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Further towards homotopy-theoretic foundations for the M5-brane model in M-theory, we have a note observing emergence of twisted string structure (“non-abelian gerbe field”) on the M5-worldvolume, derived from twisted cohomotopy-theory, assuming Hypothesis H:
This amounts to observing one big homotopy-commutative diagram, the outer part of which we had derived earlier. The observation that the inner part thus factors through $B String^{c_2}$ follows immediately. The article walks a spotlight through this big diagram, explaining all its constitutents.
Comments are welcome. Please grab the latest version of the file from behind the above link.
p. 3 ’Here we restrict attention to the subgroup … only for sake of exposition’ Is this because otherwise you’d have loads of $Sp(1)$ factors all over, and these are making little difference to the mathematics?
p. 6 The analogy between rational superspaces and topological spaces. What happens with the topology analogue of the 517-torus fibration? Do you need some form of supergeometry to generate it?
Typos
conisdering; Stringvstructure; quatization
and
In direct analogy with $Spin^c$-structure, is called $String^{c_2}$-structure…
missing ’it’.
Is this because otherwise you’d have loads of $Sp(1)$ factors all over, and these are making little difference to the mathematics?
Yes, exactly
p. 6 The analogy between rational superspaces and topological spaces. What happens with the topology analogue of the 517-torus fibration? Do you need some form of supergeometry to generate it?
I think now the combination of the two will be
$\mathbb{R}^{10,1\vert \mathbf{32}}_{exc,s} \sslash String^{c_2}_{\chi_8}$hence the “super-exceptional Poincaré $String^{c_2}$ group”, if you wish.
On this, the Hopf WZ term is going to be a universal 5-gerbe connection
$\mathbb{R}^{10,1\vert \mathbf{32}}_{exc,s} \sslash String^{c_2}_{\chi_8} \overset{\;\;\;\; \widetilde{\mathbf{\Gamma}}_7\;\;\;\;}{\longrightarrow} B^6 U(1)_{conn}$whose curvature 7-form is the M5 super-cocycle
$\mathbb{R}^{10,1\vert \mathbf{32}}_{exc,s} \sslash String^{c_2}_{\chi_8} \overset{h_3^{exc}\wedge \mu_{M2} + \mu_{M6}}{\longrightarrow} \mathbf{\Omega}^7_{cl}$from FSS19c
and whose integral class is the
$\mathbb{R}^{10,1\vert \mathbf{32}}_{exc,s} \sslash String^{c_2}_{\chi_8} \to B String^{c_2}_{\chi_8} \overset{ \widetilde \Gamma_7 \;\coloneqq\; H_3^{univ} \wedge (\widetilde \Gamma_4 + \tfrac{1}{2}p_1) + 2\Gamma_7 }{\longrightarrow} K(\mathbb{Z},7)$from FSS19b.
Something like this…
Typos
Thanks! Fixed now.
For what it’s worth, I think I’ve calculated that
$[S^5,B String^{c_2}(4)] \simeq (\mathbb{Z}/2)^3$so that if your M5-brane is topologically a sphere, there are eight possible “sectors” for $B String^{c_2}(4)$-bundles on it. Does this show up in the physics?
The relation to “the physics” is this:
1) in the bottom part of the big diagram the slicing over $B Spin(8)$ is what enforces the relation between the C-field and the field of gravity, which is the anomaly cancellation conditions, such as that $[G_4 +\tfrac{1}{4}p_1]$ is integral.
2) the precomposition of that with the Borel-equivariant Hopf fibration encodes the relative shifted trivialization of the C-field on the M5-worldvolume
The commutativity of the outer diagram follows, and hence the map to $B String^{c_2}(4)$ (for instance as an element in $[S^5, B String^{c_2}(4)]$) is determined by these two physics conditions: anomaly cancellation and relative trivialization of the C-field. These together serve to single out that twisted string class.
So what you might want to check is this:
If we fix the worldvolume topology $\Sigma$ but allow the target 8-manifold $X^8$ to vary, to which extent does the twisted String class on $\Sigma$ vary? That would be interesting to know!
anomaly cancellation and relative trivialization of the C-field. These together serve to single out that twisted string class.
hmm, that’s not as deep as I was thinking. :-)
If we fix the worldvolume topology $\Sigma$ but allow the target 8-manifold $X^8$ to vary, to which extent does the twisted String class on $\Sigma$ vary? That would be interesting to know!
Hmm, yes, that would be interesting. I guess you mean not just $X^8$ but the embedding $\phi\colon \Sigma \to \mathbb{R}^{2,1} \times X^8$? And maybe also $b$?
hmm, that’s not as deep as I was thinking. :-)
let me rephrase that. What I was thinking was that is in principle possible for someone to have done some kind of toy-model calculation for a spherical M5-brane on a specific target space, and applying the anomaly cancellation and shifted trivialisation conditions, found three independent configurations (corresponding to the three factors of $(\mathbb{Z}/2)^3)$. One might expect such configurations to have more concrete interpretations from a physics point of view. Or maybe I’m being too naive, and the sort of “realistic” target 8-manifolds don’t particularly support M5-branes of that shape.
it is in principle possible for someone to have done some kind of toy-model calculation
Yes, since it hit the arXiv 11 hours ago.
Be the first.
Isn’t the worldvolume of the M5-brane of dimension 6? Why the 5-sphere?
David R. is thinking of $\Sigma = \mathbb{R}^{0,1} \times S^5$.
I would be happy to make some small contribution, but I fear that I would be missing any crucial physical that may rule out my simplistic approach. Also, I wouldn’t quite know how to put such an idea is the language that M-theorists would be used to. Also, having a sensible and motivated target space in which to place the spherical M5-brane, along the lines of something already in the literature. I guess at the least one would want $X^8$ to have nontrivial $\pi_5$, so as to wrap an M5-brane of this form around it, and better, have a factor of something like $\mathbb{R}^6 \setminus \{0\}$. But I know $X^8$ is usually something rather rich
On the contrary. The mathematical formulation allows to run without bothering about the folklore.
Just idling about, I came across Hitchin’s SL(2) over the octonions where he mentions (p. 13) a couple of fibrations arising for quaternionic reasons that I hadn’t seen yet
$S^3 \times S^3 \to Sp(2) \to S4$, $S^3 \to Sp(2) \to S^7$.
Feels like the second at least is in the vicinity.
Oh, but the second of these is the fiber of the projection map of the coset space realization $Sp(2)/Sp(1) \simeq S^7$; and the first composes that with the quaternionic Hopf fibration.
Well, I can mock something up and run it by you, to see if you think it would pass muster in arXiv:hep-th
Re #14, oh yes, of course.
Does that idea of Atiyah and Witten to look at M-theory on $G_2$-manifolds in relation to M-theory on the 8-manifold $\mathbf{H P}^2$, in the section on confinement, point perhaps to a broader relationship between M-theory on G2-manifolds and M-theory on 8-manifolds, something that could plug into the work you’ve done here?
An insightful discussion of the relation between 7d $G_2$-structure and 8d $Spin(7)$-structure is in arXiv:1405.3698
[ I am on vacation with family for a few days, back next week ]
Thanks. Enjoy your break.
For what it’s worth, what I mentioned in #15 is now available here: Topological sectors for heterotic M5-brane charges under Hypothesis H (unstable link, ok for a few months). Thanks to Urs for lots of discussion and useful references. [edit: fixed link, it broke when updating to an edited version]
Finally here is a followup, deducing the local differential-form content complementing the previously discussed topological sector:
$\,$
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If you are interested, please grab the latest pdf from behind that link.
Comments are welcome.
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Abstract:
A new super-exceptional embedding construction of the heterotic M5-brane’s sigma-model was recently shown to produce, at leading order in the super-exceptional vielbein components, the super-Nambu-Goto (Green-Schwarz-type) Lagrangian for the embedding fields plus the Perry-Schwarz Lagrangian for the abelian self-dual higher gauge field. Beyond that, further fields emerge in the model, arising from probe M2- and probe M5-brane wrapping modes. Here we classify the full super-exceptional field content and work out some of their characteristic interactions from the rich super-exceptional Lagrangian of the model. We show that $\mathrm{SU}(2) \times U(1)$-valued scalar and vector fields emerge in the model from probe M2- and M5-branes wrapping the vanishing cycle in the $\mathrm{A}_1$-type singularity; together with a pair of spinor fields of $U(1)$-hypercharge $\pm 1$ and each transforming as $\mathrm{SU}(2)$ iso-doublets. Then we highlight the appearance of a WZW-type term in the super-exceptional PS-Lagrangian and find that on the electromagnetic field it gives the first DBI-correction, while on the iso-vector scalar field it has the form characteristic of the coupling of vector mesons to pions via the Skyrme baryon current. We discuss how this is suggestive of a form of $\mathrm{SU}(2)$-flavor chiral hadrodynamics emerging on the single ($N=1$) heterotic M5-brane, different from, but akin to, large-N holographic QCD.
A naive question regarding your comment on the ’deep M-theoretic regime’: this article examines $N=1$ branes, and you mention $N\gg 1$ is unrealistic. Is there any relation of $N$ to other things like the number of flavours? I have a vague memory of you mentioning something like this before. Or am I just conflating a bunch of random ideas?
Yes, the gauge symmetry on $N$ coincident D-branes is, in the plain vanilla case, $SU(N)$. If these are color branes then this is the usual gauge symmetry of $N = N_c$-colors. If they instead are flavor branes then this is $SU(N)$ flavor symmetry of $N = N_f$ flavors.
In the color case, a large (in fact fantastically huge) number $N$ of coincident branes becomes approximated by a classical supergravity solution and thus the corresponding $SU(N)$ color gauge theory may be described by the AdS/QCD correspondence. This approximation works remarkably well to a large extent, but is ultimately unrealistic, since in nature $N_c$ is not fantastically large but equals 3.
Moreover, people then wave their hands and claim that with a fantastically large number of color branes present, they can just as well pretend that $N_f =2$ or $=3$ flavor branes are also present without further ado. This assumption gets baptized “adding probe branes”.
One would really want to consider $N_c = 3$ color branes with $N_f =2$ or $=3$ flavor branes properly added, but it is (or was) unknown how to do this, since it means to work in the highly curved strongly coupled regime, aka M-theory. This state of affairs is what the first graphics here means to illustrate.
Unfortunately little time at the moment. But presumably we can think of (57) in relation to the diagram of maximal central extensions of super Lie algebras from the superpoint. But we don’t usually see $\mathbb{R}^{6,1| \mathbf{16}}$ there.
The 7d worldvolume of the MK6 is an intermediate step: The $\tfrac{1}{2}M5$ is the intersection of that with the MO9-plane.
This is a key subtlety that runs through the whole discussion of the heterotic M5: It’s really what some people call a “domain wall in the MK6”.
Moreover, the super-exceptional $\tfrac{1}{2}M5$-locus retains (as KK-modes) the higher vielbein modes perpendicular to the M5-locus inside the Mk6. As a result, the passage from the super-exceptional MK6 locus $\big(\mathbb{R}^{10,1\vert \mathbf{32}}_{\mathrm{ex}_s}\big)^{\mathbb{Z}_2^A}$ to the super-exceptional $\tfrac{1}{2}M5$-locus $\big(\mathbb{R}^{9,1\vert \mathbf{32}}_{\mathrm{ex}_s}\big)^{\mathbb{Z}_2^A}$ only removes one chiral half of the fermions.
For that reason the big diagram on p.4 focuses on the field content as seen on the MK6.
(We tried other versions, but they tended to become harder to read.)
Some trivial things:
understood long time ago
a long time ago’ or ’long ago’
the question on the extent
the question of the extent
supersymetric
supersymmetric
they are all meant to be fits in effective field theory to clues from experiment.
Sounds odd.
[MS15] in the bibliography comes before [MH16], and other similar.
Thanks!
Hm, the references are alphabetically ordered by author’s last names…
But, I see, MH16 should come before MS15 even then. Changing now…
As should ME11. But it’s an odd way to arrange things. I see in the text MS15 and want to look it up in the bibliography, and have to hunt all through the M’s because I don’t know what M stands for.
But with a paper like this, we should be discussing more profound issues! I was wondering whether pure mathematicians had encountered traces of related structures. Given how tightly connected are exceptional structures, I guess, in a sense obviously Yes. But are they anywhere close to ’super-exceptional geometry’?
So the big remaining mathematical question about the “super-exceptional” spacetime $\mathbb{R}^{10,1\vert \mathbf{32}}_{\mathrm{ex}_s}$ remains: What is its universal property, if any? It’s roughly something like the universal super Lie 1-algebra equipped with a map to the sugra super Lie 3-algebra $hofib(\mu_2)$ which is surjective onto the space of generators. Only that I don’t know how to make precise the word “universal” here for this to be true in a precise sense.
But I should highlight that the focus of this particular article (the third or fourth now in the series concerned with the super-exceptional spacetime) is decidedly on its physics meaning. Given that we show that the field content and parts of the interactions clearly smell of 2-flavor hadrodynamics, the big question arising here is: What about the rest?
We have notes showing (as you may have guessed from the lists of references added to the nLab as of late) that the further fermionic couplings we get match those of radiative and leptonic decays of mesons – if the super-exceptional fermions are interpreted as leptons. But shouldn’t they rather be interpreted as baryons as in the Walecka-type models? Or maybe – a wild thought – both (unified in a KK-tower, the leptons being the massless modes)? And maybe that’s not so wild after all, given the generations pattern and specifically the old idea of baryon-lepton symmetry.
If one interprets the super-exceptional fermions as leptons and passes fully to the $\tfrac{1}{2}M5$-locus, then one ends up with two pairs of chiral leptons, one of them interacting with the mesons (hence secretly via the weak force) while the other does not. That’s striking, because this is the single most important characteristic property of the standard model.
I have spent hours and hours and days and weeks now banging my head against this. (I hear something weird has happened in the world in the last weeks? Haven’t checked yet, need to catch up on the news. Everyone’s okay? ;-)
But in the end we decided to leave these and other questions for later and first write an article on just those interaction terms that we can clearly identify.
divide et impera.
Well, Newton had two lengthy plague-induced stays at Woolsthorpe Manor between summer 1665 and spring 1667, so you can always count on having some further time during the ’second wave’.
I don’t have more time to work when the world is under house arrest. Because then I have to design advanced treasure hunts through the garden for a 6 yo all day. I found the semileptonic omega-meson decays while advising a marble run setup.
But now that we had the exchange above, it occurs to me that I may have been missing the obvious regarding the first point in #30: I suppose the super-exceptional spacetime wants to be the “1-connected 1-truncated cover” of the M2 super Lie 3-algebra. Hm…
When you say ’the M2 super Lie 3-algebra’ you mean 11d supergravity Lie 3-algebra rather than M2-brane 3-algebra? And the former is distinct from but related to M-theory supersymmetry algebra?
Yes, that’s right, in all clauses.
We originally called this the $\mathfrak{sugra}$ Lie 3-algebra (on p. 54 here) but then later the $\mathfrak{m}2\mathfrak{brane}$ Lie 3-algebra (p. 12 here), to rhyme on the “string Lie 2-algebra”.
From the point of view of Hypothesis H, the underlying superspace of this super Lie 3-algebra is the rational 2-stack $\widehat{\mathbb{T}^{10,1\vert \mathbf{32}}}$ corresponding to the “extended spacetime” which is classified by the C-field in Cohomotopy (p. 44 here):
$\array{ \widehat{\mathbb{T}^{10,1\vert \mathbf{32}}} &\longrightarrow& S^7_{\mathbb{R}} \\ \big\downarrow & {}^{(pb)} & \big\downarrow^{\mathrlap{(h_{\mathbb{H})_{\mathbb{R}}}}} \\ \mathbb{T}^{10,1\vert \mathbf{32}} &\underset{ (\mu_{M2}, \mu_{M5}) }{\longrightarrow}& S^4_{\mathbb{R}} }$(The “M2-brane 3-algebra” of BLG model-origin is something very different, instead, emerging in a different corner of this story (p. 36 here); its naming was a step in the wrong direction that caused much harm in the minds of people.)
Now it’s noteworthy that the super-exceptional spacetime $\mathbb{T}^{10,1\vert \mathbf{32}}_{ex_s}$ (and hence all the detailed physics that emerges on it, as per the article above) appears as an atlas, in the sense of higher differentiable super-group stacks, of this extended spacetime:
$\array{ \mathbb{T}^{10,1\vert \mathbf{32}}_{ex_s} &\overset{\color{blue}\text{atlas}}{\longrightarrow}& \widehat{\mathbb{T}^{10,1\vert \mathbf{32}}} &\longrightarrow& S^7_{\mathbb{R}} \\ && \big\downarrow &{}^{(pb)}& \big\downarrow^{\mathrlap{(h_{\mathbb{H})_{\mathbb{R}}}}} \\ && \mathbb{T}^{10,1\vert \mathbf{32}} &\underset{ (\mu_{M2}, \mu_{M5}) }{\longrightarrow}& S^4_{\mathbb{R}} }$So is there an intrinsic reason why one might want to find the “1-connected 1-truncated cover” of a Lie 3-algebra?
Yes it becomes a groupal effective epi/atlas after looping.
So if by $\mathbb{R}^{10,1\vert \mathbf{32}}_{ex_s}$ we now mean the super-manifold which underlies the super-group that integrates the super Lie algebra of the same name,
and by $\widehat{\mathbb{R}^{10,1\vert \mathbf{32}}}$ we mean the supergeometric 2-stack that underlies the super-group 2-stack that integrates the super Lie 3-algebra of the same name,
then the morphism $\mathbb{R}^{10,1\vert \mathbf{32}}_{ex_s} \longrightarrow \widehat{\mathbb{R}^{10,1\vert \mathbf{32}}}$ is an atlas of super-group 2-stacks, and here I mean a groupal atlas, in that it is a homomorphism of group objects.
Since the group structure in the given case is called supersymmetry, we might just say that it’s a supersymmetric atlas of the extended super-spacetime super 2-stack.
It’s this respect for the group structure, hence the super-symmetry, which makes this atlas be interesting – in fact exceptionally interesting.
So what is the kernel pair of $\mathbb{R}^{10,1\vert \mathbf{32}}_{ex_s} \longrightarrow \widehat{\mathbb{R}^{10,1\vert \mathbf{32}}}$? Is it at all interesting/useful?
That kernel pair should consist of:
one ordinary super-vielbein
two copies of the extra super-exceptional vielbein components
a degree-2 generator whose differential is the difference of the two 3-forms $H_{ex}$ built from either of the two copies of super-exceptional vielbein fields.
The interesting aspect of this atlas that I am after is this:
The $\widehat{\mathbb{R}^{10,1\vert\mathbf{32}}}$ is a super rational model for the 3-sphere fibration given by pullback of the quaternionic Hopf fibration. That 3-sphere fiber is the “$SU(2)_R$” in the notation of the last articles.
But it’s hard to see how to recover the integral periods over this $SU(2)$ from Lie integration of $\widehat{\mathbb{R}^{10,1\vert\mathbf{32}}}$
On the other hand, in its cover by the super-exceptional spacetime, we see that $SU(2)_R$: it’s the space in which that exponentiated pion field takes values!
So it should be possible now to use this to build the integral (ie beyond rational) super-3-sphere fibration over 11d superspacetime that would be the target space for the complete 5-brane model: combining the global topological structure with the local super-differential form structure.
In departing from ’plain’ geometry, there are these three descriptors ’exceptional’, ’generalized’, ’super’. How does the combination play out with respect to Kapranov’s account of the ’super’ part in Supergeometry in mathematics and physics?
Recall the difference he points to:
one can almost say that mathematicians and physicists mean different things when they speak about supergeometry.
and
superspace, a concept not synonymous with “supermanifold” of mathematicians. In fact, superspace is a supermanifold with a rather special “spinor-conformal” structure,
superspace enabling “the idea of non-observable square roots”.
And all this arising from physicists deploying more of the sphere spectrum:
The existing super-mathematics uses only the first two levels (0th and 1st) of $\mathbb{S}$, with physical applications exploiting the parallelism between the 1st and the 2nd levels
Regarding the first point, this is just the fact that physicist’s “super” is shorthand specifically for: “super-Poincaré”.
Regarding the last point: It still seems to me that the way to make the rough idea work that super-geometry is secretly sphere-spectrum graded geometry is that indicated at spectral super-scheme
So I guess then my question is whether one can see the “generalized exceptional” as part of the same story as the super/sphere spectrum part. But maybe that requires working out the indication in your final sentence.
The striking observation is that the exceptional geometry arises when “approximating” the higher super-geometry which is the homotopy fiber of the M2-brane cocycle by ordinary supergeometry.
This is really what happened, way back, in Section 6 of D’Auria-Fre’ 82, only that these authors didn’t put it that way, and that it took a long time until somebody did.
It’s quite remarkable.
With the idea to use this fantastic observation as the pivot point around which to bring together current research on higher and on exceptional structures in M-theory, I had written the proposal-description for what became our Durham symposium Higher structures in M-theory based on my note From higher to exceptional geometry (schreiber).
(But if you want to make an insight public while keeping it a secret, just write it into a conference description: nobody takes note of these, people just give their talks irrespective of title or agenda of the event. ;-)
Hello and compliments for the new interesting tile in the Hypothesis H mosaic.
I wanted to ask a question which touches only partially your preprint, but which has made me think a lot (also in relation with double field theory), still without being clear to me (but at least this gives me the chance to ask!).
If we forget the fermionic coordinates, the exceptional coordinates of $\mathbb{R}^{1,10}_{\mathrm{ex},\mathrm{bos}}$ are $\{ x^a, B_{a b}, B_{a_1 \cdots a_5} \}$.
And the $3$-flux form is then $H = e_{a b} \wedge e^a \wedge e^b = \mathrm{d} B_{a b} \wedge e^a \wedge e^b$.
This is in terms of local algebra and I am happy with everything.
What I tried and couldn’t understand is how this local picture can be globalized. We can definetely glue the coordinates $\{ x^a, B_{a b}, B_{a_1 \cdots a_5} \}$ to obtain a manifold, but in this case we don’t recover the patching conditions of the 3-form and 5-form field (at least as far as I understand!). For instance we know that the 3-form field is pathced by $2$-form gauge transformations $B_i - B_j = \mathrm{d} A_{ij}$ on two-fold overlaps $U_i \cap U_j$ and etc on n-fold overlaps. How is it possible to embody this fact by gluing local $\mathbb{R}^{1,10}_{\mathrm{ex},\mathrm{bos}}$?
Or maybe I am just out of the track, thank you very much in advance!
Hi Luigi,
good, let’s talk about this. I don’t have a complete picture yet, but I can say what I think I understand. Maybe we can make some progress by chatting about it:
First, the actual object with a higher geometric aspect to it is the “extended superspacetime” $\widehat {\mathbb{R}^{10,1\vert \mathbf{32}}}$ (“M2-brane Lie 3-algebra”, (3.15) in D’Auria-Fre 82) whose super-vielbein, if you wish, has the usual 11d super-vielbein components $e^a$ (degree $(1,even)$), $\psi^\alpha$ (degree $(1,odd)$), and in addition one component $h_3$ in degree $(3,even)$, satisfying
$d h_3 \;=\; \tfrac{1}{2}\overline{\psi} \Gamma_{a b} \psi \wedge e^a \wedge e^b \,.$This is the purely fermionic part of the condition for the worldvolume 3-flux that holds on M5-brane worldvolumes probing the 11d spacetime, where $G_4^0 = \tfrac{1}{2}\overline{\psi} \Gamma_{a b} \psi \wedge e^a \wedge e^b$ is the C-field 4-flux form in the case that the bosonic 4-flux vanishes, and only this fermionic part remains, which is fixed by the “torsion constraints”.
In any case, this means that this $h_3$ is an incarnation of a curving 3-form on a twisted gerbe (twisted by the $G_4$-flux). Its genuinely higher geometric nature is embodied by the fact that it is of degree 3. For example, of we look at the bosonic “stage” where the fermions all vanish, the above algebra reduces to that generated by a single closed degree-3 element, which is $CE( \mathbf{B}^2 U(1) )$, the Chevalley Eilenberg algebra of the circle 3-group. Modeling geometry on this gives the higher degree transition functions that you expect to see.
So this is the higher geometry. But this is not yet the exceptional geometry.
The exceptional geometry is, in itself, an ordinary geometry (supergeometric but not higher geometric). What makes it connect to the actual higher geometry is that it comes equipped with a map to the higher geometry, which is an atlas in the sense of higher stacks.
Namely, if we think of the above higher geometry $\widehat{ \mathbb{R}^{10,1\vert \mathbf{32}} }$ as a 2-stacky supergeometry, then an atlas for it is a super geometry equipped with a map to this higher geometry which surjects onto its ordinary (non-higher part) (an effective epimorphism of stacks). Since everything is Spin-equivariant and supersymmetric, we want to ask that any such atlas is Spin-equivariant and supersymmetric, too, hence a homomorphism of the translational super $L_\infty$-Spin-modules on both sides.
To find such an atlas, we are faced with the following question: Find a super Lie 1-algebra $L$ and a homomorphism of super $L_\infty$-algebras $L \longrightarrow \widehat{ \mathbb{R}^{10,1\vert \mathbf{32}} }$ which is surjective onto the lowest degree elements. (In general we’d ask an atlas to be surjective on the categorical $\pi_0$, but in the case at hand that’s spanned simply by all the lowest degree elements).
Said more concretely, we are asking for a dgc-superalgebra which is generated, first of all, from the $e^a$ (degree $(1,even)$) and $\psi^\alpha$ (degree $(1,odd)$) as above, with their differentials also as above, and in addition any number of further generators in degree $(1, something)$, such that the map that embeds the original space of degree-1 generators in this enlarged space extends to a homomorphism of dgc-superalgebras from $CE\big(\widehat{ \mathbb{R}^{10,1\vert \mathbf{32}}})$. But, since the only further relation in the latter algebra is $d h_3 \;=\; \tfrac{1}{2}\overline{\psi} \Gamma_{a b} \psi \wedge e^a \wedge e^b$, and since our map is the identity on the right hand side of this equation, the design criterion on the enlarged set of degree-1 generators is:
There must be a way to form a sum of wedge products of these degree-1 generators to form a degree-3 element $H_{ex}$ (being a function of all these degree-1 generators) which satisfies the same equation that $h_3$ satisfied by construction:
$d H_{ex} \;=\; \tfrac{1}{2}\overline{\psi} \Gamma_{a b} \psi \wedge e^a \wedge e^b$If this is achieved, what have built is a super Lie 1-algebra $\mathbb{R}^{10,1\vert \mathbf{32}}_{ex}$ and a homomorphism of super $L_\infty$-algebras
$\mathbb{R}^{10,1\vert \mathbf{32}}_{ex} \overset{atlas}{\longrightarrow} \widehat{ \mathbb{R}^{10,1\vert \mathbf{32}} }$which is surjective in lowest degree, hence which serves as an atlas for the higher spacetime geometry.
Question: What are examples of such atlases?
This question – not in this language but otherwise evidently just this question – is what D’Auria and Fré asked and began to answer in Section 6 of their 82 article. There they found two solutions to this question. Later Bandos et al in arXiv:hep-th/0406020 enlarged this to a 1-paramater family of solutions, indexed by an $s \in \mathbb{R}\setminus \{0\}$.
continued…
…
Interestingly, the element $H_{ex_s}$ on the atlas $\mathbb{R}^{10,1\vert\mathbf{32}}_{ex_s}$ which appears this way is rather rich in structure. It starts out in the way you expect from the higher geometry
$H_{ex_s} \;=\; d B \;\oplus\; \cdots$but then it has many more contributions. The next one is a contraction of the wedge product of three copies of the 2-index exceptional vielbein:
$H_{ex_s} \;=\; d B \;\oplus\; e_{a_1 a_2} \wedge e^{a_2 a_3} \wedge e_{a_3}{}^{a_1} \;\oplus\; \cdots$Here the $\oplus$-notation means that prefactors (which are known analytic functions of the parameter $s$) to the monomials are not shown. Because displaying these prefactors makes a typographical mess that distracts from the structure of these terms. We tried to pretty-print the structure appearing here in in out note, so I won’t try to further do so here.
The point I just want to highlight here is that this second summand on $H_{ex_s}$ secretly carries some non-trivial global topological structure: If we restrict the whole $\mathbb{R}^{10,1\vert \mathbf{32}}_{ex_s}$ to an ADE-singularity, then this term becomes the local volume for on an $SU(2)_R \simeq S^3$ geometry (section 4.1).
(I believe the way this appears is quite unlike what has previously been considered in the exceptional geometry literature, but let me know if you have seen it before).
Just such a 3-sphere $SU(2)_R$ is indeed the fiber of the extended spacetime in the topological analog of the constructions here (top left of the big diagram on p.1 here).
So this is one question one should ask now: How does the 3-sphere appearing infinitesimally in the exceptional geometry for the extended spacetime connect to the topological 3-sphere seen on in the topological incarnation of the extended spacetime.
Is there an atlas of the $\mathfrak{m}5\mathfrak{brane}$ Lie 6-algebra of any interest? Something requiring a sum of wedge products of degree-1 generators to form a degree-6 element satisfying the equation for the differential of $c_6$ but with $c_3$ ($h_3$) replaced as above?
The group of Lauria Andrianapoli is working on that. But not done yet, apparently it’s tough.
Thank you very much for your explanation! I am personally persuaded that your idea could capture the true meaning of exceptional field theory / extended geometry.
We have an atlas (in the sense you explained) of the form
$\mathbb{R}^{1,10|\mathbf{32}}_{\mathrm{ex}} \, \xrightarrow{\;\;\phi\;\;} \, \widehat{\mathbb{R}^{1,10|\mathbf{32}}}$which can be defined in terms of $\mathrm{CE}(-)$ algebras as you explained.
But now let me see if I understand what follows in terms of global objects…
The extended super-spacetime $\widehat{\mathbb{R}^{1,10|\mathbf{32}}}$ can be globalized to a super-$2$-gerbe $\widehat{M}$ on the $(1,10|\mathbf{32})$-dimensional super-manifold $M$ (the super-spacetime).
In terms of $V$-manifolds the super-spacetime $M$ is without any doubt a $\mathbb{R}^{1,10|\mathbf{32}}$-manifold. If I understand correctly this should be enough to say that the higher bundle $\widehat{M}$ is a particular case of $\widehat{\mathbb{R}^{1,10|\mathbf{32}}}$-manifold, where $\widehat{\mathbb{R}^{1,10|\mathbf{32}}}$ is the extended super-Minkowski spacetime. So there is a well-defined sense in which the gerbe $\widehat{M}$ is obtained by gluing local copies of $\widehat{\mathbb{R}^{1,10|\mathbf{32}}}$.
Thus we should have something like an atlas (in the sense you explained here for geometric stacks) of this form:
$\bigsqcup_{i \in I} \mathbb{R}^{1,10|\mathbf{32}}_{\mathrm{ex}} \, \xrightarrow{\;\;\{\phi_i\}_{i \in I}\;\;} \, \widehat{M}$which reduces to the original $\phi:\mathbb{R}^{1,10|\mathbf{32}}_{\mathrm{ex}} \rightarrow \widehat{\mathbb{R}^{1,10|\mathbf{32}}}$ locally on any component of the $\widehat{\mathbb{R}^{1,10|\mathbf{32}}}$-cover (here I mean a $V$-cover in the $V$-manifold sense).
Does this argument make sense, at least heuristically?
EDIT: Or alternatively can somehow the atlas $\mathbb{R}^{1,10|\mathbf{32}}_{\mathrm{ex}} \, \xrightarrow{\phi} \, \widehat{\mathbb{R}^{1,10|\mathbf{32}}}$ that you constructed for the extended super-Minkowski space be globalized to some atlas $\mathcal{U} \rightarrow \widehat{M}$ on the total space of the super-2-gerbe $\widehat{M}$?
EDIT2: I read more carefully the previous posts and some references, but still sorry for my lack of precision.
$\bigsqcup_{i \in I} \mathbb{R}^{1,10|\mathbf{32}}_{\mathrm{ex}}\times_{\widehat{M}}\bigsqcup_{i \in I} \mathbb{R}^{1,10|\mathbf{32}}_{\mathrm{ex}} \begin{matrix}\rightarrow \\ \rightarrow \end{matrix} \bigsqcup_{i \in I} \mathbb{R}^{1,10|\mathbf{32}}_{\mathrm{ex}} \, \xrightarrow{\;\;\{\phi_i\}_{i \in I}\;\;} \, \widehat{M}$where the kernel pair should look like a different choice of vielbein $\{e^a, B_{a b}, B_{a b c d e}\}_k$ and $\{e^a, B_{a b}, B_{a b c d e}\}_j$ on two different local $\mathbb{R}^{1,10|\mathbf{32}}_{\mathrm{ex}}$ at $i=k$ and at $i=j$, together with a $2$-form whose differential gives the difference between $H$ on the local patch $i=k$ and the one on the local patch $i=j$. I am being super-super-imprecise and non-formal, but this may look like the solution to many problems in globalization of Exceptional Field Theory!
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