Super manifolds: an incomplete survey
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. |
This page has not been refereed. The information given here might be incomplete or provisional. |
\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
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 aTex syntax error
-graded algebra. For example, the endomorphism ring of a super vector space inherits a naturalTex syntax error
-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
Tex syntax error
-degree of a homogenous vector . ForTex syntax error
-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 .
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 satisfying the Leibniz rule: \footnote{Whenever we write formulas involving the degree of certain elements, we implicitly assume that these elements are homogenous.} \[ D(a\cdot b) = Da \cdot b + (-1)^{|D| |a|} a \cdot Db. \]
Tex syntax erroris a super Lie algebra with respect to the bracket operation \[ [D,E] := DE - (-1)^{|D| |E|}ED \] This means that the following axioms are satisfied for
Tex syntax error. \begin{defn} A {\em super Lie algebra} is a super vector space together with a {\em Lie bracket} that is skew symmetric
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
Tex syntax error
isTex syntax error
equipped with the sheafTex syntax error
of commutative superTex syntax error
-algebrasTex syntax error
. \begin{defn} A super manifold of dimensionTex syntax error
is a pairTex syntax error
consisting of a (Hausdorff and second countable) topological spaceTex syntax error
together with a sheaf of commutative superTex syntax error
-algebrasTex syntax error
that is locally isomorphic toTex syntax error
. A morphismTex syntax error
between super manifoldsTex syntax error
is defined to be a continuous mapTex syntax error
, together with a mapTex syntax error
of sheaves coveringTex syntax error
. More precisely, for every open subsetTex syntax error
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
Tex syntax error
forTex syntax error
and we denote this category of super manifolds byTex 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 spaceTex syntax error
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
andTex syntax error
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 dimensionTex syntax errorover the ordinary manifold
Tex syntax errorand
Tex syntax errorthe associated algebra bundle of alternating multilinear forms on . Then its sheaf of sections gives a super manifold
Tex syntax errorof dimension
Tex syntax error, denoted by
Tex syntax error. Marjorie Batchelor proved in [Ba] that every super manifold is isomorphic to one of this type. More precisely, let
Tex syntax errordenote the category of real vector bundles over smooth manifolds, and for
Tex syntax error, consider the vector bundle
Tex syntax errorover
Tex syntax errorwith 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
Tex syntax errorbut there are only {\em non-natural} isomorphisms
Tex syntax error, 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
Tex syntax errorinduces 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
Tex syntax errorin
Tex syntax errorthat satisfy \[ (|f_1|(s),...,|f_p|(s)) \in |M| \subseteq \R^{p} \text{ for all } s \in |S|. \] The
Tex syntax error,
Tex syntax errorare called the coordinates of
Tex syntax errordefined by \[ f_i = \phi^*(x_i) \quad \text{ and } \quad \eta_j = \phi^*(\theta_j), \] where
Tex syntax errorare 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
Tex syntax errorodd generators. We need to check that this is representable. We claim that is represented by the open super submanifold characterized by
5 Super Vector Bundles
Tex syntax errorof
Tex syntax error-modules of dimension
Tex syntax error. 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
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
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
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 Tex syntax error-graded algebra. For example, the endomorphism ring of 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 \[ \sigma=\sigma_{V,W}
Tex syntax error-degree of a homogenous vector . For
Tex syntax error-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 .
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 satisfying the Leibniz rule: \footnote{Whenever we write formulas involving the degree of certain elements, we implicitly assume that these elements are homogenous.} \[ D(a\cdot b) = Da \cdot b + (-1)^{|D| |a|} a \cdot Db. \]
Tex syntax erroris a super Lie algebra with respect to the bracket operation \[ [D,E] := DE - (-1)^{|D| |E|}ED \] This means that the following axioms are satisfied for
Tex syntax error. \begin{defn} A {\em super Lie algebra} is a super vector space together with a {\em Lie bracket} that is skew symmetric
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
Tex syntax error
isTex syntax error
equipped with the sheafTex syntax error
of commutative superTex syntax error
-algebrasTex syntax error
. \begin{defn} A super manifold of dimensionTex syntax error
is a pairTex syntax error
consisting of a (Hausdorff and second countable) topological spaceTex syntax error
together with a sheaf of commutative superTex syntax error
-algebrasTex syntax error
that is locally isomorphic toTex syntax error
. A morphismTex syntax error
between super manifoldsTex syntax error
is defined to be a continuous mapTex syntax error
, together with a mapTex syntax error
of sheaves coveringTex syntax error
. More precisely, for every open subsetTex syntax error
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
Tex syntax error
forTex syntax error
and we denote this category of super manifolds byTex 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 spaceTex syntax error
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
andTex syntax error
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 dimensionTex syntax errorover the ordinary manifold
Tex syntax errorand
Tex syntax errorthe associated algebra bundle of alternating multilinear forms on . Then its sheaf of sections gives a super manifold
Tex syntax errorof dimension
Tex syntax error, denoted by
Tex syntax error. Marjorie Batchelor proved in [Ba] that every super manifold is isomorphic to one of this type. More precisely, let
Tex syntax errordenote the category of real vector bundles over smooth manifolds, and for
Tex syntax error, consider the vector bundle
Tex syntax errorover
Tex syntax errorwith 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
Tex syntax errorbut there are only {\em non-natural} isomorphisms
Tex syntax error, 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
Tex syntax errorinduces 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
Tex syntax errorin
Tex syntax errorthat satisfy \[ (|f_1|(s),...,|f_p|(s)) \in |M| \subseteq \R^{p} \text{ for all } s \in |S|. \] The
Tex syntax error,
Tex syntax errorare called the coordinates of
Tex syntax errordefined by \[ f_i = \phi^*(x_i) \quad \text{ and } \quad \eta_j = \phi^*(\theta_j), \] where
Tex syntax errorare 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
Tex syntax errorodd generators. We need to check that this is representable. We claim that is represented by the open super submanifold characterized by
5 Super Vector Bundles
Tex syntax errorof
Tex syntax error-modules of dimension
Tex syntax error. 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
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
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
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}
Tex syntax error-graded algebra. For example, the endomorphism ring of 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 \[ \sigma=\sigma_{V,W}
Tex syntax error-degree of a homogenous vector . For
Tex syntax error-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 .
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 satisfying the Leibniz rule: \footnote{Whenever we write formulas involving the degree of certain elements, we implicitly assume that these elements are homogenous.} \[ D(a\cdot b) = Da \cdot b + (-1)^{|D| |a|} a \cdot Db. \]
Tex syntax erroris a super Lie algebra with respect to the bracket operation \[ [D,E] := DE - (-1)^{|D| |E|}ED \] This means that the following axioms are satisfied for
Tex syntax error. \begin{defn} A {\em super Lie algebra} is a super vector space together with a {\em Lie bracket} that is skew symmetric
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
Tex syntax error
isTex syntax error
equipped with the sheafTex syntax error
of commutative superTex syntax error
-algebrasTex syntax error
. \begin{defn} A super manifold of dimensionTex syntax error
is a pairTex syntax error
consisting of a (Hausdorff and second countable) topological spaceTex syntax error
together with a sheaf of commutative superTex syntax error
-algebrasTex syntax error
that is locally isomorphic toTex syntax error
. A morphismTex syntax error
between super manifoldsTex syntax error
is defined to be a continuous mapTex syntax error
, together with a mapTex syntax error
of sheaves coveringTex syntax error
. More precisely, for every open subsetTex syntax error
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
Tex syntax error
forTex syntax error
and we denote this category of super manifolds byTex 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 spaceTex syntax error
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
andTex syntax error
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 dimensionTex syntax errorover the ordinary manifold
Tex syntax errorand
Tex syntax errorthe associated algebra bundle of alternating multilinear forms on . Then its sheaf of sections gives a super manifold
Tex syntax errorof dimension
Tex syntax error, denoted by
Tex syntax error. Marjorie Batchelor proved in [Ba] that every super manifold is isomorphic to one of this type. More precisely, let
Tex syntax errordenote the category of real vector bundles over smooth manifolds, and for
Tex syntax error, consider the vector bundle
Tex syntax errorover
Tex syntax errorwith 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
Tex syntax errorbut there are only {\em non-natural} isomorphisms
Tex syntax error, 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
Tex syntax errorinduces 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
Tex syntax errorin
Tex syntax errorthat satisfy \[ (|f_1|(s),...,|f_p|(s)) \in |M| \subseteq \R^{p} \text{ for all } s \in |S|. \] The
Tex syntax error,
Tex syntax errorare called the coordinates of
Tex syntax errordefined by \[ f_i = \phi^*(x_i) \quad \text{ and } \quad \eta_j = \phi^*(\theta_j), \] where
Tex syntax errorare 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
Tex syntax errorodd generators. We need to check that this is representable. We claim that is represented by the open super submanifold characterized by
5 Super Vector Bundles
Tex syntax errorof
Tex syntax error-modules of dimension
Tex syntax error. 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
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
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
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