Super-exceptional geometry: origin of heterotic M-theory and super-exceptional embedding construction of M5

In the quest for the mathematical formulation of M-theory, we consider three major open problems: a first-principles construction of the single (abelian) M5-brane Lagrangian density, the origin of the gauge field in heterotic M-theory, and the supersymmetric enhancement of exceptional M-geometry. By combining techniques from homotopy theory and from supergeometry to what we call super-exceptional geometry within super-homotopy theory, we present an elegant joint solution to all three problems. This leads to a unified description of the Nambu-Goto, Perry-Schwarz, and topological Yang-Mills Lagrangians in the topologically nontrivial setting. After explaining how charge quantization of the C-field in Cohomotopy reveals D’Auria-Fré’s “hidden supergroup” of 11d supergravity as the super-exceptional target space, in the sense of Bandos, for M5-brane sigma-models, we prove, in exceptional generalization of the doubly-supersymmetric super-embedding formalism, that a Perry-Schwarz-type Lagrangian for single (abelian) N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = (1, 0) M5-branes emerges as the super-exceptional trivialization of the M5-brane cocycle along the super-exceptional embedding of the “half ” M5-brane locus, super-exceptionally compactified on the Hořava-Witten circle fiber. From inspection of the resulting 5d super Yang-Mills Lagrangian we find that the extra fermion field appearing in super-exceptional M-geometry, whose physical interpretation had remained open, is the M-theoretic avatar of the gaugino field.

• Typically, it is asserted that this is an open problem only for N ≥ 2 coincident M5branes, while the special case of a single M5-brane is well-known. Indeed, there is a non-covariant Lagrangian formulation [1,90,105] adapted to M5-s wrapped on the Mtheory circle fiber, as well as a covariant version at the cost of introducing an auxiliary field [12,89]. Both of these involved some ingenuity in their construction which makes them look somewhat baroque. Indeed, their double dimensional reduction reproduces the D4-brane Lagrangian, and hence the 5d super Yang-Mills + topological Yang-Mills Lagrangian, only up to an intricate field redefinition [1, section 6], [2, section 6 & appendix A].
• Such complications, already in the formulation of the base case of a theory whose expected generalization remains elusive, may indicate that the natural perspective on the problem has not been identified yet. What has been missing is a derivation of the M5-brane Lagrangian systematically from first principles of M-theory, with manifest dimensional reduction to the D4-brane.
Second open problem: Heterotic gauge enhancement. The non-perturbative completion of heterotic string theory has famously been argued [63,64] to be given by the Mtheoretic completion of 11-dimensional supergravity KK-compactified on a Z 2 -orbifolded circle fiber, where the Z 2 -action on the circle has two fixed points, hence two fixed planes as an action on spacetime: the MO9-planes.
• With an actual formulation of M-theory lacking, the argument for this is necessarily indirect, and it goes as follows. Plain 11d supergravity turns out to have a gravitational anomaly when considered on such MO9 boundaries, hence to be inconsistent in itself. Thus, if the putative M-theory completion indeed exists and hence is consistent, it must somehow introduce a further contribution to the total anomaly such as to cancel it. The form of that further anomaly contribution inferred this way is the same as that of a would-be field theory of charged chiral fermions on the MO9-planes, just as found in heterotic string theory.
• This suggests that if M-theory actually exists, it must include avatars of these super gauge field theory degrees of freedom appearing on MO9-branes. While many consistency checks for this assumption have been found, it remained open what the M-theoretic avatar of the heterotic gauge field actually is. In [63,64] the 10d SYM action on the MO9s is just added by hand to that of 11d supergravity.

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Third open problem: Super-exceptional M-geometry. The Kaluza-Klein (KK) compactifications of 11d supergravity on n-tori have a rich space of scalar moduli fields invariant under ever larger exceptional Lie groups as n increases [28], reflecting just the expected duality symmetries acting on the corresponding string theories [71]. This led to the proposal [70] (see also [9,75,87,115,116]) that M-theory is an enhancement of D = 11 supergravity to a theory of "exceptional geometry" with a "generalized tangent bundle" of the form T ex X n exceptional tangent bundle := T X n ⊕ ∧ 2 X T * X n M2 wrapping modes ⊕ ∧ 5 X T * X n M5 wrapping modes ⊕ · · · (1.1) locally encoding wrapping modes of the M2-and the M5-brane already before KKcompactification.
• While the exceptional generalized geometry enhancements of the bosonic sector of 11d supergravity is well studied (see, e.g., [38] and references therein), the inclusion of fermionic exceptional coordinates, hence a unification of supergeometry with exceptional generalized geometry to "super-exceptional generalized geometry", had remained an open problem [24, p. 39], [25, pp. 4, 7]. Arguments were given in [9,45,103,114] that the super-exceptional geometry for maximal n = 11 is to be identified with what was called the "hidden supergroup of 11d supergravity" in [4,10,32], but open questions remained. In particular, the physical meaning of had remained open.
• It may seem that supersymmetrization is but an afterthought once the bosonic sector of exceptional geometry is understood, (e.g. [20] for n = 7). But most aspects of M-theory are controlled by -and are emergent from -its local supersymmetry structure (see, e.g., [46,111]), with the bosonic sector being implied by the spin geometry, instead of the other way around. The lift of this supersymmetry first principle to exceptional generalized geometry had remained open.
The joint solution. In [45, 4.6], [103] we had already observed that a supersymmetric enhancement of n = 11 exceptional M-geometry is provided by what [32] called the "hidden supergroup" of 11d supergravity. With [47, proposition 4.31], [48, proposition 4.4], it follows that this must be the correct target space for M5-brane sigma-models, as we explain in section 3. Accordingly, in section 4 we consider super-exceptional 5-brane embeddings and find in section 5 that this induces the Perry-Schwarz Lagrangian (reviewed in section 2) and, after super-exceptional equivariantization along the M-theory circle fiber introduced in section 6, the full super-exceptional M5-brane Lagrangian, in section 7. The resulting D4-brane Lagrangian with its 5d SYM+tYM Lagrangian is manifest (remark 7.5) and

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identifies the super-exceptional fermion as the M-theoretic avatar of the heterotic gauge field (remark 5.4).
Before giving more detail in the Outline of results, we recall the two foundational principles of our development: Principle 1: Super-geometry. Despite the evident relevance of super-geometry for the foundations of M-theory, many constructions in the literature start out with the bosonic data (e.g. [1, section 2]) and relegate super-geometrization to an afterthought (e.g. [1, section 3]). Countering this tendency, the "doubly supersymmetric" approach of [13,66], reviewed under the name "super-embedding approach" in [106,107], shows that seemingly mysterious, or at least convoluted-looking, aspects of traditional constructions find their natural meaning and more elegant formulation when strictly everything is systematically internalized into super-geometry. In particular, the all-important "κ-symmetry" of super pbrane sigma-models, which, following [52], is traditionally imposed by hand onto the action principle, is revealed by the superembedding approach to be ( [108], see [106, section 4.3], [68, section 4.3]) nothing but the super-odd-graded component of the super-worldvolume super-diffeomorphism symmetry -hence a consequence of the fundamental principle of general covariance internal to super-geometry.
p-brane sigma-models NSR-type [85,92] GS-type [16,52] super embedding [13] super-exceptional embedding section 3  Indeed, all of the following has been systematically obtained from the superembedding approach: the equations of motion of the superstring [13, section 4] of the M2-brane [13, section 3] and of the M5-brane [66,67], [106, 5.2], as well as the Lagrangian density of the superstring and of the M2-brane [11,68]. But an analogous derivation of the M5-brane's Lagrangian density had remained open. Notice that it is the Lagrangian density which gives the crucial instanton contributions for these branes [15,58].
Principle 2: homotopy theory. The gauge principle of physics -read as saying that no two things (e.g. field configuratons) are ever equal or not, but that we have to ask for gauge transformations between these, and higher order gauge-of-gauge transformations between those -is mathematically embodied in homotopy theory, these days increasingly referred to as "higher structures" (see [ Homotopy theory, and more so super-homotopy theory, is extremely rich. But if, for the time being, we ignore torsion cohomology groups, homotopy theory simplifies to rational homotopy theory [91,110] (see [53,62], and see [46], [19, section 2] for review in our context). The main result here is that topological spaces, regarded up to rational weak homotopy equivalence, are encoded by their differential graded-commutative algebra of Sullivan differential forms, regarded up to quasi-isomorphism. If we suppress some technical fine-print (see [19, (8)] for the precise statement), we may schematically write this as follows: For super-homotopy theory this yields rational superspaces in rational super-homotopy theory [69, section 2] (see [46] for review). The following (Here the reader may regard the last column to be the very definition of the second and third columns.) One finds that a considerable amount of structures expected in M-theory emerge naturally in rational super-homotopy theory: • On super-geometric ∞-groupoids, the Sullivan construction of rational homotopy theory (see, e.g., [53,62]) unifies with higher super Lie integration [18, section 3.1] to exhibit super L ∞ -algebroids as models for rational super-homotopy theory. Their
• Using super-homotopy theory, we had shown [39,43] that the completion of the "old brane scan" to the full "brane bouquet" emerges from the superpoint R 0|1 as the classification of iterated universal invariant higher central extensions.

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The M2-brane in super-homotopy theory. [69, proposition 6.10] The κ-symmetric Green-Schwarz-type Lagrangian density for the M2-brane [16] looks intricate when written out in the traditional component formulation (see [34, (2.1)], [31, (3)]), but attains a highly elegant form in a fully supergeometric formulation. Indeed, promoting the M2 worldvolume itself to a super-manifold embedded into target super-spacetime, locally of the form shown on the left of the following diagram super M2-brane cocycle super spacetime super ADE-action T 10,1|32 the M2-brane's Lagrangian density L NG arises simply as the super-homotopy theoretic trivialization of the M2-brane cocycle restricted along the super-embedding. Concretely, this identifies the Lagrangian with the super-volume form These fermionic correction terms, systematically obtained here simply by expanding out the super-volume form in components, constitute the otherwise intricate-looking components of the Green-Schwarz-type Lagrangian for the M2-brane, which is thereby revealed simply as the super-Nambu-Goto Lagrangian. What had been left open in [69] is the analogous result for brane species with gauge fields on their worldvolume, notably the case of the M5-brane, which is a much richer situation (see [39]). This will be one of the main topics that we address in this paper.
Outline of results. We establish the following: (i) In section 2 we generalize the bosonic Perry-Schwarz Lagrangian L PS = F ∧ F to a coordinate-invariant expression applicable to possibly non-trivial worldvolume circle bundles.
(ii) In section 3 we recall super-exceptional M-geometry with the super-exceptional M5brane cocycle and introduce super-exceptional embedding of M-brane spacetimes.

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(iii) In section 4 we introduce specifically the super-exceptional embedding of the 1 2 M5 = MK6 ∩ MO9 brane configuration and find the super-exceptional lift of the isometry along the Hořava-Witten-circle S In the course of this identification we find that Then we show (proposition 5.9) that the super-exceptional Perry-Schwarz Lagrangian arises via a super-exceptional analog of the super-embedding mechanism as a trivialization of the S 1 HW -compactified super-exceptional M5-brane cocycle after restriction along the super-exceptional embedding of the 1 2 M5. (v) In section 6 we show that an equivariant enhancement of the super-exceptional M5cocycle with respect to super-exceptional ΩS 2 HW -action exists, where ΩS 2 is the based JHEP02(2020)107 loop space of the two-sphere. Furthermore, this unifies it with the super-exceptional Perry-Schwarz and the super-exceptional topological Yang-Mills Lagrangian (theorem 6.9).
To put this in perspective, we also explain (by proposition 6.6) how ΩS 2 HW S 1 HW refines the naive circle action by taking the super-cocycle for the little-string in 6d into account. This is a 6d analog to capturing the form fields in 11d M-theory via the Cohomotopical 4-sphere coefficient [42,102], leading to a description of type IIA in ten dimensions using a refined variant of the loop space of S 4 , namely the cyclic loop space [43,44] (see [46] for overview).

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Here and in the following the double slash denotes a homotopy quotient, which may be represented as an ordinary quotient via the Borel construction: Finally, we observe (remark 7.6) that there are two extensions of the compactified super-exceptional 1 2 M5-spacetime on which the D4 WZ-term becomes exact already before dimensional reduction: one of these implements the heterotic Green-Schwarz mechanism and the WZ-term of the heterotic NS5-brane (remark 7.8).
With this, we may sum up the whole picture elegantly in the following homotopy diagram: For ease of reference and in order to introduce notation needed in later sections, we review here the bosonic part of the Perry-Schwarz-Lagrangian from [90], re-cast in coordinateindependent Cartan calculus and generalized to possibly non-trivial circle fibrations. We try to bring out the logic that motivated the construction in [90], but below in section 6 and section 7 we re-derive the Perry-Schwarz Lagrangian systematically from first principles. Readers familiar with this material may want to skip this section and just follow pointers to it from the main text when needed. The formulation of a manifestly covariant Lagrangian for the self-dual higher gauge field without further auxiliary fields in 6 dimensions (and generally in 4k + 2-dimensions), and hence in particular for the single M5-brane sigma-model, is famously subtle, at best (see e.g. [60,83]). But if one considers breaking manifest Lorentz invariance to 5 dimensions, as JHEP02(2020)107 befits KK compactification of the theory on a circle fiber, such as for double dimensional reduction of the M5 brane to the D4-brane, then there is a Lagrangian formulation due to Perry-Schwarz [1,90,105], following [61].
This "non-covariant" formulation of self-dual higher gauge theory and specifically of the M5-brane sigma-model may be covariantized by introducing an auxiliary scalar field [88] (whose gradient plays the role of the spacetime direction which gets singled out, thus promoting this choice to a dynamical field) which yields the covariant formulation of the M5-brane sigma-model [12,89]. This comes with a corresponding auxiliary gauge symmetry that admits a gauge fixing which recovers the non-covariant formulation, rendering the two formulations equivalent, with each "about as complicated" as the other [1, p. 3].
Worldvolume and self-duality. Let (Σ 6 , g) be a Lorentzian manifold of signature (−, +, +, +, +, +), to be called (the bosonic body of) the worldvolume of an M5-brane. In this dimension and with this signature, corresponding to the metric g, the Hodge star operator on differential forms * : Ω • (Σ 6 ) Ω 6−• (Σ 6 ) squares to +1. This allows for considering on a differential 3-form H ∈ Ω 3 Σ 6 (2.1) the condition that it be self-dual We will assume that H is exact and pick a trivializing 2-form Compactification on S 1 . Consider then on the worldvolume Σ 6 the structure of an We write v 5 ∈ Γ(T Σ 6 ) for the vector field which encodes the infinitesimal S 1 -action, hence the derivative of the circle action U(1) × Σ 6 ρ − Σ 6 at the neutral element, along a chosen basis element t ∈ T e (U(1)) u(1) R: Accordingly, we write : for the Lie derivative of differential forms along the vector field (2.5), where d denotes the de Rham differential and where under the brace we are using Cartan's magic formula. Next, consider an Ehresmann connection on the S 1 -bundle (2.4), hence a differential 1-form which satisfies the Ehresmann conditions in that it is normalized and invariant:

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Here on the left we have the operation of contracting differential forms with vector fields, and on the right we have the Lie derivative from (2.6). So, in particular, the composition is a projection operator: The complementary projection is that onto horizontal differential forms with respect to the bundle structure (2.4): Observe that: Lemma 2.1 (Horizontal vs. vertical differential). If the Ehresmann connection (2.7) on the S 1 bundle is flat, in that dθ 5 = 0 , (2.10) then for any differential form ω ∈ Ω • Σ 6 we have that the vertical component of the differential of its horizontal component (2.9) is the vertical component of its full differential: Proof. By direct computation, we have: Finally, we require the vector field v 5 from (2.5) to be a spacelike isometry. This means that it interacts with the Hodge star operator as * (2.12) Self-duality after S 1 -compactification. We introduce notation for the contraction of the 3-form H and its Hodge dual with the vector field v 5 (2.5) as follows (to be called the "compactified fields", a notation that follows [90, (5), (6)]): With this, we get the following immediate but crucially important re-formulation of the self-duality condition after S 1 -compactification (extending [90, (8)]): Lemma 2.2 (Self-duality after S 1 -compactification.). Given an S 1 -bundle structure (2.4) on the worldvolume Σ 6 and any choice of Ehresmann connection (2.7), the self-duality condition (2.2) is equivalently expressed in terms of the compactified fields (2.13) as: (2.14)

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Proof. We have the following chain of equivalences: Here the first step is decomposition into horizontal and vertical components (2.9), the second step uses the isometry property (2.12) to conclude that the two resulting component equations are equivalent to each other. The last step identifies the compactified fields (2.13).
The gauge field. The contraction of the vector field v 5 from (2.5) with the 2-form potential B from (2.3) defines the 1-form potential Hence we get a decomposition of the 2-form as where on the right we have the horizontal component of B according to (2.9). We say that the 2-flux density encoded by B is the horizontal component of the exterior differential of this vector potential F := (dA) hor . (2.17) We will find in a moment that this is the 5d field strength with all higher KK-modes still included, but it is most convenient here (and in all of the following to just call it "F " already in the 6d compactification before passing to KK zero-modes. With this we have (cf. [90, (5) Proof. We compute as follows:

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Here the first step is the definition (2.13), while the second step is (2.3). The fourth step uses the definition (2.15) of the vector potential and identifies the Lie derivative (2.6). The fifth step applies vertical/horizontal decomposition (2.9) to both summands and uses (2.17) under the first brace and the expressions (2.7) and (2.15) under the third brace; while under the second brace it uses Cartan's formula (2.6), observing that ι v 5 A = 0 by definition (2.15) and by nilpotency of the contraction operation. The last step notices that this makes the second and fourth summands cancel each other.
With lemma 2.3, the self-duality condition (2.2) in the equivalent form (2.14) after S 1 -compactification says that the combination H − L v 5 B hor is horizontally exact: (2.20) Weak self-duality and PS equations of motion. In summary, we have the following implication of self-duality after S 1 -compactification (extending [90, (16)]): Proposition 2.4 (Self-duality for flat circle bundles). If the worldvolume Σ 6 is equipped with an S 1 -principal bundle structure (2.4) which is flat (2.10), then the self-duality condition H = * H from (2.2), in its equivalent incarnation on compactified fields (2.20) implies the following differential equation: By lemma 2.1, equation (2.21) may be understood as expressing "self-duality up to horizontally exact terms". The proposal of [90] is to regard (2.21), which is second order as a differential equation for B, as the defining equation of motion for a self-dual field on Σ 6 compactified on S 1 . Given this, one is led to finding a Lagrangian density whose Euler-Lagrange equation is the self-duality equation.

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At this point, as in [90, (16)], the bosonic PS Lagrangian of definition 2.5 is motivated as the evident choice that makes the following proposition 2.7 true. Below in section 5, we find a deeper origin of the Lagrangian (in the case that self-duality is imposed). Proof. We may evidently regard the PS Lagrangian as the quadratic part of the following bilinear form on differential 2-forms with values in differential 6-forms: Observe then that this bilinear form is symmetric up to a total differential: the first summand is strictly symmetric, as it is the standard Hodge pairing, while for the second summand symmetry up to a total derivative is established by a local integration by parts. Together these imply that the variational Euler-Lagrange derivative of the PS Lagrangian is twice the result of varying just the second factor of B: Hence the vanishing of the variational derivative is equivalent to (2.21).
Reduction to 5d Maxwell theory. Consider finally the special case of Kaluza-Klein compactification/double dimensional reduction, where In this case, expression (2.19) reduces to Hence, with self-duality ([90, above (16)]), we have is the Hodge star operator on the base 5-manifold. Consequently, we have extension of [90, above (16)] to the topologically nontrivial setting: Proposition 2.8 (5d Abelian Yang-Mills from Perry-Schwarz). The S 1 -dimensional reduction of the PS Lagrangian (2.24) is the Lagrangian of 5d Maxwell theory for the vector potential A from (2.15): (2.28)

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6d self-dual field as (abelian) 5d Yang-Mills with KK-modes. Due to proposition 2.8, one may regard the general bosonic Perry-Schwarz Lagrangian (definition 2.5) for the self-dual field on Σ 6 compactified on S 1 as that of (abelian) 5d Yang-Mills theory with a tower of KK-modes included, which is a perspective on the M5-brane theory later advanced in [35,76] (see [77, 3.4.3])). Here for the abelian bosonic sector, this perspective may be fully brought out by introducing the following notation: where on the left we have (2.18), as before, while on the right we are introducing notation for the remaining summand in H. With this notation, the bosonic PS Lagrangian, assuming self-duality (2.2) as in (2.24), finds the following suggestive expression: so that KK-reduction to 5d YM theory, as in proposition 2.8, is now given syntactically simply by replacing F * 5 F . We find the form (2.30) of the PS Lagrangian to be reflected by its super-exceptionalization in proposition 5.1 below; see definition 5.2 and remark 5.8.
The following, definition 3.3 and proposition 3.5, are a formulation in rational superhomotopy theory due to [45, section 4.5], of the classical supergravity results in [32, section 6], [10, section 3] (see also [4]). The definition 3.3 of super-exceptional spacetime involves a parameter s, eq. (3.2), which arises mathematically in proposition 3.5 from different possibilities of decomposing the H 3 -flux on super-exceptional spacetime [10]. We discover the physical meaning of this parameter below in section 5.
the D = 11, N = 1, n = 11 super-exceptional M-theory spacetime (T 10,1|32 ) exs over ordinary D = 11, N = 1 super Minkowski spacetime T 10,1|32 is the rational super space given dually by the following super dgc-algebra: Here the index α ranges over a linear basis of the real Pin + (10, 1)-representation 32 and the Clifford generators Γ a acting on these are as in definition 3.1; and we use Einstein summation convention with the e a 1 a 2 and e a 1 ···a 5 understood as completely antisymmetrized in their indices.

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Remark 3.4 (Super-exceptional M-theory spacetime as a supermanifold). We may alternatively regard T 10,1|32 exs from definition 3.3 as a super-manifold with canonical global coordinate functions , hence with bosonic part being the exceptional tangent bundle (1.1) for maximal n = 11: and with super-group structure such that the Chevalley-Eilenberg algebra in definition 3.3 identifies with the super de Rham dgc-algebra of left-invariant (hence translationally supersymmetric) super-differential forms: Beware the bracketing in the last two lines on the right, in contrast to (dB) a 1 ···a 3 etc. The bosonic component of the generator e a 1 a 2 is the de Rham differential of the bosonic component functions of a 2-form B : (without antisymmetryization over all three indices, at this point), instead of the component functions of the de Rham differential of a 2-form, which is instead obtained by anti-symmetrizing over all three indices as in (3.10) below. This has a crucial effect in the following discussion; see lemma 5.1.
given by and the transgression element in proposition 3.5 has the property, from (3.9), that up to terms proportional to the 5-index tensor e α 1 ···α 5 , its only dependence on dB a 1 a 2 := α 0 (s)e a 1 a 2 is through the leading term H := (dB a 2 a 3 ) ∧ dx a 2 ∧ dx a 3 ; concretely: where the last term O {e a 1 ···a 5 } denotes summands that vanish when the 5-index generators e a 1 ···a 5 are set to zero.
We will see below in proposition 5.1 (ii). that in super-exceptional M-geometry at value s = −3 the bosonic Perry-Schwarz Lagrangian appears naturally.
The point of the super-exceptional M-theory spacetime from definition 3.3 is that it is a super-manifold (via remark 3.4) which approximates the universal super 3-stack T 10,1|32 classified by the super M2-brane cocycle (1.2). We record this phenomenon (which can be traced back to the "hidden supergroup of 11d supergravity" in [32]):

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Hence we say: ∧ e a 1 ∧ · · · e a 5 ∈ CE R 10,1|32 exs . We close by commenting on the role and meaning of these constructions: (ii) However, the actual meaning or role of this "hidden supergroup" had remained open; as it was not used for the re-derivation of D = 11 supergravity in [32], but discussed as an afterthought. In particular, the meaning or role of the extra fermion field required by super-exceptional spacetime had remained open [4, p. 3] and the relation of the bosonic fields to exceptional M-geometry had remained unnoticed except in [114] and then more recently in [9, p. 6], [45,103].
(iii) But in [47, proposition 4.31], [48, proposition 4.4], we found that the M5-brane sigma model in a given C-field background is characterized as making the outermost square of the following diagram homotopy-commute, here now displayed for flat super-spacetimes instead of curved topological spacetimes:

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Any such outer square factors universally through the homotopy pullback square shown on the right, which exhibits the M2-brane extended super spacetime T 10,1|32 (1.2) as the super moduli 3-stack classifying M5-brane sigma-model fields [42]. But now with the (extended) worldvolume Σ itself a super manifold, this factorization through T 10,1|32 is to be further factorized, as shown by the dashed map, through an actual super-manifold still classifying these fields. This is the role of the superexceptional spacetime R 10,1|32 exs [45, section 4.6]: it is the actual super manifold which serves as a stand-in for the classifying super space T 10,1|32 (which is not itself a super-manifold).
(iv) The key consequence of T 10,1|32 exs being a super-manifold, is that the indecomposable degree-3 generator h 3 on T 10,1|32 (1.2), which has trivial contraction with any vector field, pulls back to the decomposable 3-form H exs on T 10,1|32 exs (by lemma 3.7), which, like any differential form on a super manifold, in general has non-trivial contraction with vector fields. Below in section 5 it is such a non-trivial contraction of H exs with a vector field on the super-exceptional spacetime, which makes the Perry-Schwarz Lagrangian appear. The analogous contraction with h 3 on T 10,1|32 would vanish, which is the reason why the embedding construction of the M5-brane does not work with extended super-spacetime, but requires passing to super-exceptional spacetime. Pin + (10, 1) acts in the evident way, while reflection ρ along the a r -axis acts dually as follows: 1 ρ * : −e a | a = a r e a | otherwise e a 1 a 2 − −e a 1 a 2 | a 1 , a 2 = a r e a 1 a 2 | otherwise e a 1 ···a 5 − −e a 1 ···a 5 | one of the a i = a r e a 1 ···a 5 | otherwise It is traditionally understood that this compactification is one of two possible ways of obtaining classes of D = 6, N = (1, 0) superconformal field theories from M-theory [33, section 6]. We make this mathematically concrete with proposition 4.4 below, whose proof shows that the particular spinor structure of the 1 2 M5-locus (definition 4.1 below) is what allows a lift of the spacetime isometry along the M-theory circle to a symmetry also of the exceptionalized super-spacetime, including the exceptionalized 3-flux density on the M5-brane.
We use this to identify in proposition 5.    1. Black branes. The appropriate names of the fixed loci depend on the duality frame: if the circle S 1 A inside the quaternion space H on which G A acts is taken for M/IIA duality, then the result is type I' and the fixed loci are named as shown on the left (see also e.g. [56, around figure 2], [59]). On the other hand, if the circle S 1 HW on which G HW acts is taken for M/IIA duality, the result is HET on a G ADE -orbifold, and labels as used in [51, p. 8] are as shown on the right. Moreover, under T-duality along S 1 B the I'-perspective turns into a configuration of a IIB NS5-brane parallel to an O9-plane (see, e.g., [57]).

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Proof. Direct inspection of the defining differential relations in (4.3) shows that the derivation L v exs a evidently vanishes on all generators except possibly on η. The action on η is computed as follows: = ι v exs a (s + 1)e a 1 Γ Γ Γ a 1 (P ψ) + e a 1 a 2 Γ Γ Γ a 1 a 2 (P ψ) + 1 + s 6 e a 1 ···a 5 Γ Γ Γ a 1 ···a 5 (P ψ) where under the brace we used the defining property (4.1) of the spinor projections of definition 4.1.
With the super-exceptional lift of the circle isometry given, we have the corresponding super-exceptional version of the projection (2.9) onto horizontal differential forms. The next definition 4.6 formalizes the 1 2 M5-locus (as in remark 4.2) inside the superexceptional MK6-spacetime (as formalized by definition 4.3) without discarding the ambient MK6 spacetime, but breaking its supersymmetry from D = 7, N = 1, to D = 6, N = (1, 0). Hence the generators of the MK6 super dgc-algebra ("FDA") are all retained, but all spinors on the right of the differential relations (4.3) get projected not just by P 16 but by P 8 (as in definition 4.1): Definition 4.6 (Super-exceptional 1 2 M5 spacetime). For s ∈ R \ {0, −6}, we say that the super-exceptional 1 2 M5-spacetime is the rational super space given dually by the same super dgc-algebra (4.3) as that of the super-exceptional MK6 of definition 4.3, but with spinor projections P 8 instead of just P 16 (definition 4.1) on the right of the differential relations: Directly analogous to definition 4.6 we may apply the heterotic spinor projection already on the full super-exceptional M-theory spacetime:

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The following is a direct consequence of the above definitions:  superspacetime Hex s n v (4.14) Remark 4.10 (Alternative brane configuration MO1 MO9). For the case when the parameter value is s = −6 after all, there is an alternative brane configuration one may consider, namely the configuration of an "M-wave" (see [69, 2.2.3]) inside an MO9-plane, i.e., with the spinor projection (4.1) replaced by the following: In this case, there is a corresponding alternative to the super-exceptional isometry (4.6) given by v exs With this alternative brane configuration and alternative super-exceptional isometry, all of the following constructions go through for all of s ∈ R \ {0}, including s = −6; but then there is no value of s for which the leading term of the super-exceptional Perry-Schwarz Lagrangian equals the original bosonic Perry-Schwarz Lagrangian, i.e., what fails is item (ii) of proposition 5.1 below. This does not mean that this alternative case is not of interest, but its interpretation will need to be discussed elsewhere.

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Here the first line is the definition of the Dirac adjoint, the second line uses that Γ Γ Γ 5 is the identity on the projected spinors, by definition. In the third step we commute Γ Γ Γ 5 = iΓ 5 with Γ 0a 1 ···an , thereby picking up a sign σ = +1 | odd number of a i s = 5 −1 | even number of a i s = 5 (4.15) Finally we use (Γ Γ Γ 5 ) † = Γ Γ Γ 5 from (3.1) to absorb the Γ Γ Γ 5 again, this time into the left spinor factor. Hence the expression we started with equals its product with σ, and so vanishes when σ = −1, hence when Γ a 1 ···an has an even number of indices differing from 5.

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Here P 16 and P 8 are the spinor projections from definition 4.1, and on the far left we used lemma 4.12 to recognize that the M2-brane cocycle on the 1 2 M5-locus retains only the summands proportional to e 5 , while the M5-brane cochain on the 1 2 M5-locus retains only the summands not proportional to e 5 . Notice that the vertical pullback is syntactically the identity, due to (3.3). This makes manifest that the vertical pullback to the exceptional spacetimes intertwines the contraction operations for v 5 (4.5) and for v exs 5 (4.6): Since we also have (4.20)

Super-exceptional Perry-Schwarz & Yang-Mills Lagrangians
In proposition 5.1 we find a natural super-exceptional pre-image of the bosonic Perry-Schwarz Lagrangian, recorded as definition 5.2 below. This allows us to extract the supercomponents (in proposition 5.3 below) and identify the super-exceptional M-theory avatar of the gaugino field (remark 5.4 below). We also find the super-exceptional lift of the topological Yang-Mills Lagrangian (definition 5.5 below) and its relation to the superexceptional Perry-Schwarz Lagrangian (lemma 5.6 below). This plays a crucial role when we unify all this super-exceptional data in section 6. Then we show (proposition 5.9 below) that the super-exceptional Perry-Schwarz Lagrangian arises as the trivialization of the super-exceptional M5-brane cocycle restricted along the super-embedding of the 1 2 M5spacetime and compactified on the M/HW-theory circle (definition 5.7 below). This is a key ingredient in the full super-embedding theorem 7.3 further below in section 7 We start with identifying the super-exceptional lift of the PS Lagrangian.
where on the left H := d(B a 2 a 3 ) ∧ dx a 2 ∧ dx a 3 denotes the plain H-flux (3.6) and F its induced 2-form flux (2.18) according to lemma 2.3; and hence on the right we find a super-exceptional pre-image F exs of the 2-form flux.
(ii) If, moreover, s = −3 (as in example 3.6), we have where on the left we have the Perry-Schwarz Lagrangian (2.24), and hence on the right we find a super-exceptional pre-image L PS exs .
Proof. By the assumption that σ * e a 1 ···a 5 = 0, and since the odd forms σ * ψ and σ * η vanish after pullback to the bosonic space R 5,1 ×R 1 , we find from (3.6) by direct computation that This proves the first statement. For the second, it is now sufficient to observe with (3.10) that, by the assumption s = −3, we have in the present case σ * H exs = H. Hence the second claim now follows directly from the first. (ii) The super-exceptional PS Lagrangian: With the exceptional pre-image of the bosonic 2-form flux identified, we find the induced supersymmetric completion, keeping in mind the notation deg = (bosonic, fermionic):  In this way, it is the extra super-exceptional fermionic coordinate η which is the avatar on the super-exceptional M-theory spacetime of what becomes the gaugino field upon compactification to heterotic M-theory on S 1 HW . Note that an approximate construction of the 11d gravitino in the context of E 8 gauge theory as a condensate of the gauge theory fields is given in [37]. Proof. We need to show that But this follows directly from the definitions and the the fact that the contraction is a graded derivation of degree (−1, even), hence in particular nilpotent. Indeed, we have In order to see the super-exceptional Perry-Schwarz Lagrangian arise from the superexceptional M5-brane cocycle, we now first consider the M5-brane sigma-model wrapped on the S  Notice that this is indeed still a cocycle, in that it is closed, d ι v exs 5 (i exs ) * dL WZ exs = 0, by (4.8) in proposition 4.4. (definition 4.6), fields on this would-be D4 may still depend on the M-theory circle direction, hence have KK-modes along the circle. In this sense, definition 5.7 exhibits the brane cocycle corresponding to the perspective on the M5-brane as a non-perturbative D4-brane with KK-modes included, as considered in [35,76] (see [77, 3.4.3]).
We now establish the following super-exceptional analog of the super-embedding mechanism.
Proof. Unravelling the definitions, we have to show that For the first summand on the left, we immediately obtain by lemma 4.12; see example 4.13. For the second summand (or rather twice the second summand, for notational convenience) we compute as follows:

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Here the first step just collects the factors. The second fact uses that pullback is an algebra homomorphism, by definition. The third step uses that contraction with v exs 5 is a graded derivation of bi-degree (−1, even). In the fourth step we commute the differential in the second summand, first with the pullback operation (using that pullback is in fact a dgalgebra homomorphism, by definition) and then with the contraction operation, using that the corresponding Lie derivative vanishes, by (4.8) in proposition 4.4. In the fifth step we use that the differential (commutes with pullback, as before, and) is a graded derivation of degree (1, even). In the sixth step we realize the presence of the projection e 5 ∧ ι exs v 5 according to lemma 4.12, in view of dH exs = (π exs ) * µ M2 (3.5). In the seventh step we again commute the differential with pullback and with contraction, as before. In the eighth step we again use the derivation property of the differential to collect a total differential, observing that de 5 = 0 holds on the super-exceptional 1 2 M5-spacetime, by lemma 4.11. In the ninth step we collect terms and identify, in the tenth step, the super-exceptional 2-flux and its dual, from definition 5.2. Finally, in the last step we split off the factor of 2 again, just for emphasis.
As a corollary we observe the following: Proof. For the first statement, we may compute as follows: where we used first lemma 5.6, then (4.8) from lemma 4.4 and then proposition 5.9, and finally, under the brace, we observe that contraction with elements in degree (1, even) is nilpotent. This nilpotency also directly implies the second statement, by lemma 5.6.

Super-exceptional equivariance along M-theory circle
We show (theorem 6.9 below) that the super-exceptional Perry-Schwarz Lagrangian and the super-exceptional topological Yang-Mills Lagrangian unify with the super-exceptional M5 WZ curvature term into the Borel-equivariant enhancement of the super-exceptional M5brane cocycle with respect to the super-exceptional S 1 HW isometry left-induced to an ΩS 2 HWaction on the super-exceptional 1 2 M5-spacetime (definition 6.8). In order to put this in JHEP02(2020)107 perspective, we first show (proposition 6.6 below) that, similarly, the little-string-extended D = 6, N = (1, 1), superspacetime carries an ΩS 2 -action whose homotopy quotient is the D = 5, N = 2, superspacetime.
To set the scene, we first recall how homotopy quotients are represented in rational cohomology by (Borel-) equivariant de Rham cohomology. From general homotopy theory we need the following two basic facts (see [86]). For any kind of higher geometric spaces (here, rational super spaces), we have: (i) Forming based loop spaces is an equivalence from pointed connected spaces to ∞groups, whose inverse is the classifying space construction 2 (ii) For X a space and G an ∞-group, an ∞-action ρ of G on X is equivalently a homotopy fiber sequence of the following form which then exhibits the space in the middle as the homotopy quotient of X by the ∞-action ρ.
To prepare for proposition 6.6 and theorem 6.9 below, we now consider a sequence of examples of homotopy quotients of rational super spaces as in (6.2). In the following diagrams we always show the systems of spaces on the left with their super dgc-algebra (FDA) models shown on the right. Throughout we use that homotopy pullbacks of super spaces are modeled by pushouts of semi-free super dgc-algebras (FDAs) as soon as the morphism pushed out along is a cofibration in that it exhibits iterated addition of generators. For more background see, for instance, [19,43,53,62]. Example 6.1 (Rational S 1 -equivariant cohomology and Cartan model). Let X be any rational super-space of finite type, hence CE lX any finitely generated super-dgc algebra, with differential to be denoted d X , and equipped with a graded derivation ι v of degree (−1, even) such that the corresponding Lie derivative vanishes identically: The homotopy quotient by the corresponding rational S 1 -action as in (6.2) is given by

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This is the algebraic structure of the Cartan model for G-equivariant Borel cohomology (see e.g. [81, section 5], [55,82]), here for G = S 1 . Example 6.2 (Complex Hopf fibration). The complex Hopf fibration h C realizes the 2sphere S 2 as the homotopy quotient of the 3-sphere by an action which is classified by the canonical comparison map from BΩS 2 to BS 1 : (6.5) Notice that the classifying map c 1 here exhibits, by (6.1), a canonical comparison homomorphism of ∞-groups This loop ∞-group of the 2-sphere, ΩS 2 , is an ∞-group very similar to but just slightly richer than the plain circle. . This means that an ∞-action by S 1 as in example 6.1 left-induces an ∞-action by ΩS 2 , with its homotopy quotient fiber sequence (6.2) given by homotopy pullback along (6.6), as shown in the following: The resulting dgc-algebra, shown in blue, is much like the Cartan model for S 1 -equivariant cohomology as in example 6.1, except that here all even powers of the generator ω 2 in bi-degree (2, even), which classifies the circle action, are trivialized in cohomology, by the new generator ω 3 in bi-degree (3, even).
Lemma 6.4 (ΩS 2 -equivariant cocycles). Rational cocycles on a homotopy quotient X ΩS 2 for an ΩS 2 -action that is left-induced according to example 6.3 from an S 1 -action as in example 6.1 are, if they are at most linear in the generator ω 2 , precisely given by pairs consisting of a cocycle α on X and a trivialization β of its contraction with v: Proof. To see that cocycles of this form are closed, we compute as follows, directly unwinding the definitions, where the three lines correspond to application of the three summands JHEP02(2020)107 Conversely, reading this same equation as a condition for the vanishing of the coefficients of the products of generators shows that every cocycle of the form on the left of (6.8) satisfies the conditions shown on the right.

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This provides the motivation to similarly form the homotopy quotient of the superexceptional N = (1, 0) spacetime T 5,1|8 × T 1 exs from definition 4.6 not just by the S 1 HWaction of flowing along the M-theory circle fiber along the super-exceptional isometry v exs 5 from definition 4.4, but the left-induced ΩS 2 -action, via example 6.3 To indicate this, we will write ΩS 2 HW to denote this ∞-group with that ∞-action understood, hence with the comparison (6.5) specifically being Hence: Definition 6.8 (Homotopy ΩS 2 HW -quotient of super-exceptional 1 2 M5 along S 1 HW ). Write R 5,1|8 × R 1 exs ΩS 2 HW for the homotopy quotient of the super-exceptional 1 2 M5 spacetime (4.6) by the rational ΩS 2 HW -action which is left-induced, via example 6.3, by the rational S 1 HW -action given by the the super-exceptional S 1 HW -flow (4.6). Hence, with (6.7), the defining super dgc-algebra (FDA) is that on the right of the following diagram: 16) where on the right α ∈ CE (T 5,1|8 × T 1 ) exs is any element in the CE-algebra of the superexceptional 1 2 M5-spacetime (definition 4.6), ι v exs 5 is contraction with the super-exceptional isometry (4.6), and d 1 2 M5 now denotes the differential on that algebra, in contrast to the new differential d defined above. Now we may state and prove the main statement of this section: Theorem 6.9 (Super-exceptional ΩS 2 HW -equivariant M5-cocycle). The super-exceptional Perry-Schwarz Lagrangian L PS exs (5.4) and the super-exceptional topological Yang-Mills Lagrangian L tYM exs (5.6) are the components that enhance the super-exceptional M5-brane cocycle dL WZ ex (3.12) restricted along the embedding i exs of the super-exceptional 1 2 M5 spacetime (4.12) to an equivariant cocycle with respect to the ΩS 2 HW -action (6.15), hence to a cocycle on the homotopy ΩS 2 HW -quotient (6.16) of the super-exceptional 1 2 M5-spacetime, as follows: Proof. By lemma 6.4 the first claim equation (6.18), is equivalent to the two statements 1. dL PS exs = ι v exs 5 (i exs ) * dL WZ exs .
2. ι v exs 5 L PS exs = L tYM exs . The first of these is the content of proposition 5.9, while the second is lemma 5.6.
The second claim (6.19) is immediate from the nature of the map (6.16).
By proposition 5.10, both summands here already vanish separately. Proof. For the first statement we compute as follows: Here the first step is the definition of the equivariant differential (6.16), while the second step uses the definition of d 1 2 M5 from (4.11). Then under the first brace we used lemma 4.11, and under the second brace we used the definition (4.6) in proposition 4.4. The second statement is directly implied by the first and by the differential relations dω 2 = 0 and dω 3 = −ω 2 ∧ ω 2 from (6.16).

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Proof. We may rewrite (6.17) as follows: Here the first line is the definition (6.17) with the observation (6.21) inserted, as shown under the brace. Then, in the first step, we use that the differential is a derivation of bi-degree (1, even) and under the brace we unwind the definition of the equivariant differential (6.16) and then used proposition 5.9 and lemma 5.6. In the last step we collect terms and identify under the brace the super-exceptional horizontal projection from definition 4.5. This means we are now reduced to showing that the first summand in the last line of (7.3) is (i exs ) * dL WZ exs horex s = dL NG exs . (P ψ)Γ a 1 ···a 5 (P ψ) ∧ e a 0 ∧ · · · ∧ e a 5 + 1 2 id − e 5 ∧ ι v exs 5 (i exs ) * H exs ∧ e 5 ∧ ι exs v 5 (i exs ) * dH exs =0 = d(e 0 ∧ e 1 ∧ e 2 ∧ e 3 ∧ e 4 ∧ e 5 ) .

(7.5)
Here the first step is unwinding the definitions. The second step is multiplying out and using, in the second summand, the fact that pullback is an algebra homomorphism. The third step observes that the first term is just those summands of µ M5 (1.2) whose indices are along the 1 2 M5-locus, hence in {0, 1, 2, 3, 4, 5 }, while in the second term we realize the presence of the projection e 5 ∧ ι exs v 5 according to lemma 4.12, in view of dH exs = (π exs ) * µ M2 from (3.5). With the projection operator up front, this makes the second term vanish, as shown under the brace. The last step is [69, lemma 6.9]. This establishes the first line in (7.2).
Remark 7.6 (Exact ΩS 2 HW -equivariant super-exceptional M5-Lagrangians). Recall that the first summand in (5.8) is exact, implying the super-exceptional embedding construction before compactification (corollary 7.4), while the second summand in (5.8) becomes exact only after dimensional reduction, implying the super-exceptional embedding construction of the D4 WZ-term (remark 7.5). It is therefore natural to ask for a pullback of the situation to a richer extended super-spacetime on which also the second summand (7.8), and hence the full ΩS 2 HW -equivariant super-embedded super-exceptional M5-brane cocycle from (5.8), become exact, ΩS 2 HW -equivariantly. Since (7.8) is the wedge product of two equivariantly closed terms, by proposition 6.10 and proposition 6.11, there are two canonical  The new term L WZ NS5 in the Lagrangian that appears this way has the form of the WZ-term for the heterotic NS5-brane [78, (1.5)].
In summary, we thus arrive at the picture shown in (1.7).

Outlook
In closing, we briefly comment on a few interconnections, issues to be addressed in the future, and some loose ends.
Combining local super-exceptional geometry with global topology. The discussion in this article focuses on the situation of vanishing bosonic 4-flux, keeping only the super-components of the 4-flux, being the M2-brane cocycle µ M2 (1.2). We had discussed the opposite case of pure bosonic flux in [48]. In that case, the subtlety is all in the global topological structure of the Hopf-Wess-Zumino term controlled by Cohomotopy cohomology theory [47,48], while here the subtlety is all in the local differential structure of the Perry-Schwarz Lagrangian, hence of 5d super Yang-Mills plus KK-modes: In a full picture of the M5-brane sigma-model, both the super-exceptional local geometry and the cohomotopical global structure are to be combined. This will be discussed elsewhere.
Non-abelian gauge enhancement. While we have only discussed abelian gauge fields here, their appearance (by the construction in section 5, section 7) via the M2/M5 supercocycle µ M2/M5 (1.2) means that the super-cohomotopical gauge enhancement mechanism found in [19] applies. Together with the constraints of half-integral flux quantization and tadpole cancellation, which we demonstrate in [47] and [104], respectively, to follow from C-field charge quantization in full Cohomotopy (8.1), this should generate the expected types of non-abelian gauge fields, both for heterotic M-theory as well as for coincident 5-branes, in the form discussed in [40,41,100,101]. While the details remain to be worked out, this opens up the possibility of a concrete strategy for systematically deriving the non-abelian D=6 M5-brane theory from first principles.

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2. The assumptions that spinor 1-forms are fixed under Γ Γ Γ 5 is what implies the technical lemma 4.12. This lemma appears crucially in various of the following proofs leading up to and including the main theorem 7.3, notably it is used in both (5.9) and (5.10) proving proposition 5.9 and then again in (7.5) proving theorem 7.3.
This makes it seem at least unlikely that this assumption could be removed while still retaining the result of a super-exceptional embedding construction.
DBI corrections in α and super-exceptional 5-form. In our analysis of the superexceptional Perry-Schwarz Lagrangian in section 5, we have restricted to the special case that the value of the 5-index tensor e a 1 ···a 5 on super-exceptional spacetime vanishes. The formulas (3.5) and (3.6) show that, without this assumption, the super-exceptional Perry-Schwarz Lagrangian picks up further correction terms. At the same time, we have studied here the expected DBI corrections to the brane Lagrangian starting at quartic order in the field strength F [49] (see [112]). It is natural to conjecture that these two effects are related, but this needs more investigation.
AKSZ sigma-model description. The article [6] discusses the local differential structure of the M5 Hopf-Wess-Zumino term in view of exceptional geometry from the point of view of AKSZ sigma-models. The development seems complementary to ours here, but it would be interesting to see if there is a connection.