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 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 morphism of ringed spaces (…).

Forgetting the graded part by projecting out the nilpotent ideal in O XO_X 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 the contravariant embedding of ordinary smooth manifolds into algebras XC (X)X \mapsto C^\infty(X) is a full and faithful functor.

As manifolds modeled on Grassman algebras

We discuss a desription 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. 2 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. 1 and as manifolds over superpoints, def. 4 are equivalent.

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



A brief survey is in

Discussion with an eye on integration over supermanifolds is in

As locally ringed spaces

  • 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

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

  • Anatoly Konechny and Albert Schwarz,

    On (kl|q)(k \oplus l|q)-dimensional supermanifolds, in: Julius Wess, V. Akulov (eds.) Supersymmetry and Quantum Field Theory (D. Volkov memorial volume) Springer-Verlag, 1998 , Lecture Notes in Physics, 509 (arXiv:hep-th/9706003)

    Theory of (kl|q)(k \oplus l|q)-dimensional supermanifolds Sel. math., New ser. 6 (2000) 471 - 486

  • Albert Schwarz, I- Shapiro, Supergeometry and Arithmetic Geometry (arXiv:hep-th/0605119)

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

  • L. Balduzzi, C. Carmeli, R. Fioresi, The local functors of points of Supermanifolds (arXiv:0908.1872)

As manifolds modeld on Grassmann algebras

  • Bryce DeWitt, Supermanifolds, Cambridge Monographs on Mathematical Physics, 1992

  • Alice Rogers


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.

Revised on October 9, 2013 11:32:20 by Urs Schreiber (