Supermanifold - Knowlpedia

Definition

A smooth real supermanifold of dimension $p|q$ is a superspace $X=(|X|,\mathcal O_X)$ that is locally isomorphic to a superdomain $U^{p|q}$ . Its morphisms are morphisms of locally superringed spaces. These objects and morphisms form the category $\mathbf{SMan}_{\mathbb R}^{\mathrm{sm}}$ .

Let $\mathcal J_X\subseteq\mathcal O_X$ be the sheaf of nilpotent elements, equivalently in this smooth model the ideal generated locally by odd functions. Then

\mathcal O_X/\mathcal J_X\cong C^\infty_{X_{\mathrm{red}}},

where $X_{\mathrm{red}}$ is an ordinary smooth $p$ -manifold with underlying topological space $|X|$ . The odd dimension $q$ records the rank of $\mathcal J_X/\mathcal J_X^2$ .

Local versus global structure

Local coordinates are written $(x^1,\ldots,x^p;\theta^1,\ldots,\theta^q)$ , with even $x^i$ and odd $\theta^\alpha$ . Transition morphisms may mix the even coordinates with even nilpotent expressions and the odd coordinates with odd expressions. Consequently, being locally an exterior-algebra model does not itself choose a global splitting.

Ordinary smooth manifolds embed fully faithfully as supermanifolds of dimension $p|0$ . Complex-analytic and algebraic supermanifolds use different local function sheaves; unlike the smooth real case, they need not split.

References

D. A. Leites, “Introduction to the theory of supermanifolds,” Russian Mathematical Surveys 35(1), 1980, 1–64. Article.
C. Carmeli, L. Caston, and R. Fioresi, Mathematical Foundations of Supersymmetry, EMS, 2011. Publisher record. Relevant: Chapter 4.