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.
I am slowly beginning to work on geometry of physics – supergeometry. So far there is nothing but the definition of super-Minkowski spacetimes and now a computation (here) that the 7-cocycle on the extension of 11d super Minkowski spacetime by its 4-cocycle descends along the rational Hopf fibration $S^7 \to S^4$ to a cocycle on 11 d super Minkowski spacetime itself with coeffcients in the rational 4-sphere.
Why is the rheonomic modality now denoted $\Re$ and so just the same as the reduction modality? Elsewhere, e.g., reduction modality, you still have the rheonomic modality as $R$.
Woops, that was a typo. Thanks for catching this, have fixed it now.
Now you have it as $Re$. I thought I saw $Rh$ somewhere. Have you settled on one thing?
And if we see old notation around the nLab, would you like us to change it? E.g., at reduction modality it still has $ʃ_{inf}$ and $\flat_{inf}$.
Should that be $2p+3$ or $2p+2$ in proposition 2? Otherwise we don’t get $b^6\mathbb{R}/b^2\mathbb{R}$ later on.
@DavidC, thanks again, I suppose the excuse is that I am jet-lagged. It should be $Rh$ throughout. I have fixed that and the other instances you pointed out now. Yes, if you see anything that is obviously a typo or outdated notation and if you have the energy, it would be great if you would fix it.
@DavidR, thanks for catching this, yes, that’s a typo at that point and at a few other instances, too. Have fixed it now here and at two other points. Thanks again.
I have now copied over some of the material from super Cartan geometry to geometry of physics – supergeometry and harmonized notation. Not done yet, though.
Did some minor further editing. Am using excerpts for course notes at Structure Theory for Higher WZW Terms (schreiber)
I am in the process of compiling detailed introductory lecture notes on superalgebra, in a new section of the same name in the entry geometry of physics – supergeometry.
Actually I started writing this as a new section “Background” in the entry Deligne’s theorem on tensor categories. But for that entry it has become too long, and I will remove it there and replace all pointer to it with the corresponding pointers to the “geometry of physics” entry.
If you do take the tensor category as primary and reconstruct a supergroup, $G$, how do you choose an $H$?
I am in the process of writing the announced lecture notes.
The first section
is stable. The next section
is under construction. The first subsection “Super coordinate systems” should be close to done.
To make “supergeometry” be on par with “superalgebra” I have renamed their joint ambient entry to “geometry of physics – supergeometry and superphysics”.
Did a bunch of further editing at geometry of physics – supergeometry. Have essentially all the material in place now that I wanted to. A good bit of exposition, too. But towards the end I still want to expand further. Not tonight, though.
So we meet “super-mathematics” apparently in two different ways:
Having (2) is very useful here as you can find a model of super synthetic geometry, find groups within it for the supergroups whose representations form tensor categories, etc.
Then the question: Is this just a happy coincidence?
Yes! I keep wondering about this, but I have no answer yet.
I can’t help but feel that it must be a hint of something deep. But I don’t see it yet.
It feels like (2) is forcing the choice of that series of four sites:
$\array{ \mathcal{C} = & \ast & CartSp & FormalCartSp & SuperFormalCartSp \\ \\ Sh(\mathcal{C}) = & Set & SmoothSet & FormalSmoothSet & SuperFormalSmoothSet } \,.$And even with variants, such as Stein manifolds, or choosing a different $\infty$-topos for values of sheaves, as at differential algebraic K-theory, localisation for the three kinds of dimension is a given.
I can’t remember if Theo Johnson-Freyd’s claims about Deligne’s theorem in Spin, statistics, orientations, unitarity were ever mentioned here:
Our main contribution here is to categorify the notion of “field” and to interpret the main theorem of [Del02] as asserting that the “categorified algebraic closure” of $\mathbb{R}$ is not $\mathbb{C}$ but rather the category $SuperVect_{\mathbb{C}}$ of complex supervector spaces.
So his difference from Kapranov:
The papers [GK14, Kap15] suggest that rather than $O(\infty)$, it is the sphere spectrum that controls supermathematics. Very low homotopy groups cannot distinguish between various important spectra. The connection with topological quantum field theory focused on in this paper provides a reason to prefer $O(\infty)$.
This paper was pointed to by Qiaochu Yuan when saying at MO:
This theorem can be intepreted as saying that the symmetric monoidal category of super vector spaces over $\mathbb{C}$ is in some sense “algebraically closed”
Thanks for these pointers. I had not been aware of these remarks until now.
Wait, in the same MO thread there is something possibly more interesting: John Rognes writes in here to support my hunch that we should regard Sagave’s construction as showing how stable homotopy theory is automatically super-graded in this generalized sense.
So Rognes’ topological logarithmic geometry, as in this talk, is presumably a brave new variant of logarithmic geometry.
So this result in Sagave-Schlichtkrull 11, between theorems 1.7-1.8 seems to be exactly the piece that was missing in the story that I was after. I have made a brief edit to my notes concerning this at infinity-groups of units – augmented definition.
Previously, all we had there was that for $E$ an $E_\infty$-ring spectrum, then its $\infty$-group of units $GL_1(R)$ (i.e. $R^\times$) is canonically $\mathbb{S}$-graded, in that there is an $\infty$-monoid homomorphism
$(GL_1(R), \cdot) \longrightarrow (\mathbb{S}, +) \,.$Now a $\mathbb{Z}$-grading on an ordinary ring $R$ is of course equivalently a monoid homomorphism
$(R,\cdot) \longrightarrow (\mathbb{Z}, +) \,.$This restricts in particular to a monoid homomorphism of the form
$(GL_1(R),\cdot) \longrightarrow (\mathbb{Z}, +) \,,$but of course it is more structure.
So the remaining question was: for $E$ an $\infty$-ring, does the $\mathbb{S}$-grading on $GL_1(R)$ extend to one on all of $(R,\cdot)$, or at least on the underlying $(\Omega^\infty R,\cdot)$?
Now I suppose this is just what the text in between theorem 1.7 and 1.8 answers positively.
The construction $\Omega^{\mathcal{J}}$ that makes this work appears in the main text as (4.4)
So that’s great.
Even if you can do surprising things with a superpoint, e.g., the brane bouquet, won’t the full $\mathbb{S}$-graded story need a richer notion of point? If we can detect kinds of point from spectra of algebras, shouldn’t we be looking for spectra of things like $E_{\infty}-rings$?
I see ’spectrum of an E-infinity ring’ is called for at A Survey of Elliptic Cohomology - E-infinity rings and derived schemes, but we don’t have it yet. On the other hand, we do have prime spectrum of a symmetric monoidal stable (∞,1)-category.
Lurie’s new work Elliptic Cohomology I, which is supposed to “carry out the details of the program” of the Survey, defines
What I don’t quite understand yet is how to see the $\mathbb{S}$-grading of any connective $E_\infty$-ring as a lift of a supercommutative superalgebra structure.
For that we somehow need to loop once (or twice). Because the “super”-part in $\mathbb{S}$-grading happens in degree 1 (or 2).
But for $E$ an $E_\infty$-ring, then $\Omega E$ is not an $E_\infty$-ring anymore, but just an $E$-module.
So what are we to make of this, regarding the idea that we naturally get supergeometry from stable homotopy theory?
So Kapranov is saying:
The Koszul sign rule arises from the groupoid of 1-dimensional supervector spaces, and the $\mathbb{Z}$-graded rule from $\mathbb{Z}$-graded 1-dimensional vector spaces:
$1-SVect: \pi_0 = \mathbb{Z}/2, \pi_1 = k^{\ast}$
$1-Vect^{\mathbb{Z}}: \pi_0 = \mathbb{Z}, \pi_1 = k^{\ast}$
Replace $k$ by $\mathbb{Z}$ for $\mathbb{Z}/2$- or $\mathbb{Z}$-graded abelian groups which are free of rank 1.
$1-SAb: \pi_0 = \mathbb{Z}/2, \pi_1 = \mathbb{Z}/2$
$1-Ab^{\mathbb{Z}}: \pi_0 = \mathbb{Z}, \pi_1 = \mathbb{Z}/2$, the free Picard groupoid generated by one formal object (symbol) $L$. This is $\tau_{[0, 1]}\mathbb{S}$
[Corollary 3.3.2. The ’S’ in $1-SAb^{\mathbb{Z}}$ shouldn’t be there.]
Mathematicians’ approaches to super-mathematics arise from $\tau_{[0, 1]}\mathbb{S}$, but physicists also look at $\Omega \tau_{[1, 2]}\mathbb{S}$ whose Picard groupoid is $1-SAb$.
Note $\Omega \tau_{[1, 2]}\mathbb{S} = (\tau_{[0, 1]}\mathbb{S})/2$.
For higher versions we should look for the Picard groupoid of $\tau_{[0, 2]}\mathbb{S}$, etc.
Could we then say that the reason physicists look at $\Omega \tau_{[1, 2]}\mathbb{S}$ and so $1-SAb$ or $1-SVect$ is that they don’t have a way to represent $\tau_{[0, 2]}\mathbb{S}$ yet?
We know from here that
there is no strict skeletal Picard 2-category whose K-theory realizes the 2-truncation of the sphere spectrum.
I know the background, but there is a gap in the proposal that I myself made. I had said that the suggestion that super-grading is to be read as grading by the first or second stable homotopy groups of spheres, in turn suggests that super-geometry ought to be a shadow of $\mathbb{S}$-graded $E_\infty$-geometry, and that this is particularly suggestive since that result by Sagave shows that every connective $E_\infty$-ring is canonically $\mathbb{S}$-graded.
The gap in this suggestion of mine is that $\mathbb{S}$-grading taken at face value categorifies $\mathbb{Z}$-grading (via $\pi_0^S$). And the induced $\mathbb{Z}$-grading of any $E_\infty$-ring $E$ is just the standard $\mathbb{Z}$-grading of its $\pi_\bullet(E)$.
Now it is true that the sign rule of this $\mathbb{Z}$-grading comes from the underlying $\mathbb{Z}/2$-grading after reduction mod 2, so that every $E_{\infty}$-scheme has underlying it what some people, following Schwarz et al., call an “N-supermanifold” (or N-superscheme). But not every supermanifold is an N-supermanifold, and in particular the crucial ones like super-spacetimes are not. And finally, there are the arguments by Kapranov that the true supergrading (not the one arising as a shadow of cohomological gradiing) happens in $\pi_1^S$ and $\pi_2^S$, not in $\pi_0^S$.
All of this suggests that if we are to see genuine super-geometry as a shadow in $E_\infty$-geometry, it should appear after some kind of looping. $Spec(\mathbb{S})$ does not seem to be want to be a higher super-point. “$Spec(\Omega\mathbb{S})$” would seem to be what we want, only that it does not make sense, since $\Omega \mathbb{S}$ is not an $E_\infty$-ring.
Are there any reasons to believe Kapranov over Theo (#17)?
Moving up the ladder in representation theory, is there a reason that $|2sLine_{\mathbb{C}}|$ shares homotopy groups from 0 to 3 with $O(\infty)$?
Are there any reasons to believe Kapranov over Theo (#17)?
I read Johnson-Freyd 15 as giving a neat reason why to consider the image of the J-homomorphism $O \to GL_1(\mathbb{S})$ in the context that we are discussing. However Kapronov’s arguments for his suggestion (the super-representation categories and the appearance of $\mathbb{Z}/24$ for the string) lie outside that image. Still, both the arguments of Kapranov and of Johnson-Freyd seem to nicely fit together, I don’t see a tension between them regarding the picture that they suggest.
However presently neither helps me see what you ask for in #22 and what I said in #23 I am still stuck with: see what the $E_\infty$-analog of supermanifolds such as the basic superpoints should be. An affine $E_\infty$-scheme simply is $\mathbb{Z}$-graded commutative in $\pi_0$. We may choose to remember just the underlying $\mathbb{Z}/2$-grading, and hence interpret this as an N-supermanifold, but the issue remains that the superpoints and superspacetimes of interest don’t lift to N-supermanifold structure.
On the other hand, maybe I am looking at it the wrong way somehow, which easily happens, as the idea we are after is so vague at the moment.
From the ring of functions side, how does one get led to manifolds and supermanifolds? There’s the idea of smooth algebras or $C^{\infty}$-rings, and then smooth superalgebras or $C^{\infty}$-superalgebras.
What happens in the $E_{\infty}$ case? I see Jos Nuiten (thesis) has smooth $E_{\infty}-rings$ as $E_{\infty}$ rings with respect to the smash product on spectra in $\mathbf{H} = Sh_{\infty}(SmMfd)$. Are there smooth super-$E_{\infty}-rings$, replacing by $\mathbf{H} = Sh_{\infty}(SmSuperMfd)$?
Is there a Fermat theory approach? We have super Fermat theories in arXiv:1211.6134:
In light of the history of $C^{\infty}$-rings and their role in synthetic differential geometry, it is natural to believe that super Fermat theories should play a pivotal role in synthetic supergeometry, but we do not pursue this in this paper.
So there should be a
super Fermat (∞,1)-algebraic theory,
and some full $\mathbb{S}$-version?
Has there been any uptake of Christopher Schommer-Pries and Nathaniel Stapleton’s ’superalgebraic cartesian sets’?
Cohesion is in the air (p.11) Rational cohomology from supersymmetric field theories.
Since the category $s\mathbf{A}$ has all finite products the category of superalgebraic cartesian sets is a cohesive topos
So then the site is infinity-cohesive:
Example 3.1. The site for a presheaf topos, hence with trivial topology, is $\infty$-cohesive, def. 2.1, if it has finite products.
In view of
Kapranov’s arguments for his suggestion (the super-representation categories and the appearance of $\mathbb{Z}/24$ for the string)
and the difficulties untangling the different instances of $\mathbb{Z}/2$, maybe it would make sense to see what the $\mathbb{Z}/24$ has to do with the superpoint.
You say at super algebra that one route is via M2 branes ending on a M9 brane, but the latter doesn’t show up on the brane bouquet.
Presumably one needs some kind of K3 technology (eg K3 cohomology), since a K3 surface minus 24 discs is the relation in framed cobordism expressing the ’mod 24’.
A possible solution to the issue that $E_\infty$-geometry is by default $\mathbb{Z}$-graded in degree 0, instead of $\mathbb{Z}/2$-graded, might be to observe that $\mathbb{Z}/2$-graded geometry sits in $\mathbb{Z}$-graded geometry by the construction that regards a $\mathbb{Z}/2$-graded commutative algebra as a periodically $\mathbb{Z}$-graded algebra (this embedding is faithful, but not full).
Hence we should maybe be looking for a “brane bouquet” starting with $Spec(E)$ for $E$ a periodic ring spectrum.
Something related to this is the “$\mathbb{Z}/2$-graded formalism” of Charles Rezk’s arXiv:0902.2499 (section 2), for $E_\infty$-algebras over even periodic $E_\infty$-rings.
So how about $E_\infty$-superpoints? Let $E$ be an even 2-periodic $E_\infty$-ring spectrum. For $N$ an $E$-module spectrum, write $\mathbb{P}N$ for the free $E_\infty$-algebra over $E$ that it generates.
Then $\mathbb{P}( \oplus_{i = 1}^q \Sigma E )$ is like an $E_\infty$-version of the Grassmann algebra over $E$ on $q$ generators, and
$E^{0\vert q} \coloneqq Spec(\mathbb{P}( \oplus_{i = 1}^q \Sigma E )) \;\;\; \in Aff_{E}$would be the corresponding $E_\infty$-theoretic super-point.
I suppose.
I’ll need to have this unpacked this a bit. Why the ’$\Sigma$’? What does $\mathbb{P}$ do more explicitly?
Here $\Sigma$ is the suspension operation on underlying spectra. On the level of homotpy groups this is simply the degree shift by +1. Hence this makes $\Sigma E$ be the analog in $E_\infty$-algebra over $E$ of $\mathbb{R}[1] = \mathbb{R} \theta$ for ordinary algebra over $\mathbb{R}$.
Then $\mathbb{P}$ forms the free commutative $E$-algebra over this module $\Sigma E$ (I was following the notation in Charles’ arXiv:0902.2499, p. 10 and later). This is the homotopy analog of saying that the Grassmann algebra $\wedge^\bullet_{\mathbb{R}} \mathbb{R}$ is the free graded-symmetric $\mathbb{R}$-algebra generated from $\mathbb{R} \theta$.
In any case, superpoint or not, I suppose the answer to my puzzlement above is: the homotopy-theoretic analog of superalgebras is not general $E_\infty$-rings, but $E_\infty$-algebras over even 2-periodic $E_\infty$-rings.
I have added a comment to this extent to geometry of physics – superalgebra, starting here.
And we might then form $E^{p \vert q}$ and then super-$E$-spaces?
Yes, but the other way around: fixing an even periodic $E_\infty$-ring $E$, then affine super-$E$-schemes would be the formal duals of $E_\infty$-algebras over $E$. Among these, there are the super-Cartesian $E$-schemes
$E^{p \vert q} \coloneqq Spec \left( Sym_E \left( \left( \vee_{i = 1}^p E \right) \vee \left( \vee_{j = 1}^q \Sigma E \right) \right) \right)$(where I am falling back to writing $Sym_E$ for what I previously denoted $\mathbb{P}$, the free $E_\infty$-symmetric $E$-algebra construction on an $E$ $\infty$-module).
Now I am thinking: by direct analogy to what we said yesterday in the other thread regarding extensions of $\mathbb{C}^{0 \vert 2}$ in the category over $\mathbb{C}$, I suppose it should follow that also the extensions of $E^{0 \vert 2}$ (say in formal duals of Hopf $E_\infty$-algebras over $E$) would be just the brane bouquet as over $\mathbb{R}$, but with all coefficients of $\mathbb{R}$ replaced by $E$ (and all degree shifts replaced by $\Sigma$). Maybe I am missing something here, but it seems to me this conclusion must follow just from the defining free property of those $E^{p \vert q}$.
So maybe more interesting would be to look for an “$E$ superpoint” given by an $E_\infty$-algebra (over $E$) that looks a little like $\mathcal{O}(E^{0 \vert 2})$, but is not just built freely this way. Maybe there is something like this that arises naturally somewhere.
Apart from this what needs thinking is how to merge the $E_\infty$-geometry with the cohesive geometry. I vaguely have the thought that we should consider as site the $\infty$-category whose objects are formal products
“$\mathbb{R}^n \times Spec(A)$”
of a Cartesian space $\mathbb{R}^n$ with an affine super-$E$-scheme $Spec(A)$ (the latter in place of an infinitesimally thickened point over $\mathbb{R}$ as in the construction of the Cahier topos), and whose morphisms are pairs, consisting of 1) a smoothly $\mathbb{R}^{n_1}$-parameterized morphism of $E_\infty$-algebras $A_2 \to A_1$ over $E$ (not sure yet though what “smoothly parameterized” should mean) and 2) a morphism $Spec( C^\infty(\mathbb{R}^{n_1}) \otimes_{\mathbb{R}} A_1 ) \to \mathbb{R}^{n_2}$.
The second component, due to the tensoring over $\mathbb{R}$, will collapse the $E_\infty$-algebra equivalently to a super-dg-algebra over $\mathbb{R}$, but the first part would retain genuine $E_\infty$-ring theoretic information.
Hm, these links lead to discussion of parameterized spectra. What should be the relevance for the present discussion?
Sorry, probably late in the day, is that $Map_{Alg_{E_\infty}(\mathcal{C})}(E_X,E'_X)$ at the end of the second answer not about maps between parameterized $E_{\infty}$-algebras (admittedly over the same $X$)?
Yes, but what’s the relation to what we are discussing? I don’t understand what you mean to point me to.
I was hoping to shed some light on your decompostion:
consisting of 1) a smoothly $\mathbb{R}^{n_1}$-parameterized morphism of $E_\infty$-algebras $A_2 \to A_1$ over $E$ (not sure yet though what “smoothly parameterized” should mean) and 2) a morphism $Spec( C^\infty(\mathbb{R}^{n_1}) \otimes_{\mathbb{R}} A_1 ) \to \mathbb{R}^{n_2}$.
and further hoping that Harpaz’s decomposition might help:
$Map_{Alg_{E_\infty}(\mathcal{C})}(E_X,E'_X) \simeq Map_{E_\infty}(X,Map_{\mathrm{Sp}}(S^0,E')) \times Map_{Alg_{E_{\infty}}(\mathrm{Sp})}(E,E')$
Probably not, then.
While I’m on the trail of likely unhelpful literature, was there something in what Sagave is doing in arXiv:1111.6731 on the second page about the bottom homotopy group being $\mathbb{Z}/2$, or is that kind of consideration the reason you brought up 2-periodicity in the first place?
Oh, now I see what you mean to say!
Right, but, you see, the problem is not in knowing what a parameterized morphism of $E_\infty$-algebras would be, parameterized over an infinity-groupoid. That’s trivial. Here the problem is to say what it would mean to parameterize smoothly over a smooth manifold.
Consider the simple case that the domain ring is the sphere spectrum. Then homomorphisms into any $A$ are just elements of $A$. And so a parameterization now is just a family of elements of $A$. But $A$ is just an $E_\infty$-algebra, there is a priori no smooth structure on it in any sense, so what would be a smooth family of elements? This needs some extra concept to make sense of.
Of course one standard answer is to just tensor the codomain with the algebra of smooth functions on the given parameter space. But if we tensor over an $\mathbb{R}$-algebra, then all the torsion in the $E_\infty$-algbra disappears, it collapses equivalently to just a dg-algebra, and so the whole point of invoking spectral geometry disappears.
So that’s why making sense of the category of objects “$\mathbb{R}^n \times Spec(A)$” needs some extra idea.
Regarding the $\mathbb{Z}/2$ that you point to: this is indeed related, but it’s not the answer to an open question.
Namely for $E$ an $E_\infty$-ring then $\pi_0(Pic(E))$ of its Picard $\infty$-groupoid is the group of “shift twists”, equivalently it classifies those $E$-modules which are just shifted copies $\Sigma^n E$ of $E$. Hence for a $p$-periodic $E$ then this is $\mathbb{Z}/p$. If we take $E$ to be even 2-periodic then this is $\mathbb{Z}/2$. corresponding to the fact already appealed to above, that we have the “$E$-lines” $E$ and $\Sigma E$.
Re #45, is there nothing to be gained from Jos’s smooth $E_{\infty}$ rings I mentioned in #29?
with the recent expansion of the chapters on categories and toposes and on smooth sets I am now streamlining the existing discussion here on supergeometry.
For instance, the proof of the super-differential-cohesive system of adjoint functors on SuperFormalSmoothSet (this prop.) used to try to prove it all from scratch. I have rewritten it now, just citing the proof of cohesion and of differential-cohesion from the earlier chapters, and instead expanding out the arguments for the super-aspect more.
1 to 48 of 48