Super manifolds: an incomplete survey
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\begin{abstract} This is for submission in the manifold atlas \end{abstract} We survey some basic notions of super geometry because we feel that some readers of our papers [HKST] or [ST] may not be familiar with these concepts and we would like a comprehensive place to refer to. Almost all the material is taken from the beautiful survey article on super manifolds by Deligne and Morgan, [DM]. Standard references also include Leites [L], Bernstein [Be], Manin [M] or Voronov [V].
Contents |
1 Super Algebra
- Let us begin by explaining briefly what {\em 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 \[ \sigma=\sigma_{V,W}
- V\otimes W \overset{\cong}{\ra} W \otimes V \] There are two standard choices, yielding two very different {\em symmetric} monoidal categories. For super vector spaces one has \[ \sigma(v\otimes w) = (-1)^{|v|\cdot |w|} w\otimes v , \] 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 .
2 Super Manifolds
- We will define super manifolds as ringed spaces following [DM]. By a morphism we will always mean a map of ringed spaces. The local model for a super manifold of dimension is
Tex syntax error
equipped with the sheafTex syntax error
of commutative superTex syntax error
-algebrasTex syntax error
. \begin{defn} A super manifold of dimension is a pairTex syntax error
consisting of a (Hausdorff and second countable) topological space together with a sheaf of commutative superTex syntax error
-algebrasTex syntax error
that is locally isomorphic toTex 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 \[ F(U) - \cO_N(U) \ra \cO_M(|f|^{-1}(U)) \] that are compatible with the restriction maps of the two sheaves. In the future we shall write for and we denote this category of super manifolds by
Tex syntax error
. \end{defn} To every super manifold there is an associated {\em reduced manifold} \[ M^{red}:=(|M|,\cO_M / \Nil) \] obtained by dividing out the ideal of nilpotent functions. By construction, this gives a smooth manifold structure on the underlying topological space and there is an inclusion of super manifoldsTex syntax error
. Note that the sheaf of idealsTex syntax error
is generated by odd functions. Other geometric super objects can be defined in a similar way. For example, replacingTex syntax error
byTex syntax error
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 superTex syntax error
-algebras that locally look likeTex 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 {\em unitary} field theory but this is not relevant for the current discussion.
Example 2.1.
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 . Marjorie Batchelor proved in [Ba] that every super manifold is isomorphic to one of this type. 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 \[ \Pi : \BunMan \to \SMan \quad \text{ and } \quad J: \SMan \to \BunMan \] come equipped with natural isomorphisms but there are only {\em 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
Tex syntax errorthan the linear bundle maps coming from
Tex syntax error. \end{example} The following proposition gives two extremely useful ways of looking at morphisms between super manifolds. We shall use the notation
Tex syntax errorfor the algebra of (global) functions on a super manifold . \begin{prop} For
Tex syntax error, the functor induces natural bijections \[ \SMan(S,M) \cong \Alg(C^\infty(M), C^\infty(S)) \] If
Tex syntax erroris an open super submanifold (a {\em domain}), \SMan(S,M) is in bijective correspondence with those in that satisfy \[ (|f_1|(s),...,|f_p|(s)) \in |M| \subseteq \R^{p} \text{ for all } s \in |S|. \] The , are called the coordinates of
Tex syntax errordefined by \[ f_i = \phi^*(x_i) \quad \text{ and } \quad \eta_j = \phi^*(\theta_j), \] where are coordinates on
Tex syntax error. Moreover, by the first part we see that
Tex syntax errorand hence
Tex syntax error. \end{prop} 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 [L].
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 \[ Y:\SMan \ra \Fun(\SMan^{op},\Set), \quad Y(M) = ( \> S \mapsto \SMan(S,M) \> ). \] This Yoneda embedding is fully faithful and identifies
Tex syntax errorwith the the category of {\em representable} functors, defined to be those in the image of . We will sometimes refer to an arbitrary functor
Tex syntax erroras a {\em generalized super manifold}. Note that Proposition [1] 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
Tex syntax error
. According to the functor of points approach, such a group object inTex syntax error
can be described by giving a functor such that the composition with the forgetful functorTex syntax error
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 : \[ (f_1,...,\eta_q) \times (h_1,...,\psi_q) \mapsto (f_1 + h_1,...,\eta_q + \psi_q) . \] The super general linear group is defined byTex syntax error
5 Super Vector Bundles
Tex syntax errorof
Tex syntax error-modules of dimension . The most basic example of a super vector bundle is the {\em tangent bundle} of a super manifold . It is the sheaf of
Tex syntax error-modules
Tex syntax errordefined by
Tex syntax error
Tex syntax erroris 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
Tex syntax errorover has a {\em total space} that comes with a projection map . It can be most easily described in terms of its -points \[ E(S)= \{ (f,g) \ \mid\ f\in\SMan(S,M), g \in f^*(\cE^{ev}(M)) \} \] So is an even global section of the pullback bundle on and the projection comes from forgetting this datum. To prove that this functor
Tex syntax erroris representable one uses the local triviality of
Tex syntax errorand Proposition [3]. It follows by construction that the typical fibre of the projection is and the structure group is . There is an important operation of {\em parity reversal} on the category of vector bundles over . It is an involution \[ \Pi : \Vect_M\ra \Vect_M \] that takes a vector bundle with grading involution to . This means that even and odd parts are exchanged. To define on morphisms it is easiest to give it as , where is the trivial bundle of dimension (aka the constant sheaf of free
Tex syntax error-modules). One can define the super Lie algebra of a super Lie group as follows. A vector field
Tex syntax erroris called left-invariant if is related to itself under the left-translation by all :
Tex syntax error
\begin{thebibliography}{EKMM} \bibitem[Ba]{Ba} M. Batchelor, {\em The structure of super manifolds.} Transactions of the A.M.S., Vol. 253, 1979, p. 329. \bibitem[Be]{Be} Bernstein, {\em Algebra and analysis with anticommuting variables.} Reidel, 1987. \bibitem[DM]{DM} P. Deligne, J. Morgan, {\em Notes on Supersymmetry (following Joseph Bernstein).} Quantum fields and strings: A course for Mathematicians, Volume 1, A.M.S.- I.A.S. 1999, p. 41--98. \bibitem[HKST]{HKST} Henning Hohnhold, Matthias Kreck, S. Stolz, P. Teichner, {\em Differential Forms and 0-dimensional Super Symmetric Field Theories}, to appear in Quantum Topology, Journal of the European Math. Soc. 2010. \bibitem[L]{L} D. Leites, {\em Introduction to the theory of supermanifolds.} Russian Math. Surveys, 35 no. 1 (1980). \bibitem[M]{M} Y. Manin, {\em Gauge fields and complex geometry.} Springer-Verlag, 1988. \bibitem[ST]{ST} S. Stolz, P. Teichner, {\em Quantum field theories and generalized cohomology.} \\ 2007 Survey, http://math.berkeley.edu/\ {}teichner/papers.html \bibitem[V]{V} T. Voronov, {\em Geometric integration theory on supermanifolds.} Harwood Publ., 1991. \end{thebibliography};
6 References
- [Ba] Template:Ba
- [Be] Template:Be
- [DM] Template:DM
- [HKST] Template:HKST
- [L] Template:L
- [M] Template:M
- [ST] Template:ST
- [V] Template:V
Cite error:
<ref>
tags exist, but no <references/>
tag was found
|1$ (aka the constant sheaf of free $\cO_M$-modules). One can define the super Lie algebra $\fg$ of a super Lie group $G$ as follows. A vector field $\xi \in \cT G$ is called '''left-invariant''' if $\xi$ is related to itself under the left-translation by all $f: S \to G$: $$ S \times G \overset{f \times \id} \ra G \times G \overset{\mu} \ra G .$$ Here we interpret $\xi$ as a vertical vector field on $S \times G$ in the obvious way. The super Lie algebra $\fg$ consists of all left-invariant vector fields on $G$. Pulling back via the unit $e: \pt\to G$ defines an isomorphism $\fg \cong T_eG$, in particular, the vector space dimension of $\fg$ is $p|q$.
\begin{thebibliography}{EKMM} \bibitem[Ba]{Ba} M. Batchelor, {\em The structure of super manifolds.} Transactions of the A.M.S., Vol. 253, 1979, p. 329. \bibitem[Be]{Be} Bernstein, {\em Algebra and analysis with anticommuting variables.} Reidel, 1987. \bibitem[DM]{DM} P. Deligne, J. Morgan, {\em Notes on Supersymmetry (following Joseph Bernstein).} Quantum fields and strings: A course for Mathematicians, Volume 1, A.M.S.- I.A.S. 1999, p. 41--98. \bibitem[HKST]{HKST} Henning Hohnhold, Matthias Kreck, S. Stolz, P. Teichner, {\em Differential Forms and 0-dimensional Super Symmetric Field Theories}, to appear in Quantum Topology, Journal of the European Math. Soc. 2010. \bibitem[L]{L} D. Leites, {\em Introduction to the theory of supermanifolds.} Russian Math. Surveys, 35 no. 1 (1980). \bibitem[M]{M} Y. Manin, {\em Gauge fields and complex geometry.} Springer-Verlag, 1988. \bibitem[ST]{ST} S. Stolz, P. Teichner, {\em Quantum field theories and generalized cohomology.} \ 2007 Survey, http://math.berkeley.edu/\ {}teichner/papers.html \bibitem[V]{V} T. Voronov, {\em Geometric integration theory on supermanifolds.} Harwood Publ., 1991. \end{thebibliography}
;
== References ==
{{#RefList:}}
[[Category:Theory]]{{Stub}}
\begin{abstract} This is for submission in the manifold atlas \end{abstract} We survey some basic notions of super geometry because we feel that some readers of our papers - Sign rule: Commuting two odd quantities yields a sign .
2 Super Manifolds
- We will define super manifolds as ringed spaces following [DM]. By a morphism we will always mean a map of ringed spaces. The local model for a super manifold of dimension is
Tex syntax error
equipped with the sheafTex syntax error
of commutative superTex syntax error
-algebrasTex syntax error
. \begin{defn} A super manifold of dimension is a pairTex syntax error
consisting of a (Hausdorff and second countable) topological space together with a sheaf of commutative superTex syntax error
-algebrasTex syntax error
that is locally isomorphic toTex 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 \[ F(U) - \cO_N(U) \ra \cO_M(|f|^{-1}(U)) \] that are compatible with the restriction maps of the two sheaves. In the future we shall write for and we denote this category of super manifolds by
Tex syntax error
. \end{defn} To every super manifold there is an associated {\em reduced manifold} \[ M^{red}:=(|M|,\cO_M / \Nil) \] obtained by dividing out the ideal of nilpotent functions. By construction, this gives a smooth manifold structure on the underlying topological space and there is an inclusion of super manifoldsTex syntax error
. Note that the sheaf of idealsTex syntax error
is generated by odd functions. Other geometric super objects can be defined in a similar way. For example, replacingTex syntax error
byTex syntax error
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 superTex syntax error
-algebras that locally look likeTex 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 {\em unitary} field theory but this is not relevant for the current discussion.
Example 2.1.
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 . Marjorie Batchelor proved in [Ba] that every super manifold is isomorphic to one of this type. 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 \[ \Pi : \BunMan \to \SMan \quad \text{ and } \quad J: \SMan \to \BunMan \] come equipped with natural isomorphisms but there are only {\em 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
Tex syntax errorthan the linear bundle maps coming from
Tex syntax error. \end{example} The following proposition gives two extremely useful ways of looking at morphisms between super manifolds. We shall use the notation
Tex syntax errorfor the algebra of (global) functions on a super manifold . \begin{prop} For
Tex syntax error, the functor induces natural bijections \[ \SMan(S,M) \cong \Alg(C^\infty(M), C^\infty(S)) \] If
Tex syntax erroris an open super submanifold (a {\em domain}), \SMan(S,M) is in bijective correspondence with those in that satisfy \[ (|f_1|(s),...,|f_p|(s)) \in |M| \subseteq \R^{p} \text{ for all } s \in |S|. \] The , are called the coordinates of
Tex syntax errordefined by \[ f_i = \phi^*(x_i) \quad \text{ and } \quad \eta_j = \phi^*(\theta_j), \] where are coordinates on
Tex syntax error. Moreover, by the first part we see that
Tex syntax errorand hence
Tex syntax error. \end{prop} 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 [L].
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 \[ Y:\SMan \ra \Fun(\SMan^{op},\Set), \quad Y(M) = ( \> S \mapsto \SMan(S,M) \> ). \] This Yoneda embedding is fully faithful and identifies
Tex syntax errorwith the the category of {\em representable} functors, defined to be those in the image of . We will sometimes refer to an arbitrary functor
Tex syntax erroras a {\em generalized super manifold}. Note that Proposition [1] 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
Tex syntax error
. According to the functor of points approach, such a group object inTex syntax error
can be described by giving a functor such that the composition with the forgetful functorTex syntax error
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 : \[ (f_1,...,\eta_q) \times (h_1,...,\psi_q) \mapsto (f_1 + h_1,...,\eta_q + \psi_q) . \] The super general linear group is defined byTex syntax error
5 Super Vector Bundles
Tex syntax errorof
Tex syntax error-modules of dimension . The most basic example of a super vector bundle is the {\em tangent bundle} of a super manifold . It is the sheaf of
Tex syntax error-modules
Tex syntax errordefined by
Tex syntax error
Tex syntax erroris 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
Tex syntax errorover has a {\em total space} that comes with a projection map . It can be most easily described in terms of its -points \[ E(S)= \{ (f,g) \ \mid\ f\in\SMan(S,M), g \in f^*(\cE^{ev}(M)) \} \] So is an even global section of the pullback bundle on and the projection comes from forgetting this datum. To prove that this functor
Tex syntax erroris representable one uses the local triviality of
Tex syntax errorand Proposition [3]. It follows by construction that the typical fibre of the projection is and the structure group is . There is an important operation of {\em parity reversal} on the category of vector bundles over . It is an involution \[ \Pi : \Vect_M\ra \Vect_M \] that takes a vector bundle with grading involution to . This means that even and odd parts are exchanged. To define on morphisms it is easiest to give it as , where is the trivial bundle of dimension (aka the constant sheaf of free
Tex syntax error-modules). One can define the super Lie algebra of a super Lie group as follows. A vector field
Tex syntax erroris called left-invariant if is related to itself under the left-translation by all :
Tex syntax error
\begin{thebibliography}{EKMM} \bibitem[Ba]{Ba} M. Batchelor, {\em The structure of super manifolds.} Transactions of the A.M.S., Vol. 253, 1979, p. 329. \bibitem[Be]{Be} Bernstein, {\em Algebra and analysis with anticommuting variables.} Reidel, 1987. \bibitem[DM]{DM} P. Deligne, J. Morgan, {\em Notes on Supersymmetry (following Joseph Bernstein).} Quantum fields and strings: A course for Mathematicians, Volume 1, A.M.S.- I.A.S. 1999, p. 41--98. \bibitem[HKST]{HKST} Henning Hohnhold, Matthias Kreck, S. Stolz, P. Teichner, {\em Differential Forms and 0-dimensional Super Symmetric Field Theories}, to appear in Quantum Topology, Journal of the European Math. Soc. 2010. \bibitem[L]{L} D. Leites, {\em Introduction to the theory of supermanifolds.} Russian Math. Surveys, 35 no. 1 (1980). \bibitem[M]{M} Y. Manin, {\em Gauge fields and complex geometry.} Springer-Verlag, 1988. \bibitem[ST]{ST} S. Stolz, P. Teichner, {\em Quantum field theories and generalized cohomology.} \\ 2007 Survey, http://math.berkeley.edu/\ {}teichner/papers.html \bibitem[V]{V} T. Voronov, {\em Geometric integration theory on supermanifolds.} Harwood Publ., 1991. \end{thebibliography};
6 References
- [Ba] Template:Ba
- [Be] Template:Be
- [DM] Template:DM
- [HKST] Template:HKST
- [L] Template:L
- [M] Template:M
- [ST] Template:ST
- [V] Template:V
Cite error:
<ref>
tags exist, but no <references/>
tag was found
|1$ (aka the constant sheaf of free $\cO_M$-modules). One can define the super Lie algebra $\fg$ of a super Lie group $G$ as follows. A vector field $\xi \in \cT G$ is called '''left-invariant''' if $\xi$ is related to itself under the left-translation by all $f: S \to G$: $$ S \times G \overset{f \times \id} \ra G \times G \overset{\mu} \ra G .$$ Here we interpret $\xi$ as a vertical vector field on $S \times G$ in the obvious way. The super Lie algebra $\fg$ consists of all left-invariant vector fields on $G$. Pulling back via the unit $e: \pt\to G$ defines an isomorphism $\fg \cong T_eG$, in particular, the vector space dimension of $\fg$ is $p|q$.
\begin{thebibliography}{EKMM} \bibitem[Ba]{Ba} M. Batchelor, {\em The structure of super manifolds.} Transactions of the A.M.S., Vol. 253, 1979, p. 329. \bibitem[Be]{Be} Bernstein, {\em Algebra and analysis with anticommuting variables.} Reidel, 1987. \bibitem[DM]{DM} P. Deligne, J. Morgan, {\em Notes on Supersymmetry (following Joseph Bernstein).} Quantum fields and strings: A course for Mathematicians, Volume 1, A.M.S.- I.A.S. 1999, p. 41--98. \bibitem[HKST]{HKST} Henning Hohnhold, Matthias Kreck, S. Stolz, P. Teichner, {\em Differential Forms and 0-dimensional Super Symmetric Field Theories}, to appear in Quantum Topology, Journal of the European Math. Soc. 2010. \bibitem[L]{L} D. Leites, {\em Introduction to the theory of supermanifolds.} Russian Math. Surveys, 35 no. 1 (1980). \bibitem[M]{M} Y. Manin, {\em Gauge fields and complex geometry.} Springer-Verlag, 1988. \bibitem[ST]{ST} S. Stolz, P. Teichner, {\em Quantum field theories and generalized cohomology.} \ 2007 Survey, http://math.berkeley.edu/\ {}teichner/papers.html \bibitem[V]{V} T. Voronov, {\em Geometric integration theory on supermanifolds.} Harwood Publ., 1991. \end{thebibliography}
- Sign rule: Commuting two odd quantities yields a sign .
2 Super Manifolds
- We will define super manifolds as ringed spaces following [DM]. By a morphism we will always mean a map of ringed spaces. The local model for a super manifold of dimension is
Tex syntax error
equipped with the sheafTex syntax error
of commutative superTex syntax error
-algebrasTex syntax error
. \begin{defn} A super manifold of dimension is a pairTex syntax error
consisting of a (Hausdorff and second countable) topological space together with a sheaf of commutative superTex syntax error
-algebrasTex syntax error
that is locally isomorphic toTex 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 \[ F(U) - \cO_N(U) \ra \cO_M(|f|^{-1}(U)) \] that are compatible with the restriction maps of the two sheaves. In the future we shall write for and we denote this category of super manifolds by
Tex syntax error
. \end{defn} To every super manifold there is an associated {\em reduced manifold} \[ M^{red}:=(|M|,\cO_M / \Nil) \] obtained by dividing out the ideal of nilpotent functions. By construction, this gives a smooth manifold structure on the underlying topological space and there is an inclusion of super manifoldsTex syntax error
. Note that the sheaf of idealsTex syntax error
is generated by odd functions. Other geometric super objects can be defined in a similar way. For example, replacingTex syntax error
byTex syntax error
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 superTex syntax error
-algebras that locally look likeTex 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 {\em unitary} field theory but this is not relevant for the current discussion.
Example 2.1.
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 . Marjorie Batchelor proved in [Ba] that every super manifold is isomorphic to one of this type. 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 \[ \Pi : \BunMan \to \SMan \quad \text{ and } \quad J: \SMan \to \BunMan \] come equipped with natural isomorphisms but there are only {\em 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
Tex syntax errorthan the linear bundle maps coming from
Tex syntax error. \end{example} The following proposition gives two extremely useful ways of looking at morphisms between super manifolds. We shall use the notation
Tex syntax errorfor the algebra of (global) functions on a super manifold . \begin{prop} For
Tex syntax error, the functor induces natural bijections \[ \SMan(S,M) \cong \Alg(C^\infty(M), C^\infty(S)) \] If
Tex syntax erroris an open super submanifold (a {\em domain}), \SMan(S,M) is in bijective correspondence with those in that satisfy \[ (|f_1|(s),...,|f_p|(s)) \in |M| \subseteq \R^{p} \text{ for all } s \in |S|. \] The , are called the coordinates of
Tex syntax errordefined by \[ f_i = \phi^*(x_i) \quad \text{ and } \quad \eta_j = \phi^*(\theta_j), \] where are coordinates on
Tex syntax error. Moreover, by the first part we see that
Tex syntax errorand hence
Tex syntax error. \end{prop} 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 [L].
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 \[ Y:\SMan \ra \Fun(\SMan^{op},\Set), \quad Y(M) = ( \> S \mapsto \SMan(S,M) \> ). \] This Yoneda embedding is fully faithful and identifies
Tex syntax errorwith the the category of {\em representable} functors, defined to be those in the image of . We will sometimes refer to an arbitrary functor
Tex syntax erroras a {\em generalized super manifold}. Note that Proposition [1] 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
Tex syntax error
. According to the functor of points approach, such a group object inTex syntax error
can be described by giving a functor such that the composition with the forgetful functorTex syntax error
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 : \[ (f_1,...,\eta_q) \times (h_1,...,\psi_q) \mapsto (f_1 + h_1,...,\eta_q + \psi_q) . \] The super general linear group is defined byTex syntax error
5 Super Vector Bundles
Tex syntax errorof
Tex syntax error-modules of dimension . The most basic example of a super vector bundle is the {\em tangent bundle} of a super manifold . It is the sheaf of
Tex syntax error-modules
Tex syntax errordefined by
Tex syntax error
Tex syntax erroris 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
Tex syntax errorover has a {\em total space} that comes with a projection map . It can be most easily described in terms of its -points \[ E(S)= \{ (f,g) \ \mid\ f\in\SMan(S,M), g \in f^*(\cE^{ev}(M)) \} \] So is an even global section of the pullback bundle on and the projection comes from forgetting this datum. To prove that this functor
Tex syntax erroris representable one uses the local triviality of
Tex syntax errorand Proposition [3]. It follows by construction that the typical fibre of the projection is and the structure group is . There is an important operation of {\em parity reversal} on the category of vector bundles over . It is an involution \[ \Pi : \Vect_M\ra \Vect_M \] that takes a vector bundle with grading involution to . This means that even and odd parts are exchanged. To define on morphisms it is easiest to give it as , where is the trivial bundle of dimension (aka the constant sheaf of free
Tex syntax error-modules). One can define the super Lie algebra of a super Lie group as follows. A vector field
Tex syntax erroris called left-invariant if is related to itself under the left-translation by all :
Tex syntax error
\begin{thebibliography}{EKMM} \bibitem[Ba]{Ba} M. Batchelor, {\em The structure of super manifolds.} Transactions of the A.M.S., Vol. 253, 1979, p. 329. \bibitem[Be]{Be} Bernstein, {\em Algebra and analysis with anticommuting variables.} Reidel, 1987. \bibitem[DM]{DM} P. Deligne, J. Morgan, {\em Notes on Supersymmetry (following Joseph Bernstein).} Quantum fields and strings: A course for Mathematicians, Volume 1, A.M.S.- I.A.S. 1999, p. 41--98. \bibitem[HKST]{HKST} Henning Hohnhold, Matthias Kreck, S. Stolz, P. Teichner, {\em Differential Forms and 0-dimensional Super Symmetric Field Theories}, to appear in Quantum Topology, Journal of the European Math. Soc. 2010. \bibitem[L]{L} D. Leites, {\em Introduction to the theory of supermanifolds.} Russian Math. Surveys, 35 no. 1 (1980). \bibitem[M]{M} Y. Manin, {\em Gauge fields and complex geometry.} Springer-Verlag, 1988. \bibitem[ST]{ST} S. Stolz, P. Teichner, {\em Quantum field theories and generalized cohomology.} \\ 2007 Survey, http://math.berkeley.edu/\ {}teichner/papers.html \bibitem[V]{V} T. Voronov, {\em Geometric integration theory on supermanifolds.} Harwood Publ., 1991. \end{thebibliography};
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