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Physicists have invented supersymmetry in which they use new variables, mathematically corresponding to Graßmann numbers (elements of some exterior algebra) and physically to "fermionic degrees of freedom". Other notions have been developped as well, like super vector spaces, super algebras, super Lie algebras, etc (that is, $\mathbb{Z}/2\mathbb{Z}$-graded objects). These structures are "quite easy" to find in the realm of Mathematics. But then the theory of supermanifolds has been invented (there are different versions of it), and I am wondering what interests mathematicians have to expand this theory beyond the needs of Physics. Well, for me, it is quite difficult to think of some aspects of differential geometry without refering to corresponding aspects of Physics (general relativity, gauge theory, string theory, etc). So my question is:

Is there any motivation to introduce in Mathematics the notion of supermanifolds? Could somebody provide some striking example? Where the notion of supermanifold appears naturally in Mathematics?

For instance, graded manifolds can be seen as a unifying concept for Lie and Courant algebroids (but I am asking for something a little bit different).


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@Benjamin: it's not just that. Apparently one can recover the de Rham complex of a smooth manifold $M$ as the space of smooth functions on $\text{hom}(\mathbb{R}^{0|1}, M)$ (the internal hom) or something like that. Again I don't really know how this works. – Qiaochu Yuan Jul 13 '11 at 19:09
You're right, it's a 'ß' -- I provided one :-) – joriki Jul 13 '11 at 19:10
I took it from this Wikipedia reference:ßmann#cite_note-0 (note 1) Now I found this:… (note 2). – joriki Jul 13 '11 at 19:29
@Alon Amit: I am sorry for this usage of the word. I mean I would like to have a nice example that clearly shows the necessity to introduce supermanifolds in Mathematics. For instance, joriki explained here… a reason why we need the abstract notion of manifold. I am searching some kind of example that will enlighten the use of supermanifolds in Mathematics. By the way, I do not speak English very well, sorry. Ah, there is a typo, thanks, I did not see it! – Benjamin Jul 13 '11 at 19:47
@Benjamin: I think it is a little strange to speak of "necessity" in mathematics. Really nothing is necessary. Any given tool we use could probably in principle be avoided by using some other tool. (I guess a counterexample to this would be an important result that has only essentially one known proof, but I think such things are rare and probably young if they exist.) – Qiaochu Yuan Jul 14 '11 at 4:53

In the early 1980s, Witten, Alvarez-Gaume, and Getzler gave short proofs of (a special case of) the Atiyah-Singer index theorem using supermanifold tools. This has applications in topology beyond physics.

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I have read the book of Berline-Getzler-Vergne and the proof by Witten too. But they do not use supermanifolds directly, but they use other concepts from supergeometry. But the core of these proofs are based on heat kernel techniques. – Benjamin Jul 14 '11 at 7:28

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