Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
1 to 32 of 32
Yesterday I had filled in some minimum regarding the idea of rheonomy at D’Auria-Fre formulation of supergravity. (I thought I had done this long ago, but found the section empty now. )
There is something immensely curious going on here, which I was wanting to formalize ever since I started looking into this. Now maybe I am getting closer, but I am still stuck with something:
So the simple beautiful idea of rheonomy is that differential forms on a supermanifold $X$ are constrained to be something like “holomorphic” in the super-direction, in that they are fully determined by their restriction along the inclusion $\Re X \longrightarrow X$ of the underlying ordinary manifold, in (vague? or good?) analogy with how holomorphic functions on $\mathbb{C}$ are determined by their restriction along $\mathbb{R} \hookrightarrow \mathbb{C}$. This rheonomy constraint turns out to be equivalent to the more popular “superspace constraints” that are used elsewhere in the SuGra literature, but is evidently conceptually a much nicer perspective.
The striking claim then is that the equations of motion of supergravity theories enode precisely nothing but the constraint on a higher super-Cartan geometry on $X$, modeled on a given extended super-Minkowski spacetime, to have higher super-vielbein fields $E$ which, as super-differential forms on $X$, are rheonomic.
So the statement is something like that a solution to supergravity is nothing but a certain $G$-structure satisfying a “holomorphicity”-like constraint.
Apart from being beautiful and remarkable in itself, this smells like it has a good chance of having an “elementary” formalization in differential cohesion. That’s what I am after. I know how to naturally say “higher super-Cartan geometry” axiomatically in differential cohesion, but I don’t know yet how to say “rheonomy” in this way.
In fact I am pretty much in the dark about it at the moment, but from the above there are some evident guesses as to what one has to consider.
So given some manifold $X$ modeled on a framed space $V$, such as some extended super-Minkowski spacetime, then we are simply looking at an orthogonal structure exhibited by a diagram of the form
$\array{ X && \longrightarrow && \mathbf{B} O(V) \\ & \searrow &\swArrow_{E}& \swarrow \\ && \mathbf{B}GL(V) }$where the homotopy $E$ is (for some value of “is”) the vielbein, i.e. in the running example it is the super-vielbein together with the relevant higher form fields.
Now we may restrict this Cartan geometry to the underlying ordinary (reduced) manifold, simply by precomposing with the unit of the reduction modality
$\left( \array{ \Re X && \longrightarrow && \mathbf{B} O(V) \\ & \searrow &\swArrow_{\Re E}& \swarrow \\ && \mathbf{B}GL(V) } \right) \;\;\;\; \coloneqq \;\;\;\; \left( \array{ \Re X &\longrightarrow& X && \longrightarrow && \mathbf{B} O(V) \\ && & \searrow &\swArrow_{E}& \swarrow \\ && && \mathbf{B}GL(V) } \right)$So rheonomy is supposed to be some constraint on $E$ that makes it be fully determined by its restriction $\Re E$.
When one expresses $E$ in terms of actual differential forms with values in a super-$L_\infty$-algebra, then the constraint simply says that the curvature forms of this super $L_\infty$-algebra valued differential form are such that their components with incdices in directions perpendicular to $\Re X$ in $X$ are linear combinations of the components with all indices parallel to $X$.
So if I allowed myself to speak of components of $L_\infty$-algebra valued differential forms I’d be done. But I am suspecting that there is a more fundamental way to express what’s going on here, in terms of some general abstract differential cohesion yoga applied to the above diagrams.
And it looks like some kind of formal étaleness condition, or maybe formal smoothness condition on $E$. Hm…
Hm, so it’s something about a subspace of the space of squares
$\array{ \Re X &\longrightarrow& \mathbf{B}O(V) \\ \downarrow &\swArrow& \downarrow \\ X &\longrightarrow& \mathbf{B}GL(V) }$(with the bottom and the vertical morphisms the fixed ones) such that there is at least one lift
$\array{ \Re X &\longrightarrow& \mathbf{B}O(V) \\ \downarrow &\nearrow& \downarrow \\ X &\longrightarrow& \mathbf{B}GL(V) }$and such that this lift satisfies some further constraint.
(Because where above I said “the external components of the curvatures are to be linear combinations of the internal components” I suppose it must read “are to be specified linear combinations”, i.e. with fixed coefficients. That is the specific differential equation to be solved.)
Hm…
What is curious here is that all these problems disappear for 11-dimensional supergravity: Howe 97 shows that here the equations of motion are already equivalent to just the torsion constraint. Hence superspacetimes solving the 11-dimensional supergravity equations of motion are equivalent simply to first-order integrable higher super-Cartan geometries modeled on the M2-extended 11d super-Minkowski spacetime, with its canonical torsion.
This it is possible to say axiomatically. Whereas for lower dimensional and less supersymmetries theories one would need answers to the above questions, and so I am stuck with axiomatizing it.
So even though it sounds mad, the following is true:
Question: Of all variants of theories of gravity, which one has a structure that has an elementary axiomatization in homotopy type theory with six modalities added?
Answer: 11-dimensional supergravity. Aka low energy M-theory.
for lower dimensional and less supersymmetries theories one would need answers to the above questions
If the questions have fully disappeared for 11-d supergravity, do they partially disappear for other dimensions in a systematic way? Can one see ’why’ they disappear at 11-d?
It seems that for the maximal supersymmetric compactifications of 11d SuGra it remains true that the torsion constraint implies all the rest. So for instance for $N=8$ SuGra in $d = 4$, but not for $N \lt 8$. (This is claimed in some articles, but I still have to find the source that gives the computation.)
But otherwise, that it comes out this way is a bit of a miracle. One computes the implications of the torsion constraint, and it just so happens that in high enough dimension with enough supersymmetry, it implies everything. Of course that’s not the only miracly about 11d SuGra. It’s generally a very exceptional object.
Regarding the “why” in “why is 11d sugra special”: so you know that there is the suggestion that the answer is: because the relevant exceptional super Lie algebra cocycle for SuGra in this dimension is controlled by the structure of the octonions.
What I have trouble making up my mind about is whether this intimate relation between exceptional super Lie cocycles and the division algebras “explains” the exceptionality of the former via that of the latter — or the other way around.
See what I mean? We might wonder: “why” is there the sequence $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, $\mathbb{O}$? Since supersymmetry is nicely “geometric” we might think of supersymmetry as giving a nice geometric “explanation” of “why $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$, $\mathbb{O}$”.
The literature seems to usually take the other point of view, that the division algebras “explain” the exceptional supersymmetry structures. Not sure.
This might matter at some point for the formulation internal to a topos. We recently had long discussion about the different internal incarnations of $\mathbb{R}$ in a topos and the relation of these to cohesion. Maybe one should think about internal incarnations of $\mathbb{O}$…
Interesting. Is your hunch then to consider the non-standard direction? The existence of certain exceptional super Lie cocycles is tied to expressibility with modalities (#5), and these account for the division algebras.
Maybe one should think about internal incarnations of $\mathbb{O}$…
Should we build up to this by looking at internal incarnations of $\mathbb{C}$ and $\mathbb{H}$, perhaps internalizing the Cayley–Dickson construction? Or maybe that’s the wrong direction too.
Is there any way to form a ’String(n, 1)’, or does non-compactness of ’Spin(n, 1)’ prevent this?
The existence of certain exceptional super Lie cocycles is tied to expressibility with modalities (#5), and these account for the division algebras.
Yes, presently the formulation of cocycles and supersymmetry tends to lend itself nicely to elementary axiomatization. But of course from this I don’t really get the particular cocycles without looking at models. Not sure yet.
@David C #8
I’ve wondered this too, for the specific case of $Spin(5,1) = SL(2,\mathbb{H})$.
EDIT: It could arise from considerations for Clifford algebras…
Had missed #8. There is
Would one expect something octonion-ic about $String(9, 1)$, quaternion-ic about $String(5, 1)$, etc.?
Because where above I said “the external components of the curvatures are to be linear combinations of the internal components” I suppose it must read “are to be specified linear combinations”, i.e. with fixed coefficients. That is the specific differential equation to be solved.
Do you mean that there are some particular specified coefficients, which you haven’t mentioned, that determine the notion in question? Or that different solutions might have different coefficients?
I’d like to help, but I’d need more of a refresher on how we say “differential form” in differential cohesion and how it relates to classical Lie-algebra-valued forms.
Yes, there are specific fixed linear combinations whose coefficients I haven’t mentioned, which are characteristic of the given supergravity in question (i.e. it’s dimension, number of symmetries and possibly content of scalar fields).
But the thing is, that I am not after formalizing these differential form equations. I have come to believe that for fields of gravity the fundamental description is by $G$-structure. Given a $G$-structure, we might ask for a compatible connection expressed in terms of differential forms, but it seems that the elementary axiomatization of that will be way more messy than that of just the $G$-structure.
So I think for the moment I don’t have a further question here. There is now a beautiful story to be told about 11d sugra, and that’s what I am looking into.
From this point of view maybe the main remaining question is to which extent we may characterize supergeometric cohesion abstractly. It’s an instance of differential cohesion such that the infinitesimals contracted away by the reduction modality “are graded commuting”. One should try to either find a direct axiomatization of this, or else find some other good condition that implies it (such as that twists of generalized cohomology theories have a particularly nice form, maybe).
What happened to the super-geometry should be $\mathbb{S}$-graded geometry idea of Kapranov here?
Yes, that’s what I was alluding to in my last sentence. in #14, as at super algebra – Abstract Idea
The most definite hook into this, that I have available is that the “geometric” twists of $KU$ are precisely super line 2-bundles. And I know how $KU$ plays a key part in the story, being the linear type such that secondary integral transforms through cohesive prequantum correspondences with coefficients in $KU$ reproduces the correct quantum mechanics.
In that story the twists $Twists \to GL_1(KU)$ appear naturally as the “superposition principle”, namely as the analog of $U(1) \to GL_1(\mathbb{C})$ in quantum mechanics.
So that’s what I meant when I said in #14 “some other good condition that implies it”: maybe characterizing $KU$ abstractly (maybe via Snaith’s theorem) and then demanding that the moduli object of 2-lines is equivalent to the 3-tuncation of $GL_1(KU)$ might characterize super-cohesion, given that it seems a strong condition and we know at least that super-cohesion realizes it.
To test this in the present case, one should ask: given just the information that the moduli type of 2-lines is equivalent to the 3-truncation of $KU$, does it follow that one finds types which are like super-Minkowski supergroups and carry those exceptional cocycles?
It seems to be a story worth thinking about, but has still large gaps to be filled in.
I have this vague question which either may not make sense or perhaps does, but you’ve already told me if only I could listen properly. Here goes.
It seems like we’ve made an effort to shift from ordinary Minkowski space to a super-version, and we’ve realised that this is one step of un-truncating from a larger $\mathbb{S}$ account, which gives us the rationale of the super-grading rule, etc. Now we’re wondering if details of the truncation provide an intrinsic way to characterise super-cohesion.
So the question: Why don’t we find ourselves talking about the next untruncated layer in the same kind of way? Why isn’t there a super-duper Minkowski space, a super-duper grading rule, and a super-duper cohesion?
super algebra – Abstract Idea goes on to talk about $tmf$-grading, and so on. Is there possibly a higher form of cohesion to do with the $tmf$ equivalent of
demanding that the moduli object of 2-lines is equivalent to the 3-truncation of $GL_1(KU)$?
And I guess the ultimate question, why should the Hegelian Idea care for these truncations anyway when it sets about forming the universe?
I wish I knew answers to this. I may only hope to slowly make progress on this.
Regarding the role of super-Minkowski spacetime though, I find the brane bouquet very suggestive.
At its root is just the superpoint $\mathbb{R}^{0|q}$. If we consider the natural relative cohesion of supergeometric-homotopy types over bare super-homotopy types then this is just a point.
This $\mathbb{R}^{0|q}$ happens to have group structure and this happens to carry a 2-cocycle. The extensions induced by this cocycle are the super-Minkowski spacetimes. These happen to carry in turn higher cocycles themselves. The extensions induced by these are the extended super-Minkowski spacetimes. These happen to again carry higher cocycles. The WZW terms of these are those of the super-branes with tensor multiplet fields, the D-branes and the M5.
It could go on like this, the bouquet spreading out ever more. But it seems that at least as far as the exceptional cocycles are concerned, it stops at this stage. (Though I don’t have a proof for this.)
I should say that here I am thinking of remark 4.7 in The brane bouquet (schreiber).
Where would one even think to go looking for other bouquets?
Could there be a variant of Grassmann algebras to help find a root?
Where would one even think to go looking for other bouquets?
I wish I knew. But maybe we could turn this around and find criteria that characterize supercohesion as one of probably only few possibilities to have the points in cohesion be equipped with interesting group structure carrying interesting cocycles. (Where “interesting” must be something like non-trivial and non-reducible, of sorts.)
See what I mean? Let’s demand a fundamental bouquet to start on an object as small as possible, literally a tiny object. Then in most models of cohesion this will be the root of a trivial bouquet. Conversely, those models of cohesion where bouquets rooted in tiny objects are interesting will be singled out. Smooth supercohesion is among them, but for instance plain smooth cohesion or plain complex analytic cohesion is ruled out, as are their infinitesimal thickenings with commuting infinitesimals. Of course I don’t know what happens for various flavours of tangent cohesion, these might well have immensely rich tiny objects, since the spectra over the point in tangent cohesion are infinitesimal.
Is one piece of evidence that there are unlikely to be many interestingly rich bouquets that we don’t have candidates for the branches to hand? I mean, physics had already found many of the parts of the super-bouquet before we knew there was such a thing.
Maybe, hard to say. Physics went through a long process of finding and partially accepting supergeometry as a variant of traditonal geoemtry. It’s still regarded as something pretty exotic for most researchers. (Last week in Srni there were lectures on compact Riemannian spaces admitting a Killing spinor – the kind of thing that physicists ask for in the context of supersymmetry – but when I asked the lecturers if there shouldn’t be a natural derivation of the classification of these inside supergeometry, and if they knew of any, the answer showed that they found this a somewhat scary question.)
So now here we are thinking about scanning the space of all exotic flavors of geometries encoded by cohesive toposes. It may seem like an evident question to us, but I think it would be a stretch to argue that physicists will have seen hints of much of the possibilities here already. Unless of course there is some constraint that really singles out supergeometry. I don’t know, seems hard to tell. But something to keep thinking about.
I didn’t mean it had to be physicists necessarily who had found these things. Couldn’t pure mathematicians also stumble across such things?
I didn’t know when writing #17 that it was an ’official’ term, but I see Nora Ganter in her research description for the Hamburg conference say
Kapranov’s conjectural super-duper symmetry formalism.
She also mentions a contrast in her piece
Our methodology is different from Fiorenza, Rogers and Schreiber’s.
Is there an easy way to understand the difference?
[I seem to like ’super-duper’. I was using it back here too.]
Thanks for the pointer.
I have talked with Matt Ando about this over a year back, and I am currently in contact with Eric Sharpe about related matters, who in turn may be in contact with Matt, but I may not know which “different methodology” the above refers to. Is that “pre-release preprint” available online?
Is that “pre-release preprint” available online?
not that I’m aware of, and I get updates on some of these things. Nora explained the approach to me a bit over a year ago, so I may be able to reconstruct some of it, but not in a hurry.
re #28 Thanks, DavidR. I might just ask Matt again. But of course I’d be happy to learn about whatever you get hold of.
re #26: Regarding “super-duper”: I feel hesitant about this. Already the terminology “super” didn’t serve the subject too well, it seems to me. Just recently a maths colleague of mine voiced the suspicion that physicists used the “super”-terminology just to make their ideas appear in shiny light.
In discussions such as above we are talking about reasons that might make supergeometry more immediate than ordinary geoemtry. Imagine at some point in the future a nice sober fundamental theory that has to constantly be referred to as “super-duper”. I’d rather not.
I was, of course, only joking about “super-duper”. But what would be a good term? $\mathbb{S}$-graded geometry is a mouthful.
Oh, okay. Sorry, I could entirely imagine people making this proposal.
Now “$\mathbb{S}$-graded geometry” does not seem that awkward to me. But if I could revert history, I would change the convention to something like “fermionic geometry”. For it is one of the sad outcomes of all the debates about super-symmetry that the important fact that the presence of fermions already shows that the world is built on super-geometry is widely under-appreciated.
The 2 pages of Ganter’s research description (#26) are expanded in the description of a research program with Matthew Ando, Subtle symmetries and the refined Monster, which I’ve mentioned at Nora Ganter.
1 to 32 of 32