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 \cite{Deligne&Morgan1999}. Standard references also include \cite{Leites1980}, \cite{ | + | 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 \cite{Deligne&Morgan1999}. Standard references also include \cite{Leites1980}, \cite{Berezin1987}, \cite{Manin1988} or \cite{Voronov1991}. The material below is a prerequisite to our papers \cite{Hohnhold&Kreck&Stolz&Teichner2010} and \cite{Stolz&Teichner2008}. |
==Super Algebra== | ==Super Algebra== |
Revision as of 17:05, 27 June 2014
An earlier version of this page was published in the Bulletin of the Manifold Atlas: screen, print. You may view the version used for publication as of 09:56, 1 April 2011 and the changes since publication. |
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
Tex syntax error-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
Tex syntax error-graded algebra. For example, the endomorphism ring
Tex syntax errorof a super vector space inherits a natural
Tex syntax error-grading from that of . The distinction between these notions only arises from the choice of symmetry operators
Tex syntax error
There are two standard choices, yielding two very different symmetric monoidal categories. For super vector spaces one has
Tex syntax erroris the
Tex syntax error-degree of a homogenous vector
Tex syntax error. For
Tex syntax error-graded vector spaces the signs would be omitted. This basic difference is sometimes summarized as the
Tex syntax errorSign rule: Commuting two odd quantities yields a sign . As a consequence, a super algebra is commutative if for all homogenenous
Tex syntax errorwe have
Tex syntax error
Tex syntax error-graded algebra. The standard examples of commutative super algebras are the exterior algebras
Tex syntax error. As we shall see, the generators of
Tex syntax erroryield the so-called odd coordinates on super manifolds; these anti-commute and hence are useful when trying to describe physical systems involving Fermions. Super algebras also arise naturally in algebraic topology: for every space , the cohomology ring
Tex syntax erroris a commutative super algebra. Let be a commutative super algebra. The derivations of are endomorphisms
Tex syntax errorsatisfying the Leibniz rule: [1]
Tex syntax erroris a super Lie algebra with respect to the bracket operation
Tex syntax error.
Note that we cyclically permuted the 3 symbols and put down the signs according to the above sign rule.
2 Super Manifolds
Tex syntax errorequipped with the sheaf
Tex syntax errorof commutative super
Tex syntax error-algebras
Tex syntax error.
Tex syntax errorconsisting of a (Hausdorff and second countable) topological space together with a sheaf of commutative super
Tex syntax error-algebras
Tex syntax errorthat is locally isomorphic to
Tex syntax error. 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
Tex syntax error
Tex syntax error.
Tex syntax error. Note that the sheaf of ideals
Tex syntax erroris generated by odd functions. Other geometric super objects can be defined in a similar way. For example, replacing
Tex syntax errorby and by analytic functions one obtains 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
Tex syntax error. 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, letTex syntax errordenote the category of real vector bundles over smooth manifolds, and for
Tex syntax error, consider the vector bundle over with sheaf of sections
Tex syntax error. Then the functors
Tex syntax error
Tex syntax errorthan the linear bundle maps coming from
Tex syntax error.
Tex syntax errorfor the algebra of (global) functions on a super manifold .
Proposition 2.3.
ForTex syntax error, the functor induces natural bijections
Tex syntax error
Tex syntax erroris an open super submanifold (a ),
Tex syntax erroris in bijective correspondence with those in that satisfy
Tex syntax error
Tex syntax errordefined by
Tex syntax error. Moreover, by the first part we see that
Tex syntax errorand 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 error, where varies over all super manifolds . More formally, embed the category
Tex syntax errorof super manifolds in the category of contravariant functors from
Tex syntax errorto
Tex syntax errorby
Tex syntax error
Tex syntax errorwith the the category of 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
Tex syntax error. According to the functor of points approach, such a group object in
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 . The most basic example of a super vector bundle is the tangent bundle of a super manifold . It is the sheaf of
Tex syntax error-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 involutionTex syntax error-modules).
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.
Tex syntax error-graded algebra. For example, the endomorphism ring
Tex syntax errorof a super vector space inherits a natural
Tex syntax error-grading from that of . The distinction between these notions only arises from the choice of symmetry operators
Tex syntax error
There are two standard choices, yielding two very different symmetric monoidal categories. For super vector spaces one has
Tex syntax erroris the
Tex syntax error-degree of a homogenous vector
Tex syntax error. For
Tex syntax error-graded vector spaces the signs would be omitted. This basic difference is sometimes summarized as the
Tex syntax errorSign rule: Commuting two odd quantities yields a sign . As a consequence, a super algebra is commutative if for all homogenenous
Tex syntax errorwe have
Tex syntax error
Tex syntax error-graded algebra. The standard examples of commutative super algebras are the exterior algebras
Tex syntax error. As we shall see, the generators of
Tex syntax erroryield the so-called odd coordinates on super manifolds; these anti-commute and hence are useful when trying to describe physical systems involving Fermions. Super algebras also arise naturally in algebraic topology: for every space , the cohomology ring
Tex syntax erroris a commutative super algebra. Let be a commutative super algebra. The derivations of are endomorphisms
Tex syntax errorsatisfying the Leibniz rule: [1]
Tex syntax erroris a super Lie algebra with respect to the bracket operation
Tex syntax error.
Note that we cyclically permuted the 3 symbols and put down the signs according to the above sign rule.
2 Super Manifolds
Tex syntax errorequipped with the sheaf
Tex syntax errorof commutative super
Tex syntax error-algebras
Tex syntax error.
Tex syntax errorconsisting of a (Hausdorff and second countable) topological space together with a sheaf of commutative super
Tex syntax error-algebras
Tex syntax errorthat is locally isomorphic to
Tex syntax error. 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
Tex syntax error
Tex syntax error.
Tex syntax error. Note that the sheaf of ideals
Tex syntax erroris generated by odd functions. Other geometric super objects can be defined in a similar way. For example, replacing
Tex syntax errorby and by analytic functions one obtains 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
Tex syntax error. 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, letTex syntax errordenote the category of real vector bundles over smooth manifolds, and for
Tex syntax error, consider the vector bundle over with sheaf of sections
Tex syntax error. Then the functors
Tex syntax error
Tex syntax errorthan the linear bundle maps coming from
Tex syntax error.
Tex syntax errorfor the algebra of (global) functions on a super manifold .
Proposition 2.3.
ForTex syntax error, the functor induces natural bijections
Tex syntax error
Tex syntax erroris an open super submanifold (a ),
Tex syntax erroris in bijective correspondence with those in that satisfy
Tex syntax error
Tex syntax errordefined by
Tex syntax error. Moreover, by the first part we see that
Tex syntax errorand 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 error, where varies over all super manifolds . More formally, embed the category
Tex syntax errorof super manifolds in the category of contravariant functors from
Tex syntax errorto
Tex syntax errorby
Tex syntax error
Tex syntax errorwith the the category of 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
Tex syntax error. According to the functor of points approach, such a group object in
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 . The most basic example of a super vector bundle is the tangent bundle of a super manifold . It is the sheaf of
Tex syntax error-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 involutionTex syntax error-modules).
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.
Tex syntax error-graded algebra. For example, the endomorphism ring
Tex syntax errorof a super vector space inherits a natural
Tex syntax error-grading from that of . The distinction between these notions only arises from the choice of symmetry operators
Tex syntax error
There are two standard choices, yielding two very different symmetric monoidal categories. For super vector spaces one has
Tex syntax erroris the
Tex syntax error-degree of a homogenous vector
Tex syntax error. For
Tex syntax error-graded vector spaces the signs would be omitted. This basic difference is sometimes summarized as the
Tex syntax errorSign rule: Commuting two odd quantities yields a sign . As a consequence, a super algebra is commutative if for all homogenenous
Tex syntax errorwe have
Tex syntax error
Tex syntax error-graded algebra. The standard examples of commutative super algebras are the exterior algebras
Tex syntax error. As we shall see, the generators of
Tex syntax erroryield the so-called odd coordinates on super manifolds; these anti-commute and hence are useful when trying to describe physical systems involving Fermions. Super algebras also arise naturally in algebraic topology: for every space , the cohomology ring
Tex syntax erroris a commutative super algebra. Let be a commutative super algebra. The derivations of are endomorphisms
Tex syntax errorsatisfying the Leibniz rule: [1]
Tex syntax erroris a super Lie algebra with respect to the bracket operation
Tex syntax error.
Note that we cyclically permuted the 3 symbols and put down the signs according to the above sign rule.
2 Super Manifolds
Tex syntax errorequipped with the sheaf
Tex syntax errorof commutative super
Tex syntax error-algebras
Tex syntax error.
Tex syntax errorconsisting of a (Hausdorff and second countable) topological space together with a sheaf of commutative super
Tex syntax error-algebras
Tex syntax errorthat is locally isomorphic to
Tex syntax error. 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
Tex syntax error
Tex syntax error.
Tex syntax error. Note that the sheaf of ideals
Tex syntax erroris generated by odd functions. Other geometric super objects can be defined in a similar way. For example, replacing
Tex syntax errorby and by analytic functions one obtains 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
Tex syntax error. 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, letTex syntax errordenote the category of real vector bundles over smooth manifolds, and for
Tex syntax error, consider the vector bundle over with sheaf of sections
Tex syntax error. Then the functors
Tex syntax error
Tex syntax errorthan the linear bundle maps coming from
Tex syntax error.
Tex syntax errorfor the algebra of (global) functions on a super manifold .
Proposition 2.3.
ForTex syntax error, the functor induces natural bijections
Tex syntax error
Tex syntax erroris an open super submanifold (a ),
Tex syntax erroris in bijective correspondence with those in that satisfy
Tex syntax error
Tex syntax errordefined by
Tex syntax error. Moreover, by the first part we see that
Tex syntax errorand 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 error, where varies over all super manifolds . More formally, embed the category
Tex syntax errorof super manifolds in the category of contravariant functors from
Tex syntax errorto
Tex syntax errorby
Tex syntax error
Tex syntax errorwith the the category of 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
Tex syntax error. According to the functor of points approach, such a group object in
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 . The most basic example of a super vector bundle is the tangent bundle of a super manifold . It is the sheaf of
Tex syntax error-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 involutionTex syntax error-modules).
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.