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I am planning to write various $n$Lab pages on super-structures. But I need to get a better overview of the literature and the approaches.
Let $SuperPoint = GrAlg^{op}$ be the category of $\mathbb{Z}_2$-graded Grassmann algebras. Then much of superalgebra etc takes place in the presheaf topos $Sh(SuperPoint) := [SuperPoint^{op}, Set]$.
This perspective goes by the name of somebody or somebody’s book, right? I forget, who is it?
As we add smooth structure, typically people pass to sheaves over the large site of supermanifolds. But much smaller sites should do.
Of course I am thinking of $CartSp_{super}$, the full subcategory of supermanifolds on super-Cartesian spces $\mathbb{R}^{p|q}$s.
But I have a question about something else: it seems not entirely unnatural to proceed hierarchically from the previous step: assume we agreed that $Sh(SuperPoint)$ is out new base topos that replaces $Sh(Point) = Sh(*) = Set$. Then it would make sense to say that the smooth version is the topos
$Sh(SmoothMfd, Sh(SuperPoint)) \to Sh(SuperPoint)$over $Sh(SuperPoint)$, hence
$\simeq Sh(SmoothMfd \times SuperPoint) \,.$Notice that the category of all supermanifolds is the “semidirect product of categories”
$SuperManifold = SmthMfd \ltimes SuperPoint$(in the sense used as in the literature referenced at Cahiers topos).
I am wondering about whether I should think of supergeometry as taking place over the direct product site or the semidirect product site. Is this considered anywhere in the literature?
Found the answer at least to my first question:
the person is Albert Schwarz. Apparently he first wrote about looking at the presheaf topos $Sh(SuperPoint)$ in
A. Schwarz. On the de notion of superspace. Teor. Mat. Fiz. 60 (1984), 37; English transl. Theoret. and Math. Phys. 60 (1984).
apparently independently of this, around the same time, the same proposal was promoted in
V. Molotkov., Infinite-dimensional ℤ 2 k-supermanifolds ICTP preprints, IC/84/183, 1984.
Comprehensive discussion is in Christoph Sachse’s thesis
apparently 1984 was the magic year for this idea. Also Alexander Voronov is credited for having come up with it then.
I am listing all the references that I have collected so far at super infinity-groupoid.
So I am happy, it seems that a critical mass of experts says (not quite in these words, but that’s what they mean to say): super-mathematics should be regarded as taking place relative to the cohesive $\infty$-topos of $\infty$-sheaves over superpoints.
That’s good, because it seems that the “rheonomy” constraint in D’Auria-Fre formulation of supergravity is exactly the (on-shell) $\infty$-Ehresmann condition on $\infty$-connections if we work in the cohesive smooth $\infty$-topos relative over that
$Smooth Super\infty Grpd := Sh_{\infty}(CartSp, Super \infty Grpd) \stackrel{\Gamma}{\to} Super \infty Grpd := Sh_{\infty}(SuperPoint)$Molotkov from 3. is the Bulgarian physicist who came up with the glutos idea. Rosenberg had a different idea about the spectra of superspaces: one should as open sets not only localizations in usual space but also kind of skew ones combining odd end even coordinates. The spectrum in his sense could be more interesting in comparison to usual presentations by manifold with a sheaf including also the Grassman numbers.
Rosenberg had a different idea about the spectra of superspaces: one should as open sets not only localizations in usual space but also kind of skew ones combining odd end even coordinates.
Is there a motivating application for this?
If I recall correctly, there are also competing variants even of the standard definition at least of super conformal surface. I forget the details, but I seem to recall hearing Rogers talk about different moduli spaces given by different main definitions of super-surfaces. Does anyone have the details readily available?
Coming back to the approach of working over the base topos on superpoints: the big issue that all the authors concentrate on is (in my words) how to fix the fact that the emebdding of $SuperManifold$ into $Sh(Manifold \times Superpoint)$ is, while faithful, not full.
But need it be? I have come to this question by looking at what D’Auria-Fré say about supergravity. There the point is that we want spacetime to be an ordinary manifold, but that the supersymmetry on the field content is most naturally expressed if spacetime were a supermanifold, for then supersymmetry is just translation in the odd directions. So they say: make spacetime a supermanifold but then impose “rheonomy” in order to ensure that the field content is effectively determined by its value on the reduced ordinary manifold.
If one looks at this, it seems all they really need is some oddd variables adjoined in a Cartesian-product fashion. More general supermanifolds are not actually necessary, and hence the site $SmthMfd \times SuperPoint$ seems to be well-adapted to the situation.
More concretely, it seems to me that what they are saying is most naturally formalized like this:
fix a super-$L_\infty$-algebra $\mathfrak{g}$ Then the $\infty$-groupoid of $\mathfrak{g}$-valued field configurations is the presheaf on $SmoothMfd$, with values in presheaves on Super points, with values in simplicial sets given by
$\left( U \mapsto \left( \mathbb{R}^{0|q} \mapsto \left( [k] \mapsto \left\{ \Omega^\bullet(U \times \mathbb{R}^{0|q} \times \Delta^k) \leftarrow W(\mathfrak{g}) : A \;|\; F_A satisfies some horizontality \right\} \right) \right) \right)$and where the nature of the horizontality condition is naturally understood by regarding this as a presheaf on ordinary manifolds with values in super $\infty$-groupoids.
The motivation is representation theory: the points of spectra are useful in study of representation theoretic questions. In supercase there is very little done so far, for example the story with Borel-Weil-Bott theorem is more complicated than the non-super case (supercase studied by Penkov, in 1980s). But eventually, what, supercommutativity is just one very special kinds of braided noncommutativity and one should use general methods, which work for braided commutative geometry in any braided monoidal category, a la in the worls of Shahn Majid, where the constructions in supergeometry come as special cases, anyons are some other special cases.
It may be that physicist in their current models put emphasis on some constructions which use only the split structure as you sharply observed, but this does not mean that the math of braided geometry should be limited to the features of particular setup supergravity may like.
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