# 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

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**Definition 2.1.**A

*super manifold*of dimension

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*reduced manifold*

Tex syntax errorand there is an inclusion of super manifolds

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*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

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*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.**

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*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

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*non-natural*isomorphisms

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**Proposition 2.3.**

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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

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

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*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

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Tex syntax errorcan 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

Tex syntax error-modules of dimension

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*tangent bundle*of a super manifold . It is the sheaf of

Tex syntax error-modules defined by

Tex syntax error: 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

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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 :

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## 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.