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I have created an entry spectral symmetric algebra with some basics, and with pointers to Strickland-Turner’s Hopf ring spectra and Charles Rezk’s power operations.
In particular I have added amplification that even the case that comes out fairly trivial in ordinary algebra, namely $Sym_R R$ is interesting here in stable homotopy theory, and similarly $Sym_R (\Sigma^n R)$.
I am wondering about the following:
In view of the discussion at spectral super scheme, then for $R$ an even periodic ring spectrum, the superpoint over $R$ has to be
$R^{0 \vert 1} \;=\; Spec(Sym_R \Sigma R) \simeq Spec\left( R \wedge \left( \underset{n \in \mathbb{N}}{\coprod} B\Sigma(n)^{\mathbb{R}^n} \right)_+ \right) \,.$This of course is just the base change/extension of scalars under Spec of the “absolute superpoint”
$\mathbb{S}^{0\vert 1} \simeq Spec(Sym_{\mathbb{S}} (\Sigma \mathbb{S}))$(which might deserve this notation even though the sphere spectrum is of course not even periodic).
This looks like a plausible answer to the quest that David C. and myself were on in another thread, to find a plausible candidate in spectral geometry of the ordinary superpoint $\mathbb{R}^{0 \vert 1}$, regarded as the base of the brane bouquet.
Unfortunately, rather than follow you on this, I have a stack of philosophy of medicine essays to mark.
But, were one to look to start bouquet building, is there a natural choice of $R$? In the ordinary case, you chose $\mathbb{R}$ over $\mathbb{C}$. Does that suggest $KO$ sooner than $KU$ (at K-theory spectrum)? Is there an even-periodic version of MO, as MP is to MU? Do you just sum over even suspensions?
Or did I mean the $MR$ of MR cohomology theory, and its possible periodic version?
Unfortunately, rather than follow you on this
Luckily I am progressing on geological time scales with this, so you will have an easy time catching up later.
is there a natural choice of $R$
It seems the only natural choice is $R = \mathbb{S}$. That of course is not 2-periodic. So the story might be this: down in “absolute geometry”, we start with $Spec(Sym_{\mathbb{S}}\Sigma \mathbb{S})$. Trying to understand this will make us want make us look at it locally, first by passing to the cover $Spec(MU) \to Spec(\mathbb{S})$ and then further localize to Morava E-theory $Spec(E) \to Spec(MU)$ or to some other complex oriented cohomology theory. The “restriction” of $Spec(Sym_{\mathbb{S} \Sigma \mathbb{S}})$ to this $Spec(E)$ is the spectral superscheme $Spec(Sym_E \Sigma E)$, our actual specral superpoint.
Something like this.
But all this is vain speculation. To make progress we need to find some actual spectral analog of a branch in the bouquet. I bet once we see this, the rest will fall into place.
The thing I understand now (whence the edits in spectral symmetric algebra) and which I did not appreciate before, is that $Sym_E \Sigma E$ is not a boring analog of $Sym_{\mathbb{R}} \mathbb{R}[1]$, but an interesting one.
You may remember that I had suggested $Spec(Sym_E \Sigma E)$ in some earlier discussion here as the correct spectral superpoint, but then I said that it looks like this just yields the theory as over $\mathbb{R}$, with the $E$ coefficients just running along.
What I didn’t realize before is that there is this coefficient of the Thom spaces of the universal bundles over the classifying spaces of the symmetric groups involved (here) coming from the fact that the permutaion action (that makes the graded-symmetric algebra $Sym$) is much more interesting on smash powers of a module spectrum then on tensor powers of a vector space.
It is easy to see some grand speculations growing out of this: If we take our ground ring $E$ to be MU, then
$MU^{0 \vert 1} = Spec(Sym_{MU} \Sigma MU) = Spec( MU \wedge (\underset{n \in \mathbb{N}}{\coprod} B\Sigma(n)^{\tau_n} )_+ )$Now since $B \Sigma_n \simeq Embed(\{1,\cdots, n\}, \mathbb{R}^\infty)/\Sigma(n)$ and using that $\pi_0(MU \wedge X_+)$ is cobordism classes of closed even dimensional manifolds (with stable almost complex structure) in $X$, we see that
$\pi_0 \left( MU \wedge (\underset{n \in \mathbb{N}}{\coprod} B\Sigma(n)^{\tau_n} )_+\right)$is something like a Fock space of brane worldvolumes.
To make progress we need to find some actual spectral analog of a branch in the bouquet.
Is that largely a matter of choosing the right spectral (group scheme?) cohomology?
To make progress we need to find some actual spectral analog of a branch in the bouquet.
Is that largely a matter of choosing the right spectral (group scheme?) cohomology?
It’s about computing a “maximal invariant higher central extension” of one of these spectral group schemes and checking wether that proceeds at all in analogy with the rational bouquet story.
So if we start with the spectral superpoint $R^{0 \vert 1} = Spec( Sym_E \Sigma E)$, first of all we need to regard it somehow a spectral group scheme, hence regard $Sym_E \Sigma E$ as a Hopf ring spectrum. That ought to work as in Strickland-Turner, with the coproduct being on shifted generators $\theta \mapsto \theta_1 + \theta_2$.
Second then we need to determine the maximal central extension of this group scheme, and then see what that has to do with the super-translation super Lie algebra $\mathbb{R}^{1 \vert 1}$ (which is the maximal central extension of $\mathbb{R}^{0 \vert 1}$).
Or maybe even better than such explicit computations would be a general argument that if we start with a spectral super group scheme which somehow corresponds to some super $L_\infty$-algebra, that then its maximal higher invariant central extensions as super group schemes corresponds somehow to the maximal higher central extension of that super $L_\infty$-algebra. Hence an argument that the construction of “super equivariant Whitehead towers” (or whatever they are to be called) is somehow preserved by rationalization (or by something like this).
Won’t spectral exterior algebras need to feature?
Sure, that for the superpoint is an example: $Sym_R (\Sigma R)$. This is the spectral analog of the exterior algebra on one generator, which is $Sym_k (k[1])$. More generally, the spectral analog of the Grassmann algebra on $n$ generators $Sym_k (k^n[1]) = Sym_k (k^{\oplus^n}[1]) = Sym_k \left((k[1])^{\oplus n}\right)$ is $Sym_R \left((\Sigma R)^{\oplus^n}\right)$, where the direct sum of spectra is their wedge sum $\oplus = \vee$.
For some reason I’d never seen a free graded-commutative algebra over $k$ written like that.
Does a periodic version of $\mathbb{S}$ ever appear?
$\mathbb{S} P = \vee_{n \in \mathbb{Z}} \Sigma^{2 n} \mathbb{S}$I don’t think there’s any way to give it an $E_\infty$-ring structure.
According to Adeel in another thread (here), there is discussion of spectral symmetric algebras in Jacob Lurie’s opus. That’s what I’d expect, but all I found so far is a brief remark below prop. 2.20 in “Spectral Schemes” (here).
It’s natural to expect that the Hopf cosemiring spectrum structure on symmetric algebras that Strickland-Tuner 97 find on $Sym_{\mathbb{S}} \mathbb{S}$ lifts to the corresponding $E_\infty$-structure and generalizes to other spectral symmetric algebras, reflecting the additive and multiplicativ structure that one expects to see on any kind of affine line.
Is this discussed anywhere?
The spectral affine line over the sphere spectrum is denoted $\mathbf{I}$ in Adeel’s Brave new motivic homotopy theory I (3.2.1).
Thanks for the pointer. I have included this and some discussion at spectral symmetric algebra.
Presumably we’re in linear HoTT territory, the !-modality instantiating the bosonic Fock space construction. I wonder if there is a possible modal treatment of the fermionic Fock space.
There’s an unbound $n$ on the left, but it would appear not in the expressions on the right.
There should be $Spec$ continuing on the right? So also at spectral super-scheme.
There’s an unbound $n$ on the left, but it would appear not in the expressions on the right.
Thanks, fixed now. It needs to be like so
$\begin{aligned} Sym_R(\Sigma^n R) & \simeq R \wedge \left( \underset{k \in \mathbb{N}}{\coprod} S^k/\Sigma(k) \right)_+ \\ & \simeq R \wedge \left( \underset{k \in \mathbb{N}}{\coprod} (B \Sigma(k))^{n \tau_k} \right) \end{aligned} \,,$e.g. slide 4 of Charles’ pdf.
There should be $Spec$ continuing on the right? So also at spectral super-scheme.
Yes, thanks. Also fixed now.
There must be now an $n$ missing from the upper right term. Should that be $S^{n k}/\Sigma(k)$?
Yes. Good that you are paying closer attention than I am! I am fixing it in the entry.
Since it came up again on g+, what is the difficulty in doing what Urs asks for in #7? What does it mean to find the cohomology of $R^{0 \vert 1} = Spec( Sym_E \Sigma E)$ as a spectral group scheme? Is such a cohomology representable? If so, what plays the role of $\mathbf{B}^1 \mathbb{R}$ as it appears here?
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