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Thanks. I have reshuffled to avoid that items in the list of technical reviews seemingly get to sit in a list of philosophical discussion. Also cross-linked with philosophy of physics.
I haven’t had a chance for a proper look. In view of our discussion about my Friedman talk, I’m interested in what it’s possible to do usefully philosophically.
He writes
As Friedman argues [14], physical theories are embedded in a package of background formal and conceptual assumptions, which can be hard to see explicitly, or appear matters of necessity when they are seen. SUSY places current spacetime and particle physics in a broader logical landscape, revealing hidden assumptions and contingencies. In this regard, its epistemic value is independent of whether or not SUSY is realised in nature.
I wonder how far one can usefully pursue this strategy:
In this paper, I choose to focus only on SUSY, divorcing it from the context of string theory, but it is useful to bear this motivation in mind throughout.
I will therefore mostly ignore its quantum field theoretic roots, and present it as a generalisation of a spacetime symmetry
Some items one could highlight:
supergeometry is a great example of internalization at work
superspace and supersymmetry are logically distinct:
Just as not every interesting construction on space is necessarily Poincaré invariant,
so not every interesting construction on superspace is necessarily supersymmetric (namely: super Poincaré-invariant);
in particular, the phase space of every field theory with fermions is a superspace, whether its supersymmetric or not
e.g. the phase spaces of realistic models of nature are all superspaces, since fermions exist: the superspace nature reflects the skew-symmetry of fermion variables established ever since Stern-Gerlach 1921
which should make one wonder what actually is more “natural”: a superspace with non-super symmetry or one with super-symmetry.
(what we currently see (super phase spaces with non-supersymmetry) is analogous having a smooth manifold whose symmetry group is a topological group without Lie group structure. It can happen, but is a little odd.)
To check, where does the local/global supersymmetry distinction come into what you’ve just written?
This is the distinction between Klein- (global) and Cartan- (local) geometry.
For any kind of subgroup inclusion, a coset space is a “globally symmetric geometry” while a manifold with Cartan-connection is a “locally symmetric” space.
For Lorentz group inside Poincaré group this gives Minkowski spacetime (globally symmetric) and pseudo-Riemannian manifolds (locally symmetric), the latter also called configurations of the field of gravity (possibly “off-shell”, ie no EOMs at this point).
For Lorentz group inside super-Poincaré group this gives super-Minkowski spacetime (globally symmetric) and super-Riemannian manifolds (locally symmetric), the latter also known as configurations in supergravity (off-shell).
On top of this we can ask that the Cartan-connection is torsion-free. This makes its local symmetry be “global to first infinitesimal order”.
A miracle happens: For 10+1 d such super-Cartan geometry which is “locally super-symmetric but globally so to first infinitesimal order” corresponds exacty to those fields of supergravity which are on-shell after all.
It looks like no such local supersymmetry is observed currently (except for a small hint that the observationally preferred Starobisnky model of cosmic inflation seems to prefer local supersymmetry). But it also looks like something unexplained is happening in B-meson decays, the best current answer to which is currently a Randall-Sundrum-like GUT model. That’s broadly the kind of models found from 11d local supersymmetry.
Thanks! I don’t recall this expression “locally super-symmetric but globally so to first infinitesimal order” before. It’s presumably on nLab pages in other terms.
I haven’t used this precise combination of words before, but the idea I am describing is that briefly indicated at integrability of G-structures. It is an old observation due to Guillemin 1965 the formalization of which, following what I have in dcct v2 (p. 556), was the aim of Felix’s thesis (here).
Namely Guillemin’s observation was this: Since coset spaces carry a canonical $G$-structure, obtained by left translation of frames, we may say that a $G$-structure on a manifold locally modeled on these coset spaces is “integrable to $k$th order” if around each of its points the restriction of its $G$-structure to the order-$k$ infinitesimal neighbourhood is isomorphic to the canonical $G$-structure on the coset space.
Integrablity to first order is what is called “torsion -free”, integrability to infinite order is called “flat”. (This terminology follows the case of $O(n)$-structure, which are equivalently Riemannian metrics and where these two cases are equivalently torsion-freeness and flatness of their affine connections, as traditionally defined). A good review of this case in classical diff geo language is on p. 4 of Lott’s arXiv:0108125.
So torsion-freeness aka “integrability to first infinitesimal order” says that the “global symmetry” holds in each tangent space (on each first order infinitesimal neighbourhood). Again for $O(n)$-structure, this is essentially the embodiment of Einstein’s “equivalence principle” in gravity.
Lott’s article above notices that partial vanishing conditions on super-torsion generically appear (as the spacetime dimension and the number of supersymmetries varies) implied by equations of motion of supergravity, and then looks for a way to fit this into Guillemin’s scheme. But Lott’s article only goes up to dimension $D = 6$ and did not seem to be aware that in $D = 11$ the situation is verbatim that of Guillemin’s old article.
Good, thanks. I was just thinking it must mean torsion-free as in Felix’s work.
Of course, everything is there for the 11d case in super-Cartan geometry, torsion constraints in supergravity, D’Auria-Fre formulation of supergravity, On curved spacetime and supergravity equations of motion,…
Thanks, right, I forget where I wrote about this. :-)
It’s a great topic, I think, still waiting to be further explored. One loose end in Felix’s thesis was the choice of order $k$ in the above. As written, that choice was left implicit in the choice of model of the syntax (depending on which order of infinitesimals were included in the site of definition of the differential-cohesive $\infty$-topos which interprets the syntax). But that choice should instead be manifest in the syntax.
Back then I thought that the primate of first order flatness (torsion-freeness) ought to be explained by the supergeometry, since a hallmark of supergeometry is that the odd-graded infinitesimals are necessarily first order. So, I think, one should find a way to intrinsically extract from the progression of adjoint modalities in super-cohesion the first order infinitesimal disks on which to require $G$-structures to coincide with the canonical one. This seems to work, I had made notes on this somewhere, but now I forget where…
…but now I forget where…
Looks like something here.
Thanks! Yes, that’s it. I thought I had cleaned that up and recorded it in some $n$Lab entry, too. But maybe I didn’t.
Presumably there’s then the question of how these considerations play out in the context of the sphere spectrum approach to superalgebra, e.g., what does the sphere spectrum know of different orders of infinitesimal?
Perhaps a Goodwillie calculus for spectral super-geometry.
So the idea is that the equations of motion on SuGra are implied via Hypothesis H by demanding that the rationalization of the underlying super-cocycle is equivalent to the canonical one on each first order neighbourhood.
This is indicated in.., Bu the nLab is going down right now…
(now that we are back…)
This is indicated briefly on p. 7-8 of arXiv:1805.05987. (But looking at it again now, I see it’s rather too brief. Need to come back to this at some point…)
At Green-Schwarz action functional it says
the effective target space fields exhibit local supersymmetry (i.e. “high energy supersymmetry”, different from “low energy supersymmetry” that the LHC was looking for).
How then does observability operate with this graded scale from locally super-symmetric to locally super-symmetric but globally so to first infinitesimal order to locally super-symmetric but globally so to $k^{th}$ infinitesimal order to globally super-symmetric?
Through the vanishing of torsion components.
0th order means: There is a gravitational field.
1st order means: It has no torsion.
higher order means: It is flat.
Here in supergeometry we would be speaking about super-torsion, but if you have not seen this before first check out torsion in plain gravity: In ordinary Einstein-gravity the torsion component of the metric vanishes. If it doesn’t that implies that the equation for parallel transport of spinning particles gets modified, in particular that fermions don’t quite obey the “equivalence principle”.
People have been looking for such effects for ages, and there is a vast literature (which however is hard to navigate due to differing terminology), famously going back to (at least) work by Friedrich Hehl in the early 1970s, Keywords are “Einstein-Cartan theory”, “spin-torsion coupling” etc.
When I was a graduate student I had a literature list on this stuff (because a befriended professor tried to make me connect the Hehl-ian theory he had been brought up with with B-field effects in string theory ) but I would have to recompile this. Would certainly be worth an $n$Lab entry.
So I don’t have the best references at hand, but looking around, this one here seems to give a good impression:
A few years back these investigations received a sudden spike in attention when a group claimed that the apparently observed anomaly seen in the anaomalous magnetic moment of the electron $(g_e - 2)$ could be explained by non-vanishing torsion in the gravitational field. As far as I remember, the very next day a couple of preprints appeared explaining that no torsion in the gravitational field could possibly have this particular effect. So that claim was bogus, but the general point that there could be some (other) observable effect of torsion-ful gravity remains, and some people keep looking for it.
Now in 11d supergravity, it is the supertorsion which vanishes. Since this is the sum of the bosonic torsion with a gravitino contribution, this means that the ordinary bosonic torsion is in fact predicted to be non-zero, at least through quantum effects (classically there would have to be a gravitino vev, which is not so plausible).
Thanks again!
So this account via first-order flatness can also be expressed via that $Diff(\Sigma)$-equivariant account as written out at general covariance,
$[\Sigma//Diff(\Sigma),\; \mathbf{Fields}] \simeq [\Sigma,\; \mathbf{Fields}]//Diff(\Sigma) \,.$I hadn’t considered that the $\Sigma$ could be a superspace (although the generality is explained here).
Then the magical properties of 11d are what? That for superspaces, $\Sigma$, based on $\mathbb{R}^{10,1|32}$, then covariance forces fields of supergravity to be on-shell?
What happens as you work up the brane bouquet with the requirement of torsion-freeness? Does what is special about the 11d case emerge progressively?
It’s not the covariance that forces them on-shell, but the torsion-freeness. The fields here are a $G$-structure and the condition which forces them on-shell is that restricted to each tangent space this $G$-structure be isomorphic to a canonical one.
Some condition is certainly required to put the general and generally equivariant fields on-shell. The striking aspect of the torsion constraint in 11d sugra is that it has an elementary fundamental formulation in terms of just basic notions of (synthetic) differential geometry and can be axiomatized straightforwardly in differential cohesion.
In contrast, away from the situation of 11d supergravity, that on-shell constraint is “Einstein’s equations” in their respective form, and, nice as these may be, they are not as elementary as a torsion constraint.
This is in fact the wall that a whole subcommunity of mathematical physics has been and is banging there heads against: People get exited about how the field of gravity is naturally a Cartan geometry, but then struggle to find a perspective that would bring in the Einstein equations in an equally natural way.
Right. I ought to have absorbed this material better. I’ve just looked over again those Bristol talk slides. A couple of questions occur to me:
Does the fact the ’miracle’ happens in 11d have anything to do with its position in the brane bouquet?
When spacetime becomes a higher Cartan geometry modeled on a homotopy 3-type extension of $\mathbb{R}^{10,1|32}$, is there a higher super-torsion to set to $0$?
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