nLab
supermanifold

Definition

A supermanifold $X$ of dimension $p\mid q$ is a ringed space $(\mid X\mid ,O_X)$ where

• the topological space $\mid X\mid$ is second countable space, Hausdorff space,

• $O_X$ is a sheaf of commutative super algebras that is locally on small enough open subsets $U\subset \mid X\mid$ isomorphic to one of the form $C^{\mathrm{\infty }}(U)\otimes {\wedge }^{•}{ℝ}^q$.

Forgetting the graded part by projecting out the nilpotent ideal in $O_X$ yields the underlying ordinary smooth manifold $X_{\mathrm{red}}$.

One just writes $C^{\mathrm{\infty }}(X)$ for the super algebra $O_X(X)$ of global sections.

With the obvious morphisms of ringed space this forms the category SDiff of supermanifolds.

Examples

Example For $E\to X$ a smooth finite-rank vector bundle the manifold $X$ equipped with the Grassmann algebra over $C^{\mathrm{\infty }}(X)$ of the sections of the dual bundle

$$O_X(U):=\Gamma ({\wedge }^{•}(E^{*}))$$O_X(U) := \Gamma (\wedge^\bullet(E^*))

is a supermanifold. This is usually denoted by $\Pi E$.

Example In particular, let ${ℝ}^{p+q}\to {ℝ}^p$ be the trivial rank $q$ vector bundle on ${ℝ}^p$ then one writes

$${ℝ}^{p\mid q}:=\Pi ({ℝ}^{p+q}\to {ℝ}^p)$$\mathbb{R}^{p|q} := \Pi (\mathbb{R}^{p+q} \to \mathbb{R}^p)

for the corresponding supermanifold.

Properties

Theorem (Batchelor’s theorem) Every supermanifold is isomorphic to one of the form $\Pi E$ where $E$ is an ordinary smooth vector bundle.

Warning Still, the category of supermanifolds is far from being equivalent to that of vector bundles because a morphism of vector bundles translates to a morphism of supermanifolds that is strictly homogeneous in degrees, while a general morphism of supermanifolds need not be of this form.

But we have the following useful characterization of morphisms of supermanifolds:

Theorem

• There is a natural bijection

  $$\mathrm{SDiff}(X,Y)\simeq \mathrm{SAlgebras}(C^{\mathrm{\infty }}(Y),C^{\mathrm{\infty }}(Y)),$$SDiff(X,Y) \simeq SAlgebras(C^\infty(Y), C^\infty(Y)),

  so the contravariant embedding of supermanifolds into superalgebra is a full and faithful functor.

• Composition with the standard coordinate functions on ${ℝ}^{p\mid q}$ yields an isomorphism

  $$\mathrm{SDiff}(X,{ℝ}^{p\mid q})\simeq (C^{\mathrm{\infty }}(X{)}^{\mathrm{ev}}\times \cdots p\mathrm{times}\cdots \times C^{\mathrm{\infty }}(X{)}^{\mathrm{ev}})\times (C^{\mathrm{\infty }}(X{)}^{\mathrm{odd}}\times \cdots q\mathrm{times}\cdots \times C^{\mathrm{\infty }}(X{)}^{\mathrm{odd}}).$$SDiff(X, \mathbb{R}^{p|q}) \simeq (C^\infty(X)^{ev} \times \cdots p times \cdots \times C^\infty(X)^{ev}) \times (C^\infty(X)^{odd} \times \cdots q times \cdots \times C^\infty(X)^{odd}).

Proof

The first statement is a direct extension of the classical fact that the contravariant embedding of ordinary smooth manifolds into algebras $X\mapsto C^{\mathrm{\infty }}(X)$ is a full and faithful functor.

Related entries

• NQ-supermanifold

• integration over supermanifolds

• ringed site

• super vector bundle

• superconnection

• superdifferential form

Some references

• F. A. Berezin, D. A. Leĭtes, Supermanifolds, (Russian) Dokl. Akad. Nauk SSSR 224 (1975), no. 3, 505–508; English transl.: Soviet Math. Dokl. 16 (1975), no. 5, 1218–1222 (1976).

• I. L. Buchbinder, S. M. Kuzenko, Ideas and methods of supersymmetry and supergravity; or A walk through superspace

• Bryce de Witt, Supermanifolds, Cambridge Monographs on Mathematical Physics, 1984, 1992

• Yu. I. Manin, Topics in noncommutative geometry, Princeton Univ. Press 1991.

• P. Deligne, P. Etingof, D.S. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison and E. Witten, eds. Quantum fields and strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)

• V. S. Varadarajan, Supersymmetry for mathematicians: an introduction, AMS and Courant Institute, 2004.

There are many books in physics on supersymmetry (most well known is by Wess and Barger from 1992), but they emphasise more on the supersymmetry algebra?s rather than on (the superspace and) supermanifolds. They should therefore rather be listed under entry supersymmetry. One should also note that there are two different definitions of a supermanifold which are not equivalent (some examples intersect but not all); they are sometimes connected and even named according to the main proponents of the two approaches: Leites (via sheaves) and de Witt (via supernumber?s).