), this appearance of additional bosonic representations and their interplay with supersymmetry may play an essential role.

Appendices A E 8p8q conventions
The E 8p8q generators t M split into 120 compact ones X IJ "´X JI and 128 non-compact ones Y A , with SOp16q vector indices I, J,¨¨¨P 16 , spinor indices A, P 128, and the collective label M " prIJs, Aq. The conjugate SOp16q spinors are labeled by dotted indices 9 A, 9 B, . . . . In this SOp16q basis the totally antisymmetric E 8p8q structure constants f MN K possess the nonvanishing components: with components η AB " δ AB and η IJ KL "´2δ IJ KL . When summing over antisymmetrized index pairs rIJs, an extra factor of 1 2 is always understood. We will also need the projector onto the adjoint representation We refer to [26,39] for other useful E 8p8q identities.

B Gamma matrix identities
In this appendix, we give some of the SO(16) gamma matrices identities we have used to rewrite the curvature R A in a more compact form. We started with 14 terms quadratic in the Cartan forms, where a simple counting gives only 12 independent terms. Then using an explicit representation of the SOp16q gamma matrices together with the section constraints (2.27), we were able to write R A with 7 independent terms quadratic in the Cartan forms.
The main identities behind this simplification are the following