# Super manifolds: an incomplete survey

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We present an incomplete survey on some basic notions of super manifolds which may serve as a short introduction to this subject. Almost all the material is taken from the beautiful survey article on super manifolds [Deligne&Morgan1999]. Standard references also include [Leites1980], [Berezin1987], [Manin1988] or [Voronov1991]. The material below is a prerequisite to our papers [Hohnhold&Kreck&Stolz&Teichner2010] and [Stolz&Teichner2008].

## Contents |

## 1 Super Algebra

Let us begin by explaining briefly what *super* means in an algebraic context, working with the ground field of real numbers. The monoidal category of *super vector spaces*, with tensor products, is by definition the same as the monoidal category of -graded vector spaces, with the graded tensor product. As a consequence, a super algebra is simply a monoidal object in this category and is hence the same thing as a -graded algebra. For example, the endomorphism ring of a super vector space inherits a natural -grading from that of . The distinction between these notions only arises from the choice of symmetry operators

There are two standard choices, yielding two very different *symmetric* monoidal categories. For super vector spaces one has

where is the -degree of a homogenous vector . For -graded vector spaces the signs would be omitted. This basic difference is sometimes summarized as the

**Sign rule:** Commuting two odd quantities yields a sign .

*commutative*if for all homogenenous we have

Let be a commutative super algebra. The *derivations* of are endomorphisms satisfying the Leibniz rule: ^{[1]}

**Definition 1.1.**A

*super Lie algebra*is a super vector space together with a

*Lie bracket*that is skew symmetric

*Jacobi identity*

Note that we cyclically permuted the 3 symbols and put down the signs according to the above sign rule.

## 2 Super Manifolds

We will define super manifolds as ringed spaces following [Deligne&Morgan1999]. By a morphism we will always mean a map of ringed spaces. The local model for a super manifold of dimension is equipped with the sheaf of commutative super -algebras .

**Definition 2.1.**A

*super manifold*of dimension is a pair consisting of a (Hausdorff and second countable) topological space together with a sheaf of commutative super -algebras that is locally isomorphic to . A morphism between super manifolds is defined to be a continuous map , together with a map of sheaves covering . More precisely, for every open subset there are algebra maps

*reduced manifold*

*complex (analytic) super manifolds*. There is also an important notion of

*cs manifolds*. These are spaces equipped with sheaves of commutative super -algebras that locally look like . One relevance of

*cs*manifolds is that they appear naturally as the smooth super manifolds underlying complex analytic super manifolds. In our work, cs manifolds are essential to define the notion of a

*unitary*field theory but this is not relevant for the current discussion.

**Example 2.2.**
Let be a real vector bundle of fiber dimension over the ordinary manifold and the associated algebra bundle of alternating multilinear forms on . Then its sheaf of sections gives a super manifold of dimension , denoted by . In the current *smooth* setting, Marjorie Batchelor proved in [Batchelor1979] that every super manifold is isomorphic to one of this type (this is not true for analytic super manifolds). More precisely, let denote the category of real vector bundles over smooth manifolds, and for , consider the vector bundle over with sheaf of sections . Then the functors

come equipped with natural isomorphisms but there are only *non-natural* isomorphisms , coming from a choice of a partition of unity. In other words, these functors induce a bijection on isomorphism classes of objects and inclusions on morphisms but they are not equivalences of categories because there are many more morphims in than the linear bundle maps coming from .

The following proposition gives two extremely useful ways of looking at morphisms between super manifolds. We shall use the notation for the algebra of (global) functions on a super manifold .

**Proposition 2.3.**
For , the functor induces natural bijections

If is an open super submanifold (a ), is in bijective correspondence with those in that satisfy

where are coordinates on . Moreover, by the first part we see that and hence .

The proof of the first part is based on the existence of partitions of unity for super manifolds, so it is false in analytic settings. The second part always holds and is proved in [Leites1980].

## 3 The Functor of Points

Tex syntax errorby

Tex syntax error

*representable*functors, defined to be those in the image of . We will sometimes refer to an arbitrary functor

Tex syntax erroras a

*generalized super manifold*. Note that Proposition 2.3 makes it easy to describe the morphism sets

Tex syntax error. We'd also like to point out that the functor of points approach is closely related to computations involving additional odd quantities (the odd coordinates of as opposed to those of ) in many physics papers.

## 4 Super Lie Groups

These are simply group objects in . According to the functor of points approach, such a group object in can be described by giving a functor such that the composition with the forgetful functor is representable.

**Example 4.1.**
The simplest super Lie group is the additive group structure on . It is given by the following composition law on , obviously natural in :

The *super general linear group* is defined by

This follows directly from proposition 2.3 using that a map between super algebras is invertible if and only if it is invertible modulo nilpotent elements.

## 5 Super Vector Bundles

A (super) vector bundle over a super manifold is a locally free sheaf of -modules of dimension . The most basic example of a super vector bundle is the *tangent bundle* of a super manifold . It is the sheaf of -modules defined by

is locally free of dimension : If are local coordinates on , then a local basis is given by . Note that there is also a linear fibre bundle with structure group , where is a super manifold of dimension . More generally, any vector bundle over has a *total space* that comes with a projection map . It can be most easily described in terms of its -points

So is an even global section of the pullback bundle on and the projection comes from forgetting this datum. To prove that this functor is representable one uses the local triviality of and Proposition 2.3. It follows by construction that the typical fibre of the projection is and the structure group is .

There is an important operation of*parity reversal*on the category of vector bundles over . It is an involution

One can define the super Lie algebra of a super Lie group as follows. A vector field is called *left-invariant* if is related to itself under the left-translation by all :

Here we interpret as a vertical vector field on in the obvious way. The super Lie algebra consists of all left-invariant vector fields on . Pulling back via the unit defines an isomorphism , in particular, the vector space dimension of is .

## 6 Footnotes

- ↑ Whenever we write formulas involving the degree of certain elements, we implicitly assume that these elements are homogenous.

## 7 References

- [Batchelor1979] M. Batchelor,
*The structure of supermanifolds*, Trans. Amer. Math. Soc.**253**(1979), 329–338. MR536951 (80h:58002) Zbl 0413.58002 - [Berezin1987] F. A. Berezin,
*Introduction to algebra and analysis with anticommuting variables*, Reidel 1987. - [Deligne&Morgan1999] P. Deligne and J. W. Morgan,
*Notes on supersymmetry (following Joseph Bernstein)*, Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), 41–97, Amer. Math. Soc., Providence, RI, (1999), 41–97. MR1701597 (2001g:58007) Zbl 1170.58302 - [Hohnhold&Kreck&Stolz&Teichner2010] H. Hohnhold, M. Kreck, S. Stolz and P. Teichner,
*Differential Forms and 0-dimensional Super Symmetric Field Theories*, to appear in Quantum Topology, Journal of the European Math. Soc. (2010).

- [Leites1980] D. A. Leites,
*Introduction to the theory of supermanifolds*, Russian Math. Surveys 35 No 1 (1980), 3–57. MR0565567 (81j:58003) Zbl 0439.58007 - [Manin1988] Y. I. Manin,
*Gauge field theory and complex geometry*, Springer-Verlag, Berlin, 1988. MR954833 (89d:32001) Zbl 0884.53002 - [Stolz&Teichner2008] S. Stolz and P. Teichner,
*Supersymmetric Euclidean field theories and generalized cohomology*, Survey (2008). Available at http://math.berkeley.edu/~teichner/Papers/Survey.pdf. - [Voronov1991] T. Voronov,
*Geometric integration theory on supermanifolds*, Harwood Academic Publishers, Chur, 1991. MR1202882 (95b:58023) Zbl 0839.58014

## 8 External links

- The Wikipedia page on Super manifolds.

There are two standard choices, yielding two very different *symmetric* monoidal categories. For super vector spaces one has

where is the -degree of a homogenous vector . For -graded vector spaces the signs would be omitted. This basic difference is sometimes summarized as the

**Sign rule:** Commuting two odd quantities yields a sign .

*commutative*if for all homogenenous we have

Let be a commutative super algebra. The *derivations* of are endomorphisms satisfying the Leibniz rule: ^{[1]}

**Definition 1.1.**A

*super Lie algebra*is a super vector space together with a

*Lie bracket*that is skew symmetric

*Jacobi identity*

Note that we cyclically permuted the 3 symbols and put down the signs according to the above sign rule.

## 2 Super Manifolds

We will define super manifolds as ringed spaces following [Deligne&Morgan1999]. By a morphism we will always mean a map of ringed spaces. The local model for a super manifold of dimension is equipped with the sheaf of commutative super -algebras .

**Definition 2.1.**A

*super manifold*of dimension is a pair consisting of a (Hausdorff and second countable) topological space together with a sheaf of commutative super -algebras that is locally isomorphic to . A morphism between super manifolds is defined to be a continuous map , together with a map of sheaves covering . More precisely, for every open subset there are algebra maps

*reduced manifold*

*complex (analytic) super manifolds*. There is also an important notion of

*cs manifolds*. These are spaces equipped with sheaves of commutative super -algebras that locally look like . One relevance of

*cs*manifolds is that they appear naturally as the smooth super manifolds underlying complex analytic super manifolds. In our work, cs manifolds are essential to define the notion of a

*unitary*field theory but this is not relevant for the current discussion.

**Example 2.2.**
Let be a real vector bundle of fiber dimension over the ordinary manifold and the associated algebra bundle of alternating multilinear forms on . Then its sheaf of sections gives a super manifold of dimension , denoted by . In the current *smooth* setting, Marjorie Batchelor proved in [Batchelor1979] that every super manifold is isomorphic to one of this type (this is not true for analytic super manifolds). More precisely, let denote the category of real vector bundles over smooth manifolds, and for , consider the vector bundle over with sheaf of sections . Then the functors

come equipped with natural isomorphisms but there are only *non-natural* isomorphisms , coming from a choice of a partition of unity. In other words, these functors induce a bijection on isomorphism classes of objects and inclusions on morphisms but they are not equivalences of categories because there are many more morphims in than the linear bundle maps coming from .

The following proposition gives two extremely useful ways of looking at morphisms between super manifolds. We shall use the notation for the algebra of (global) functions on a super manifold .

**Proposition 2.3.**
For , the functor induces natural bijections

If is an open super submanifold (a ), is in bijective correspondence with those in that satisfy

where are coordinates on . Moreover, by the first part we see that and hence .

The proof of the first part is based on the existence of partitions of unity for super manifolds, so it is false in analytic settings. The second part always holds and is proved in [Leites1980].

## 3 The Functor of Points

Tex syntax errorby

Tex syntax error

*representable*functors, defined to be those in the image of . We will sometimes refer to an arbitrary functor

Tex syntax erroras a

*generalized super manifold*. Note that Proposition 2.3 makes it easy to describe the morphism sets

Tex syntax error. We'd also like to point out that the functor of points approach is closely related to computations involving additional odd quantities (the odd coordinates of as opposed to those of ) in many physics papers.

## 4 Super Lie Groups

These are simply group objects in . According to the functor of points approach, such a group object in can be described by giving a functor such that the composition with the forgetful functor is representable.

**Example 4.1.**
The simplest super Lie group is the additive group structure on . It is given by the following composition law on , obviously natural in :

The *super general linear group* is defined by

This follows directly from proposition 2.3 using that a map between super algebras is invertible if and only if it is invertible modulo nilpotent elements.

## 5 Super Vector Bundles

A (super) vector bundle over a super manifold is a locally free sheaf of -modules of dimension . The most basic example of a super vector bundle is the *tangent bundle* of a super manifold . It is the sheaf of -modules defined by

is locally free of dimension : If are local coordinates on , then a local basis is given by . Note that there is also a linear fibre bundle with structure group , where is a super manifold of dimension . More generally, any vector bundle over has a *total space* that comes with a projection map . It can be most easily described in terms of its -points

So is an even global section of the pullback bundle on and the projection comes from forgetting this datum. To prove that this functor is representable one uses the local triviality of and Proposition 2.3. It follows by construction that the typical fibre of the projection is and the structure group is .

There is an important operation of*parity reversal*on the category of vector bundles over . It is an involution

One can define the super Lie algebra of a super Lie group as follows. A vector field is called *left-invariant* if is related to itself under the left-translation by all :

Here we interpret as a vertical vector field on in the obvious way. The super Lie algebra consists of all left-invariant vector fields on . Pulling back via the unit defines an isomorphism , in particular, the vector space dimension of is .

## 6 Footnotes

- ↑ Whenever we write formulas involving the degree of certain elements, we implicitly assume that these elements are homogenous.

## 7 References

- [Batchelor1979] M. Batchelor,
*The structure of supermanifolds*, Trans. Amer. Math. Soc.**253**(1979), 329–338. MR536951 (80h:58002) Zbl 0413.58002 - [Berezin1987] F. A. Berezin,
*Introduction to algebra and analysis with anticommuting variables*, Reidel 1987. - [Deligne&Morgan1999] P. Deligne and J. W. Morgan,
*Notes on supersymmetry (following Joseph Bernstein)*, Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), 41–97, Amer. Math. Soc., Providence, RI, (1999), 41–97. MR1701597 (2001g:58007) Zbl 1170.58302 - [Hohnhold&Kreck&Stolz&Teichner2010] H. Hohnhold, M. Kreck, S. Stolz and P. Teichner,
*Differential Forms and 0-dimensional Super Symmetric Field Theories*, to appear in Quantum Topology, Journal of the European Math. Soc. (2010).

- [Leites1980] D. A. Leites,
*Introduction to the theory of supermanifolds*, Russian Math. Surveys 35 No 1 (1980), 3–57. MR0565567 (81j:58003) Zbl 0439.58007 - [Manin1988] Y. I. Manin,
*Gauge field theory and complex geometry*, Springer-Verlag, Berlin, 1988. MR954833 (89d:32001) Zbl 0884.53002 - [Stolz&Teichner2008] S. Stolz and P. Teichner,
*Supersymmetric Euclidean field theories and generalized cohomology*, Survey (2008). Available at http://math.berkeley.edu/~teichner/Papers/Survey.pdf. - [Voronov1991] T. Voronov,
*Geometric integration theory on supermanifolds*, Harwood Academic Publishers, Chur, 1991. MR1202882 (95b:58023) Zbl 0839.58014

## 8 External links

- The Wikipedia page on Super manifolds.

There are two standard choices, yielding two very different *symmetric* monoidal categories. For super vector spaces one has

where is the -degree of a homogenous vector . For -graded vector spaces the signs would be omitted. This basic difference is sometimes summarized as the

**Sign rule:** Commuting two odd quantities yields a sign .

*commutative*if for all homogenenous we have

Let be a commutative super algebra. The *derivations* of are endomorphisms satisfying the Leibniz rule: ^{[1]}

**Definition 1.1.**A

*super Lie algebra*is a super vector space together with a

*Lie bracket*that is skew symmetric

*Jacobi identity*

Note that we cyclically permuted the 3 symbols and put down the signs according to the above sign rule.

## 2 Super Manifolds

We will define super manifolds as ringed spaces following [Deligne&Morgan1999]. By a morphism we will always mean a map of ringed spaces. The local model for a super manifold of dimension is equipped with the sheaf of commutative super -algebras .

**Definition 2.1.**A

*super manifold*of dimension is a pair consisting of a (Hausdorff and second countable) topological space together with a sheaf of commutative super -algebras that is locally isomorphic to . A morphism between super manifolds is defined to be a continuous map , together with a map of sheaves covering . More precisely, for every open subset there are algebra maps

*reduced manifold*

*complex (analytic) super manifolds*. There is also an important notion of

*cs manifolds*. These are spaces equipped with sheaves of commutative super -algebras that locally look like . One relevance of

*cs*manifolds is that they appear naturally as the smooth super manifolds underlying complex analytic super manifolds. In our work, cs manifolds are essential to define the notion of a

*unitary*field theory but this is not relevant for the current discussion.

**Example 2.2.**
Let be a real vector bundle of fiber dimension over the ordinary manifold and the associated algebra bundle of alternating multilinear forms on . Then its sheaf of sections gives a super manifold of dimension , denoted by . In the current *smooth* setting, Marjorie Batchelor proved in [Batchelor1979] that every super manifold is isomorphic to one of this type (this is not true for analytic super manifolds). More precisely, let denote the category of real vector bundles over smooth manifolds, and for , consider the vector bundle over with sheaf of sections . Then the functors

come equipped with natural isomorphisms but there are only *non-natural* isomorphisms , coming from a choice of a partition of unity. In other words, these functors induce a bijection on isomorphism classes of objects and inclusions on morphisms but they are not equivalences of categories because there are many more morphims in than the linear bundle maps coming from .

The following proposition gives two extremely useful ways of looking at morphisms between super manifolds. We shall use the notation for the algebra of (global) functions on a super manifold .

**Proposition 2.3.**
For , the functor induces natural bijections

If is an open super submanifold (a ), is in bijective correspondence with those in that satisfy

where are coordinates on . Moreover, by the first part we see that and hence .

The proof of the first part is based on the existence of partitions of unity for super manifolds, so it is false in analytic settings. The second part always holds and is proved in [Leites1980].

## 3 The Functor of Points

Tex syntax errorby

Tex syntax error

*representable*functors, defined to be those in the image of . We will sometimes refer to an arbitrary functor

Tex syntax erroras a

*generalized super manifold*. Note that Proposition 2.3 makes it easy to describe the morphism sets

Tex syntax error. We'd also like to point out that the functor of points approach is closely related to computations involving additional odd quantities (the odd coordinates of as opposed to those of ) in many physics papers.

## 4 Super Lie Groups

These are simply group objects in . According to the functor of points approach, such a group object in can be described by giving a functor such that the composition with the forgetful functor is representable.

**Example 4.1.**
The simplest super Lie group is the additive group structure on . It is given by the following composition law on , obviously natural in :

The *super general linear group* is defined by

This follows directly from proposition 2.3 using that a map between super algebras is invertible if and only if it is invertible modulo nilpotent elements.

## 5 Super Vector Bundles

A (super) vector bundle over a super manifold is a locally free sheaf of -modules of dimension . The most basic example of a super vector bundle is the *tangent bundle* of a super manifold . It is the sheaf of -modules defined by

is locally free of dimension : If are local coordinates on , then a local basis is given by . Note that there is also a linear fibre bundle with structure group , where is a super manifold of dimension . More generally, any vector bundle over has a *total space* that comes with a projection map . It can be most easily described in terms of its -points

So is an even global section of the pullback bundle on and the projection comes from forgetting this datum. To prove that this functor is representable one uses the local triviality of and Proposition 2.3. It follows by construction that the typical fibre of the projection is and the structure group is .

There is an important operation of*parity reversal*on the category of vector bundles over . It is an involution

One can define the super Lie algebra of a super Lie group as follows. A vector field is called *left-invariant* if is related to itself under the left-translation by all :

Here we interpret as a vertical vector field on in the obvious way. The super Lie algebra consists of all left-invariant vector fields on . Pulling back via the unit defines an isomorphism , in particular, the vector space dimension of is .

## 6 Footnotes

- ↑ Whenever we write formulas involving the degree of certain elements, we implicitly assume that these elements are homogenous.

## 7 References

- [Batchelor1979] M. Batchelor,
*The structure of supermanifolds*, Trans. Amer. Math. Soc.**253**(1979), 329–338. MR536951 (80h:58002) Zbl 0413.58002 - [Berezin1987] F. A. Berezin,
*Introduction to algebra and analysis with anticommuting variables*, Reidel 1987. - [Deligne&Morgan1999] P. Deligne and J. W. Morgan,
*Notes on supersymmetry (following Joseph Bernstein)*, Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), 41–97, Amer. Math. Soc., Providence, RI, (1999), 41–97. MR1701597 (2001g:58007) Zbl 1170.58302 - [Hohnhold&Kreck&Stolz&Teichner2010] H. Hohnhold, M. Kreck, S. Stolz and P. Teichner,
*Differential Forms and 0-dimensional Super Symmetric Field Theories*, to appear in Quantum Topology, Journal of the European Math. Soc. (2010).

- [Leites1980] D. A. Leites,
*Introduction to the theory of supermanifolds*, Russian Math. Surveys 35 No 1 (1980), 3–57. MR0565567 (81j:58003) Zbl 0439.58007 - [Manin1988] Y. I. Manin,
*Gauge field theory and complex geometry*, Springer-Verlag, Berlin, 1988. MR954833 (89d:32001) Zbl 0884.53002 - [Stolz&Teichner2008] S. Stolz and P. Teichner,
*Supersymmetric Euclidean field theories and generalized cohomology*, Survey (2008). Available at http://math.berkeley.edu/~teichner/Papers/Survey.pdf. - [Voronov1991] T. Voronov,
*Geometric integration theory on supermanifolds*, Harwood Academic Publishers, Chur, 1991. MR1202882 (95b:58023) Zbl 0839.58014

## 8 External links

- The Wikipedia page on Super manifolds.