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-Fre'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) 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 Horava-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.


Introduction
An actual formulation of M-theory remains a fundamental open problem from physical and mathematical points of view (see [Mo14,Sec. 12][HSS18, Sec. 2]). We had initiated in [FSS13b] a program of attacking this problem, based on universal constructions in super-homotopy theory (see [FSS19a] for review), and used this to find firstpriniciples derivations of various aspects expected of M-theory (see [FSS16b][FSS19b] [FSS19c]). In this paper we look to carry this further and consider the following three major sub-problems: 1. Provide a systematic construction of M5-brane Lagrangians.
2. Identify M-theory avatar degrees of freedom of the gauge field and gaugino field appearing on MO9-planes.
3. Extend exceptional M-geometry to the supergeometric setting in a natural and constructive manner.
We present a unified approach which leads to an elegant joint solution to all three at once, using the two principles, super-geometry and super-homotopy theory, explained below.
• Typically, it is asserted that this is an open problem only for N ≥ 2 coincident M5-branes, while the special case of a single M5-brane is well-known. Indeed, there is a non-covariant Lagrangian formulation [PS97] [Sc97] [APPS97a] adapted to M5-s wrapped on the M-theory circle fiber, as well as a covariant version at the cost of introducing an auxiliary field [PST97] [BLNPST97]. 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 [APPS97a, Sec. 6][APPS97b, Sec. 6 & App. 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 [HW95][HW96] to be given by the M-theoretic 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 [HW95][HW96] the 10d SYM action on the MO9s is just added by hand to that of 11d supergravity.
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 [CJ79], reflecting just the expected duality symmetries acting on the corresponding string theories [HT95]. This led to the proposal [Hul07] (see also [KNS00][We03] [PW08][We11] [Ba17]) 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) locally encoding wrapping modes of the M2-and the M5-brane already before KK-compactification.
• While the exceptional generalized geometry enhancements of the bosonic sector of 11d supergravity is well studied (see, e.g., [Fe18] 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 [Ce14,p. 39] [CEK13,pp. 4,7]. Arguments were given in [Va07][Ba17] [FSS18][SS18] that the super-exceptional geometry for maximal n = 11 is to be identified with what was called the "hidden supergroup of 11d supergravity" in [D'AF82] [BAIPV04][ ADR16], but open questions remained. In particular, the physical meaning of (a) the extra fermion field η on super-exceptional spacetime (see Def. 3.3 below), (b) the parameter s ∈ R \ {0} for decompositions of the C-field (see Prop. 3.5 below) had remained open. • It may seem that supersymmetrization is but an afterthought once the bosonic sector of exceptional geometry is understood, (e.g. [BSS18] for n = 7). But most aspects of M-theory are controlled by -and are emergent from -its local supersymmetry structure (see, e.g., [To97][FSS19a]), 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 [FSS18,4.6] [SS18] we had already observed that a supersymmetric enhancement of n = 11 exceptional M-geometry is provided by what [D'AF82] called the "hidden supergroup" of 11d supergravity. With [FSS19b,Prop. 4.31] [FSS19c,Prop. 4.4], it follows that this must be the correct target space for M5-brane sigmamodels, as we explain in §3. Accordingly, in §4 we consider super-exceptional 5-brane embeddings and find in §5 that this induces the Perry-Schwarz Lagrangian (reviewed in §2) and, after super-exceptional equivariantization along the M-theory circle fiber introduced in §6, the full super-exceptional M5-brane Lagrangian, in §7. The resulting D4-brane Lagrangian with its 5d SYM+tYM Lagrangian is manifest (Remark 7.5) and 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. [APPS97a, Sec. 2]) and relegate supergeometrization to an afterthought (e.g. [APPS97a,Sec. 3]). Countering this tendency, the "doubly supersymmetric" approach of [BPSTV95] [HS97], reviewed under the name "super-embedding approach" in [So99][ So01], 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 supergeometry. In particular, the all-important "κ-symmetry" of super p-brane sigma-models, which, following [GS84], is traditionally imposed by hand onto the action principle, is revealed by the superembedding approach to be ( [STV89], see [So99,Sec. 4.3][HS05, Sec. 4.3]) nothing but the super-odd-graded component of the superworldvolume super-diffeomorphism symmetry -hence a consequence of the fundamental principle of general covariance internal to super-geometry.
p-brane sigma-models NSR-type [NS71] [R71] GS-type [GS84] [BST87] super embedding [BPSTV95]  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 [BSS18,Sec. 2] for a lightning introduction and pointers to details, and see [JSSW19] for a gentle invitation). Combining this with super-geometry yields super-homotopy theory where super-geometric ∞-groupoids (super-∞-stacks) unify super moduli spaces for higher super gauge fields with super-orbifolds appearing as super-spacetimes.  [GM13], and see [FSS19a][BSS18, Sec. 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 gradedcommutative algebra of Sullivan differential forms, regarded up to quasi-isomorphism. If we suppress some technical fine-print (see [BSS18,(8)] for the precise statement), we may schematically write this as follows:

Physics
One finds that a considerable amount of structures expected in M-theory emerge naturally in rational superhomotopy theory: • On super-geometric ∞-groupoids, the Sullivan construction of rational homotopy theory (see, e.g., [Hes06] [GM13]) unifies with higher super Lie integration [BM18,Sec. 3.1] to exhibit super L ∞ -algebroids as models for rational super-homotopy theory. Their Chevalley-Eilenberg algebras are the "FDA"s as known in the supergravity literature [vN82][D'AF82] [CDF91].
• Using super-homotopy theory, we had shown [FSS13b][FSS16a] 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.
• This process culminates [FSS19a,p. 12] in the D = 11, N = 1 (hence N = 32) super-Minkowski spacetime, carrying the super M2-brane cocycle µ M2 and, on the corresponding higher extension, the super M5-brane cocycle [FSS13b] which is the curvature of the M5 Wess-Zumino (WZ) term [BLNPST97,(8)] [FSS15] (see [FSS19a] for review): on the super M2 worldvolume. Here e a := dx a + θ Γ a dθ denotes the vielbein 1-forms which are left-invariant "li" with respect to the translational supersymmetry action of R 2,1|8·2 on itself (see Remark 3.4 below). Consequently, the super-volume form (5) has as bosonic component the ordinary volume form vol 2+1 ∈ Ω 3 R 2,1 , to which the fermionic components are added, ensuring overall supersymmetry-invariance of 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 [HSS18] 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 [FSS13b]). This will be one of the main topics that we address in this paper.
Outline of results. We establish the following: (i) In §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 §3 we recall super-exceptional M-geometry with the super-exceptional M5-brane cocycle and introduce super-exceptional embedding of M-brane spacetimes.
(iii) In §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 1 HW : super-exceptional lift of...   In the course of this identification we find that i) the putative parameter s of the super-exceptional geometry is fixed to s = −3 (Prop. 5.1 ii)) ii) the super-exception fermion η is the M-theory avatar of the heterotic gaugino field (Prop. 5.3, Rem. 5.4).
Then we show (Prop. 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 M5brane cocycle after restriction along the super-exceptional embedding of the 1 2 M5.
This is a partial analog for the M5-brane of the super-embedding construction of the M2-brane (4). To get the full statement we need not just compactify but equivariantize along S 1 HW : (v) In §6 we show that an equivariant enhancement of the super-exceptional M5-cocycle with respect to superexceptional ΩS 2 HW -action exists, where ΩS 2 is the based 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 Prop. 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 [Sa13] [FSS15], 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 [FSS16a][FSS16b] (see [FSS19a] for overview).
(vi) In §7 we put all the pieces together and establish (Cor 7.4) the full super-exceptional embedding construction of the M5-brane Lagrangian as a sum of the super-exceptional Nambu-Goto Lagrangian and the superexceptional Perry-Schwarz Lagrangian, the analogue of the M2 brane construction (4): Finally, we observe (Remark 7.6) that there are two extensions of the compactified super-exceptional 1 2 M5spacetime 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 elegantly sum up the whole picture in the following homotopy diagram: 2 Perry-Schwarz Lagrangian for M5 on S 1 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 [PS97], re-cast in coordinate-independent Cartan calculus and generalized to possibly non-trivial circle fibrations. We try to bring out the logic that motivated the construction in [PS97], but below in §6 and §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), an hence in particular for the single M5brane sigma-model, is famously subtle, at best (see e.g. [Mo12] [HR18]). But if one considers breaking manifest Lorentz invariance to 5 dimensions, as 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 [PS97][Sc97] [APPS97a], following [HT88].
This "non-covariant" formulation of self-dual higher gauge theory and specifically of the M5-brane sigmamodel may be covariantized by introducing an auxiliary scalar field [PST96] (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 [PST97] [BLNPST97]. 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 [APPS97a, p. 3].
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 S 1 = U(1)-principal bundle 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 (12), 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 (11), hence a differential 1-form which satisfies the Ehresmann conditions in that it is normalized and invariant: 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 (13). So, in particular, the composition is a projection operator: The complementary projection is that onto horizontal differential forms with respect to the bundle structure (11): Observe that: Lemma 2.1 (Horizontal vs. vertical differential). If the Ehresmann connection (14) on the S 1 bundle is flat, in that then for any differential form ω ∈ Ω • Σ 6 we have that the vertical component of the differential of its horizontal component (16) is the vertical component of its full differential: Proof. By direct computation, we have: Finally, we require the vector field v 5 from (12) to be a spacelike isometry. This means that it interacts with the Hodge star operator as 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 (12) as follows (to be called the "compactified fields", a notation that follows [PS97, (5), (6)]): With this, we get the following immediate but crucially important re-formulation of the self-duality condition after S 1 -compactification (extending [PS97, (8)]): Lemma 2.2 (Self-duality after S 1 -compactification.). Given an S 1 -bundle structure (11) on the worldvolume Σ 6 and any choice of Ehresmann connection (14), the self-duality condition (9) is equivalently expressed in terms of the compactified fields (20) as: Proof. We have the following chain of equivalences: Here the first step is decomposition into horizontal and vertical components (16), the second step uses the isometry property (19) to conclude that the two resulting component equations are equivalent to each other. The last step identifies the compactified fields (20).
The gauge field. The contraction of the vector field v 5 from (12) with the 2-form potential B from (10) 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 (16). We say that the 2-flux density encoded by B is the horizontal component of the exterior differential of this vector potential 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 following0 to just call it "F" already in the 6d compactification before passing to KK zero-modes.
Proof. We compute as follows: Here the first step is the definition (20), while the second step is (10). The fourth step uses the definition (22) of the vector potential and identifies the Lie derivative (13). The fifth step applies vertical/horizontal decomposition (16) to both summands and uses (24) under the first brace and the expressions (14) and (22) under the third brace; while under the second brace it uses Cartan's formula (13), observing that ι v 5 A = 0 by definition (22) 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 (9) in the equivalent form (21) after S 1 -compactification says that the combination H − L v 5 B hor is horizontally exact: Weak self-duality and PS equations of motion. In summary, we have the following implication of self-duality after S 1 -compactification (extending [PS97, (16)]): Proposition 2.4 (Self-duality for flat circle bundles). If the worldvolume Σ 6 is equipped with an S 1 -principal bundle structure (11) which is flat (17), then the self-duality condition H = * H from (9), in its equivalent incarnation on compactified fields (27) implies the following differential equation: By Lemma 2.1, equation (28) may be understood as expressing "self-duality up to horizontally exact terms". The proposal of [PS97] is to regard (28), 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.
The PS Lagrangian density. As in [PS97,(17)], we say: Definition 2.5 (Bosonic PS Lagrangian). The bosonic Perry-Schwarz-Lagrangian for a 2-form field (10) on a worldvolume Σ 6 compactified on a flat (17) S 1 -bundle (11) is: Remark 2.6 (Equivalent descriptions). (i) Equivalently, using the isometry property (19) the bosonic PS Lagrangian (29) reads where in the second line we inserted the expression for H from (20); (ii) and if the self-duality condition (9) is imposed, the bosonic PS Lagrangian (30) becomes At this point, as in [PS97,(16)], the bosonic PS Lagrangian of Def. 2.5 is motivated as the evident choice that makes the following Prop. 2.7 true. Below in §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 (28).
Reduction to 5d Maxwell theory. Consider finally the special case of Kaluza-Klein compactification/double dimensional reduction, where L v 5 B hor = 0 for KK-reduction to D4.
In this case, expression (26) reduces to Hence, with self-duality ([PS97, above (16)]), we have where now * 5 : is the Hodge star operator on the base 5-manifold. Consequently, we have extension of [PS97,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 (31) is the Lagrangian of 5d Maxwell theory for the vector potential A from (22): 6d self-dual field as (abelian) 5d Yang-Mills with KK-modes. Due to Prop. 2.8, one may regard the general bosonic Perry-Schwarz Lagrangian (Def. 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 [Do10][LPS10] (see [La19,3.4.3])). Here for the abelian bosonic sector, this perspective may be fully brought out by introducing the following notation: where on the left we have (25), 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 (9) as in (31), finds the following suggestive expression: so that KK-reduction to 5d YM theory, as in Prop. 2.8, is now given syntactically simply by replacing F * 5 F. We find the form (37) of the PS Lagrangian to be reflected by its super-exceptionalization in Prop. 5.1 below; see Def. 5.2 and Remark 5.8.
We find this arise from the super-exceptional embedding construction below in Theorem 7.3, see Remark 7.5.
3 Super-exceptional M-geometry  [FSS19c] identify this as the correct local target for M5-brane sigma-models.
For definiteness, we make explicit our spinor and Clifford algebra convention: . We write 32 for the irreducible real representation of Pin + (10, 1), via a Clifford-algebra representation {Γ a Γ a Γ a } 10 a=0 whose generators satisfy the following relations: . We will use " Γ a " for exhibiting expressions manifestly compatible with the literature, and " Γ Γ Γ a " for emphasizing the actual real Pin + (10, 1)-action.
The following, Def. 3.3 and Prop. 3.5, are a formulation in rational super-homotopy theory due to [FSS18,Sec. 4.5], of the classical supergravity results in [D'AF82, Sec. 6][BAIPV04, Sec. 3] (see also [ADR16]) . The Definition 3.3 of super-exceptional spacetime involves a parameter s eq. (40), which arises mathematically in Prop. 3.5 from different possibilities of decomposing the H 3 -flux on super-exceptional spacetime [BAIPV04]. We discover the physical meaning of this parameter below in §5.
the D = 11, N = 1, n = 11 super-exceptional M-theory spacetime (T 10,1|32 ) ex s 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: deg=(1,even) e a 1 a 2 0≤a 1 <a 2 ≤10 deg=(1,even) e a 1 ···a 5 0≤a 1 <···<a 5 ≤10 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 Def. 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.
Remark 3.4 (Super-exceptional M-theory spacetime as a supermanifold). We may alternatively regard T 10,1|32 ex s from Def. 3.3 as a super-manifold with canonical global coordinate functions , hence with bosonic part being the exceptional tangent bundle (1) for maximal n = 11: and with super-group structure such that the Chevalley-Eilenberg algebra in Def. 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 := B a 1 a 2 dx a ∧ dx b d(B a 1 a 2 ) = dx µ ∂ µ B a 1 a 2 (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 (48) below. This has a crucial effect in the following discussion; see Lemma 5.1.
We will see below in Prop. 5.1 (ii). that in super-exceptional M-geometry at value s = −3 the bosonic Perry-Schwarz Lagrangian appears naturally.
Remark 3.9 (Super-exceptional spacetime as target for M5-brane sigma-models). (iii) However, in [FSS19b,Prop. 4.31] [FSS19c,Prop. 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: Any such outer square factors universally through the homotopy pullback square shown on the right, which exhibits m2brane as the super Lie 3-algebra underlying the universal moduli super 3-stack of M5-brane sigma-model fields [FSS15]. But now with the (extended) worldvolume Σ itself a super-manifold, this factorization through m2brane 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 super-exceptional spacetime R 10,1|32 ex s [FSS18, Sec. 4.6].
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 [DHTV14, Sec. 6]. We make this mathematically concrete with Prop. 4.4 below, whose proof shows that the particular spinor structure of the 1 2 M5-locus (Def. 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 Prop. 5.1 the exceptional pre-image of the bosonic 2-flux density F. Finally, we show in Prop. 5.3 that this induces the super 2-flux of super Yang-Mills theory thereby, in particular, identifying the extra fermion field on exceptional super-spacetime with the M-theoretic avatar of the gaugino field in 10d heterotic string theory.  Fig. 2][HKLTY15]). On the other hand, if the circle 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 [GKST01,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., [HZ99]).
2. Fundamental (sigma-model) brane. The last line in the above diagram indicates the extension of the actual M5-brane worldvolume for which we are to construct a Lagrangian density. This is not required to be fixed by Γ Γ Γ 5 and hence may stretch along the 5 -5-plane/torus "at an angle", partially wrapping around the S 1 HW , partially running parallel to it [Wi97,p. 9], corresponding to D4/NS5 bound states in IIA [Wi97, Sec. 2] and (p, q)-5brane webs in IIB [AH97], or rather their further intersection with an O-plane [HZ99] (see [HKLTY15,around Fig. 10]).
and for all a ∈ {1, 2, 3, 4, 5 , 5}, the infinitesimal superspace symmetry v a v a ∈ Γ T R 6,1|16 , ι v a : for any odd-graded constants (Pχ) α Proof. Direct inspection of the defining differential relations in (53) 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: where under the brace we used the defining property (51) of the spinor projections of Def. 4.1.
The following is a direct consequence of the above definitions: superspacetime  In this case, there is a corresponding alternative to the super-exceptional isometry (56) given by v ex s 9 := δ a 9 v a − (1 + s) 1 2 ε a 1 a 2 01 v a 1 a 2 + χ α v α . 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 Prop. 5.1 below. This does not mean that this alternative case is not of interest, but its interpretation will need to be discussed elsewhere.
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 ···a n , thereby picking up a sign σ = +1 | odd number of a i s = 5 −1 | even number of a i s = 5 (65) Finally we use (Γ Γ Γ 5 ) † = Γ Γ Γ 5 from (39) 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 ···a n has an even number of indices differing from 5.
For the M-brane cochains, and in terms of super-exceptional embedding, this means the following: with ι v 5 given in (55). Hence, for the super-exceptional M-brane cocycles (Def. 3.8), we have Proof. By Lemma 4.8, the pullbacks along i and i ex s act on the spinors by applying the projection P 8 . Hence Lemma 4.11 applies to the spinor bilinears after restriction and says that all summands in the pullbacks of the M-brane cochains that do not contain an index = 5 (for the M2-brane cocycle) or do contain an index = 5 (for the M5-brane cochain) vanish. This is just what is expressed by the projection operators in (66).

Super-exceptional Perry-Schwarz & Yang-Mills Lagrangians
In Prop. 5.1 we find a natural super-exceptional pre-image of the bosonic Perry-Schwarz Lagrangian, recorded as Def. 5.2 below. This allows us to extract the super-components (in Prop. 5.3 below) and identify the superexceptional M-theory avatar of the gaugino field (Remark 5.4 below). We also find the super-exceptional lift of the topological Yang-Mills Lagrangian (Def. 5.5 below) and its relation to the super-exceptional Perry-Schwarz Lagrangian (Lemma 5.6 below). This plays a crucial role when we unify all this super-exceptional data in §6. Then we show (Prop. 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 M5-spacetime and compactified on the M/HW-theory circle (Def. 5.7 below). This is a key ingredient in the full super-embedding Theorem 7.3 further below in §7 We start with identifying the super-exceptional lift of the PS Lagrangian.
Proposition 5.1 (Super-exceptional lift of bosonic 2-flux and PS Lagrangian). Consider Σ bos := R 5,1 × R 1 and a section σ of the super-exceptional 1 2 M5-brane spacetime projection (62) : where on the left H := d(B a 2 a 2 ) ∧ dx a 2 ∧ dx a 3 denotes the plain H-flux (44) and F its induced 2-form flux (25) according to Lemma 2.3; and hence on the right we find a super-exceptional pre-image F ex s 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 (31), and hence on the right we find a super-exceptional pre-image L PS ex s . 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 (44) by direct computation that This proves the first statement. For the second, it is now sufficient to observe with (48) that, by the assumption s = −3, we have in the present case σ * H ex s = 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 [ES03].
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 Notice that this is indeed still a cocycle, in that it is closed, d ι v exs 5 (i ex s ) * dL WZ ex s = 0, by (58) in Prop. 4.4.
Remark 5.8 (Cocycle for M5 /S 1 as D4 + KK). From the type I perspective on the 1 2 M5-locus (as in Remark 4.2) the compactified M5-cocycle of Def. 5.7 would be that of a D4-brane inside a 1 2 NS5 = NS5 ∩ O8, by the general rules of dimensional reduction of brane cocycles [FSS16a, Sec. 3][FSS16b, Sec. 3][BSS18, Sec. 2.2]. However, since we need not consider double dimensional reduction, in that the super-exceptional coordinate functions along v 5 are still present in the normally thickened 1 2 M5 locus R 5,1|8 × R 1 ex s (Def. 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, Def. 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 [Do10][LPS10] (see [La19,3.4

.3]).
We now establish the following super-exceptional analog of the super-embedding mechanism.
Proposition 5.9 (PS Lagrangian trivializes compactified M5-cocycle along super-exceptional embedding). The compactification of the restriction of the super-exceptional M5-brane cocycle (Def. 5.7) along the embedding of the super-exceptional 1 2 M5 spacetime (Def. 4.6) is trivialized by the super-exceptional PS Lagrangian (74): 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: 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 ex s 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 dg-algebra homomorphism, by definition) and then with the contraction operation, using that the corresponding Lie derivative vanishes, by (58) in Prop. 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 ex s v 5 according to Lemma 4.12, in view of dH ex s = (π ex s ) * µ M2 (43). 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 Def. 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 (58) from Lemma 4.4 and then Prop. 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.
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 M5-brane cocycle with respect to the super-exceptional S 1 HW isometry leftinduced to an ΩS 2 HW -action on the super-exceptional 1 2 M5-spacetime (Def. 6.8). In order to put this in perspective, we first show (Prop. 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 [NSS12]). 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 (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 Prop. 6.6 and Theorem 6.9 below, we now consider a sequence of examples of homotopy quotients of rational super spaces as in (82). 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, [Hes06] 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 (82) is given by Example 6.2 (Complex Hopf fibration). The complex Hopf fibration h C realizes the 2-sphere 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 : Notice that the classifying map c 1 here exhibits, by (81), a canonical comparison homomorphism of ∞-groups ΩS 2 This loop ∞-group of the 2-sphere, ΩS 2 , is an ∞-group very similar to but just slightly richer than the plain circle.
Example 6.3 (Left-induced ΩS 2 -equivariant cohomology). 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 (82) given by homotopy pullback along (86), 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 of the differential 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 (88) satisfies the conditions shown on the right.
We next observe, in Prop 6.6 below, that an ∞-action by the ∞-group ΩS 2 (86) characterizes the D = 6, N = (1, 1), super-spacetime fibered over the D = 5, N = 2, superspacetime and exhibits the little-string cocycle in D = 6 as coming from a 2-sphere-valued super-cocycle in D = 5 (hence in rational Cohomotopy in degree 2). To put this in perspective, we first recall the analogous situation in D = 11: (2) constitute a super rational S 4 -valued cocycle, i.e., a cocycle in rational Cohomotopy in degree 4.
(i) Conversely, the total rectangle on the left of (91) exhibits, by (82), the D = 5 super-spacetime R 4,1|8+8 as the homotopy ΩS 2 -quotient of the little string-extended D = 6, N = 2, super-spacetime: (ii) To make this explicit as an ΩS 2 -quotient in the form of (87), we may (co)fibrantly resolve the bottom instead of the right morphism of the rectangle in (91). This yields the following: In conclusion, D = 6, N = (1, 1), superspacetime extended by its brane content reduces to D = 5 superspacetime by enhancing the ∞-action by S 1 to an ∞-action by ΩS 2 (86).
This provides the motivation to similarly form the homotopy quotient of the super-exceptional N = (1, 0) spacetime R 5,1|8 × R 1 ex s from Def. 4.6 not just by the S 1 HW -action of flowing along the M-theory circle fiber along the super-exceptional isometry v ex s 5 from Def. 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 (85) 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 ex s 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 leftinduced, via Example 6.3, by the rational S 1 HW -action given by the the super-exceptional S 1 HW -flow (56). Hence, with (87), the defining super dgc-algebra (FDA) is that on the right of the following diagram: where on the right α ∈ CE (R 5,1|8 × R 1 ) ex s is any element in the CE-algebra of the super-exceptional 1 2 M5spacetime (Def. 4.6), ι v exs 5 is contraction with the super-exceptional isometry (56), and d 1 /2M5 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: in that (i) this is indeed a cocycle with respect to the differential d = d1 /2M5 + ω 2 ∧ ι v exs 5 + d S 2 from (96): (ii) it does enhance the super-exceptional M5-brane cocycle, in that it extends it through the homotopy quotient projection q ΩS 2 HW (96): Proof. By Lemma 6.4 the first claim equation (98), is equivalent to the two statements 1. dL PS ex s = ι v exs 5 (i ex s ) * dL WZ ex s . 2. ι v exs 5 L PS ex s = L tYM ex s . The first of these is the content of Prop. 5.9, while the second is Lemma 5.6.
The second claim (99) is immediate from the nature of the map (96).
By Prop. 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 (96), while the second step uses the definition of d1 /2 M5 from (61). Then under the first brace we used Lemma 4.11, and under the second brace we used the definition (56) in Prop. 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 (96).

Super-exceptional M5 Lagrangian from super-exceptional embedding
Finally we discuss the super-exceptional embedding construction of the M5-brane Lagrangian. We consider the super-exceptional Nambu-Goto Lagrangian for the 1 2 M5-brane (Def. 7.1) below and prove (Corollary 7.4 below) that the sum of super-exceptional Nambu-Goto Lagrangian and the super-exceptional Perry-Schwarz Lagrangian arise as relative trivialization of the super-exceptional M5-brane cocycle along the super-exceptional embedding of the 1 2 M5-brane. We go further and and consider the ΩS 2 HW -equivariant enhancement of this statement, corresponding to KK-compactification to the D4-brane, and prove (Theorem 7.3 below) that this corrects the relative trivialization by a summand proportional to the super-exceptional topological Yang-Mills term that constitutes the D4-brane WZ term (Remark 7.5 below). While a priori this further summand is exact only after compactification on S 1 HW , as befits the nature of the D4 arising form the M5, we observe (Remark 7.6 below) that this D4 term, too, does become genuinely exact after a natural completion of the ΩS 2 HW -action on the super-exceptional 1 2 M5 spacetime.
In direct generalization of the super Nambu-Goto Lagrangian (5), we set: Definition 7.1. The super-exceptional Nambu-Goto Lagrangian for the 1 2 M5-brane (Remark 4.2) is the super volume form of the 1 2 M5-locus, hence, with (42), the left-invariant completion of the bosonic volume form under translational supersymmetry, hence is the element L NG ex s := svol 5+1 ex s := e 0 ∧ e 1 ∧ e 2 ∧ e 3 ∧ e 4 ∧ e 5 = (π ex s ) * e 0 ∧ e 1 ∧ e 2 ∧ e 3 ∧ e 4 ∧ e 5 ∈ CE R 5,1|8 × R 1 ex s in the CE-algebra of the super-exceptional 1 2 M5-spacetime (Def. 4.6.) Remark 7.2 (S 1 B vs. S 1 HW directions). Beware that the last wedge factor in (103) is e 5 and that e 5 does not appear (see Remark 4.2 for discussion of the 5-5 -plane). Mathematically, this comes out from the MO9-projection in the last step (107) in the proof of Theorem 7.3 below.
where under the brace we show an equivalent re-formulation in terms of the element C 1 from (90).
Proof. We may rewrite (97) as follows: Here the first line is the definition (97) with the observation (101) 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 (96) and then used Prop. 5.9 and Lemma 5.6. In the last step we collect terms and identify under the brace the super-exceptional horizontal projection from Def. 4.5. This means we are now reduced to showing that the first summand in the last line of (105) is We compute as follows: (i ex s ) * dL WZ ex s hor exs = id − e 5 ∧ ι v exs 5 (i ex s ) * (π ex s ) * µ M5 + 1 2 H ex s ∧ dH ex s = id − e 5 ∧ ι v exs 5 (i ex s ) * (π ex s ) * µ M5 + 1 2 id − e 5 ∧ ι v exs 5 (i ex s ) * H ex s ∧ (i ex s ) * dH ex s = 1 5! ∑ a i ∈{0,1,2,3,4,5 } (Pψ)Γ a 1 ···a 5 (Pψ) ∧ e a 0 ∧ · · · ∧ e a 5 + 1 2 id − e 5 ∧ ι v exs 5 (i ex s ) * H ex s ∧ e 5 ∧ ι ex s v 5 (i ex s ) * dH ex s =0 = d(e 0 ∧ e 1 ∧ e 2 ∧ e 3 ∧ e 4 ∧ e 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 (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 ex s v 5 according to Lemma 4.12, in view of dH ex s = (π ex s ) * µ M2 from (43). With the projection operator up front, this makes the second term vanish, as shown under the brace. The last step is [HSS18, Lemma 6.9]. This establishes the first line in (104).
Finally, to see equality to the expression shown in (104) under the brace, observe that where the first step is (101), the second step is (108) and the third step is (100). This, therefore, establishes also the identification under the brace in (104). (i ex s ) * dL WZ ex s = d L WZ ex s + L PS ex s ∧ e 5 .
Proof. The statement is the first component (the one independent of the equivariance generators ω 2 and ω 3 ) of the equivariant statement in Theorem 7.3. More formally, equation (109) is the pullback of (13) along the homotopy quotient homomorphism q Remark 7.8 (The heterotic 5-brane). The heterotic super-exceptional 1 /2M5-spacetime T 5,1|16 × T 1 het ex s is the homotopy fiber of the wedge square of the super-exceptional 2-flux (73) on the super-exceptional 1 /2M5-spacetime (Def. 4.6) .
The new term L WZ NS5 in the Lagrangian that appears this way has the form of the WZ-term for the heterotic NS5brane [Le10,(1.5)].
In summary, we thus arrive at the picture shown in (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 (2). We had discussed the opposite case of pure bosonic flux in [FSS19c]. In that case, the subtlety is all in the global topological structure of the Hopf-Wess-Zumino term controlled by Cohomotopy [FSS19c] [FSS19b], 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: Single M5-brane sigma-model Aspect Global topological Local differential 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. However, as explained in [HSS18,, in the vein of understanding supergravity as super Cartan geometry [Lo90][EE12], we are to think of the super-exceptional Minkowski spacetimes considered here, such as the superexceptional 1 2 M5-spacetime illustrated in (52), as being the infinitesimal moving frames on a super-exceptional curved Cartan geometry. Furthermore, the covariant form of this geometry is to be obtained systematically by actually moving the frames, subject only to the condition of vanishing super-torsion, which, by [CL94][Ho97] (see [CGNT04,Sec. 2.4]), is equivalently the condition that the equations of motion of 11d supergravity hold.
Thus super-Cartan geometry provides a systematic and precise mechanism for globalizing/covariantizing all the local brane constructions given here, as well as the related constructions in [HSS18][BSS18], reviewed in [FSS19a]. While this provides the algorithm, it still has to be run, the details still have to be worked out elsewhere.
Necessity of the 1 2 M5 configuration? The string theory folklore claims (e.g. [DHTV14]) that there are two ways of geometrically engineering D = 6 N = (1, 0) theories, one of them being the 1 2 M5-brane configuration of Remark 4.2. Since our rigorous construction confirms this expectation, it would be interesting to precisely classify this -on a mathematical basis -and possibly determine further available choices for the construction. This needs further thinking. At this point, we may at least highlight precisely where the spinor projections of Def. 4.1, which define the 1 2 M5-locus, enter our proofs: 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 super-exceptional Perry-Schwarz Lagrangian in §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 (43) and (44) 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 [FT85] (see [Ts00]). It is natural to conjecture that these two effects are related, but this needs more investigation.
AKSZ sigma-model description. The article [Ar18] 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.