nLab supermanifold

Contents

supersymmetry

Applications

Manifolds and cobordisms

Definitions

Genera and invariants

Classification

Theorems

Contents

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

is popular. The definition

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 (Berezin & Leites 1975).

Definition

Definition

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

• the topological space $|X|$ is second countable space, Hausdorff space,

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

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

Forgetting the graded part by projecting out the nilpotent ideal in $O_X$ (i.e. applying the bosonic modality) yields the underlying ordinary smooth manifold $X_{red}$.

One just writes $C^\infty(X)$ 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 $C^\infty(X)$ of the sections of the dual bundle

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

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

Example

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

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

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.

(Batchelor 1979 reviewed in Batchelor 1984, §1.13; Rogers 2007, §8.2)

Remark

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:

Theorem
• There is a natural bijection

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

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

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

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

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 2007).

(…)

As manifolds over the base topos on superpoints

Let $SuperPoint$ be the category of superpoints. And $Sh(SuperPoint) = PSh(SuperPoint)$ 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.

Definition

Definition

Let

$SuperSet := Sh(SuperPoint)$

be the sheaf topos over superpoints. Let

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

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.

Definition

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

This appears as (Sachse, def. 4.6).

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

Write SmoothMfd for the category of ordinary smooth manifolds.

Definition

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

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

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

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. 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.

References

Via functorial geometry

Discussion from the point of view of functorial geometry:

As locally ringed spaces

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

• 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 $\mathbb{Z}_2^k$-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, Monographs on Mathematical Physics, Cambridge University Press (1984) [doi:10.1017/CBO9780511564000]

• Alice Rogers, Supermanifolds: Theory and Applications, World Scientific (2007) [doi:10.1142/1878]

(Rogers 2007, Ch. 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.)

• Alice Rogers, Aspects of the Geometrical Approach to Supermanifolds, in: Mathematical Aspects of Superspace, NATO ASI Series 132, Springer (1984) 135-148 [doi:10.1007/978-94-009-6446-4_5]

Other

Discussion with an eye towards supergravity:

Discussion with an eye towards integration over supermanifolds:

Discussion of global properties: