Great to hear that, Urs, looking forward to it too.

So I suppose in principle the discussion in spectral super-scheme applies more generally starting from E-infinity geometry, in the sense that one can truncate the latter in several ways to get every geometry corresponding to the elements in the table of the homotopy groups of spheres, and somehow that in particular gives a tower of brane bouquets. Are there any particular objects we expect to emerge in these bouquets, say in the one that corresponds to $\pi_3=\mathbb{Z}_{24}$?

]]>Hi Alonso, I have just landed. Looking forward to chatting tomorrow.

I think the right answer is a slight variant of Kapranov’s suggestion, namely the definition that I recorded at

However, it’s still waiting to be applied in interesting ways (beyond Rezk’s work cited there).

(I was hoping that out of a spectral super-point would emerge a spectral form of the brane bouquet. But still don’t know.)

]]>What ever happened to this point by Kapranov? Is it clearer now how that would come about? Does one end up replacing the Super- in the site of FormalSuperCartesianSp with something more refined using S? Are there any interesting consequences of this?

]]>Presumably there could be a ’Jordan’-like form of higher superalgebra.

]]>Thanks, I hadn’t seen that . That’s a funny question. Regarding Kapranov’s insight, it is asking:

]]>Are there some references other than Kapranov

A recent MO question on this.

]]>Yes, Mikhail Kapranov further amplified the issue in #31 after his talk, to various people who would listen.

]]>Maybe that counting issue Urs is explaining to me in #31 came from my confusion with respect to this:

]]>One thing is worth noticing. Supergeometry, as understood by mathematicians, tackles only the first two columns of Table 1. A a similar-sounding concept (supersymmetry) used by physicists, dips into the third column as well: fermions are always wedded to spinors in virtue of the Spin-Statistics Theorem.

I see Kapranov has constructed a table in his New Spaces article

]]>Table 1 (which expands, somewhat, a table from the online encyclopedia nLab)

The question (how does superalgebra conceptually relate to K-theory and other generalized cohomology) has today also come up on MO, here.

]]>I’d need some more info on the homotopy type of $gl_1(tmf)$.

According to Hopkins here

The above square ends up giving us enough understanding of the homotopy type of $gl_1(tmf )$ [we refer the reader to [1] for the actual computations].

[1] is Ando, Hopkins and Rezk Multiplicative orientations of KO-theory and of the spectrum of topological modular forms. Can’t say I can see that actual computations easily.

]]>then isn’t it the 1-truncation of the sphere spectrum that we want?

There is a bit of repetition and ambiguity in that spectrum, regarding supersymmetry:

degree 0 and 1 together encode $\mathbb{Z}$-grading and the Koszul-sign rule on that;

degree 1 and 2 together encode $\mathbb{Z}_2$-grading (of super vector spaces, say) and the Koszul-sign rule on that;

degree 1 and 2 and 3 together match to the grading on (global sections of) super 2-vector spaces; (one Koszul sign rule on super 2-vector spaces, another one on their super 2-line automorphisms)

From what you say then, if superalgebra is about abelian 2-groups, then isn’t it the 1-truncation of the sphere spectrum that we want?

Why should geometric relevance stop at a certain truncation level? What is the next after super-$n$-lines? spin-$n$-lines then string $n$-lines?

Mike Hopkins probably gives enough information here (pp. 7-9) and here(p. 5) for $gl_1(tmf)$. I’ll see.

]]>Hi David,

concerning the counting: I might have a glitch in the text somewhere, then I’ll fix it, but generally a homotopy n-type with grouplike $E_\infty$-structure (“abelian $\infty$-group structure”) is an $(n+1)$-group. Compare: a 0-type (= set) with group structure is a 1-group.

Concerning the other issue: probably there are more twists than in super-$n$-lines. I need to see. But I’d suspect they give the geometrically relevant ones.

I’d need some more info on the homotopy type of $gl_1(tmf)$. I know that there is a $\mathbb{Z}$ in degree 3. What else is in lower degree?

]]>And,

To see how ordinary superalgebra arises this way, consider the case of $KU$. While there is the canonical map of abelian ∞-groups

$2Line \simeq B^2 U(1) \to B gl_1(KU)$from line 2-bundles, this does not hit all homotopy groups on the right. But refining to supergeometry and replacing the left with super line 2-bundles (see there) we do hit everything:

$s2Line \to bgl_1^\ast(KU).$

you’re calling ’$s2Line$’ what you call ’$2\mathbf{sLine}$’ at super line 2-bundle?

Were we to look to take it up a notch, how would it go?

]]>To see how ordinary ??algebra arises this way, consider the case of $tmf$. While there is the canonical map of abelian ∞-groups

$3Line \simeq B^3 U(1) \to B gl_1(tmf)$from line 3-bundles, this does not hit all homotopy groups on the right. But refining to (??)geometry and replacing the left with (??) line 3-bundles we do hit everything:

$??3Line \to bgl_1^\ast(tmf).$

On super algebra above the table, are the numbers quite right? It talks about 2-truncation of the free abelian $\infty$-group on one generator, so the free abelian 2-group, and then also of homotopy for $n = 1, 2$. However, in the table under $n = 2$, it has abelian 3-group.

]]>My fault. I kept thinking of adding an alert here, but then never got around to. Thanks for pushing me.

On the other hand, at the beginning when Zoran brought it up there I kept thinking that this discussion is off-topic over at “Motivic quantization”. But the pleasant surprise now to me is that in fact it is right on topic. Something nice is happening here…

]]>Ah, ok. I guess maybe I can’t afford to ignore any topic with ’motivic’ in its name any more. )-:

]]>Yes, exactly, for some reason we had that discussion brached off here. But the entry on superalgebra - Abstract idea also meanwhile reflects this.

]]>Where does ’under Spec S’ come from? Wouldn’t an affine scheme under Spec S be a ring with a *ring map* to S, instead of a monoid map from its multiplicative monoid (or the units therein) to the additive monoid of S, which is what I would think of a grading as being?

Hey David,

yes, that’s what I mean.

I still need to absorb this myself. I wholeheartedly agree that this is a very interesting line to further pursue. I’ll try to come back to it soon. (If only the day had more hours…! )

]]>So, Morava’s notes? There’s a sequel here. Not sure I’m much the wiser however.

]]>These are all good and right questions and suggestions, yes. I don’t really know any detailed answer.

But as Thomas Nikolaus rightly amplified to me in Halifax, Kapranov’s suggestion comes down to saying that we should study $E_\infty$-algebraic geometry *under* $Spec \mathbb{S}$, where $\mathbb{S}$ is the sphere spectrum.

Interested readers are supposed to ask Google for the way “To the left of the sphere spectrum.”

(The dark side of the moon…)

]]>