nLab supermanifold

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Contents

Context

Supergeometry

superalgebra and (synthetic ) supergeometry

Background

Introductions

Superalgebra

Supergeometry

Supersymmetry

supersymmetry

Supersymmetric field theory

Applications

Manifolds and cobordisms

manifolds and cobordisms

cobordism theory, Introduction

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

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 (Berezin & Leites 1975).

Definition

Definition

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.

Examples

Example

For E→XE \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.

Example

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

ℝ p|q:=Π(ℝ p+q→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 ΠE\Pi E where EE 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)≃SAlgebras(C ∞(Y),C ∞(X)), 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 ℝ p|q\mathbb{R}^{p|q} yields an isomorphism

    SDiff(X,ℝ 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}
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 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.

Definition

Definition

Let

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

be the sheaf topos over superpoints. Let

ℝ∈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.

Definition

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.

Definition

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

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

U× X′U′ →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).

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 dual superalgebra

Via the formally dual superalgebra of the super-function algebras on supermanifolds

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 ℤ 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

  • Katsumi Yagi: Super manifolds, Osaka J. Math. 25 4 (1988) 909-932 [euclid:ojm/1200781174]

  • Bryce DeWitt, Supermanifolds, Monographs on Mathematical Physics, Cambridge University Press (1984, 1992) [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 integration over supermanifolds:

Discussion of global properties:

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

See also pdf

Supergeometry of fermion fields

Discussion of the classical mechanics of the spinning particle or of classical field theory with fermion fields (possibly but not necessarily super-symmetric) as taking place in supergeometry:

via (possibly infinite-dimensional) supermanifolds:

and more generally via smooth super sets:

Discussion with focus on supersymmetry:

and specifically in the context of super- string theory (regarding worldsheets as super Riemann surfaces):

Last revised on November 9, 2024 at 12:28:36. See the history of this page for a list of all contributions to it.

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