nLab supermanifold




Manifolds and cobordisms



A supermanifold is a space locally modeled on Cartesian spaces and superpoints.

There are different approaches to the definition and theory of supermanifolds in the literature. The definition

is popular. The definition

has been argued to have advantages, see also the references at super ∞-groupoid.

As locally representable sheaves on super Cartesian spaces

See at geometry of physics – supergeometry the section Supermanifolds.

As locally ringed spaces

We discuss a description of supermanifolds that goes back to (BerezinLeites).



A supermanifold XX of dimension p|qp|q is a ringed space (|X|,O X)(|X|, O_X) where

A morphism of supermanifolds is a homomorphism of ringed spaces (…).

Forgetting the graded part by projecting out the nilpotent ideal in O XO_X (i.e. applying the bosonic modality) yields the underlying ordinary smooth manifold X redX_{red}.

One just writes C (X)C^\infty(X) for the super algebra O X(X)O_X(X) of global sections.

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



For EXE \to X a smooth finite-rank vector bundle the manifold XX equipped with the Grassmann algebra over C (X)C^\infty(X) of the sections of the dual bundle

O X(U):=Γ( (E *)) O_X(U) := \Gamma (\wedge^\bullet(E^*))

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


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

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

for the corresponding supermanifold.



(Batchelor’s theorem)

Every supermanifold is isomorphic to one of the form ΠE\Pi E where EE is an ordinary smooth vector bundle.


Nevertherless, the category of supermanifolds is far from being equivalent to that of vector bundles: 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:

  • There is a natural bijection

    SDiff(X,Y)SAlgebras(C (Y),C (X)), SDiff(X,Y) \simeq SAlgebras(C^\infty(Y), C^\infty(X)),

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

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

    SDiff(X, p|q)(C (X) ev××C (X) ev) ptimes×(C (X) odd××C (X) odd) qtimes SDiff(X, \mathbb{R}^{p|q}) \simeq \underbrace{ (C^\infty(X)^{ev} \times \cdots \times C^\infty(X)^{ev})}_{p\; times} \times \underbrace{ (C^\infty(X)^{odd} \times \cdots \times C^\infty(X)^{odd})}_{q\; times}

The first statement is a direct extension of the classical fact that smooth manifolds embed into formal duals of R-algebras.

As manifolds modeled on Grassman algebras

We discuss a description of supermanifolds that goes back to (DeWitt 92) and (Rogers).


As manifolds over the base topos on superpoints

Let SuperPointSuperPoint be the category of superpoints. And Sh(SuperPoint)=PSh(SuperPoint)Sh(SuperPoint) = PSh(SuperPoint) its presheaf topos.

We discuss a definition of supermanifolds that characterizes them, roughly, as manifolds over this base topos. See (Sachse) and the references at super ∞-groupoid.

See also this post at Theoretical Atlas.




SuperSet:=Sh(SuperPoint) SuperSet := Sh(SuperPoint)

be the sheaf topos over superpoints. Let

Ring(SuperSet) \mathbb{R} \in Ring(SuperSet)

be the canonical continuum real line under the restricted Yoneda embedding of supermanifolds and equipped with its canonical internal algebra structure, hence by prop. the presheaf of algebras which sends a Grassmann algebra to its even subalgebra, as discussed at superalgebra.


A superdomain is an open subfunctor (…) of a locally convex 𝕂\mathbb{K}-module.

This appears as (Sachse, def. 4.6).

We now want to describe supermanifolds as manifolds in SuperSetSuperSet modeled on superdomains.

Write SmoothMfd for the category of ordinary smooth manifolds.


A supermanifold is a functor X:SuperPoint opSmoothMfdX : SuperPoint^{op} \to SmoothMfd equipped with an equivalence class of supersmooth atlases.

A morphism of supermanifolds is a natural transformation f:XXf : X \to X', such that for each pair of charts u:UXu : U \to X and u:UXu' : U' \to X' the pullback

U× XU p U p 1 u U u X f X \array{ U \times_{X'} U' &&\stackrel{p'}{\to}&& U' \\ {}^{\mathllap{p_1}}\downarrow & & && \downarrow^{\mathrlap{u'}} \\ U &\stackrel{u}{\to}& X &\stackrel{f}{\to}& X' }

can be equipped with the structture of a Banach superdomain such that p 1p_1 and p 2p_2 are supersmooth (…)

This appears as (Sachse, def. 4.13, 4.14).



The categories of supermanifolds defined as locally ringed spaces, def. and as manifolds over superpoints, def. are equivalent.

This appears as (Sachse, theorem 5.1). See section 5.2 there for a discussion of the relation to the DeWitt-definition.


Via functorial geometry

Discussion from the point of view of functorial geometry:

As locally ringed spaces

  • Felix 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

A more general variant of this in the spirit of locally algebra-ed toposes is in

  • Alexander Alldridge, A convenient category of supermanifolds (arXiv:1109.3161)

As manifolds over superpoints

The observation that the study of super-structures in mathematics is usefully regarded as taking place over the base topos on the site of super points has been made around 1984 in

and in

  • V. Molotkov., Infinite-dimensional 2 k\mathbb{Z}_2^k-supermanifolds , ICTP preprints, IC/84/183, 1984.

A summary/review is in the appendix of

A review with more emphasis on the relevant category theory/topos theory is in

The site of formal duals not just to Grassmann algebras but to all super-infinitesimally thickened points is discussed in (Konechny-Schwarz) above and also in

As manifolds modelled on Grassmann algebras

  • Bryce DeWitt, Supermanifolds, Cambridge Monographs on Mathematical Physics, 1984

  • Alice Rogers, Supermanifolds: Theory and Applications, World Scientific, (2007)

Alice Rogers claims, in Chapter 1, that the smooth-manifold-of-(infinite-dimensional)-Grassmann-algebras approach (the “concrete approach”) is identical to the sheaf-of-ringed-spaces approach (the “algebro-geometric” approach) and that this equivalence is shown in Chapter 8. DeWitt seems unsure of this, but is writing more than 20 years earlier, before the ringed-space approach has been fully developed.


Discussion with an eye towards supergravity is in

Discussion with an eye on integration over supermanifolds is in

Global properties are discussed in

  • Louis Crane, Jeffrey M. Rabin, Global properties of supermanifolds, Comm. Math. Phys. Volume 100, Number 1 (1985), 141-160. (Euclid)

See also:

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 algebras rather than on (the superspace and) supermanifolds. They should therefore rather be listed under entry supersymmetry.

See also pdf

Last revised on February 28, 2023 at 20:55:01. See the history of this page for a list of all contributions to it.