Geometric calculations using Grassmann variables

Question

Physicists seem to get huge computational value by introducing Grassmann variables and Grassmann integration into differential geometric calculations.

See: http://en.wikipedia.org/wiki/Grassmann_number

Can someone here motivate these techniques mathematically, and include the simplest pure-math example where their use and value can be illustrated. I have thought they were invented for computing volumes of constrained moduli spaces, although physicists did not originally describe them this way. Do any mathematicians use them rigorously to actually solve problems?

Answer 1

Grassman variables are a neat and handy way to talk about exterior algebras and the geometric objects built on them, like forms and spinors. The physics notation in particular gets a lot of mileage out of Grassman calculations that look formally like Gaussian integrals and behave remarkably like them. The best intro to that, and the bet purely mathematical application I know, is Mathai and Quillen's paper where they get an explicit representation of the Thom form that is quite powerful. They can be a little intimidating because physicists speak about them in language grating to mathematicians and tend to use them in the vicinity of highly nonrigorous thinking, but they themselves can be worked with mathematically without difficulty.

Answer 2

I don't know nearly enough about the subject to give you a particularly intelligent answer, but can tell you that the formalism of Grassmann variables is sometimes used in heat-kernel-based treatments of the Atiyah-Singer index theorem and its generalizations--see for instance Berline-Getzler-Vergne's book "Heat kernels and Dirac Operators" and the work of Bismut which it's partly based on. Grassmann variables can be given rigorous meaning as coordinates on a supermanifold, and in particular there's no problem in rigorously defining the Berezin integral.

Answer 3

They're used for instance in the construction of the Hilbert scheme, which in turn is used to construct all sorts of moduli spaces (e.g. moduli of genus g curves). Dan describes this a bit in his answer here; a good reference for this application (including the Hilbert scheme construction) is Harris and Morrison's Moduli of Curves.