nLab
supermanifold

Idea

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

• As locally ringed spaces

is popular. The definition

• As manifolds over superpoints

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).

Definition

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 $C^{\mathrm{\infty }}(U)\otimes {\wedge }^{•}{ℝ}^q$.

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

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.

As manifolds modeled on Grassman algebras

We discuss a desription of supermanifolds that goes back to (deWitt) and (Rogers).

(…)

As manifolds over the base topos on superpoints

Let $\mathrm{SuperPoint}$ be the category of superpoints. And $\mathrm{Sh}(\mathrm{SuperPoint})=\mathrm{PSh}(\mathrm{SuperPoint})$ 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.

Definition

Definition

Let

$$\mathrm{SuperSet}:=\mathrm{Sh}(\mathrm{SuperPoint})$$SuperSet := Sh(SuperPoint)

be the sheaf topos over superpoints. Let

$$𝕂\in \mathrm{SuperSet}$$\mathbb{K} \in SuperSet

be the canomnical commutative ring object, as discussed at superalgebra.

Definition

A superdomain is an open subfunctor (…) of a locally convex $𝕂$-module.

This appears as (Sachse, def. 4.6).

We now want to describe supermanifolds as manifolds in $\mathrm{SuperSet}$ modeled on superdomains.

Write SmoothMfd for the category of ordinary smooth manifolds.

Definition

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

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

$$\begin{array}{ccccc}U{\times }_{X\prime }U\prime & & \stackrel{p\prime }{\to }& & U\prime \\ {}^{p_1}\downarrow & & & & {\downarrow }^{u\prime }\\ U& \stackrel{u}{\to }& X& \stackrel{f}{\to }& X\prime \end{array}$$\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 $p_1$ and $p_2$ are supersmooth (…)

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

Properties

Proposition

The categories of supermanifolds defined as locally ringed spaces, def. 1 and as manifolds over superpoints, def. 2 are equivalent.

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

Related concepts

• integration over supermanifolds

• super vector bundle

• superconnection

• superdifferential form

• NQ-supermanifold

References

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 proposal that the study of super-structures in mathematics should be regarded as taking place over the base topos on the site of super points has been made around 1984 in

• Albert Schwarz, On the definition of superspace Teoret. Mat. Fiz., 1984, Volume 60, Number 1, Pages 37–42 (Mi tmf5111), (russian original)

and in

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

and in

• Alexander Voronov, Maps of supermanifolds , Teoret. Mat. Fiz., 60(1):43–48, 1984.

See also

• Anatoly Konechny and Albert Schwarz,

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

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

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

A comprehensive discussion of the situation over the site of superpoints is given in

• Christoph Sachse, A Categorical Formulation of Superalgebra and Supergeometry (arXiv:0802.4067)

The site of formal duals not just to Grassmann algebras but to all super-infinitesimally thickened points is discussed in

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

As manifolds modeld on Grassmann algebras

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

• Alice Rogers

Other

• Yuri 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.

• Alberto S. Cattaneo, Florian Schaetz, Introduction to supergeometry, arxiv/1011.3401

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.