Superspace - Knowlpedia

Definition

In smooth real supergeometry, a superspace is a pair

(|X|,\mathcal O_X)

consisting of a topological space $|X|$ and a sheaf $\mathcal O_X=\mathcal O_{X,\bar 0}\oplus\mathcal O_{X,\bar 1}$ of supercommutative real superalgebras whose stalks are local superalgebras. Thus each stalk has a unique maximal homogeneous ideal, and its residue field is $\mathbb R$ .

A morphism $f:X\to Y$ is a continuous map $|f|:|X|\to |Y|$ together with a parity-preserving local morphism of sheaves

f^\sharp:\mathcal O_Y\longrightarrow |f|_*\mathcal O_X.

The direction of $f^\sharp$ is contravariant: functions on the target pull back to functions on the source.

Scope of the term

This is the locally superringed-space usage. A supermanifold is a superspace satisfying a finite-dimensional local model condition. In physics, “superspace” often means an affine supermanifold carrying a supersymmetry action, and especially super-Minkowski space. That narrower usage should not be substituted for the general definition.

Complex-analytic, algebraic, DeWitt, and Rogers superspaces have related but different structure sheaves or test categories. They are not silently identified with the smooth real Berezin–Leites/Kostant model used here.

References

V. S. Varadarajan, Supersymmetry for Mathematicians: An Introduction, Courant Lecture Notes 11, American Mathematical Society, 2004. Publisher record. Relevant: Chapters 3–4.
C. Carmeli, L. Caston, and R. Fioresi, Mathematical Foundations of Supersymmetry, EMS, 2011. Publisher record. Relevant: Chapters 3–4.