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A thought.
On the one hand, one may see (discussed at Structure Theory for Higher WZW Terms (schreiber)) that 11d supergravity is the first-order integrable higher super Cartan geometry for the higher WZW terms of the M2 and the M5, the latter twisted by the former.
On the other hand there have long been hints that 11d sugra should have a natural formalization in terms of exceptional generalized geometry.
Here is an observation of a possible connection:
The central result in the last section of
D’Auria, Fre, Geometric Supergravity and its Hidden Supergroup (pdf)
of which a more comprehensive analysis is in
Azcarraga et al, On the underlying gauge group structure of D = 11 supergravity (arXiv:hep-th/0406020)
is that the ordinary 1-Cartan geometry on which the M2-brane twist trivializes is locally modeled on the extension of super-Minkowski tangent spaces by the 2-forms $\wedge^2 T^\ast$ and by the 5-forms $\wedge^5 T^\ast$.
In fact these authors realize this extension as a super Lie algebra extension of the supersymmetry Lie algebra in 11d. But disregarding this super Lie structure for a moment, it is of course precisely the exceptional generalized tangent bundle considered originally in
where it was observed that the E-series of the exceptional groups canonically acts on this extension, at least up do d = 7.
So this seemes to give a plausible path connecting the higher geometry of M-theory with its exceptional geometry. Since the exceptional geometry here may be thought of as cover of spacetime over which the M2 2-gerbe trivializes, there may be a chance that the exceptional geometry may be understood as well-adapted local data for the higher geometry in some sense.
I am typing out this thought in a tad more detail in a file here: pdf
With this last step to exceptional geometry added, the Proceß as I am seeing it now looks as follows:
Here the lower steps are as surveyed at Modern Physics formalized in Modal Homotopy Type Theory (schreiber)
Here is maybe a clearer way to state what I am suggesting is the connection between the higher and the exceptional geometry formulation of supergravity.
So the higher geometric formulation is about structure induced by definite globalizations of WZW terms, hence by higher principal connections on some manifold that are tangent-space-wise equivalent to first infinitesimal order to a fixed reference higher connection.
Hence by definition, we are talking about an object equipped patchwise with an equivalence to a known object, and so the information of the global object is equivalent to knowing these gauge transformations on each patch.
The idea is then simply that that the section of an exceptional tangent bundle that corresponds to a higher definite WZW term provides the data of that equivalence.
This is a very natural thing to consider.
To put this in perspective, notice what this means in the traditional case where we are just talking about a definite form. Here knowledge of the definite form is clearly equivalent to an open cover with a chart-wise tangent bundle isomorphism that turns the definite form fiberwise into the fixed reference form. This is precisely the way in which one proves that for a manifold to carry a form definite on some reference form is equivalent to its frame bundle having a reduction to the stabilizer subgroup of that definite form.
Now when we lift that story from forms to higher connections with these forms as their curvature, then there is one extra piece of data in the local equivalences that make the global definite structure patchwise equivalent to the reference structure: namely in addition to the $GL(d)$-transformation that adjusts the tangent spaces such that the curvature matches the reference curvature, there is in addition a gauge transformation that turns the connection into the fixed reference connection. It is that extra data of the gauge transformation that is captured by a section of the exceptional tangent bundle.
So for the M2-brane we are talking about a 3-form connection, and hence that extra data is patchwise a 2-form, and hence after trivializing the tangent bundle locally we are still to specifiy a section of $\wedge^2 T^\ast$. That’s how the exceptional generalized geometry comes in.
Finally, the gauge transformation is given locally by a potential of a closed 3-form $A$ (the difference of the restriction of the global 3-form connection and the reference 3-connection, which by construction have equal 4-curvature in the given local trivialization), so the gauge transformation in question is, by the standard proof of the Poincare lemma, in each first-order neighbourhood the 2-form which at $\epsilon_v$ is propotional to $\iota_v A$. This is the just formula at the heart of exceptional generalized geometry for how the 3-form components in $E_d$ act on vectors to produce sections of the exceptional tangent bundle.
Now with that understood, then it is interesting to note the nature of the first term in (6.1) of DAuria-Fre 82, pdf, or (28) in arXiv:hep-th/0406020. This is a left-invariant 3-form on the extended supergroup whose bosonic part is $\mathbb{R}^{10,1} \oplus \wedge^2 (\mathbb{R}^{10,1})^\ast$ of the form
$e_a \wedge e_b \wedge B^{a b}\,,$where $\{e_a\}$ is the canonical basis of left-invariant 1-forms on (super-)Minkowski spacetime and $\{B^{a b}\}$ is the canonical basis of the left-invariant forms on $\wedge^2 (\mathbb{R}^{10,1})^\ast$.
Now what is noteworthy about this form is that this is a cousin of the tradtional Liouville-Poincaré+1-form in that if we pull it back to Minkowski space via the section $v \mapsto (v, \iota_v A)$ discussed above, then it reproduces the 3-form $A$.
So that gives a maybe more pronounced way of saying what’s going on here:
we have that M2-WZW term on some 11d superspacetime $X$, whose curvature is compatible with the super Lie algebra structure on the tangent fibers, but whose connection is not (cannot be, since on these fibers the curvature is a nontrivial Lie algebra cocycle). But then we form the exceptional tangent bundle over $X$. The WZW term pulled back to there now does have 3-connection compatible with the super Lie algebra structure, namely here the 4-form curvature becomes a trivializable Lie algebra cocycle. And finally, then, by the above, the original 3-form connection, the non-equivariant one, induces a canoncial section of the exceptional tangent bundle and the pullback of the equivariantized 3-connection along that section recovers the original non-equivariant 3-form connection.
This maybe sheds a new light on what to think of the free choice of the parameter $s$ in (30) of arXiv:hep-th/0406020: because if we want that nice Liouville-Poincaré-like property to hold for the full super 3-form in (28) then the term with coefficient $\alpha_1$, which is
$B_{a b} \wedge B^b{}_c \wedge B^{c a}$
has to vanish. (Or else it needs some interpretation, but I don’t presently see what that would be). Setting $\alpha_1 = 0$ means setting $s= -3$ and hence fixes the 1-parameter ambiguity. (Here I am speaking for the case that the M5-WZW term is taken to vanish for the moment. I’d rather postpone taking that into the present discussion until I see fully clearly what the story for the M2 alone is here).
To put that all together so far, I gather the picture is this:
first we started with a globalized higher WZW term $\mathbf{L} : X \longrightarrow \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn}$, pulled it back to the first order infinitesimal disk bundle $T_{inf} X$ and asked it to be a definite parameterization there. This makes $T_{inf} X$ in particular a first order infinitesimal group bundle (i.e. Lie algebra bundle).
Now we ask for a bundle extension $\mathcal{T}$ of bundles of Lie algebras with fiberwise linear (not not Lie) local splitting $\sigma$
$\array{ && \mathcal{T} &\stackrel{\mathbf{L}_0}{\longrightarrow}& \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn} \\ &{}^{\mathllap{\sigma}}\nearrow& \downarrow^{e} \\ T_{inf} U &\stackrel{T_{inf} p}{\longrightarrow}& T_{inf} X \\ \downarrow && \downarrow^{\mathrlap{ev}} \\ U &\stackrel{p}{\longrightarrow} & X &\stackrel{\mathbf{L}}{\longrightarrow}& \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn} }$where $\mathbf{L}_0$ is a completely left-invariant parameterized WZW term (i.e. in addition to the curvature form, which is always left-invariant by definition of WZW terms, also the connection $(p+1)$ form is left invariant, hence there is no information in the choice of $\mathbf{L}_0$);
such that (under the relevant differential concretification which I’ll suppress notationally)
In such a situation we have effectively traded in the connective data on the original WZW term for that splitting $\sigma$. Now that splitting is what in (exceptional) generalized geometry defines the (exceptional) “generalized metric”, so we have traded the connective data into generalized metric data. And that’s of course supposed to be the crux of it.
Is $U$ here a local chart? Is it $\mathcal{T}\to T_{inf}X$ that is the extension of Lie algebra bundles?
Yes!
Best to take $U \to X$ to be the whole cover, but if you want to think of it as a single chart, there is no loss at the moment. The gluing along patches, however, of the section $\sigma$ by more than just the structure group of the bundle $\mathcal{T}$ is going to be important for the full story. That’s the bit that in texts about (exceptional or not) generalized geometry remains mysterious, here we see how it comes about from the fact that we are really re-writing the local data of something that is a higher gerbe to begin with.
So at the heart of it all is the following very elementary situation in Lie theory, and I am wondering if this has made an appearance under some name elsewhere before, simple as it may be.
Given a germ of a Lie group $G$, consider a left-invariant closed $n$-form $\omega$ on $G$. By Poincaré it will have a potential form $A$ with $d A = \omega$, but of course $A$ itself won’t be left-invariant unless $\omega$ comes from a trivial Lie algebra cocycle.
But then we may want to parameterize the space of choices of non-left-invariant $A$ by something that is left-invariant.
So we look for an extension $p \colon \hat G \longrightarrow G$ of germs of Lie groups such that $\hat G$ carries a left-invariant form $\hat A$ with
$d \hat A = p^\ast \omega$.
Then for every splitting of the underlying bundle of the extension, hence for every section $\sigma$ of $p$ at the level of just (germs of) manifolds, the pullback $\sigma^\ast \hat A$ is a potential for $\omega$.
That possibility of parameterizing non-left-invariant potentials for left-invariant forms $\omega$ by splittings of an extension along which the left-invariant $\omega$ trivializes left-invariantly, that seems to be what is at the very heart of the appearance of exceptional geometry in supergravity, by the reasoning in the above messages.
But this is a very simple, elementary thing to consider in itself. Simple as it may be, does this have a name? Does this appear elswhere in any interesting context?
Oh, now I see it: this is a higher analog of the Atiyah sequence!
I have written that out in more detail at From higher to exceptional geometry (schreiber)
It sounds weak in hindsight, but I thought it was something like a Lie algebra splitting, but felt that that was probably too obvious, and had no good argument ending anywhere.
That a linear splitting of Lie algebra extensions is involved is the content of #6 above, but I was still unsure as to why that is the thing to consider here.
Now I think it’s clear: these splittings parameterize the local choices of 3-form connections for a fixed left invariant 4-form curvature. The key point that we are to ask for an extension that allows a left-invariant trivialization of the 4-form is that it is only due to that left-invariance that what is apriori a space of affine splittings become equivalent to linear splittings.
But if I still have your attention, don’t walk away yet, as there are more questions to be answered still:
So we understood that given a fixed definite 4-form curvature for a 2-gerbe on spacetime, then locally the choice of 3-form connections is neatly parameterized by a linear splitting of a Lie extension of the tangent bundle.
Okay. But now the exceptional groups are to be brought in. The picture is, I think, that the U-duality group is the automorphism group of $\hat g$ preserving some extra structure, and the corresponding compact subgroup is then the stabilizer of any given splitting.
Something like this. I still need to match the traditional way this is motivated to the higher-Atiyah-sequence reinterpretation that we have now.
And then the big mystery to solve is this: if we consider a spacetime manifold that is an orbifold, then at the orbifold singularities part of this global U-duality symmetry is supposed to become gauged, i.e. we are supposed to see not just a global action of these groups on the field content, but an actual gauge field for these groups is to appear.
I tried to chase literature for how this is suppsed to happen, but it seems all that the literature provides is “because some extra BPS states become massless as a 2-cylce degenerates to a singularity” and “because similar gauge enhancement has been argued for in heterotic string theory” and “because Witten said so”. I’d like to understand this in more detail, but I don’t yet.
I meant the content of #9, regarding splittings of germs.
if we consider a spacetime manifold that is an orbifold, then at the orbifold singularities part of this global U-duality symmetry is supposed to become gauged, i.e. we are supposed to see not just a global action of these groups on the field content, but an actual gauge field for these groups is to appear.
so something like sections of the bundle of groups given by the inertia groupoid? This seems like something that should act (without thinking it through) on splittings of a related algebroid. I couldn’t offer any physics-speak reasons for why this might be so!
regarding #12, one may observe the following (this must be in the literature, but right now I don’t see anyone say it):
In view of the presence of the Lorentz metric on $\mathbb{R}^{10,1}$ then of course there is a singled out linear isomorphism
$\wedge^2 (\mathbb{R}^{10,1})^\ast \stackrel{\simeq_{lin}}{\longrightarrow} \mathfrak{so}(10,1)$.
Under this ismorphism then a linear splitting of the exceptional tangent space projection
$\mathbb{R}^{10,1} \oplus \wedge^2 (\mathbb{R}^{10,1})^\ast \longrightarrow \mathbb{R}^{10,1}$
is equivalently an $\mathfrak{so}(10,1)$-valued linear 1-form.
Curiously, with this interpretation then the second term, $B_{a b} \wedge B^b{}_c \wedge B^{c a}$, mentioned in #5 above suddenly obtains geometric meaning: pulling that back along this splitting now is the “non-kinetic” part of the Chern-Simons 3-forms of that $\mathfrak{so}(10,1)$-valued linear 1-form.
This would be exactly the contribution that is supposed to show up, both in heterotic compactifications and for M2-instantons.
But I need to understand how this re-writing, that is certainly possible, genuinely affects the dynamics…
While in the expression
$B^{a_1 a_2}\wedge e_{a_1}\wedge e_{a_2} + \alpha_1(s)B_{a b} \wedge B^b{}_c \wedge B^{c a} + \cdots$for the left-invariant reference potential the second Chern-Simons like term never appears without the first Liouville-Poincaré-type term for the allowed values $s \neq 0$, in the limit that $s \to 0$ the second term diverges while the first term goes to 1, hence up to rescaling the second term does exist by itself in the limit. (By inspection of (28) and (30) in arXiv:hep-th/0406020)
Maybe this means something. In that same limit the extension algebra $\hat {\mathfrak{g}}$ becomes an “expansion” (extension followed by contraction) of the orthosymplectic super Lie algebra $\mathfrak{osp}(1|32)$. That is the result of arXiv:1504.05946.
Here is a cleaner version to say general abstractly what’s going on.
So previously I used to say that a definite globalization of a WZW-term $\mathbf{L} : V \longrightarrow \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn}$ over a $V$-manifold $X$ is a term $\mathbf{L}^X \colon X \longrightarrow \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn}$ such that its restriction to infinitesimal disks incarnated as a section
$\array{ X && \stackrel{\sigma}{\longrightarrow} && [\mathbb{D}^V , \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn}]/GL(V) \\ & \searrow && \swarrow \\ && \mathbf{B}GL(V) }$has the property that there is a cover $U \to X$ and a factorization
$\array{ U && \longrightarrow && \ast \\ \downarrow && \swArrow_{\simeq} && \downarrow^{\mathrlap{\mathbf{L}^{\mathbb{D}^V}}} \\ X && \stackrel{\sigma}{\longrightarrow} && [\mathbb{D}^V , \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn}]/GL(V) \\ & \searrow && \swarrow \\ && \mathbf{B}GL(V) }$That makes the global term be on each tangent space equivalent to the fixed one. Hence it makes both its curvature and its connection locally equivalent to the fixed ones.
But more generally we may want to only make the curvature definite, but leave the connection otherwise unconstrained. This is in fact all that is necessary for WZW terms of super $p$-brane sigma models, and it will be this freedom of choice that gives rise to the exceptional geometry degrees of freedom.
To axiomatize this via the above, we assume now that we have an extension
$p \colon \widehat {\mathbb{D}^V} \longrightarrow \mathbb{D}^V$of the infinitesimal disk, and have a reference WZW term
$\hat \mathbf{L} : \widehat {\mathbb{D}^V} \longrightarrow \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn}$
such that all possible WZW terms on $\mathbb{D}^V$ with given fixed curvature are obtained by pullback of that along sections
$\sigma \colon \mathbb{D}^{V} \longrightarrow \widehat{\mathbb{D}^V}$.
(By the nature of infinitesimal disks, these sections will automatically be the linear splittings discussed in the previous messages.)
With that data we may then naturally relax the above condition for definiteness. Namely we now demand a homotopy square as in the top part of the following diagram
$\array{ U &&\stackrel{}{\longrightarrow} && \Gamma_{\mathbb{D}^V}(\widehat{\mathbb{D}^V})/GL(V) \times\ast \\ \downarrow && \swArrow_{\simeq}&& \downarrow^{\mathrlap{(id,\hat\mathbf{L})}} \\ &&&& \Gamma_{\mathbb{D}^V}(\widehat{\mathbb{D}^V}) \times [\mathbb{D}^V,\mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn}]/GL(V) \\ \downarrow &&&& \downarrow^{\mathrlap{ev}} \\ X && \stackrel{\sigma}{\longrightarrow} && [\mathbb{D}^V , \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn}]/GL(V) \\ & \searrow && \swarrow \\ && \mathbf{B}GL(V) }$In total this describes a WZW term on a manifold whose curvature is definite, but whose connection is not otherwise constrained. Equivalently then the very top horizontal morphism now encodes the exceptional generalized geometry.
Regarding the issue of gauge group enhancement at ADE singularities, here is a moonshiny observation:
According to
the finite group quotients of the quaternionic Hopf fibration which make $S^7/\Gamma$ be smooth, spin, and with at least four Killing spinors are the “binary” extensions of the symmetry groups of the Platonic solids, hence are the finite groups $\Gamma$ that make ADE orbifolds locally of the form $something \times \mathbb{R}^4/\Gamma$.
So here is the moonshine: in The WZW term of the M5-brane (schreiber) we observed that the cohomology theory in which the M2/M5 system takes values is rationally twisted cohomotopy with coefficients in the quaternionic Hopf fibration. It is hence natural to consider it to actually be the quaternionic Hopf fibration, not just rationally. But of course we may at least allow finite quotients of that. Requiring these to be sufficiently well-behaved is what the above article is about.
In other words, it is natural to consider the M2/M5 WZW term to take values in $S^7/\Gamma \to S^4 /\Gamma$. But this then has a remarkable consequence: it means that on any smooth local patch there is globally and disregarding the differential form data only the trivial configuration. But as we move to an ADE-orbifold patch $\simeq \mathbb{R}^7 \times (\mathbb{R}^4/\Gamma)$, then suddenly the possible choices of gauge configurations becomes rich.
In any case, it seems plausible that the gauge enhancement should be visible from analysis of the generalized cohomology refinement of the M-brane charges, because down in 10d this is what does happen for the string: that coincident D-branes have gauge enhancement to $SU(n)$ may be seen by looking at the “microscopic” degrees of freedom of strings stretched between them, but mathematically one also finds it by saying that D-brane charge is in K-theory. Of course historically the latter was first suggested by the former, but abstractly it need not.
But as we move to an ADE-orbifold patch $\simeq \mathbb{R}^7 \times (\mathbb{R}^4/\Gamma)$, then suddenly the possible choices of gauge configurations becomes rich.
What particular about the patch entails this? Anything to do with exotic structures?
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