I was reading on super geometry and how it is used to model fermions and supersymmetry in classical field theory.

In texts like [1] or [2] the authors introduced auxiliary Grassmann odd variables to get anticommuting fields.

Take for example a super field $$ \phi : \mathbb{R}^{m|1} \rightarrow \mathbb{R}^{n|0} $$ mapping a super spacetime to some configuration space. It can be written as $$ \phi = x + \theta \, \psi $$ where $x$ should be a bosonic field, $\psi$ a fermionic one and $\theta$ is the odd cordinate on $\mathbb{R}^{m|1}$.

In [1] it is argued that in this formulation $\psi$ is not an anticommuting (odd) quantity. To fix this we actually need to take $\phi$ as a map $$ \phi : \mathbb{R}^{m|1} \times \mathbb{R}^{0|L}\rightarrow \mathbb{R}^{n|0} \;. $$ We denote with $\eta$ the odd coordinates of $\mathbb{R}^{0|L}$ and they serve as auxiliary Grassmann variables. $x$ is now even in those and $\psi$ odd. Therefore $\psi$ is now indeed an anticommuting field.

I have two questions regarding those auxiliary variables $\eta$:

  1. Do they have an “intuitive” or geometric interpretation or are they just formalism that in the end give us anticommuting fields?

  2. How do we get rid of them when calculating observables? I know that we Berezin integrate over the super spacetime coordinates $\theta$ when calculating the action for example. Do we do the same for the $\eta$?

[1] Hélein, Frédéric. “An Introduction to Supermanifolds and Supersymmetry”, p. 15. (He introduces the term “super manifold with flesh” for adding auxiliary variables)

[2] end of section 3 of nlab on Supergeometry. (Here the Dirac field serves as an example)


1 Answer 1

  1. The odd "flesh/fuzz" $\eta$ can be viewed as a realization of the functor of points. In a nutshell it is introduced to be able to consider Grassmann-odd indeterminates, viz. souls.

  2. Grassmann-odd and Grassmann-even souls are removed via integration. See e.g. this Phys.SE post.

  • $\begingroup$ Thanks Qmechanic. In your answer in the linked Phys.SE post (I can't comment there), where is the definition of the integral over a supernumber coming from. It's not the Berezin integral, is it? $\endgroup$ Commented Oct 28, 2018 at 15:41
  • $\begingroup$ Only in the Grassmann-odd case. $\endgroup$
    – Qmechanic
    Commented Oct 28, 2018 at 15:47
  • $\begingroup$ Does this mean that one just drops the Grassmann-even terms of the soul? And for the odd terms we Berezing integrate over the top monomial? $\endgroup$ Commented Oct 28, 2018 at 16:35
  • $\begingroup$ What is it in the Grassmann-even case then? $\endgroup$ Commented Dec 15, 2018 at 12:15
  • $\begingroup$ ${}$ See the link. $\endgroup$
    – Qmechanic
    Commented Dec 15, 2018 at 12:32

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