Contents

supersymmetry

# 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 (BerezinLeites).

### 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.

###### 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

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

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

• Composition with the standard coordinate functions on $\mathbb{R}^{p|q}$ 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 desription of supermanifolds that goes back to (DeWitt 92) and (Rogers).

(…)

## 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

• 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 $\mathbb{Z}_2^k$-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.

### Other

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)