Superdomain - Knowlpedia

Definition

Let $U\subseteq\mathbb R^p$ be open. The superdomain $U^{p|q}$ is the superspace with underlying space $U$ and structure sheaf

\mathcal O_{U^{p|q}}(W) =C^\infty(W)\otimes_{\mathbb R} \Lambda\big((\mathbb R^q)^*\big), \qquad W\subseteq U\ \text{open}.

Its dimension is $p|q$ . The ordinary coordinate functions $x^1,\ldots,x^p$ are even, while a basis $\theta^1,\ldots,\theta^q$ of $(\mathbb R^q)^*$ gives odd coordinates. Every section has a unique finite expansion

f(x,\theta)=\sum_I f_I(x)\theta^I

with smooth coefficient functions $f_I$ .

The ideal generated by the odd sections is nilpotent. Quotienting the structure sheaf by this ideal gives $C^\infty_U$ , so the reduced space is the ordinary smooth domain $U$ .

Morphisms in coordinates

A morphism of superdomains is a morphism of locally superringed spaces, not merely a continuous or smooth map of the underlying open sets. Locally it is determined by the pullbacks of target coordinates: even target coordinates pull back to even sections and odd target coordinates to odd sections, subject to the condition that the reduced even coordinate map lands in the target domain.

For $q=0$ , $U^{p|0}$ is the superspace associated to the ordinary smooth domain $U$ .

References

F. A. Berezin, Introduction to Superanalysis, D. Reidel, 1987. Relevant: Chapter 2.
V. S. Varadarajan, Supersymmetry for Mathematicians: An Introduction, Courant Lecture Notes 11, American Mathematical Society, 2004. Publisher record. Relevant: Section 4.2.