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Presumably we can take stable objects in the world of super smooth $\infty$ groupoids. Can we call them ‘super-spectra’? And then these generate generalized super-cohomologies.
We did once mention ’group super-cohomology theory’ over here. We didn’t seem to resolve there if it amounted to what we’d expect in the relevant $\infty$-topos.
Can anything of the story of the differential cohomology diagram be told passing up through the modalities to the fermionic ones? I mean not just as a cohesive $\infty$-topos, but fracturing according to two of the modalities in the top square of oppositions.
Presumably we can take stable objects in the world of super smooth $\infty$ groupoids. Can we call them ‘super-spectra’? And then these generate generalized super-cohomolo
That would actually match various conventions. But maybe “supergeometric spectra” might be more descriptive. Or maybe “stable supergeometric homotopy types”, or the like.
We did once mention ’group super-cohomology theory’ over here. We didn’t seem to resolve there if it amounted to what we’d expect in the relevant $\infty$-topos.
When I go to the article that you point me to there, I experience a very low ratio of gain over effort to unravel the notation. If we had some interesting question for which cohomology with coefficients in genuine supergroups would be the answer, that would motivate me to think about it more.
Can anything of the story of the differential cohomology diagram be told passing up through the modalities to the fermionic ones? I mean not just as a cohesive $\infty$-topos, but fracturing according to two of the modalities in the top square of oppositions.
This I was thinking about, too. First of all, it follows that the fracturing simply exists, so we have that for all stable supergeometric homotopy types $X$ a homotopy Cartesian diagram
$\array{ && \e(X) \\ & \nearrow & & \searrow \\ X && && \stackrel{\rightrightarrows}{\e(X)} \\ & \searrow && \nearrow \\ && \stackrel{\rightrightarrows}{X} }$exhibiting each such $X$ as the fibered direct sum
$X \simeq \stackrel{\rightrightarrows}{X}\underset{\stackrel{\rightrightarrows}{\e(X)}}{\oplus} \e(X)$of its even part $\stackrel{\rightrightarrows}{X}$ with its fermionic part, the negative of its bosonic part $\e \coloneqq \overline{\rightsquigarrow}$.
The other square expresses $X$ as a tensor product of its bosonic part $\stackrel{\rightsquigarrow}{X}$ with the looping of the negative of the even part.
These statements seem satisfactory. But I am still looking for an interesting application.
Spectra are like abelian groups, so superspectra are like super abelian groups: they are Z/2-graded spectra equipped with a braided monoidal product, whose braiding adds a sign (i.e., takes the additive inverse in a spectrum) when exchanging two odd degree components.
Do any interesting new phenomena appear in supergeometric cohomology? Presumably there’s a difference between ’super’ and ’super smooth’, as in super formal smooth infinity-groupoid.
I see here in the case of superschemes,
…a suitable generalization of crystalline cohomology to superschemes…agrees with the usual crystalline cohomology of the underlying even scheme under some hypothesis.
So take a generalized cohomology like K-theory. Is there then a superdifferential refinement, like there is a differential refinement differential K-theory? And if so, does it give anything more than the differential version on the underlying even space?
On a not totally unrelated topic, is there a reason why $\Omega_{cl}^{n + 1}(X)$ appears in the $\flat_{dR}$ position in the differential cohomology diagram for ordinary cohomology, and that $0|1-TFT_n (X) \cong \Omega^n_{cl}(X)$ in Supersymmetric field theories and generalized cohomology (p. 7). Should we be surprised that supergeometry has appeared already?
Is there a reason that Stolz-Teichner program doesn’t feature much in dcct? It sounds like there should be bridges.
We remark that passing to concordance classes forgets ‘geometric information’ while retaining ‘homotopical information’.
Isn’t that a little shape-modality-like?
For what it’s worth, it looks like taking concordance classes $0|1-TFT^n(X) \to 0|1-TFT^n[X]$ is just applying the shape modality to $\flat_{dR}$.
David,
good to see you being enthusiastic about these questions. They are very interesting questions. In as far as you are hoping for more substantial replies from me, I have to warn you that I am pretty absorbed with some other tasks and unlikely to spend much thinking on these particular questions here right now. Not because I don’t find them interesting, but because I am finite.
Regarding #4: I still don’t have a good example of a cohomology theory where the coefficients is genuinely super-graded, though there will be loads of examples and of course one may readily make up a few, but I don’t think I have anything particular to say right now.
Regarding the issue of the underlying bosonic manifold/scheme: so in solid cohesion then for $X$ a $V$-manifold (hence a supermanifold in general) it follows from the Aufhebung of the elastic level that the underlying bosonic space $\stackrel{\rightsquigarrow}{X}$ is a $\stackrel{\rightsquigarrow}{V}$-manifold (prop. 5.4.4 in the dcct pdf). It follows by adjunction that cohomology with coefficients in $Rh$-modal objects will depend only on the bosonic manifold underlying the supermanifold.
Regarding #5: first of all the trivial but important remark should be highlighted that by progression of the modalities, every differential refinement is also a superdifferential refinement in that it exists in the supergeometric infinity-topos, too. Beyond that, your question here would need to say first which super-version of K-theory you’d have in mind that is not just ordinary (smooth or not) K-theory embedded as a bosonic object in the supergeometric infinity-topos.
Regarding #6: yes, concordance is encoded by shape, this is a brief remark in section 5.2.4 of dcct.
I think the focus on 0-dimensional field theory (and be it $0\vert q$-dimensional) is too restrictive to deduce much from. Notice that the field theory being 0-dimensional means that “space” is “(-1)-dimensional”, hence this case is a drastic oversimplification of quantum mechanics, which itself is already a drastic simplification of field theory proper.
…but because I am finite.
A central theme of Heidegger.
Regarding the first part of #6, I see David Ben-Zvi answers a similar question by André Henriques, and sees it ultimately in terms of a Koszul duality.
Bringing Urs’s comment back over here from another thread, since it refers to my #6 above,
I may have misunderstood your aim there. It seemed to me that you were wondering there about the role of “$(0\vert 1)$-Euclidean field theory” in this business, to which I replied that I find it a red herring in the present context.
That the $\mathbb{Z}/2\mathbb{Z}$-graded algebra of differential forms on a (super-)manifold $X$ is $C^\infty([\mathbb{R}^{0\vert 1},X])$ is a standard fact of supergeometry, and the observation that the natural $Aut(\mathbb{R}^{0\vert 1})$-action on this gives the refinement to $\mathbb{Z}$-grading as well as the differential originates in Kontsevich 97, and was long amplified by Severa, see also at odd line – the automorphism supergroup.
In the article on gorms and worms (arXiv:math/0307303) they generalize this by replacing $\mathbb{R}^{0|1}$ by $\mathbb{R}^{0|q}$.
My thought was something like
Isn’t it surprising that a neat way of capturing differential forms on a real manifold relies on extending the setting to supermanifolds, to allow maps from $\mathbb{R}^{0|1}$, automorphisms of this, etc.?
Maybe this is similar to my surprise (here) that Morse theory could be thought about in terms of supersymmetry, when it feels like a phenomenon about real manifolds. But I guess there’s a history of these kinds of observations, like Hadamard’s “The shortest path between two truths in the real domain passes through the complex plane.”
It is certainly something to take note of. That was the content of that MO question that you linked to in #9. (By the way, the first perspective asked for in that question is the one that my student Hermann Stel wrote out in arXiv:1310.7407, following Anders Kock’s observations on “combinatorial differential forms”).
What Ben-Zvi recalled in that MO discussion is that one way of embedding the statement $\Omega^\bullet(X) \simeq C^\infty([\mathbb{R}^{0|1},X])$ into a more general perspective is to see it as at least analogous to the Hochschild-Kostant-Rosenberg theorem and yet more generally as an instance of the story of how Hochschild cohomology relates to differential forms.
I am not sure if this really answers the question, or rather puts it into a bigger perspective. I know that Mathieu Anel has been thinking much about the relation between the synthetic and the superistic differential forms and maybe he’ll have more to say.
In any case, what is certainly true is that as we progress from cohesion – where non-closed differential forms only appear via a choice of Hodge filtration of the de Rham coefficient objects – to solid cohesion, then it is possible to require that the previously chosen Hodge filtrations are compatible with the concept of differential forms as seen by the supergeometry, hence it is possible to sharpen the axioms and leave less to human choice, more to the gods. However, making the connection formal seems not to be particularly elegant, as it involves going to function algebras etc. That’s why I am hesitant of phrasing it as an additional axiom.
Re Urs #8,
I think the focus on 0-dimensional field theory (and be it $0\vert q$-dimensional) is too restrictive to deduce much from.
So the move to $1\vert 2$-dimensional is in the right direction presumably, in Perturbative N=2 supersymmetric quantum mechanics and L-theory with complex coefficients:
We construct L-theory with complex coefficients from the geometry of 1|2-dimensional perturbative mechanics…
Our long-term goal is to leverage an understanding of the 1|2-dimensional case to gain traction on the more complicated 2|2- and 3|2-dimensional theories. This is in analogy with Stolz and Teichner’s approach to a geometric model for elliptic cohomology: they are motivated in large part by the relation between 1|1-dimensional EFTs and Dirac operators on Riemannian spin manifolds [HST10, ST11]. In the footsteps of G. Segal [Seg88] they argue by analogy that 2|1-EFTs ought to capture structures related to Dirac operators on loop spaces.
Historically, this program started (here) with the observation that there is a relation between (1,1)-dimensional Euclidean field theories and K-theory, essentially given by the fact that a suitable smooth super-representation of the 1d super-translation Lie supergroup is determined by a choice of Dirac-like operator $D$, via $(t,\theta) \mapsto \exp(-t D^2 + \theta D)$, which when regarded as a kind of Fredholm operator may be thought of as giving a point in the classifying space for topological K-theory.
The conjecture was that an analog of this argument in 2d should similarly be related not to K-theory, but to some kind of elliptic cohomology or tmf. The hint for this is nothing but Witten’s argument for the Witten index.
But when it turned out hard to prove this conjecture, the idea arose that maybe the whole logic of the conjecture should first be explored more in toy examples of lower dimensions, i.e. $d = 0$ and even $d = -1$, I think, as well as further variants in $d = 1$. Hence there is now exploration of variants of the story in 1d, such as what you cite, but also equivariant versions, etc. pp.
As far as I am aware (but I’d be happy to hear of progress, please let me know), to date there has not been progress on the 2d story that would genuinely go beyond what Witten already had in his original article.
Except of course in that the string orientation of tmf was constructed, which is an actual lift of Witten’s construction to tmf-cohomology. But this is part of another “program”.
But this is part of another “program”.
I see Dylan Wilson has recently put up a research statement which positions his work in relation to that program.
If one can lift the Witten genus, can one ’lift’ its physical description, “the large volume limit of the partition function of the superstring”?
Exactly, that is the question: if the string orientation of tmf is what the Witten genus really wants to be, then where in there is the 2d field theory that Witten considered?
One observation in this direction is on the bottom of p. 12 in arXiv:1402.7041.
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