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started working on superalgebra. But have to interrupt now.
okay, I have spent a bit of time with superalgebra.
in the first part I briefly discuss some standard classical notions (monoids in the symmetric monoidal category of super vector spaces)
in the second part I set up the topos of super sets, identify the even line object $\mathbb{K}$ in there, and dicuss – following Sachse’s account – how internal $\mathbb{K}$-modules capture super-vector spaces and how internal algebras over $\mathbb{K}$ capture super algebras.
added to superalgebras some basic Related notions (center, simple algebras, Azumay algebras, etc.) and some basic Properties (Brauer group, Picard 2-groupoid, etc.)
At superalgebra in the section In the topos over superpoints – The line object I have tried to make more explicit that the line object discussed there is just the standard real line under the restricted Yoneda embedding of supermanifolds into sheaves on superpoints.
added to superalgebra a lightning account of the point of view expressed in
(currently in the Idea-section there. This needs to be expanded on, but not time right now.)
Presumably the second column in the table after Proposition 3 should have $\mathbb{R}$ rather than $\mathbb{C}$.
What’s the relation with the etale cohomology groups here?
So this material is related to all that stuff which used to be talked about in TWF and on the Cafe about Clifford’s clock and higher clifford algebras.
I can’t even guess about what that ? could be as the ’meaning’ of $\pi_6(\mathbb{S}) = \mathbb{Z}_2$.
Presumably the second column in the table after Proposition 3 should have ℝ rather than ℂ.
Thanks for catching that! Fixed now.
Concerning higher Clifford algebra: I do suppose that Bartels-Dounglas-Henriques monoidal 3-category of fermionic nets hosts the $\mathbb{Z}_{24}$-piece.
Concerning the entries 6 and 7: maybe D4 and M5. Hisham Sati knows more.
Hi David,
just arrived in Chicago from Pittsburgh. Quick reply:
Yes, to see how the degrees work out you need to use relations like
$H^n_{et}(R, \mathbb{G}_m) = H^0_{et}(R, \mathbf{B}^n \mathbb{G}_m) \,.$I was about to add a comment along these lines to Brauer group, but the Lab was down.
[ ah, now it seems to be back… ]
What’s a ’metric graded line’?
In the middle row ’meaning’ of that table, the ’string’ and ’spin’ refer to their respective geometries. What does the column 0 ’degree’ mean in the same context? Is that some kind of geometry? And what might go in the final column for the full, untruncated theory?
A line equipped with both a grading - an integer - and a metric. The latter is so that the automorphisms are $O(1)$ rather than $GL(1)$.
The degree is the degree map (already represented by the winding number of a circle around a circle) from stable homotopy theory.
Thanks.
The degree is the degree map
I got that. But is there a way to motivate it as the zeroth entry, where the fourth entry is about
The differential geometry of manifolds with spin structure is called spin geometry. It studies spin group-principal bundles, spin-representations and the corresponding associated bundles over spin manifolds. Their spaces of sections notably support Dirac operators,
and the fifth entry is about
The analog of spin geometry as one passes from spinning particles to spinning strings. To some extent the step from the spin group to the string 2-group?
Re Urs @8
maybe D4 and M5,
is there any reason that of the worldvolume entries in this table, the D4 Khovanov homology and M5 6d (2,0)-superconformal QFT entries are ones where there’s some tie to arithmetic (from khovanov and the Jones Polynomila via the knot/number field analogy, and from 6d (2,0)-superconformal QFT to geometric Langlands to ordinary Langlands)?
@David C #13
well, let’s work backwards. The entry that is currently described as “boson/fermion super-degree”, I would in fact call that “orientation”, because that’s precisely the analogue of spin/string structures. But then we sort of run into a wall, namely that going string -> spin -> orientation, we hit the bottom of the Whitehead tower of $O(n)$. Perhaps we need to then consider something like a metric/inner product, since that is what reduces us from $GL(n)$ to $O(n)$. But why this should in some sense be quantised, I have no idea. Or perhaps we’re seeing dimension (or virtual dimension)?
Urs 5: Kapranov gave a similar talk at Alexander Rosenberg memorial conference a bit earlier, in March (schedule), at KSU, Manhattan, Kansas.
Hi,
had been out of action for a day. Needed to recover…
The whole story is still a bit vague. But:
the degree=1 entry is labeled “boson/fermion superdegree” because the looping of the free abelian 3-group on a single generator sits inside the category of $\mathbb{Z}_2$-graded vector spaces equipped with the symmetric braiding that introduces a minus sign whenever an odd-graded element is commuted with an odd graded element. Hence it’s the category of super-vector spaces and under this identification the first $\mathbb{Z}_2$ in that table is identified with the grading group of super vector spaces.
the degree=2 entry is labeled “spin” because its Postnikov piece is a $\mathbf{B}^2 \mathbb{Z}_2$ hence classifies $\mathbb{Z}_2$-extensions.
the degree=3 pice is labeled “string” because some kind of 24-grading is known to re-appear in different guises throughout superstrig theory. Probably the way to make this more precise is to identify it as some order in the 3-category of fermionic conformal nets.
But I think you shouldn’t read this table as a completed story, but as a hint towards what kind of story there ought to be. After sketching this hint in the first part of his talk, Kapranov goes on in the second half to point out where the symmetric monoidal 2-categories that involve the first three stages of the table have been hiding all along. More such un-covering discussion would be required for moving further up the table.
If we follow the table, shouldn’t we expect there to be analogues for $n = 1$ column, where lives
superalgebra as dual to supergeometry done in a cohesive $(\infty, 1)$-topos over superpoints?
Shouldn’t there be
spin algebra as dual to spin geometry done in a cohesive $(\infty, 1)$-topos over spin-points, capturing spin-cohesion,
and likewise for ’string’? Not just the equipping of ordinary manifolds with spin and string structures, but getting at the abelian n-groups that Kapranov mentions.
In fact, should one even talk of spin and string? Isn’t it more like
orientation : super :: spin : ?? :: string : ??,
where finding out about the ?? is doing the “un-covering” you mention, and finding the right symmetric monoidal n-categories? Could they have an associated cohesion?
Is that what
I think you shouldn’t read this table as a completed story, but as a hint towards what kind of story there ought to be
is pointing to? With stability at the end of the story?
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…)
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…! )
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?
Yes, exactly, for some reason we had that discussion brached off here. But the entry on superalgebra - Abstract idea also meanwhile reflects this.
Ah, ok. I guess maybe I can’t afford to ignore any topic with ’motivic’ in its name any more. )-:
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…
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.
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:
$??3Line \to bgl_1^\ast(tmf).$
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?
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.
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)
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.
The question (how does superalgebra conceptually relate to K-theory and other generalized cohomology) has today also come up on MO, here.
I see Kapranov has constructed a table in his New Spaces article
Table 1 (which expands, somewhat, a table from the online encyclopedia nLab)
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.
Yes, Mikhail Kapranov further amplified the issue in #31 after his talk, to various people who would listen.
A recent MO question on this.
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
Presumably there could be a ’Jordan’-like form of higher superalgebra.
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