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Supergeometry and more broadly supermathematics has been around for few decades. Since its introduction by physicists, there has been an some mathematical interest in them.

Although interesting in its own, a classical mathematician (say someone who is only interested in smooth manifolds, complex geometry, representation theory, or number theory etc...) can say that these new super-objects are merely formal analogues that are only of interest to physicists, but not much to them.

One way to motivate such a classical mathematician to be interested in super-mathematics is to exploit some applications of the super world in the non-super world, and here comes my question:

What are the known examples of advances*/accomplishments in non-super mathematics in which super-mathematics has been used in an essential way?

I am mostly interested in examples coming from Geometry (algebraic and differential), number theory (in the broad sense) and representation theory (e.g. Lie algebras or algebraic groups), but I am open to hear about other areas as well.

Here, "advances" is meant in the broadest sense. It can mean a new perspective on an existing result/field, a new proof of a known fact or a totally new discovery etc...

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    $\begingroup$ "there has been an immense mathematical interest in them" I would recommend to be more careful when saying how interesting something is. $\endgroup$ Mar 1 at 16:40
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    $\begingroup$ I hope the term "super-mathematics" does not find popularity. It strikes me as a gimmicky abuse of the English language. "Graded Lie algebra" is not bedazzled enough? $\endgroup$ Mar 1 at 17:17
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    $\begingroup$ Derived algebraic geometry and other derived geometries in characteristic $0$ are essentially the study of $Q$-manifolds and their stacky generalisations, but with functions having $\mathbb{N}_0$-gradings rather than $\mathbb{Z}/2$-gradings. $\endgroup$ Mar 1 at 18:17
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    $\begingroup$ @RyanBudney your concern about the use of "super-" is more than 40 years too late (see e.g. numdam.org/item/AIHPA_1977__27_4_335_0 or zbmath.org/?q=an:0331.58005 ) $\endgroup$ Mar 2 at 19:27
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    $\begingroup$ @RyanBudney I don't know if this was a joke you were obliquely referring to, but finite group theorists are familiar with "extra-special p-groups" en.wikipedia.org/wiki/Extra_special_group $\endgroup$
    – Yemon Choi
    Mar 4 at 2:25

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An example that was motivating for me is the paper Vanishing Theorems for constructible Sheaves on Abelian Varieties by Thomas Krämer and Rainer Weissauer which uses a description of a certain category of complexes of sheaves as a Tannakian category, in particular as a category of representations of a super-group, to prove vanishing theorems for the cohomology of perverse sheaves on abelian varieties, which was in particular the missing step to describe a suitable category of perverse sheaves as a neutral Tannakian category, i.e. as a category of representations of a group.

So here, at least by this method, one needed to pass through the super case to get to the ordinary case.

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In "Unimodality and Lie superalgebras" (citation below), Stanley uses the theory of Lie superalgebras to deduce some interesting combinatorial consequences about the unimodality of certain combinatorially defined sequences of numbers, and the Sperner property for certain finite posets.

Stanley, Richard P., Unimodality and Lie superalgebras, Stud. Appl. Math. 72, 263-281 (1985). ZBL0614.17004.

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In probability/statistical mechanics there are many interesting models that are best understood via equivalences with certain "superprobability" models that are easier to study.

The best known example is probably weakly self-avoiding walk: If we choose a path in the lattice of length $n$, weighted by $\exp(-\beta \cdot\#\{$self intersections}), what does such a path typically look like? When $\beta=\infty$ this is the usual (strictly) self-avoiding walk, which cannot visit the same vertex twice, whereas for finite $\beta$ this constraint is encouraged but not enforced.

An interesting conjectural feature of both this model and the usual strictly self-avoiding walk is that in sufficiently high dimensions the self-avoidance constraint is "only felt locally", and the self-avoiding walk looks much like an ordinary simple random walk on large scales (e.g. it still converges to Brownian motion under rescaling). The critical dimension for this simple ("mean-field") behaviour to kick in is believed to be $4$, and the model in four dimensions is expected to exhibit particularly delicate behaviour, with polylogarithmic corrections to the scaling of various interesting quantities compared to the mean-field predictions.

All the best rigorous results on this (mostly due to Bauerschmidt, Brydges, and Slade — see e.g. Critical two-point function of the 4-dimensional weakly self-avoiding walk) go through an equivalence between weakly self-avoiding walk and a certain supersymmetric field theory.

Recently a few other interesting combinatorial/probability models (including edge-weighted random spanning forests and the "vertex reinforced jump process") have been analysed via similar equivalences with supersymmetric field theories. A nice overview is given in Andrew Swan's PhD thesis "Superprobability on Graphs".

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A very interesting connection between supergeometry and 'classical mathematics' is the Stolz-Teichner programme. They have shown deep connections between moduli spaces of supersymmetric quantum field theories and cohomology theories. In $0$d, we can recover de Rham cohomology and in $1$d we recover K-theory. In $2$d it is conjectured that we recover TMF.

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Etingof-Gelaki's classification of semisimple and cosemisimple triangular Hopf algebras (arxiv, journal), goes via Deligne's theorem about the existence of super fiber functors.

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$\mathbb{Z}$-graded supersymmetry, more often (almost always?) referred to as graded commutativity appears in a number of places in mathematics. Here one replaces $\mathbb{Z}_2$ by $\mathbb{Z}$, so we end up with a a $\mathbb{Z}$-graded ring or algebra $\{A_k\}_{k\in\mathbb{Z}}$ satisfying the law $$ab=(-1)^{\deg(a)\deg(b)}ba.$$ Examples of graded commutative algebras include exterior algebras and cohomology rings in algebraic topology.

There are also differential graded commutative algebras (DGCAs), which satisfy the same graded commutative law while being additionally endowed with a differential satisfying further axioms, and have a number of applications as well. One famous example of them is the algebraic de Rham complex, which is in fact the free DGCA (see MSE1436212 for more details).

Recently graded commutative algebras have also been applied in the setting of homotopy theory in the on-going work of Hesselholt–Pstrągowski, started by Dirac geometry I: Commutative algebra and followed up by Dirac geometry II: Coherent cohomology, with the main example and motivation (as far as I understand) being that the $\pi_*$ of a commutative algebra in spectra is a $\mathbb{Z}$-graded commutative algebra (Example 2.2).


On a different direction, there's also Kapranov's Supergeometry in mathematics and physics, which discusses in section 3 a conjectural relationship between supersymmetry and gradings by the sphere spectrum. See also the following MO question:

MO236266, What is the relation between the sphere spectrum and supersymmetry?

Incidentally, one specific realisation of this conjectural relationship is that if you take the sphere spectrum $\mathbb{S}$ and consider $\mathbb{S}$-graded rings in the sense of Bunke–Nikolaus (which turn out to be the same thing as rings graded by the 1-truncation $\tau_{\leq1}\mathbb{S}$ of the sphere spectrum), you obtain $\mathbb{Z}$-graded supercommutative rings back as a special case, although $\mathbb{S}$-graded rings are more general, and include also ordinary commutative rings as another special case, for instance. See

MO403355, Generalising supercommutativity as a grading by the $1$-truncated sphere spectrum

(a question I asked a while ago) for more details.

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    $\begingroup$ $\mathbb{Z}$-graded supersymmetry also goes under the name of graded commutativity. While not so systematically developed earlier as in Hesselholt--Pstragowski, this notion has been considered for ages: see e.g. the graded-commutativity of the cup product on singular cohomology. $\endgroup$ Mar 16 at 20:32
  • $\begingroup$ @LennartMeier I should definitely have mentioned the other examples of graded commutativity showing up in practice. I've now edited the question to talk a bit about them. Thanks! $\endgroup$
    – Emily
    Mar 18 at 18:11
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Superalgebras have been used in various questions of algebra in a very striking way. To give some instances:

  1. Kemer's proof of the fact that, over a field of zero characteristic, every system of identities expressible in terms of the product in associative algebra follows from finitely many of them (Finite basis property of identities of associative algebras).

  2. Nilpotence results by various authors, see, for instance, Zelmanov's celebrated proof of global nilpotence of Engel Lie algebras (Engelian Lie algebras), or the survey of Vaughan-Lee indicating some other directions (Superalgebras and Dimensions of Algebras).

  3. Shestakov's elegant examples and counterexamples in different varieties of nonassociative algebras (Superalgebras and counterexamples).

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    $\begingroup$ @LSpice : thank you for adding names of articles to the references I posted. Please note that the spelling of Zelmanov's name with an apostrophe only arises in translations (and with the extra apostrophe in "'s" looks truly absurd). I changed this back to the way it was; please do not edit it further. $\endgroup$ Mar 2 at 20:21
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    $\begingroup$ Re, I put the apostrophe in Zel'manov's name because it was there in the linked article. (For that matter, I know it is not really an apostrophe (right?), but do not know how to typeset the true character.) I trust you to know proper Russian transliteration better than I do, but I trust the author better than either of us … although it is not necessarily the author's final say what appears on that byline. But, of course, I would not edit to restore a change that you removed. $\endgroup$
    – LSpice
    Mar 2 at 20:42
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    $\begingroup$ @LSpice you honestly assumed that the authors had a say about anything relating to the English translation of their work back in the USSR? :) In any case, a quick look at Zelmanov's homepage should be convincing enough. $\endgroup$ Mar 2 at 20:58
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    $\begingroup$ Re, good point. To be clear, I was not meaning to argue the point, only to explain, but, as you say, Zelmanov's homepage is surely as authoritative a source for how he spells his own name as I could wish. $\endgroup$
    – LSpice
    Mar 2 at 22:30
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I think this is not really answering the question you asked, but I think it would be remiss not to mention the impact of Witten’s mathematical work which is motivated by supersymmetry, even if the supersymmetric aspect is not always present in the mathematical formulation (ie it is mathematics inspired by supersymmetry). He reproved the positive mass theorem based on ideas of supersymmetry, reformulated Morse theory using supersymmetry which led to his formulation of a Topological Quantum Field Theory, introduced the Seiberg-Witten equations based on ideas of supersymmetry and string theory, interpreted Langlands duality in terms of superconformal gauge theories, as well as many other mathematical physics ideas that may not yet have found a rigorous mathematical formulation.

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