# Grassmann numbers & supermanifolds

I'm asking this question because I'm currently trying to learn about Super Symmetry but I'm having trouble understanding the concept of super-space and super-manifold.

I read that in super-spaces you have 2 Grassmann numbers for each coordinate.

1. Could anyone explain to me what these Grassmann numbers are?

2. And then, what's the difference between a regular manifold and a supermanifold?

• are you asking about what is a Grassmann number in general, or about their use in SUSY? for the latter read Weinberg's QFT, Vol. III. It's the best source for SUSY IMHO (not that I have read many more books about it anyway) – AccidentalFourierTransform Nov 29 '16 at 18:31
• Yes I'm asking in general. I probably should mention I'm an engineering undergrad and therefore have know previous exposure to these concepts. Of course if you can tell me what they're used for in SUSY would be great, but I'll be more than pleased enough with a comprehensive description of the numbers. – S V Nov 29 '16 at 19:11
• Related MO.SE question: mathoverflow.net/q/100675/13917 – Qmechanic Nov 29 '16 at 21:49

1. Supernumbers and their weirdness are e.g. discussed in this Phys.SE post.

2. The next logical step is to learn the notion of $(n|m)$ super vector spaces, which have $n$ Grassmann-even and $m$ Grassmann-odd dimensions.

3. Moreover, we will assume that the reader are familiar the definition of an ordinary $n$-dimensional $C^{\infty}$-manifold, which is covered by an atlas of local coordinate charts $U\subseteq \mathbb{R}^n$.

4. Finally let's discuss $(n|m)$ supermanifolds, which is technically a sheaf of $(n|m)$ super vector spaces. Heuristically and oversimplified, a supermanifold is a generalization of the notion of a manifold (3) where the local coordinate charts now are subsets of $(n|m)$ super vector spaces.

References:

1. Bryce DeWitt, Supermanifolds, Cambridge Univ. Press, 1992.

2. Pierre Deligne and John W. Morgan, Notes on Supersymmetry (following Joseph Bernstein). In Quantum Fields and Strings: A Course for Mathematicians, Vol. 1, American Mathematical Society (1999) 41–97.

3. V.S. Varadarajan, Supersymmetry for Mathematicians: An Introduction, Courant Lecture Notes 11, 2004.

• I think there is a typo in point (4) since an $n$ dimensional manifold is not the same as a space equipped with a sheaf of rank $n$ vector spaces. – Sean Pohorence Dec 13 '16 at 2:28

There is a very detailed discussion of supernumbers, supermanifolds, and other superstuff in DeWitt "The Global Approach to Quantum Field Theory". If you are more mathematically-minded, look at this book on mathematics of QFT and String Theory.

Here are detailed online lecture notes that introduce Grassmann coordinates, supergeoemtry, supermanifolds etc.: