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Motivated by SUSY, mathematicians began to study $\mathbb{Z}_2$-graded mathematics, or super mathematics. In particular, one can formulate supergeometry just following Grothendieck style (even) algebraic geometry.

Besides its applications in physics theory, I want to know if there are math motivations to study supergeometry. I've seen some papers and monographs on supergeometry, but most of them are trying to follow classical even geometry: they are just trying to establish super analogues of theorems in even geometry.

I want to ask:

  • Are there any results in even geometry that are no longer true in the super case?
  • Are there any new phenomena that only happen in the super (purely odd or mixed) case?
  • Are there any results in geometry that were first discovered in the super setting?
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  • $\begingroup$ This doesn't exactly answer your question but I recommend section 3 of Kapranov's paper arxiv.org/abs/1512.07042 $\endgroup$
    – DamienC
    Commented Aug 26, 2023 at 9:49
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    $\begingroup$ The most classical answer is probably the sign rule for differential forms, discovered well before SUSY in physics. Given an even topological space $X$, the cohomology $H^*(X)$ is a superring and $Spec H^*(X)$ is an affine superscheme! $\endgroup$
    – Enclitic Sarcool
    Commented Aug 26, 2023 at 14:00
  • $\begingroup$ @EncliticSarcool Do you know any literature concerning the geometry of Spec H^*(X)? $\endgroup$
    – Estwald
    Commented Aug 27, 2023 at 14:13
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    $\begingroup$ Here's a comment concerning Spec of a graded commutative ring. If $x$ has odd degree then $x^2=-x^2$ so $2x$ is nilpotent. For similar reasons, for all $y$, $(2x)y$ and $y(2x)$ are also nilpotent, so $2x$ is in the nil radical. So $x$ is congruent to $-x$ modulo the nil radical. The upshot is that a graded commutative ring modulo its nil radical is strictly commutative. Since every prime ideal contains the nil radical, as far as Spec is concerned a graded commutative ring might as well be strictly commutative. Of course, the structure sheaf then gives away the game here... $\endgroup$
    – Dave Benson
    Commented Aug 27, 2023 at 21:41
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    $\begingroup$ Regarding $Spec H^\ast(X)$ -- note that this ring is supercommutative in the $\mathbb Z$-graded sense, not just in the $\mathbb Z/2$-graded sense. Hesselholt and Pstragowski call $\mathbb Z$-graded mathematics with the Koszul sign rule "Dirac geometry". $\endgroup$
    – Tim Campion
    Commented Aug 29, 2023 at 15:24

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In my view, one motivation for the study of supergeometry (though historically backwards) is Deligne's theorem to the effect that any symmetric tensor (abelian) category "of moderate growth" over an algebraically closed field of characteristic zero fibres over super vector spaces. This implies, via the Tannakian perspective, that it is equivalent to a suitable category of representations of a (pro) affine supergroup scheme. For details, see his 2002 paper "Catégories Tensorielles". So in characteristic zero, for many purposes we only have ordinary symmetry and supersymmetry.

In prime characteristic, the situation is more complicated. There are finite symmetric tensor categories that do not fibre over super vector spaces. In Benson, Etingof and Ostrik, "New incompressible symmetric tensor categories in positive characteristic" (2023), an infinite number of incompressible ones are constructed in each prime characteristic. These are nested, and the union, sometimes denoted $\mathop{\sf Ver}_{p^\infty}$ to indicate the relation to Verlinde categories, may be a good candidate for a replacement for super vector spaces in possible generalisations of Deligne's theorem to prime characteristic. Some progress on this has recently been made in a paper of Coulembier, Etingof and Ostrik, "Incompressible tensor categories" (preprint, 2023).

The upshot of this is that in my opinion, the study in characteristic $p$ of group schemes over $\mathop{\sf Ver}_{p^\infty}$ is an important new area that is almost completely untouched at this point.

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    $\begingroup$ According to Johnson-Freyd, one can push Deligne's theorem a bit further to show that $sVect_{\mathbb C}$ is injective with respect to certain "embeddings" of tensor categories, so that $sVect_{\mathbb C}$ is the "algebraic closure" of a certain class of tensor categories. I suspect this implies in the usual way that $sVect_{\mathbb C}$ is characterized uniquely up to (nonunique) symmetric monoidal exact equivalence by these properties. There's also something about being the separable closure in positive characteristic. $\endgroup$
    – Tim Campion
    Commented Aug 29, 2023 at 15:36
  • $\begingroup$ Yes, I think in characteristic $p$ this is Victor Ostrik's point of view. The Verlinde categories we construct for prime powers are presumably not separable in this sense. We also compute their cohomology, at least up to inseparable isogeny, and we give a conjecture for the exact answer, that we've verified in a lot of small cases. The generating functions for the dimensions are related to the Minc partition function and Rogers-Ramanujan type identities. $\endgroup$
    – Dave Benson
    Commented Aug 29, 2023 at 16:33
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A computational motivation for the introduction of odd (Grassmann) variables arises in the context of random matrix theory, when one seeks to represent the determinant of a matrix (rather than its reciprocal) as a Gaussian integral:
For a vector $\chi$ of odd variables and a matrix $M$ one has $$\int d\chi d\bar{\chi}e^{-\bar{\chi}M\chi}=\det M.$$ (For even variables the right-hand-side would be $1/\det M$.)
Depending on whether one in is interested in the average of $\det M$ or of $1/\det M$, one would work with odd or even variables.

This simple observation is at the basis of Efetov's work on supersymmetry in disorder and chaos.

I should emphasise that none of this assumes that supersymmetry is a physical symmetry. It is purely introduced as a computational tool.

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Two examples in hand:

  1. Any flag manifold for a complex reductive Lie group is projective by Bruhat decomposition and $\mathscr{D}$-affine by Beilinson--Bernstein, which is no longer true even for a simple Lie supergroup, cf. [Penkov--Skornyakov, 1985].
  2. It's a classical result that the standard complex structure on a flag manifold allows no small deformation, which is not always the case for flag supermanifolds.
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