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Usefulness of the functor of points approach to supermanifolds to supersymmetric calculations in physics?

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In the Notes on Supersymmetry of Deligne and Morgan it is claimed at the beginning of chapters 2.8-2.9  about the functor of points approach to supermanifolds on page 28 that this is closest to the way physicists make computations (in supersymmetric theories). 

But to be honest I have never ever encountered any functors of points in physics texts about supersymmetry so far ...

So can somebody explain to me why the functor of points approach to supermanifolds is indeed the most useful point of view for calculations in supersymmetric physics and at best also give an example such that I can see how it actually works?

asked Jun 1 in Mathematics by Dilaton (4,295 points) [ revision history ]
edited Jun 5 by Arnold Neumaier

1 Answer

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The functor of points approach (see around remark 2.3 in geometry of physics -- supergeometry) says that we understand a supermanifold \(X\) (or any other richer or more general kind of super-space) by remembering (in particular)

1. for each superpoint \(\mathbb{R}^{0\vert q}\)the set of maps  \(Hom(\mathbb{R}^{0|q}, X)\) from \(\mathbb{R}^{0|q}\) to \(X\)

2. for each map of superpoints \(f : \mathbb{R}^{0|q} \to \mathbb{R}^{0|q'}\)  the corresponding restriction map  \(f^\ast : Hom(\mathbb{R}^{0|q'},X) \to Hom(\mathbb{R}^{0|q},X)\) given by precomposing any \(\mathbb{R}^{0|q} \to X\)  with the given \(f\).

Now such maps of supermanifolds \(\mathbb{R}^{0|q} \to X\) are dually homomorphisms of their super-algebras of functions

\(C^\infty(X) \to C^\infty(\mathbb{R}^{0\vert q})\).

But the algebra of super-functions on the superpoint R^0|q, that is just the Grassmann algebra \(\wedge^\bullet \langle \theta_1, \cdots, \theta_q\rangle\) on q odd generators theta.

Hence such an algebra homomorphism is schematically of the form

\(f_0 \mapsto f_0(x) + \theta_1 \theta_2 f'_0(x) + \cdots\)

\(f_1 \mapsto \theta_1 f_1(x) + \theta_1 \theta_2 \theta_3 f'_1(x) + \cdots\)

where \(f_0\) is of even degree in \(C^\infty(X)\) and \(f_1\) of odd degree.

In other words, this is just the expansion of super-fields in terms of auxiliary Grassmann variables. This is indeed effectively the only way that supergeometry is presented in physics texts.

You see, the point is really to give a supply of Grassmann coordinates. In some texts on supergravity, there is a comment at the beginning saying something like "we fix once and for all an infinite-dimensional Grassmann algebra and assume that we may draw elements theta form it as need be". But there are pitfalls to this approach via "one single fixed infinite Grassmann algebra". These problems are discussed in 

Christoph Sachse, "A Categorical Formulation of Superalgebra and Supergeometry" (arXiv:0802.4067)

The functor of points picture fixes this: instead of postulating one single infinite-Grassmann algebra, it says "your formulas need to make sense for Grassmann variables drawn from any finite Grassmann algebra" but (that's the functoriality) your formulas must be covariant under changing the chosen Grassmann algebra, i.e. under "Grassmann coordinate transformations".

In mathematical terms this: "work on all finite dimensional Grassmann algebras such that all your formulas are covariant under change of Grassmann coordinates", that just says that all the mathemaitcal objects you are dealing with are functors on the category of super-points.

answered Jun 8 by Urs Schreiber (5,735 points) [ revision history ]
edited Jun 8 by Urs Schreiber

Many thanks Urs!

Will study your reply (and the Sachse paper) in detail and come up with follow up questions if needed ...

I have added a more to-the-point discussion of what you probably like to see: the equivalence between the concept of "superfields" and the functor of points. This is now example 3.7 in the notes.

There would be more to say. I am busy otherwise, but if you give me feedback on which particulars you like to see expanded on further, I'll try to look into it.

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