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I see super-Cartan geometry is taking shape. Will Clifford algebras make an appearance in the The super-Klein geometry: super-Minkowski spacetime section?
Is there a higher super-Cartan way of thinking about what is at 3-category of fermionic conformal nets, about the String 2-group and superstrings, as here about the spin group and fermions.
Yes, I am working on it, was going to announce it when I have at least the first main subsection essentially done.
Notice that Clifford algebras are the quantization of Grassmann algebras, in direct super-analogy of how Heisenberg algebras are the quantization of commutative algebras. Hence Clifford algebras appear in super-Cartan geometry once we (pre-)quantize: the “classical” fermion fields are anti-commuting, after quantization they satisfy Clifford relations.
In this vein those fermionic nets are expected to somehow be a “localized” (“extended”, “multi-tiered”) quantization of fermionic strings. But I am not aware that anyone has made real progress is making this intuition precise.
Right, so I should say to everyone:
I am working on an entry super-Cartan geometry. Aspects of this have been scattered throughout the nLab, but now I’d like to bring it all together in one place.
First, there is a section
which surveys existing literature and comments on the fact that a topic that deserves to be called “super-Cartan geometry” is implicit in a large body of work, even if only a tiny number of references (maybe $1\frac{1}{2}$) make this terminology explicit. Physicists have their way of speaking about the “superspace approach” to supergravity instead. Generally, as we discussed elswhere, all of supergravity in the “superspace approach” follows Cartan’s original articles, and terms like “Cartan frames” etc. are common in the physics literature, but for some reason few physicists ever say the words “Cartan geometry” or “Cartan connection”. However, we should distinguish a concept from its name and the concept is clearly there.
Next then I am in the process of writing a section
which is supposed to be a gentle expository introduction of the technical details in the spirit of the series of entries on geometry of physics. Indeed my plan is to eventually copy the material to a chapter entry geometry of physics – supergeometry.
Here I have been adding scattered pieces as placeholders and am working now on turning these into flow-text. So far the first subsection
has, while not at all polished yet, at least most of the intended content in it.
Now I need a coffee first…
Here is an entertaining (or sad, depending on mood) bit of trivia regarding that issue of “(non-)existent literature”:
I see that there are actually these two old articles that are fully explicit about the Cartan geometry:
N. S. Baaklini, Spin 3/2 Field and Cartan’s Geometry, Letters in Mathematical Physics August 1977, Volume 2, Issue 1, pp 43-47
N. S. Baaklini Cartan’s Geometrical Structure of Supergravity, Lett. Math. Phys. 2 (1977) 115.
But they are essentially uncited. Each has a single non-self citation and neither of these actually refers to the article. Notably the first one is referenced in the list of citations in the seminal old survey
but that citation is not even pointed to from the actual text there. It seems that Nieuwenhuizen just listed related literature without comment.
But he got to work with Abdus Salam, here.
Curiously, that, too, is: essentially uncited.
Why does he now publish on viXra?
Well, in either case there is not much to point to in these articles from 1977. The meat of the super-Cartan geometry in supergravity is in (D’Auria-Fre-Castellani 91). But if we are looking for the history of authors who used the explicit words “Cartan geometry” in the context of supergravity, then it looks like these articles form 1977 are the first, by some margin. The next one that I am aware of is Egeileh-Chami 2013.
But, as I said, all this is just about terminology. Whether one speaks about “Cartan geometry” or “Einstein-Cartan theory” doesn’t make a difference regarding the actual mathematics. Which is the same in either case.
Here we have smooth0Type-variations being defined as sheave topoi in CartSp and CartSp+InfPoint, etc.
What is the a priori heuristic regarding how big the domain category can be?
What bad things happen if we take a big topos H and consider Sh(H)?
Generally one may consider large sites and over them their very large (infinity,1)-sheaf (infinity,1)-topos. This all behaves as in the small case, but now in one universe level higher.
Specifically for a topos $\mathbf{H}$ itself regarded as a site, if it is regarded so with its canonical topology then (as discussed there) $Sh_{can}(\mathbf{H}) \simeq \mathbf{H}$ (true generally for 1-toposes and for $\infty$-toposes at least under some further niceness conditions).
Ah okay thanks I’ll read that. My question was older but just now triggered again. Could you get physics/differential geometry/cohesion by looking at the sheave topos from CartSp to very unrelated topoi? I was wondering, in particular, about cases where the codomain of the sheaves are not at all coming from geometry, but rather from housekeeping aout syntax. E.g. CartSp–>some topos associated with some silly type theory.
Unrelated, there is something strange here:
http://i.imgur.com/2bJ4adc.png
Regarding your question: could you try to say that again, I am not sure if I understand what you have in mind. Sorry.
Regarding the something strange: woops! Looks like somebody hit a wrong key combination at the wrong point and a copy-and-paste operation accidentally took place. Thanks for alerting me, I hadn’t even noticed. We’ll need to change that…
Say I am a computer scientist with a favorite type theory (but not HoTT) that happens to be associated with a topos T like CCCs are associated with the lambda calculus. Say he knows about R^n’s and smooth maps there - he knows the things relevant to formalize abstract coordinate transformations. If someone suggests to him to look at the category of contravariant functors from CartSp to T, could he accidentally find himself in a setting with enough structure in the this topos so that he can do interesting physics without adding more?
Thanks for expanding. So I gather this is more or less about whether a topos of the form $Sh(CartSp,T)$ over a base topos $T$ is rich enough to do some synthetic physics in.
If one proceeds with a “modalities only” approach the way I have been going on about, then for interesting physics to appear it is crucial that one considers a higher topos, because one needs at least objects like $\mathbf{B}U(1)$ to do something along the lines of Classical field theory via Cohesive homotopy types (schreiber).
However, depending on taste or application, one may be happy with something less synthetic. For instance if you considered $Sh(CartSp\rtimes InfPoint, T)$, i.e. the Cahiers topos (over $T$ if you like) and remember that there is an object $\mathbb{D}$ in there such that $T(-)\coloneqq [\mathbb{D},-]$ produces tangent bundles, then it is possible to speak about differential equations. Hence in as far as you are interested in physics qua differential equations (“equations of motion”), then there you go. This is why Lawvere called such toposes that model synthetic differential geometry “Toposes of laws of motion”.
This is related to an issue I am discussing with Mike in another thread (even though Mike and I seem to tend to get stuck on the simplest issues of terminology): in principle, of course, in view of the fact that every topos provides a “universe for mathematics”, it also provides a universe for all of mathematical physics. Hence if you have enough patience to phrase physics textbook definitions internally, then of course every topos allows to “do” theoretical physics.
So the question is really whether a given topos allows to do this more elegantly, “more synthetically” than by explicitly writing it all out the tedious way.
With respect to this an SDG topos like the Cahiers topos admits a more succinct formulation of everything related to differential equations, as it already “knows synthetically” what differentiating it, we don’t have to internalize textbook definitions of converging sequences of difference quotients in order to get that.
With the modalic axiomatics of cohesion one may get what I feel is a still “more synthetic” formulation. But for that to go through it is crucial that one has not just a cohesive 1-topos, but a cohesive higher topos. Whence the title of a book that I have in preparation.
Okay yes, I understand that in principle you got a logic with a topos and could implement math from scratch there - of course the topos viewed as geometric framework (rather than a logical one) can allow for a more direct “natural” implementation.
Regarding the physics formulations - I have read the discussion and entries about the Nichts to Werden formalization via unit and co-unit that comes with the adjoints, which are a core part of the definition of cohesion, but atm. an axiom saying “there is an ajoint to this functor” )or definitions of functors as adjoints) are not intuitive enough. I know that having an adjoint pair brings about a lot of structure, but so much that I get lost what the implications really are.
I think at the moment I’m orienting myself along the lines of “if there is an adjoint functor pair, then their is a monad and hence a pair of nice natrual transformations” (and then e.g. in your case the Nothing X->X->Werden X constructions.) Monads are simpler to visualize because I know some examples from programming where they are actually relevant and I don’t get away with viewing them as mere reformulations of already existing mathematics.
You should think of most of the adjunctions here as a means to produce classifying objects (moduli stacks).
Take $(\int \dashv \flat)$. The interpretation of $\int$ is that it sends each space $X$ to its fundamental $\infty$-groupoid of paths $\int X = \Pi_\infty(X)$. This means that morphisms $\int X \to \mathbf{B}A$ are flat $A$-valued connections: they send paths in $X$ to elements of $A$, send disks in $X$ to homotopies between elements of $A$, and so forth.
So then you might as if there is an object $\mathbf{B}A_{flat conn}$ which represents/modulates such flat connections, in that maps $X \to \mathbf{B}A_{flat conn}$ are equivalent to maps $\int X \to A$. And yes, by adjunction there is, we have
$\mathbf{B}A_{flat conn} = \flat \mathbf{B}A \,.$(That’s where the notation “$\flat$” pronounced “flat” and the name flat modality comes from).
Similar story, in turn, for $(\flat \dashv \sharp)$. For $X$ a space then $\flat X$ is its collection of points. Hence a map $\flat X\to A$ is just a map from each point of $X$ to $A$. Again you may ask if such maps have a representing/modulating object, and yes, they do, by adjunction it is $\sharp A$.
Have been editing further at super-Cartan geometry. The Introduction-section (maybe this needs another title, not sure) now has three subsections:
Of these the first two I have now in a form with, at least, continuously flowing text. The third one is still just keywords, formulas and related material thrown in, still needing to be turned into text. I will work on that tomorrow.
Have been working further at super-Cartan geometry, now on that remaining subsection 3 Super-Cartan geometry for Supergravity. It’s essentially there, but will further polish more tomorrow morning.
This is now in different style. Sections 1) and 2) I had prepared for the audience consisting of category theorists and supergravity theorists which attended Jiří Rosický’s group seminar today. Now section 3) is for an audience consisting of classical Cartan geometers and supergravity theorists, attending the “Central European Seminar” (or so I gather it’s called) tomorrow. So section 3 has no topos theory in there, but is all graded algebra.
Eventually there is to be a further section that puts this together and discusses how that graded algebra is indeed a model for the previous topos theory. That’s to be done in one of the following weeks.
Why the hat in $\mathfrak{Iso}(\widehat{\mathbb{R}}^{10,1\vert \mathbf{32}})$?
And after def 24, where it says
The quotient of that by the spin group is super-Minkowski spacetime
the symbols refer neither to spin nor to group.
Thanks for spotting this.
Regarding the first, I have added a brief remark that the hat in the last row of that table refers to extended super Minkowski spacetime, the topic of higher Cartan geometry. This is meant as an outlook not to be discussed right there in the entry.
Regarding the second, thanks, I have harmonized notation. We have super-Minkowski spacetime on the one hand regarded as a super-translation group and on the other hand regarded just as a super Lie algebra. The former is the quotient of th Poincaré group by the spin group, the latter is the quotient of the Poincaré Lie algebra by the Lorentz Lie algebra.
Worked a little on the section that I hadn’t yet got to last week: Definite superforms.
Added the proposition and proof that the joint stabilizer of the supersymmetry bracket and the 3-cocycle for the super 1-brane is $Spin(d-1,1)$.
Thanks to John Huerta for amplifying this to me!
This is a really neat statement. It says that if we regard those cocycles on the underlying super Lie algebra of super-Minkowski spacetime as directly analogous to, say, the associative 3-form, then we recover Lorentzian geometry (in analogy with $G_2$-structure) without specifying it by hand.
added missing cross-link to super Klein geometry
added pointer to today’s
Presumably this still suffers from the sort of problems you mentioned faced by other loop quantum gravity approaches.
Despite the advertisement, this article doesn’t get into the funny quantization step (just some hints on how they imagine to proceed on p. 41.) It deals just with the (super-) Ashtekar variables and the corresponding “canonical” (meaning: non-covariant) description of the classical phase space, all of which is legal and done, with different terminology, also outside the LQG community (e.g. Cattaneo-Schiavina 17a).
I see, thanks.
added pointer to this article, from today:
added pointer to today’s
added publication data for
Konstantin Eder, Super Cartan geometry and the super Ashtekar connection, Ann. Henri Poincaré 24 (2023) 3531–3599 [arXiv:2010.09630, doi:10.1007/s00023-023-01290-5]
Konstantin Eder, Super fiber bundles, connection forms and parallel transport, J. Math. Phys. 62 063506 (2021) [arXiv:2101.00924, doi:10.1063/5.0044343]
added pointer to:
added pointer also to:
John Huerta, Urs Schreiber, p. 7-8 in: M-theory from the Superpoint, Lett. Math. Phys. 108 (2018) 2695–2727 [doi:10.1007/s11005-018-1110-z, arXiv:1702.01774]
John Huerta, Hisham Sati, Urs Schreiber, p. 6-7 in: Real ADE-equivariant (co)homotopy and Super M-branes 371 (2019) 425–524 [doi:10.1007/s00220-019-03442-3, arXiv:1805.05987]
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