Supermanifold

Definition

Let $p, q \ge 0$ be integers. Let $\Lambda_q = \Lambda[\theta_1, \dots, \theta_q]$ be the real exterior algebra on $q$ generators. It is graded by parity: a product of an even number of generators is even, and a product of an odd number is odd. For an open set $U \subseteq \R^p$ , let $\mathcal{O}_{p|q}(U) = C^\infty(U) \otimes_\R \Lambda_q$ . The pair $\R^{p|q} = (\R^p, \mathcal{O}_{p|q})$ is the superdomain of dimension $p|q$ .

A smooth supermanifold of dimension $p|q$ is a pair $\mathcal{M} = (\abs{\mathcal{M}}, \mathcal{O}_\mathcal{M})$ with these properties.

$\abs{\mathcal{M}}$ is a Hausdorff, second-countable topological space.
$\mathcal{O}_\mathcal{M}$ is a sheaf of supercommutative superalgebras on $\abs{\mathcal{M}}$ ; supercommutative means $ab = (-1)^{\abs{a}\abs{b}} ba$ for homogeneous $a, b$ of parities $\abs{a}, \abs{b}$ .
Every point of $\abs{\mathcal{M}}$ has an open neighbourhood $V$ with the following property. As a space with a sheaf of superalgebras, $(V, \mathcal{O}_\mathcal{M}|_V)$ is isomorphic to $(U, \mathcal{O}_{p|q}|_U)$ for some open $U \subseteq \R^p$ .

A morphism of supermanifolds is a morphism of locally ringed spaces whose map on sheaves preserves parity.

Remarks

Let $\mathcal{J} \subseteq \mathcal{O}_\mathcal{M}$ be the ideal generated by odd elements. Then $(\abs{\mathcal{M}}, \mathcal{O}_\mathcal{M}/\mathcal{J})$ is an ordinary smooth manifold of dimension $p$ , the underlying manifold. Let $E$ be a vector bundle of rank $q$ over a manifold $M$ , and let $\Gamma(\Lambda E^*)$ be the sheaf of sections of its exterior algebra bundle. By Batchelor's theorem, every smooth supermanifold is isomorphic, not canonically, to some $(M, \Gamma(\Lambda E^*))$ .

Example: $\R^{p|q}$ itself, and more generally $(M, \Gamma(\Lambda E^*))$ for a vector bundle $E \to M$ of rank $q$ . Over a trivialising open set $U$ of $E$ with $U$ a chart domain, $\Gamma(\Lambda E^*)|_U \cong C^\infty(U) \otimes \Lambda_q$ , which is clause 3.

Non-example: $(\R, \mathcal{O})$ with $\mathcal{O}(U) = C^\infty(U)[\varepsilon]/(\varepsilon^2)$ and $\varepsilon$ declared even. Every element is even, so the sheaf is not locally isomorphic to $C^\infty \otimes \Lambda_q$ with $q \ge 1$ , which has odd elements. It is not locally isomorphic to $C^\infty$ , the case $q = 0$ , either, since $C^\infty$ has no nonzero nilpotents. Clause 3 fails.

Tag	`TQNG7T2RQ6`
URI	`m2://TQNG7T2RQ6`
Kind	definition
Writing	W0 stub
MSC	58A50
Defines	Supermanifold
Uses	Topological space, Sheaf, Superalgebra, Exterior algebra, Locally ringed space, Smooth manifold, Vector bundle

Supermanifold

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here