Split supermanifold - Knowlpedia

Definition

Let $M$ be a smooth manifold and let $E\to M$ be a real vector bundle of rank $q$ . The split supermanifold associated to $E$ , often denoted $\Pi E$ , is

\Pi E=\left(M,\mathcal O_{\Pi E}\right), \qquad \mathcal O_{\Pi E}(U) =\Gamma\!\left(U,\Lambda E^*|_U\right)

for every open set $U\subseteq M$ . It has dimension $\dim M|q$ , and its nilpotent ideal $\mathcal J$ satisfies the sheaf isomorphism

\mathcal J/\mathcal J^2\cong\mathcal E^*,

where $\mathcal E^*$ is the sheaf of smooth sections of $E^*$ .

The notation $\Pi E$ indicates that fiber-linear functions have odd parity; it does not change the underlying topological space, which remains $M$ .

What a splitting chooses

A splitting of a supermanifold $X$ is an isomorphism

\mathcal O_X\cong \Lambda_{\mathcal O_{X_{\mathrm{red}}}}\! \left(\mathcal J_X/\mathcal J_X^2\right)

compatible with reduction. It identifies all higher nilpotent terms with exterior powers of the degree-one part. Such an isomorphism contains more information than the underlying reduced manifold and odd vector bundle.

The Batchelor theorem supplies such a splitting in the finite-dimensional smooth real category, but supplies no canonical or functorial choice.

References

M. Batchelor, “The structure of supermanifolds,” Transactions of the American Mathematical Society 253, 1979, 329–338. Article.
J. Monterde and O. A. Sánchez-Valenzuela, “Existence and uniqueness of solutions to superdifferential equations,” Journal of Geometry and Physics 10(4), 1993, 315–343. Article90021-T). Relevant: split smooth models and parity reversal.