# Super-manifold

A generalization of the concept of a manifold, on which the functions take values in a commutative superalgebra. The structure of a super-manifold on a differentiable manifold $M$ with structure sheaf ${\mathcal O} _ {M}$ is defined by a sheaf ${\mathcal O} _ {\mathcal M}$ of commutative superalgebras over the sheaf ${\mathcal O} _ {M}$, whereby any point $p \in M$ possesses a neighbourhood $U$ such that the ringed space $( U, {\mathcal O} _ {\mathcal M} \mid _ {U} )$ is isomorphic to $( U, ( {\mathcal O} _ {M} \mid _ {U} ) \otimes \Lambda ( \mathbf R ^ {m} ))$, where $\Lambda ( \mathbf R ^ {m} )$ is the exterior algebra with $m$ odd generators. Analytic super-manifolds are defined in the same way. The differentiable (or analytic) super-manifolds form a category whose morphisms are the morphisms of ringed spaces that are even on the structure sheaves. The pair ( $\mathop{\rm dim} M, m$) is called the dimension of the super-manifold. A super-manifold of the form $( U, {\mathcal O} _ {U} \otimes \Lambda ( \mathbf R ^ {m} ))$, where $( U, {\mathcal O} _ {U} )$ is an open submanifold in $\mathbf R ^ {n}$, is called a super-domain of dimension $( n, m)$. Every super-manifold is locally isomorphic to a super-domain.

If $E$ is a vector bundle over $M$, then the sheaf of sections $L _ {\Lambda E }$ of the bundle $\Lambda E$ defines the structure of a super-manifold on $M$. Every differentiable super-manifold is isomorphic to a super-manifold of the form $( M, L _ {\Lambda E } )$; in the complex analytic case this is not true. At the same time there are more morphisms in the category of super-manifolds than in the category of vector bundles.

A super-manifold ${\mathcal M}$ can be defined by a functor $\underline {\mathcal M}$ from the category of commutative superalgebras into the category of sets; this functor assigns to the superalgebra $C$ the set ${\mathcal M} ( C) = \mathop{\rm Mor} ( \mathop{\rm Spec} C, {\mathcal M} )$, where $\mathop{\rm Spec} C$ is the set of prime ideals in $C$, endowed with the natural sheaf of superalgebras (see Representable functor).

The basic concepts of analysis on differentiable manifolds are also applied to super-manifolds.

The concept of a super-manifold also arose in theoretical physics; it enables one to join particles with Bose–Einstein statistics and Fermi–Dirac statistics into single multiplets, and also enables one to join the internal and dynamic symmetries of gauge theories in a single super-group.

#### References

 [1] F.A. Berezin, "Introduction to superanalysis" , Reidel (1987) (Translated from Russian) [2] D.A. Leites (ed.) , Seminar on supermanifolds , Kluwer (1990)