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 $\eta$:
Do they have an “intuitive” or geometric interpretation or are they just formalism that in the end give us anticommuting fields?
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)
