On exceptional QP-manifolds

The connection between two recent descriptions of tensor hierarchies - namely, infinity-enhanced Leibniz algebroids, given by Bonezzi&Hohm and Lavau&Palmkvist, the p-brane QP-manifolds constructed by Arvanitakis - is made precise. This is done by presenting a duality-covariant version of latter. The construction is based on the QP-manifold $T^\star[n]T[1]M \times \mathcal{H}[n]$, where $M$ corresponds to the internal manifold of a supergravity compactification and $\mathcal{H}[n]$ to a degree-shifted version of the infinity-enhanced Leibniz algebroid. Imposing that the canonical Q-structure on $T^\star[n] T[1] M$ is the derivative operator on $\mathcal{H}$ leads to a set of constraints. Solutions to these constraints correspond to $\frac{1}{2}$-BPS p-branes, suggesting that this is a new incarnation of a brane scan. Reduction w.r.t. to these constraints reproduces the known p-brane QP-manifolds. This is shown explicitly for the SL(3)$\times$SL(2)- and SL(5)-theories. Furthermore, this setting is used to speculate about exceptional 'extended spaces' and QP-manifolds associated to Leibniz algebras. A proposal is made to realise differential graded manifolds associated to Leibniz algebras as non-Poisson subspaces (i.e. not Poisson reductions) of QP-manifolds similar to the above. Two examples for this proposal are discussed: generalised fluxes (including the dilaton flux) of O(d,d) and the 3-bracket flux for the SL(5)-theory.


Introduction
Gauge theories are a corner stone of modern physics.Typically, these are based on some underlying Lie algebra structure.The natural curiosity of a mathematical physicist but also certain evidence in string theory lead to the question, whether this can this be stretched to more general algebraic structures.The study of supergravity compactifications, in particular gauged supergravities [1][2][3][4], gave raise gauge theories based on Leibniz (or Loday) algebras [5,6], whereas Lie 3-algebras appeared in the study of 6d superconformal field theories [7,8].Former in turn require p-form hierarchies of field strengths, gauge fields and gauge transformations.
The natural language for ungauged supergravity is the one of exceptional field theory and E d(d) generalised geometry [9][10][11][12][13][14][15][16][17][18], see [19] for a recent review.Gauged supergravity, in addition, needs the introduction of tensor hierarchies and generalised Cartan calculus [20][21][22][23][24][25][26].This is wellunderstood from a target space perspective.A topic that did not get a lot of attention is how such Leibniz gauge fields and the associated tensor hierarchies couple or relate to world-volumes of p-branes.For conventional p-form gauge fields, it is clear that the natural object, to which they couple, are p-dimensional world-volumes, basically via the pull-back of theses gauge fields to the world-volume.This article aims to contribute in this direction.
E d(d) -covariant p-brane world-volume theories have been constructed via actions [27][28][29][30][31][32] or via Hamiltonian formulations [10,[33][34][35][36][37][38][39].Topological terms and topological field theories to such p-branes, that are the dynamical objects in type II and eleven-dimensional supergravities, have been considered, partly including tensor hierarchies [40][41][42][43][44], but not in an E d(d) -covariant way.In these cases the underlying E d(d) -structure is rather hidden, and also the construction of the appearing manifolds is somewhat ad hoc.On the other hand, for O(d, d) generalised geometry topological σ-models, namely Courant σ-models (topological strings), were established in [45][46][47].These are based on the AKSZ-construction [48] and concrete connections to generalised geometry were used for example in [49,50].In related contexts for O(d, d) generalised geometry in [51][52][53][54], supergeometric methods were applied.One aim of this article is to present a way to derive similar objects for p-branes from an underlying object, an E d(d) -covariant QP-manifold.duality group O(d, d) SL(3) × SL(2) SL( 5) SO (5,5)  For the purposes of this article a tensor hierarchy will be the sequence where R p are representations of some underlying Lie algebra g, the Lie algebra of the duality groups E d(d) or O(d, d).The representations relevant here have been collected in table 1.
∂ : R p → R p−1 is a derivative operator with ∂ 2 = 0. Also there is a product • : R p × R q → R p+q .
Recently, Bonezzi & Hohm [55,56] and Lavau & Palmkvist [57], building on earlier results [58][59][60][61], understood the algebraic structure of a tensor hierarchy as a differential graded Lie algebra.There the tensor hierarchy is realised on a graded vector space where the •-product corresponds to the graded Lie bracket and the derivative ∂ is eponymous for the differential graded Lie algebras.In comparison, the construction by Arvanitakis [41,43] is based on a QP-manifolds M of degree n, for which functions of degree n − p ∼ R p (1.3) the graded Poisson structure (P-structure) corresponds to the •-product and a Hamiltonian Qstructure to the derivative ∂.Apart from that, many features in these two constructions are strikingly similar.For example, a Leibniz product resp.a generalised Lie derivative between objects in R 1 can be defined as a derived bracket In section 2 it is presented how these two points of view -the differential graded Lie algebra and QP-manifold ones -can be connected.The central underlying object will be leads to a hierarchy of of constraints of the schematic form (suppressing the index structure): where t (p) are coordinates on M ⋆ corresponding to generators of R (p) and ξ are the fibre coordinates on T[1]M.Their explicit form is given in (2.12).Hence, the striven for QP-manifolds are Solutions to the constraints Φ (p) will reproduce all the QP-manifolds constructed in [41,43].Consequently, this procedure could be considered as a new kind of brane scan.This is shown explicitly for the low-dimensional exceptional field theories SL(3)×SL(2) and SL (5).Some questions remain open already for E 6 (6) and even more fundamentally for E 7 (7) and E 8 (8) , as a more intricate interplay of T ⋆ [n]T [1]M and the tensor hierarchy or a more elaborate tensor hierarchy (then the one based on the representations in table 1) might need to be considered.All of the above, in some sense, plays on the algebroid level, -i.e. the derivatives ∂ of the tensor hierarchy are, in the examples here, related to derivatives w.r.t. to some underlying manifold.For Lie algebras, there is well-known statement that the g [1] associated to a Lie algebra g can be characterised by a cohomological vector field because Q 2 = 0 is equivalent to the Jacobi identity of g.Section 3 mainly deals with the question whether such an equivalence can be found for Leibniz algebras as well.
One aim is to generalise the following statement for the O(d, d)-setup [62].On T ⋆ [2]T [1]M, the underlying QP-manifold of the Courant σ-model, we can choose a different Hamiltonian where where [AB] and S A,B C correspond to the skew-symmetric and symmetric parts of the Leibniz algebra structure constants and t (p) A , t(p)A are generators of R p resp.Rp .z A , ξ A are some functions on R 1 resp.R1 .In order to proceed, two identifications need to be performed.Schematically, these are t (1) ∼ zd and t (1) ∼ ξ • t (2) .Latter is similar to the constraints in section 2. But here these conditions seem not to be implementable as constraints without introducing new negative degree coordinates.The results of all calculations, for example the master equation, are simply projected onto this constraint surface, i.e. without performing symplectic reduction {Θ, Θ} constraints = 0 ⇐ Bianchi identity. (1.10) The two examples on which this conjecture is demonstrated are the constant generalised fluxes for O(d, d), including the dilaton flux F A , and constant Q-flux, corresponding to a Nambu bracket structure constants, in the SL(5)-theory.
On the way to this conjecture the question of exceptional extended space in the context of the QP-manifolds pops up.The conclusion is that the inclusion of the extended coordinates seem to require the relaxation of either graded Jacobi identity of the P-structure or of the requirement of absence of no negative degree coordinates.
2 From enhanced Leibniz algebroids to p-brane QP-manifolds

The construction of the underlying QP-manifold and the constraint
In [55,56] (infinity-)enhanced Leibniz algebras have been defined via a quadruple (H, •, ∂, •).For the purpose of this paper, we let H be a graded vector space that is composed of a direct (graded) sum of representations R i of a Lie algebra g, corresponding to a Lie group G So, in contrast to [56,57], the grading is chosen such that R 1 has degree n − 1.
2. the •-product is identified with a degree −n graded Poisson structure (P-structure) of functions of degree less n, So far, this degree n is arbitrary.In the following, we call this graded Poisson manifold, given by (H, •), the (n-shifted) hierarchy algebra H Let it be noted that (H[n], •) is, in general, not a graded symplectic manifold but, by construction, only a graded Poisson manifold.
4. the generalised Lie derivative for a Λ = Λ M t (1) For Λ i ∈ R 1 this also gives the Leibniz product on the degree (n − 1)-objects, The generalised Cartan calculus follows from the graded Jacobi identity: The properties of (H, •, ∂, •), as described in [55,56], follow from the graded Jacobi identity of the underlying graded Poisson structure and the master equation {Θ, Θ} = 0.This is analogous to the treatment as differential graded Lie algebra in [56,57].
The underlying QP-manifold.There are some obvious issues with the above: • This differential graded manifold setting should be much more restrictive than the differential graded Lie algebra picture from [56,57].I.e. with such manifold we need to identify conventional products of functions, like U (p) V (q) , with some element W (p+q−n) .This seems to imply a plethora of additional conditions, moreover even depending on n, none of with are a part of the original data of the infinity enhanced Leibniz algebroid.
• So far, the Hamiltonian that should realise the chain of derivatives, ∂ = {Θ, •}, is not defined.Also, it seems speculative that all of the derivatives ∂ p : R p+1 → R p can come from the same Hamiltonian.For the differential graded Lie algebra this was explained in [64].
Moreover, for the examples of the duality hierarchies considered here, the derivative ∂ is typically connected to some derivative on some underlying manifold, which did not appear so far: For SL(3)×SL(2) and SL(5) these D α p β p−1 ,M have been collected in this article's conventions in appendix A. One way to solve these issues will be based on the following QP-manifold over a smooth D-dimensional1 manifold.The relevant coordinates and their degrees are , latter given explicitly in appendix A and the canonical Hamiltonian on (2.8) In order to keep g-covariance (and independence of the choice of section), the coordinates on , were extended to be full R 1 resp.R1 representations.On a Ddimensional solution to the section condition, i.e. ∂ M = (∂ µ , 0), µ, ν, ... = 1, ...D, only the coordinates (x µ , ξ µ , z µ , π µ ) will appear explicitly in all the calculations that are relevant -in particular, this is the case for the derived brackets {{Θ, u}, v} for deg u, v ≤ n, that really are of interest here.
Charge conditions and the reduced QP-manifold.The interplay of the two ingredients is twofold in order to address the above issues: • On a given solution of the section condition ∂ M = (∂ µ , 0), the components t (1) As a constraint, we write: or, in a duality covariant way, • The key restriction on the interplay of the two ingredients H[n] and for p > 1, that fixes parts of the generators t (p−1) in terms of t (p) and ξ.Moreover, we expect them to fix n as well.Following the literature [30,38], where such conditions appeared first in the construction of duality covariant p-brane σ-models, these constraints are referred to as charge conditions.For G = E d(d) (shown explicitly for d ≤ 4), their solutions correspond to the 1 2 -BPS p-branes, as shown in the following subsection.
Hence, the QP-manifold M ⊂ M ⋆ that we aimed for, fulfilling all the properties 1.-4.from the previous paragraph and consequently useful to define a infinity enhanced Leibniz algebroid, is i.e. the submanifold M ⊂ M ⋆ being subject to a Poisson reduction w.r.t. the constraints Φ (p) ≈ 0.
Constraint algebra.It should be checked whether the charge conditions lead to any secondary constraints or their application requires the Dirac procedure.For SL(3)×SL(2) and SL (5), one finds that they are first class, i.e. they Poisson commute weakly.This is easy enough to show explicitly for any of the solutions of these constraints (the p-brane charges introduced in the next subsection).Without solving the constraints (and the applying the section first), it possible to be proven in a duality covariant manner for the SL(5)-theory.Using the standard conventions from appendix A, we can write the constraints in the following explicit form,2 N N ′ ∂ LL ′ ≈ 0, Using the canonical Poisson brackets on and the section condition, the corresponding constraint algebra is: These constraints close into each other nicely.The other brackets, i.e. those involving Φ (4) or Φ (5) on either the left or right side, do not seem to commute weakly and generate secondary constraints, i.e. t (5) = 0 and t (4)M ∂ M = 0.As we will see in the next subsection, these conditions are fulfilled for any solution in the SL(5)-theory. 3In other words, t (5) = 0 and t (4)M ∂ M = 0 are implied already by the primary constraints (2.12).

E d(d) and solutions for d ≤ 4
For SL (5) in the standard conventions of appendix A, the charge conditions (2.14) take the following form: • M-theory section: ) • type IIb section: µλ ) (2.17) The corresponding solutions are: For the D3-brane there is the remaining free coordinates qm with { qm , qn } = ϵ mn q D3 .Each solution (associated to a p-brane) is characterised by an element q in R p+1 , taking account to the fact that the charge conditions (2.12) are linear in the generators t (p) , i.e. solutions for same n could be simply added.Moreover, there is always a solution with t (p) = 0 for p > 1 and t M = (z µ , 0).This solution corresponds to neglecting the hierarchy completely and will be dubbed point particle solution in the following.
It is straightforward to check that the same solutions can be obtained from SL(3)×SL(2), starting from the charge conditions (2.12).Technical details, necessary for the calculation, can be found in appendix A. Some general comments.The solutions correspond exactly to the QP-manifolds for the 1  2 -BPS branes, already found by Arvanitakis [41,43].Hence, this construction could be considered as new version of a brane scan -in particular one that, given a similar construction exists for E d(d) with d ≥ 7, could give rise to world-volume incarnations of exotic branes, as well.The severely constrained form of the QP-manifold in comparison to the infinity-enhanced Leibniz algebra resp.differential graded Lie algebras might be surprising.One physical interpretation is the following: whereas latter were introduced at the level of the (gauged) supergravity, QP-manifolds are used describe (the topological part of) world-volume theories.These are not expected to contain the full information, as has been already observed on the level of actions [30] and the current algebra [38].
So far, the charge conditions (in different settings) (2.12) have appeared only as relations between R 1 -and R 2 -representations [30,38].It was only possible to solve these equations, because an identification like z M ∼ η K,LM ξ L Q K was made.This requires an extended spaces, i.e. components of ξ L other than ξ λ appear.
The form of the reduced QP-manifold to a given p-brane is universal, in sense that it does not depend w.r.t. which E d(d) it is defined (if dimensionality allows this p-brane).Of course, the dimension of M must change.Typically, a solution of the charge condition (2.12) does not fix the n of the QP-structure.The above choice -a p-brane -is associated to the n = p + 1 is fixed by the characterising 'charges' q M2 , q D3 , ... to be degree 0. In these solutions for d ≤ 4, this ensures that • functions of degree (n − p) ∼ R p , for p > 0, as observed in [30].This solves the above mentioned issue of the additional multiplicative structure (in comparison the a differential graded Lie algebra), i.e. that U (p) V (q) ∼ W (p+q−n) .
• the hierarchy terminates.There are no non-negative degrees.I.e.t (p) ≡ 0, for p > n for all the found solutions.
• the reduced manifold M ⊂ M ⋆ actually becomes a graded symplectic manifold.In principle, the procedure does not impose this.In general, M is only a graded Poisson manifold.
As a last remark on the applicability of the SL(3)×SL(2)-and SL(5)-hierarchy to the present setup let us note that, in principle, there is a violation of the axioms of the graded Poisson structure of the hierarchies from the SL(3)×SL(2)-and SL(5)-theories -graded skew-symmetry resp.graded Jacobi identity for certain elements.These violations are not relevant on the p-brane charge solutions, see appendix B for details, but are expected to appear at the end of the hierarchy as the dual graviton becomes relevant at R (9−d) for E d(d) .
The following convention for the •-product and derivative ∂ for representations R 1 = 27, R 2 = 27, R 3 = 1 of E 6(6) with generators t (1) M , t (3) are used to define the Qand P-structure: {t M t (1) where d LMN is the symmetric invariant of E 6 (6) .Details on how this invariant looks in the index decompositions M-theory : type IIb : with µ, ν, ... = 1, ...5 resp.µ, ν, ... = 1, ..., 4 and m, n, ... = 1, 2, can be found, for example, in [38,65].The corresponding solutions of the charge conditions (2.12) (beyond those for the lower dimensional branes that also exist here and are the same as above) are: assuming n = 6.q m and q M5 are the NS5/D5 resp.M5 'charges', qm and q are some degree 5 resp.4 not fixed by solving the charge conditions (2.12).Latter have some similarities to the M2 and F1/D1 charges.Despite the fact that (for q M5 ) this M5-solution takes the same form regarding the spaces of coordinates/functions as the QP-manifold associated to the M5 in [41], simply solving the charge condition does not lead to the same P-structure.E.g., here { q, q} = 0, in comparison to { q, q} = 1 in [41].In particular, this leads to inconsistencies like {t (1)µµ ′ , t (1)νν for the M5-brane solution, and one arrives at two issues arising when applying this papers procedure for E 6(6) that are left for future research: • For E 6(6) the constraint algebra requires the Dirac procedure, as some of the constraints do not Poisson commute weakly, e.g. in the M-theory section • In order, to arrive at the M5-brane QP-manifold constructed by Arvanitakis [41,43], and also correctly reproduce, one needs to have {t (3) , t (3) } ̸ = 0. Even, when applying the Dirac procedure, this is not possible because for the E 6(6) -hierarchy, as introduced above, t (3) is a central part of algebra.
A part of the solution to these problems might be not to impose the constraint , so the one involving the end of the tensor hierarchy, would change nothing for the calculations for SL(5) and SL(3)×SL( 2), which would correspond to d = 3 or 4. Also, for 5-branes the appearing hierarchy of functions is longer than the length of the hierarchy, which ends here at R 3 .The other ones are part of the de Rham-complex T[1]M.In the end, a modified approach, similar to the E 8(8)theory [22] where even the generalised Lie derivative needs to be adjusted in order to accomodate the dual graviton.Supposedly, one has to use a different hierarchy or there has to be an additional non-trivial interplay of the hierarchy and the de Rham-complex.

Towards differential graded manifolds from Leibniz algebras
As commented on in [41] one can consider more general degree n + 1 functions as Hamiltonian for the resulting p-brane QP-manifolds.Namely, there are twists of the canonical Hamiltonian: Θ = ξ µ π µ + fluxes.This is generalising the statement for T ⋆ [2]T[1]M, that exact Courant algebroids are classified by H 3 (M), i.e. there the most general Hamiltonian looks like Θ = ξ µ π µ + H µνρ ξ µ ξ ν ξ ρ .In this article's language, the most general proper twist, considered in [30], takes the following form on the reduced QP-manifold M ⊂ M ⋆ from the previous section: K . (3.1) The contribution from f κ µν ξ µ ξ ν t (1) κ = f κ µν ξ µ ξ ν z κ is not a proper twist and can be eliminated by coordinate redefinition: ξ µ → ξ ′µ = e µ ν (x)ξ ν .In this section first steps will be undertaken to find a duality-covariant version of (3.1), as the above expression obviously depends on the choice of section but also on the type of section solution, i.e.M-theory or type IIb.In particular, the inclusion of the 'non-geometric' fluxes and 'non-geometric' coordinates ξ µν is to be expected.Moreover, instead of going to the above twists of algebroid structure, we will start here by considering Hamiltonians associated to algebra structures.Differential graded manifolds associated to Leibniz algebras.Famously, the QP-manifold T ⋆ [2]g [1] associated to a Lie algebra g has the following Hamiltonian Θ = f c ab ξ a ξ b z c whose master equation {Θ, Θ} = 0 is equivalent to the Jacobi identity of g (with {z a , ξ b } = δ b a and {ξ a , ξ b } = 0 = {z a , z b }.In analogy, a similar statement is true for the 'Courant algebra' with ξ A = (ξ a , z a ) the degree 1 coordinates from T ⋆ [2]T [1]M and constant F ABC .With {ξ A , ξ B } = η AB the master equation {Θ, Θ} = 0 implies the Bianchi identity of the constant generalised fluxes F ABC , In the following, the aim is to generalise such a construction to obtain a Hamiltonian Θ on some differential graded manifold which is characterised by a Leibniz algebra, s.t.
• for some appropriate degree (n − 1) functions z A the Leibniz algebra is obtained as a derived bracket {{Θ, • the master equation {Θ, Θ} = 0 is implied by the Leibniz identity.
In comparison to the two example above the mixed symmetry of the Leibniz algebra structure constants will pose a challenge.

Q-flux in the SL(5)-theory.
The focus here will lie on Leibniz algebras corresponding to supergravity gaugings, and particularly those for the SL( 5) exceptional field theory.But the results will be valid generally.Particular attention will be given to the 'Q-flux', as the easiest non-standard twist.The generalised fluxes In the M-theory section these components decompose into the following SL(4)-representations [66]: and locally non-geometric contributions T a,b and U b that will play no further role here.Twists by the F 4 -and the geometric f -flux are standard and can be included in the standard form (3.1) and have been discussed already in [41].
What is dubbed Q-flux here has recently been of considerable interest in the context of nonabelian U-duality, exceptional Drinfeld algebras and the construction of new uplifts to lower dimensional to 11-dimensional supergravity recently [36,[67][68][69][70][71][72][73][74][75][76][77][78][79][80].The trace-free resp.trace components of the Q-flux correspond to the symmetric resp.skew-symmetric SL(5)-tensors: It has a natural interpretation as structure constants fa b 1 b 2 b 3 of a three-bracket.If the generalised fluxes (3.4) are constant, the Bianchi identity they have to obey is the Leibniz identity: If only this flux is turned on, the (Bianchi) Leibniz identity (3.8) implies that constant Q-flux corresponds to structure constants of a three-bracket that fulfils the fundamental identity of a Nambu bracket [81]:

Twist by the Weitzenb öck connection
The first try, how two realise generalised fluxes as a derived bracket, is simply by considering the derived bracket of generalised frames E A M ∈ E d(d) , treated as elements of R 1 , in the setup of section 2. Naturally, the generalised fluxes (3.4) appear in generalised Lie derivative, the Leibniz bracket of the fields t (1) M .In terms of some generalised frame fields in The Poisson brackets in H -of t A , t (2)A = E A M t (2)M , ... -stay invariant 4 , e.g.{t A , t The derived bracket will have the striven for form A }, t where a charge condition relating R 1 and R 2 appeared in the form F . (3.12) The Hamiltonian stays the same.So, in particular, the master equation {Θ, Θ} = 0 is fulfilled trivially.The consistency condition on the Weitzenb öck connection, and hence also on the generalised fluxes Going from algebroid to algebra.One incentive was to have some QP-structure that represents a Leibniz algebra structure.In principle, this seems to be achieved here as the twist of the Pstructure, characterised by an arbitrary generalised flux.But there are two points that indicate that this is not what we look for here.Coupling the tensor hierarchy to a whole algebroid T ⋆ [n]T [1]M seemed fairly excessive and also the twist is characterised by the Weitzenb öck connection and not the generalised fluxes.Nevertheless, some algebraic identities can be derived like that.For example, one define the action of the derivative ∂ : R p → R p−1 .
where S D BC = F C (AB) .This can be derived from {Θ, {t A , t C .Naturally, the Weitzenb öck connection does not share the same properties as the generalised flux.Wheras we can describe a twist by the F 4 for example without problem, or geometric flux f c ab to some Lie group.Nevertheless, the above construction shows that we can represent these known cases also by a twist of the Poisson structure and not only by additional terms in the Hamiltonian.For other standard components of AB is, as shown in the following.
Example: the Q-flux background.Let us take the SL(5)-theory in the M-theory section ∂ M = (∂ m , 0), where we decomposed indices like X M = (x m , x mm ′ ).Let us work in the non-geometric frame 4 As usual, the generalised frame in different representations are related by the invariance condition of the E d(d)invariants.For example and take Latter vanishes when the fd abc are structure constants of a 3-bracket that satisfies the fundamental identity The Weitzenb öck connection on the other hand is not constant.In particular it has the non-vanishing components Hence, one needs to refer to coordinates of M to describe this QP-structure and the aim to represent the Leibniz algebra without this can't be achieved by this naive and straightforward procedure.

Speculation on extended exceptional QP-manifolds
So far, the underlying QP-manifold was phrased in a duality-covariant manner but the section condition (2.9) was imposed immediately.In principle, there are two ways how to generalise this.One is to really consider dependences on extended coordinates X M = (x µ , ...) -this is not the aim here.Alternatively, one could consider extended degree 1 coordinates ξ M .This seems logical, as the constructions in [41] and also in the present paper deal with completing the ξ's conjugate coordinate z µ into R 1 -representation t M .Hence, we aim to introduce degree 1 coordinates that are dual to some t M .Such that for example in the M-theory decomposition {t (1)νν ′ , ξ µµ ′ } = δ νν ′ µµ ′ .In section 3 it is shown, that assuming {t (1) M , ξ N } = δ N M will be too restrictive for the purposes there.
Why should this be useful?One application of this QP-manifold picture is a derivation of the current algebra of world-volume theories [43].For these it was shown in [38] that the worldvolume currents corresponding to t (1) M could be written as exterior products of world-volume differentials of the coordinates, dX M , which correspond to ξ M in the QP-manifold language.
What to generalise from O(d, d).For O(d, d) extending ξ µ to ξ M is natural.The degree 1 coordinates (ξ µ , z µ ) on T ⋆ [2]T [1]M can be phrased as ξ M = (ξ µ , z µ ).With this one can consider Hamiltonians like Θ = F ABC ξ A ξ B ξ C corresponding to a constant generalised flux background.
One way to construct the corresponding Poisson structure from an underlying extended QPmanifold is the following.This procedure was performed in [82] on the level of the current algebra.Start with the canonical T * [2]T [1]M ext. on the extended (doubled) base manifold M ext.with coordinates and canonical Poisson brackets { , } ′ : The standard (unextended) QP-manifold is given by • reducing to a solution of the section condition The corresponding Dirac brackets are Together with a solution of the section condition, so reducing X M → (x µ , 0)/π M → (π µ , 0).The result basically corresponds to the standard supermanifold M, T ⋆ [2]T[1]M.
Problems for the exceptional case.In analogy to the O(d, d)-case one would consider In analogy to (2.11) one identifies t M with z M Φ (1) but now for the full R 1 -index, not only on a section.As will be shown later, A is some linear operator preserving the SL(4)-structure.This constraint is second class: {Φ M , Φ N } = η L,MN t (2)L .5Besides difficulties in inverting the operator it would lead to the necessity of negative degree coordinates.Latter can also be seen coming, in analogy to the O(d, d) case, where {ξ M , ξ N } ̸ = 0, the grading of the P-structure implies that deg{ξ M , ξ N } < 0 for n > 2. The analogy to the constraint N in the O(d, d)-case is the following ansatz that appeared already in [38]: for some p (which coincidently will correspond to the spatial dimension of the corresponding brane).In [38] it was also shown that this ansatz together with the constraint Φ (2) from (2.12), in explicit terms as a constraint has the same kinds of solutions, as the hierarchy of constraints Φ (p) presented in section 2, i.e. the 1 2 -BPS p-branes.In general, the expression (3.21) fixes more than the the constraints (2.12), namely also the components t µ .For example, a solution corresponding to the M2-brane could be given as follows: • allowing for (new) negative degree coordinates.
• relaxing the requirements on the P-structure, namely one could allow for violations of the the Jacobi identity.
In the next subsection a proposal is made following latter option, as it is closer to the construction in the previous section.

A proposal
The proposal put forward here is that parts of the constraints (3.20) & (3.22) are not being subjected to a Dirac procedure, but all calculations are done in an embedding (graded Poisson) space and only the results are only projected -i.e.not Poisson reduced -onto the constraint surface.Naturally, this resulting space would not be a graded Poisson manifold in general anymore and the Jacobi identity would be violated.Going the other way, the unconstrained space would be a symplectic completion of the almost Poisson manifold that is conjectured here as the extended p-brane QP-manifold.
The proposal for a Leibniz algebra QP-manifold is based on an extended differential graded manifold In contrast to H[n] from the previous subsection the •-product structure is not imposed and conjectured to follow only on the constraint surface. 6Hence, this proposal is not a straight-forward generalisation of the setting in the previous section.
The dual pairs of coordinates, ξ M , z M and t M , t(1)M , are not independent of each other.For example in the M-theory section of the SL(5)-theory, the following ansatz for their relation preserving the SL(4)-structure of the M-theory section is made: A, B being some real constants. 7In analogy with The symbols C AB C and S A,B C correspond to skew-symmetric resp.symmetric part of the F AB C : The physical subspace M ⊂ M ⋆ is conjectured to be the space subject to the constraints: and a suitable choice of identification (3.26) of (t M , t(1)M ) with (z M , ξ M ).So, in contrast to the conservative construction in section 2, here a charge solution for a particular flux configuration will be characterised • by a charge solution from (3.28).Solutions to these will be the same p-brane charges as in section 2, with additional identifications of t µ with products of the ξ M as in (3.23).
• additionally by an identification t The Bianchi identities of constant O(d, d) generalised fluxes including the dilaton flux F A [63], for the case that F ABC F ABC = 0,8 can be reproduced from the master equation: (3.32)

Twist by the Q-flux in SL(5) M-theory section
As a non-trivial example from the exceptional field theory hierarchies, let us consider the Q-flux (three-bracket) in the SL(5)-theory in the M-theory.This is a popular example recently, as it appears as dual three-bracket structure constants in the discussion of exceptional Drinfeld algebras.
The following underlying QP-manifold, based on the one of the M2-brane, is used.degree 0 1 2 3 ξ A , t(1)A z A , t A The t (2) or other generators are not need for the simple example considered here.The only nonvanishing Poisson brackets are {z A , ξ B } = δ B A = {t  Let us note that, strictly speaking, the tensor hierarchies of the SL(3)×SL(2)-and SL(5)-theories do violate these axioms: • For SL(3)×SL(2), there is a violation of the graded skew-symmetry for C i ∈ R 3 : • For SL (5), there is a violation of the graded Jacobi identity for A ∈ R 1 and But, in both cases, these violations are not relevant for the reduced p-brane QP-manifolds.After specifying to solutions to the charge conditions (2.12), discussed in section 2, one notices that: • for SL(3)×SL( 2), on a solution of the constraints (2.12): t (6) = 0, so • for SL (5), on a solution of the constraints (2.12): t (5) = 0, so Hence, the resulting reduced spaces M ⊂ M ⋆ are QP-manifolds.
which is the Bianchi identity for generalised fluxes[63].This is not very surprising, as the constant generalised fluxes from O(d, d) generalised geometry correspond to structure constant of a Lie algebra.For E d(d) generalised fluxes this is not the case.There constant generalised fluxes correspond to structure constants of a Leibniz algebra.It is easy to reproduce a derived bracket of the form {{Θ, z A }, z B } = F C AB z C for a Leibniz algebra.But the second point -a relation between the master equation and the Bianchi identity of generalised fluxes (the Leibniz identity) -is more difficult.The conjecture put forward in section 3 is the following.The underlying manifold is T ⋆ [n]H[n].A natural choice of Hamiltonian which contains information of a Leibniz algebra is the following(3.27)

. 23 )
Of course, this expression is fairly useless when aiming to compute Poisson brackets, as the illusive Poisson brackets {ξ M , ξ N } are needed.Two ways out of this problem and the problem of the second-class nature of the constraint (3.20) could be:

( 1 ) 3 . 2 . 2
M ∼ A M N z N , as in(3.26), that will depend on the flux configuration at hand.The conjecture is that, for each flux configuration F C AB , a identification like (3.26) exists, s.t.{Θ, Θ}| M = 0 ⇐ Leibniz identity for F C AB .(3.29)This conjecture is illustrated in two examples.The O(d, d) generalised fluxes including the generalised flux F A associated to the representation R 2 = 1, and the Q-flux in SL(5) exceptional field theory.The O(d, d) generalised fluxes including the dilaton flux Surprisingly, this proposal gives also something new even in the O(d, d)-case.Applying the procedure can perform the reduction w.r.t. to the constraints (3.28) explicitly and the remaining free coordinates are: degree 0 1 2 h ξ A h h resp.h are the degree shifted generators of R 2 = 1 resp.R2 .The non-vanishing Poisson brackets are {ξ A , ξ B } = η AB , { h, h} = −{h, h} = 1.The Hamiltonian (3.27) takes the general form:

( 1 )b 1 b 2 b 3 = fa b 1 b 2 b 3 = 1 4 ϵ b 1 b 2 b 3 b
A , t(1)B }.As mentioned earlier, the Q-flux is characterised by the s ab and a ab alone, Q a (s ab + 2a ab ).s ab corresponds to the traceless part of Q, whereas fa abc = a bc .The simplest case is the presence of only s ab ̸ = 0, then the candidate Hamiltonian (3.27) is:

Table 1 :
The representations relevant for the tensor hierarchy and generalised Cartan calculus of duality groups O(d, d) and E d(d) for d ≤ 6.
1))In this article, we aim for the application to the Tand low dimensional U-duality groups.R 1 will be denoted by indices K, L, M, ..., R 2 by K, L, M, ... and, for a general R p , the indices are α p , β p , ....The other objects in the quadruple (H, •, ∂, •) are a product • : R p × R q → R p+q , a Leibniz product • : R 1 × R 1 and the derivative ∂ : R p+1 → R p with ∂ 2 = 0.