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{Berezin1987}, \cite{Manin1988} or \cite{Voronov1991}. The material below is a prerequisite to our papers \cite{Hohnhold&Kreck&Stolz&Teichner2010} and \cite{Stolz&Teichner2008}. | |
− | We present an incomplete survey on some basic notions of super manifolds which may serve as a short introduction to this subject. | + | |
==Super Algebra== | ==Super Algebra== | ||
− | <wikitex include="TeXInclude:Super_manifolds: | + | <wikitex include="TeXInclude:Super_manifolds:_an_incomplete_survey">; |
− | Let us begin by explaining briefly what ''super'' means in an algebraic context, working with the ground field of real numbers. The monoidal category of ''super vector spaces'', with tensor products, is by definition the same as the monoidal category of $\Z/2$-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 $\Z/2$-graded algebra. For example, the endomorphism ring $\End(V)$ of a super vector space $V$ inherits a natural $\Z/2$-grading from that of $V$. The distinction between these notions only arises from the choice of symmetry operators | + | Let us begin by explaining briefly what ''super'' means in an algebraic context, working with the ground field of real numbers. The monoidal category of ''super vector spaces'', with tensor products, is by definition the same as the monoidal category of $\Z/2$-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 $\Z/2$-graded algebra. For example, the endomorphism ring $\End(V)$ of a super vector space $V$ inherits a natural $\Z/2$-grading from that of $V$. The distinction between these notions only arises from the choice of symmetry operators |
$$ \sigma=\sigma_{V,W} : V\otimes W \overset{\cong}{\longrightarrow} W \otimes V.$$ | $$ \sigma=\sigma_{V,W} : V\otimes W \overset{\cong}{\longrightarrow} W \otimes V.$$ | ||
− | There are two standard choices, yielding two very different ''symmetric'' monoidal categories. For super vector spaces one has $$ \sigma(v\otimes w) = (-1)^{|v|\cdot |w|} w\otimes v , $$ where $|v|$ is the $\Z/2$-degree of a homogenous vector $v\in V$. For $\Z/2$-graded vector spaces the signs would be omitted. This basic difference is sometimes summarized as the | + | There are two standard choices, yielding two very different ''symmetric'' monoidal categories. For super vector spaces one has |
+ | $$ \sigma(v\otimes w) = (-1)^{|v|\cdot |w|} w\otimes v , $$ | ||
+ | where $|v|$ is the $\Z/2$-degree of a homogenous vector $v\in V$. For $\Z/2$-graded vector spaces the signs would be omitted. This basic difference is sometimes summarized as the | ||
$\bullet$ '''Sign rule:''' Commuting two odd quantities yields a sign $-1$. | $\bullet$ '''Sign rule:''' Commuting two odd quantities yields a sign $-1$. | ||
− | As a consequence, a super algebra is ''commutative'' if for all homogenenous $a,b \in A$ we have $$ ab = (-1)^{|a| |b|} ba , $$ a very different notion than a commutative $\Z/2$-graded algebra. The standard examples of commutative super algebras are the exterior algebras $\Lambda^*(\R^q)$. As we shall see, the generators of $\Lambda^*(\R^q)$ yield 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 $X$, the cohomology ring $H^*(X;\R)$ is a commutative super algebra. | + | As a consequence, a super algebra is ''commutative'' if for all homogenenous $a,b \in A$ we have $$ ab = (-1)^{|a| |b|} ba , $$ a very different notion than a commutative $\Z/2$-graded algebra. The standard examples of commutative super algebras are the exterior algebras $\Lambda^*(\R^q)$. As we shall see, the generators of $\Lambda^*(\R^q)$ yield 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 $X$, the cohomology ring $H^*(X;\R)$ is a commutative super algebra. |
− | Let $A$ be a commutative super algebra. The ''derivations'' of $A$ are endomorphisms $D \in \End(A)$ satisfying the Leibniz rule: <ref>Whenever we write formulas involving the degree | + | Let $A$ be a commutative super algebra. The ''derivations'' of $A$ are endomorphisms $D \in \End(A)$ satisfying the Leibniz rule: |
+ | </wikitex><ref>Whenever we write formulas involving the degree <tex>|.|</tex> of certain elements, we implicitly assume that these elements are homogenous.</ref><wikitex> | ||
$$ D(a\cdot b) = Da \cdot b + (-1)^{|D| |a|} a \cdot Db. $$ | $$ D(a\cdot b) = Da \cdot b + (-1)^{|D| |a|} a \cdot Db. $$ | ||
− | $\Der A$ is 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 $L=\Der A$. | + | $\Der A$ is 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 $L=\Der A$. |
− | {{beginrem|Definition}}<label>def:sLie</label> A ''super Lie algebra'' is a super vector space $L$ together with a ''Lie bracket'' $[\cdot, \cdot]: L\otimes L\to L$ that is skew symmetric $$ [ D,E ] + (-1)^{|D| |E|} [ E,D ] = 0$$ and satisfies the ''Jacobi identity'' | + | {{beginrem|Definition}}<label>def:sLie</label> A ''super Lie algebra'' is a super vector space $L$ together with a ''Lie bracket'' $[\cdot, \cdot]: L\otimes L\to L$ that is skew symmetric $$ [ D,E ] + (-1)^{|D| |E|} [ E,D ] = 0$$ and satisfies the ''Jacobi identity'' |
$$ [D, [ E,F ]] + (-1)^{|D| (|E| + |F|)} [ E, [F,D ]] + (-1)^{|F| (|D| + |E|)} [ F, [ D,E ]] = 0.$$ | $$ [D, [ E,F ]] + (-1)^{|D| (|E| + |F|)} [ E, [F,D ]] + (-1)^{|F| (|D| + |E|)} [ F, [ D,E ]] = 0.$$ | ||
Note that we cyclically permuted the 3 symbols and put down the signs according to the above sign rule. | Note that we cyclically permuted the 3 symbols and put down the signs according to the above sign rule. | ||
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==Super Manifolds== | ==Super Manifolds== | ||
− | <wikitex include="TeXInclude:Super_manifolds: | + | <wikitex include="TeXInclude:Super_manifolds:_an_incomplete_survey">; |
− | We will define super manifolds as ringed spaces following \cite{Deligne&Morgan1999}. By a morphism we will always mean a map of ringed spaces. The local model for a super manifold of dimension $p|q$ is $\R^p$ equipped with the sheaf $\cO_{\R^{p|q}}$ of commutative super $\R$-algebras $U \mapsto C^\infty(U) \otimes \Lambda^*(\R^q)$. | + | We will define super manifolds as ringed spaces following \cite{Deligne&Morgan1999}. By a morphism we will always mean a map of ringed spaces. The local model for a super manifold of dimension $p|q$ is $\R^p$ equipped with the sheaf $\cO_{\R^{p|q}}$ of commutative super $\R$-algebras $U \mapsto C^\infty(U) \otimes \Lambda^*(\R^q)$. |
− | {{beginrem|Definition}} A ''super manifold'' $M$ of dimension $p|q$ is a pair $(|M|,\cO_M)$ consisting of a (Hausdorff and second countable) topological space $|M|$ together with a sheaf of commutative super $\R$-algebras $\cO_M$ that is locally isomorphic to $(\R^{p},\cO_{\R^{p|q}})$. A morphism $f=(|f|,F)$ between super manifolds $M,N$ is defined to be a continuous map $|f|:|M|\to |N|$, together with a map $F$ of sheaves covering $|f|$. More precisely, for every open subset $U\subseteq |N|$ 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 $f^*$ for $F$ and we denote this category of super manifolds by $\SMan$. | + | {{beginrem|Definition}} A ''super manifold'' $M$ of dimension $p|q$ is a pair $(|M|,\cO_M)$ consisting of a (Hausdorff and second countable) topological space $|M|$ together with a sheaf of commutative super $\R$-algebras $\cO_M$ that is locally isomorphic to $(\R^{p},\cO_{\R^{p|q}})$. A morphism $f=(|f|,F)$ between super manifolds $M,N$ is defined to be a continuous map $|f|:|M|\to |N|$, together with a map $F$ of sheaves covering $|f|$. More precisely, for every open subset $U\subseteq |N|$ 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 $f^*$ for $F$ and we denote this category of super manifolds by $\SMan$. |
{{endrem}} | {{endrem}} | ||
− | To every super manifold $M$ there is an associated ''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 $|M|$ and there is an inclusion of super manifolds $M^{red} \into M$. Note that the sheaf of ideals $\Nil \subset \cO_M$ is generated by odd functions. Other geometric super objects can be defined in a similar way. For example, replacing $\R$ by $\C$ and $C^\infty$ 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 $\C$-algebras that locally look like $\cO_{\R^{p|q}}\otimes \C$. 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. | + | To every super manifold $M$ there is an associated ''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 $|M|$ and there is an inclusion of super manifolds $M^{red} \into M$. Note that the sheaf of ideals $\Nil \subset \cO_M$ is generated by odd functions. Other geometric super objects can be defined in a similar way. For example, replacing $\R$ by $\C$ and $C^\infty$ 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 $\C$-algebras that locally look like $\cO_{\R^{p|q}}\otimes \C$. 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. |
{{beginrem|Example}} | {{beginrem|Example}} | ||
− | Let $E$ be a real vector bundle of fiber dimension $q$ over the ordinary manifold $X^p$ and $\Lambda^*(E^*)$ the associated algebra bundle of alternating multilinear forms on $E$. Then its sheaf of sections gives a super manifold $(X, \Gamma (\Lambda^*E^*))$ of dimension $p|q$, denoted by $\Pi E$. In the current ''smooth'' setting, Marjorie Batchelor proved in \cite{Batchelor1979} that every super manifold is isomorphic to one of this type (this is not true for analytic super manifolds). More precisely, let $\BM$ denote the category of real vector bundles over smooth manifolds, and for $M\in \SMan$, consider the vector bundle $J(M)$ over $M^{red}$ with sheaf of sections $\Nil / \Nil^2$. Then the functors | + | Let $E$ be a real vector bundle of fiber dimension $q$ over the ordinary manifold $X^p$ and $\Lambda^*(E^*)$ the associated algebra bundle of alternating multilinear forms on $E$. Then its sheaf of sections gives a super manifold $(X, \Gamma (\Lambda^*E^*))$ of dimension $p|q$, denoted by $\Pi E$. In the current ''smooth'' setting, Marjorie Batchelor proved in \cite{Batchelor1979} that every super manifold is isomorphic to one of this type (this is not true for analytic super manifolds). More precisely, let $\BM$ denote the category of real vector bundles over smooth manifolds, and for $M\in \SMan$, consider the vector bundle $J(M)$ over $M^{red}$ with sheaf of sections $\Nil / \Nil^2$. Then the functors |
$$ \Pi : \BM \to \SMan \quad \text{ and } \quad J: \SMan \to \BM$$ | $$ \Pi : \BM \to \SMan \quad \text{ and } \quad J: \SMan \to \BM$$ | ||
− | come equipped with natural isomorphisms $J \circ \Pi(E) \cong E$ but there are only ''non-natural'' isomorphisms $\Pi\circ J(M) \cong M$, 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 $\SMan$ than the linear bundle maps coming from $\BM$. | + | come equipped with natural isomorphisms $J \circ \Pi(E) \cong E$ but there are only ''non-natural'' isomorphisms $\Pi\circ J(M) \cong M$, 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 $\SMan$ than the linear bundle maps coming from $\BM$. |
{{endrem}} | {{endrem}} | ||
− | The following proposition gives two extremely useful ways of looking at morphisms between super manifolds. We shall use the notation $C^\infty(M):=\cO_M(M)$ for the algebra of (global) functions on a super manifold $M$. | + | The following proposition gives two extremely useful ways of looking at morphisms between super manifolds. We shall use the notation $C^\infty(M):=\cO_M(M)$ for the algebra of (global) functions on a super manifold $M$. |
{{beginthm|Proposition|}}<label>prop:coords</label> | {{beginthm|Proposition|}}<label>prop:coords</label> | ||
For $S,M\in \SMan$, the functor $C^\infty$ induces natural bijections | For $S,M\in \SMan$, the functor $C^\infty$ induces natural bijections | ||
$$ \SMan(S,M) \cong \Alg(C^\infty(M), C^\infty(S)).$$ | $$ \SMan(S,M) \cong \Alg(C^\infty(M), C^\infty(S)).$$ | ||
− | If $M \subseteq \R^{p|q}$ is an open super submanifold (a $\text{domain}$), $\SMan(S,M)$ is in bijective correspondence with those $(f_1,...,f_p,\eta_1,...,\eta_q)$ in $(C^\infty(S)^{ev})^p \times (C^\infty(S)^{odd})^q$ that satisfy | + | If $M \subseteq \R^{p|q}$ is an open super submanifold (a $\text{domain}$), $\SMan(S,M)$ is in bijective correspondence with those $(f_1,...,f_p,\eta_1,...,\eta_q)$ in $(C^\infty(S)^{ev})^p \times (C^\infty(S)^{odd})^q$ that satisfy |
$$ (|f_1|(s),...,|f_p|(s)) \in |M| \subseteq \R^{p} \text{ for all } s \in |S|. $$ | $$ (|f_1|(s),...,|f_p|(s)) \in |M| \subseteq \R^{p} \text{ for all } s \in |S|. $$ | ||
− | The $f_i$, $\eta_j$ are called the coordinates of $\phi\in \SMan(S,M)$ defined by $$ f_i = \phi^*(x_i) \quad \text{ and } \quad \eta_j = \phi^*(\theta_j), $$ | + | The $f_i$, $\eta_j$ are called the coordinates of $\phi\in \SMan(S,M)$ defined by $$ f_i = \phi^*(x_i) \quad \text{ and } \quad \eta_j = \phi^*(\theta_j), $$ |
− | where $x_1,...,x_p,\theta_q,...,\theta_p$ are coordinates on $M\subseteq \R^{p|q}$. Moreover, by the first part we see that $f_i \in C^\infty(S)^{ev} = \SMan(S,\R)$ and hence $|f_i| \in \Man(|S|, \R)$. | + | where $x_1,...,x_p,\theta_q,...,\theta_p$ are coordinates on $M\subseteq \R^{p|q}$. Moreover, by the first part we see that $f_i \in C^\infty(S)^{ev} = \SMan(S,\R)$ and hence $|f_i| \in \Man(|S|, \R)$. |
{{endthm}} | {{endthm}} | ||
− | 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 <nowiki>[</nowiki>[[#{{anchorencode:Leites1980}}|Leites1980]]<nowiki>]</nowiki>{{#RefAdd:Leites1980}}. | + | 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 <nowiki>[</nowiki>[[#{{anchorencode:Leites1980}}|Leites1980]]<nowiki>]</nowiki>{{#RefAdd:Leites1980}}. |
</wikitex> | </wikitex> | ||
==The Functor of Points== | ==The Functor of Points== | ||
− | <wikitex include=" | + | <wikitex include="Super_manifolds:_an_incomplete_survey">; <label>sec:functor</label> |
− | Since sheaves are generally difficult to work with, one often thinks of super manifolds in terms of their $S$-points, i.e. instead of $M$ itself one considers the morphism sets $\SMan(S,M)$, where $S$ varies over all super manifolds $S$. More formally, embed the category $\SMan$ of super manifolds in the category of contravariant functors from $\SMan$ to $\Sset$ by | + | Since sheaves are generally difficult to work with, one often thinks of super manifolds in terms of their $S$-points, i.e. instead of $M$ itself one considers the morphism sets $\SMan(S,M)$, where $S$ varies over all super manifolds $S$. More formally, embed the category $\SMan$ of super manifolds in the category of contravariant functors from $\SMan$ to $\Sset$ by |
$$ Y:\SMan \ra \Fun(\SMan^{op},\Sset), \quad Y(M) = ( \> S \mapsto \SMan(S,M) \> ). $$ | $$ Y:\SMan \ra \Fun(\SMan^{op},\Sset), \quad Y(M) = ( \> S \mapsto \SMan(S,M) \> ). $$ | ||
− | This Yoneda embedding is fully faithful and identifies $\SMan$ with the the category of ''representable'' functors, defined to be those in the image of $Y$. We will sometimes refer to an arbitrary functor $F:\SMan^{op}\to \Sset$ as a ''generalized super manifold''. Note that Proposition \ref{prop:coords} makes it easy to describe the morphism sets $\SMan (S,M)$. 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 $S$ as opposed to those of $M$) in many physics papers. | + | This Yoneda embedding is fully faithful and identifies $\SMan$ with the the category of ''representable'' functors, defined to be those in the image of $Y$. We will sometimes refer to an arbitrary functor $F:\SMan^{op}\to \Sset$ as a ''generalized super manifold''. Note that Proposition \ref{prop:coords} makes it easy to describe the morphism sets $\SMan (S,M)$. 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 $S$ as opposed to those of $M$) in many physics papers. |
</wikitex> | </wikitex> | ||
==Super Lie Groups== | ==Super Lie Groups== | ||
− | <wikitex include="TeXInclude:Super_manifolds: | + | <wikitex include="TeXInclude:Super_manifolds:_an_incomplete_survey">; |
− | These are simply group objects in $\SMan$. According to the functor of points approach, such a group object in $\SMan$ can be described by giving a functor $G : \SMan^{op} \to \Group$ such that the composition with the forgetful functor $\Group \to \Sset$ is representable. | + | These are simply group objects in $\SMan$. According to the functor of points approach, such a group object in $\SMan$ can be described by giving a functor $G : \SMan^{op} \to \Group$ such that the composition with the forgetful functor $\Group \to \Sset$ is representable. |
{{beginrem|Example|}} <label>ex:Lie</label> | {{beginrem|Example|}} <label>ex:Lie</label> | ||
− | The simplest super Lie group is the additive group structure on $\R^{p|q}$. It is given by the following composition law on $\SMan(S,\R^{p|q})$, obviously natural in $S$: | + | The simplest super Lie group is the additive group structure on $\R^{p|q}$. It is given by the following composition law on $\SMan(S,\R^{p|q})$, obviously natural in $S$: |
$$ (f_1,...,\eta_q) \times (h_1,...,\psi_q) \mapsto (f_1 + h_1,...,\eta_q + \psi_q) . $$ | $$ (f_1,...,\eta_q) \times (h_1,...,\psi_q) \mapsto (f_1 + h_1,...,\eta_q + \psi_q) . $$ | ||
The ''super general linear group'' $GL(p|q)$ is defined by | The ''super general linear group'' $GL(p|q)$ is defined by | ||
$$ GL(p|q)(S) := \text{Aut}_{\cO_S}(\cO_S^{p|q}) \cong \Aut_{C^\infty(S)} (C^\infty(S)^{p|q}) ,$$ | $$ GL(p|q)(S) := \text{Aut}_{\cO_S}(\cO_S^{p|q}) \cong \Aut_{C^\infty(S)} (C^\infty(S)^{p|q}) ,$$ | ||
− | where $A^{p|q}$ denotes the $A$-module freely generated by $p$ even and $q$ odd generators. We need to check that this is representable. We claim that $GL(p|q)(\_)$ is represented by the open super submanifold $G \subset \R^{p^2 + q^2| 2pq}$ characterized by $$ |G| = \{ \> x \in \R^{p^2 + q^2} \> | \> x \in GL_p \times GL_q \> \}.$$ | + | where $A^{p|q}$ denotes the $A$-module freely generated by $p$ even and $q$ odd generators. We need to check that this is representable. We claim that $GL(p|q)(\_)$ is represented by the open super submanifold $G \subset \R^{p^2 + q^2| 2pq}$ characterized by $$ |G| = \{ \> x \in \R^{p^2 + q^2} \> | \> x \in GL_p \times GL_q \> \}.$$ |
− | This follows directly from proposition \ref{prop:coords} using that a map between super algebras is invertible if and only if it is invertible modulo nilpotent elements. | + | This follows directly from proposition \ref{prop:coords} using that a map between super algebras is invertible if and only if it is invertible modulo nilpotent elements. |
{{endrem}} | {{endrem}} | ||
</wikitex> | </wikitex> | ||
==Super Vector Bundles== | ==Super Vector Bundles== | ||
− | <wikitex include="TeXInclude:Super_manifolds: | + | <wikitex include="TeXInclude:Super_manifolds:_an_incomplete_survey">; <label>sec:vb</label> |
A (super) vector bundle over a super manifold $M$ is a locally free sheaf $\mathcal{E}$ of $\cO_M$-modules of dimension $p|q$. The most basic example of a super vector bundle is the ''tangent bundle'' of a super manifold $M^{p|q}$. It is the sheaf of $\cO_M$-modules $\mathcal{T} M$ defined by | A (super) vector bundle over a super manifold $M$ is a locally free sheaf $\mathcal{E}$ of $\cO_M$-modules of dimension $p|q$. The most basic example of a super vector bundle is the ''tangent bundle'' of a super manifold $M^{p|q}$. It is the sheaf of $\cO_M$-modules $\mathcal{T} M$ defined by | ||
$$ \mathcal{T} M(U) := \Der {\cO_M (U)} .$$ | $$ \mathcal{T} M(U) := \Der {\cO_M (U)} .$$ | ||
− | $\mathcal{T} M$ is locally free of dimension $p|q$: If $x_1,...,\theta_q$ are local coordinates on $M$, then a local basis is given by $\partial_{x_1},...,\partial_{\theta_q}$. Note that there is also a linear fibre bundle $TM\to M$ with structure group $GL(p|q)$, where $TM$ is a super manifold of dimension $2p|2q$. More generally, any vector bundle $\mathcal{E}$ over $M$ has a ''total space'' $E\in \SMan$ that comes with a projection map $\pi:E\to M$. It can be most easily described in terms of its $S$-points | + | $\mathcal{T} M$ is locally free of dimension $p|q$: If $x_1,...,\theta_q$ are local coordinates on $M$, then a local basis is given by $\partial_{x_1},...,\partial_{\theta_q}$. Note that there is also a linear fibre bundle $TM\to M$ with structure group $GL(p|q)$, where $TM$ is a super manifold of dimension $2p|2q$. More generally, any vector bundle $\mathcal{E}$ over $M$ has a ''total space'' $E\in \SMan$ that comes with a projection map $\pi:E\to M$. It can be most easily described in terms of its $S$-points |
$$ E(S)= \{ (f,g) \ \mid\ f\in\SMan(S,M), g \in f^*(\mathcal{E}^{ev}(M)) \}. $$ | $$ E(S)= \{ (f,g) \ \mid\ f\in\SMan(S,M), g \in f^*(\mathcal{E}^{ev}(M)) \}. $$ | ||
− | So $g$ is an even global section of the pullback bundle on $S$ and the projection $\pi$ comes from forgetting this datum. To prove that this functor $\SMan^{op}\to \Sset$ is representable one uses the local triviality of $\mathcal{E}$ and Proposition \ref{prop:coords}. It follows by construction that the typical fibre of the projection $\pi$ is $\R^{p|q}$ and the structure group is $GL(p|q)$. | + | So $g$ is an even global section of the pullback bundle on $S$ and the projection $\pi$ comes from forgetting this datum. To prove that this functor $\SMan^{op}\to \Sset$ is representable one uses the local triviality of $\mathcal{E}$ and Proposition \ref{prop:coords}. It follows by construction that the typical fibre of the projection $\pi$ is $\R^{p|q}$ and the structure group is $GL(p|q)$. |
− | There is an important operation of ''parity reversal'' on the category of vector bundles over $M$. It is an involution $$ \Pi : \Vect_M\ra \Vect_M $$ that takes a vector bundle $E$ with grading involution $\alpha$ to $(E,-\alpha)$. This means that even and odd parts are exchanged. To define $\Pi$ on morphisms it is easiest to give it as $\Pi(E)=\epsilon_{0|1}\otimes E$, where $\epsilon_{0|1}$ is the trivial bundle of dimension $0|1$ (aka the constant sheaf of free $\cO_M$-modules). | + | There is an important operation of ''parity reversal'' on the category of vector bundles over $M$. It is an involution $$ \Pi : \Vect_M\ra \Vect_M $$ that takes a vector bundle $E$ with grading involution $\alpha$ to $(E,-\alpha)$. This means that even and odd parts are exchanged. To define $\Pi$ on morphisms it is easiest to give it as $\Pi(E)=\epsilon_{0|1}\otimes E$, where $\epsilon_{0|1}$ is the trivial bundle of dimension $0|1$ (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 $\xi \in \mathcal{T} G$ is called ''left-invariant'' if $\xi$ is related to itself under the left-translation by all $f: S \to G$: | + | One can define the super Lie algebra $\fg$ of a super Lie group $G$ as follows. A vector field $\xi \in \mathcal{T} G$ is called ''left-invariant'' if $\xi$ is related to itself under the left-translation by all $f: S \to G$: |
$$ S \times G \stackrel{f \times \id}{\longrightarrow} G \times G \overset{\mu} \longrightarrow G .$$ | $$ S \times G \stackrel{f \times \id}{\longrightarrow} G \times G \overset{\mu} \longrightarrow G .$$ | ||
− | 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 $p|q$. | + | 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 $p|q$. |
</wikitex> | </wikitex> | ||
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[[Category:Theory]] | [[Category:Theory]] | ||
+ | [[Category:Forgotten in Textbooks]] |
Latest revision as of 17:05, 20 July 2017
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We present an incomplete survey on some basic notions of super manifolds which may serve as a short introduction to this subject. Almost all the material is taken from the beautiful survey article on super manifolds [Deligne&Morgan1999]. Standard references also include [Leites1980], [Berezin1987], [Manin1988] or [Voronov1991]. The material below is a prerequisite to our papers [Hohnhold&Kreck&Stolz&Teichner2010] and [Stolz&Teichner2008].
Contents |
1 Super Algebra
Let us begin by explaining briefly what super means in an algebraic context, working with the ground field of real numbers. The monoidal category of super vector spaces, with tensor products, is by definition the same as the monoidal category of -graded vector spaces, with the graded tensor product. As a consequence, a super algebra is simply a monoidal object in this category and is hence the same thing as a
-graded algebra. For example, the endomorphism ring
of a super vector space
inherits a natural
-grading from that of
. The distinction between these notions only arises from the choice of symmetry operators
![\displaystyle \sigma=\sigma_{V,W} : V\otimes W \overset{\cong}{\longrightarrow} W \otimes V.](/images/math/b/5/2/b5225adb50a186e30cbf2c87e4307b45.png)
There are two standard choices, yielding two very different symmetric monoidal categories. For super vector spaces one has
![\displaystyle \sigma(v\otimes w) = (-1)^{|v|\cdot |w|} w\otimes v ,](/images/math/9/0/b/90b2b2d5e7ce6e31da08d2a17e358b95.png)
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
.
![a,b \in A](/images/math/b/b/0/bb073fbf397263f5084987bb9f914301.png)
![\displaystyle ab = (-1)^{|a| |b|} ba ,](/images/math/2/d/7/2d7c5aafcef828633f287a675f62ae99.png)
![\Z/2](/images/math/8/0/f/80fcb476110ebdddc90b1c369907d774.png)
![\Lambda^*(\R^q)](/images/math/e/8/1/e818262f1a90c11020608836480b5a0e.png)
![\Lambda^*(\R^q)](/images/math/e/8/1/e818262f1a90c11020608836480b5a0e.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![H^*(X;\R)](/images/math/5/5/8/55824b1d976b061f9c1f90dc95038a81.png)
Let be a commutative super algebra. The derivations of
are endomorphisms
satisfying the Leibniz rule: [1]
![\displaystyle D(a\cdot b) = Da \cdot b + (-1)^{|D| |a|} a \cdot Db.](/images/math/d/9/3/d93a69e37edea871d12e70d49818ea35.png)
![\Der A](/images/math/0/4/4/04469f8c80f959e7b5d42016524a8500.png)
![\displaystyle [D,E] := DE - (-1)^{|D| |E|}ED](/images/math/f/a/b/fabce79898f93e07c6541fe82853532a.png)
![L=\Der A](/images/math/0/a/5/0a54b497d5bb97843b2d8fb16bc96a89.png)
![L](/images/math/9/3/4/934cf4f6e8d65e941a602d24451533b6.png)
![[\cdot, \cdot]: L\otimes L\to L](/images/math/6/1/5/615ad7472d37b5dff1fb18d6b71dbc62.png)
![\displaystyle [ D,E ] + (-1)^{|D| |E|} [ E,D ] = 0](/images/math/6/c/1/6c1f534b9f85a41f1abe28df4b859fc4.png)
![\displaystyle [D, [ E,F ]] + (-1)^{|D| (|E| + |F|)} [ E, [F,D ]] + (-1)^{|F| (|D| + |E|)} [ F, [ D,E ]] = 0.](/images/math/f/5/9/f595128d405d34d21d951cb31b74b3c6.png)
Note that we cyclically permuted the 3 symbols and put down the signs according to the above sign rule.
2 Super Manifolds
We will define super manifolds as ringed spaces following [Deligne&Morgan1999]. By a morphism we will always mean a map of ringed spaces. The local model for a super manifold of dimension is
equipped with the sheaf
of commutative super
-algebras
.
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![p|q](/images/math/5/2/0/520f8c4c02bc48931b2e0c4fc1333758.png)
![(|M|,\cO_M)](/images/math/9/6/b/96b7d4143994651dfca0ed82a4557998.png)
![|M|](/images/math/b/2/8/b28417c7d7758277bf6c14675ba97e64.png)
![\R](/images/math/d/8/3/d8312bf230adacd916d047420a399944.png)
![\cO_M](/images/math/8/c/f/8cf9b61f635637652a23517889d233a9.png)
![(\R^{p},\cO_{\R^{p|q}})](/images/math/c/6/2/c625a39b9ae3c2e93b1bbdb4b9f1513e.png)
![f=(|f|,F)](/images/math/e/b/d/ebdf15dc3eafd35fec4aee8efcb3dd7b.png)
![M,N](/images/math/1/8/9/18955b9072fcd747963b5d25257dc4b9.png)
![|f|:|M|\to |N|](/images/math/7/2/2/722aef6fbf7b8799c6ebf6a4290d2309.png)
![F](/images/math/7/9/8/79851a1fc5f19464a229ccdf66c8beb2.png)
![|f|](/images/math/2/8/a/28ab75c8a710b11c4496377a5a5d8158.png)
![U\subseteq |N|](/images/math/7/b/7/7b729af206e273173e38239e205c8c56.png)
![\displaystyle F(U): \cO_N(U) \ra \cO_M(|f|^{-1}(U))](/images/math/6/a/c/6acc2644f70770f9890f42df0e4c729a.png)
![f^*](/images/math/d/4/6/d4676db1e00cb90d2833cf440d088bea.png)
![F](/images/math/7/9/8/79851a1fc5f19464a229ccdf66c8beb2.png)
![\SMan](/images/math/e/b/8/eb874ffbbcafe98b2bc469dbc0cdf12b.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\displaystyle M^{red}:=(|M|,\cO_M / \Nil)](/images/math/b/7/a/b7a4eb6d29379e02f01ae4be83b5b5f5.png)
![|M|](/images/math/b/2/8/b28417c7d7758277bf6c14675ba97e64.png)
![M^{red} \into M](/images/math/2/e/5/2e5beb54ba53e730106ee569daf34f8e.png)
![\Nil \subset \cO_M](/images/math/b/5/5/b55761b052cd5c80834c401e39fd5fc1.png)
![\R](/images/math/d/8/3/d8312bf230adacd916d047420a399944.png)
![\C](/images/math/c/2/e/c2ed9023772e6a40d82075d0095fe7bb.png)
![C^\infty](/images/math/3/a/3/3a3602d2e793dd07042cb34d8945b2af.png)
![\C](/images/math/c/2/e/c2ed9023772e6a40d82075d0095fe7bb.png)
![\cO_{\R^{p|q}}\otimes \C](/images/math/f/b/5/fb525bc6a8f0cbc061a215a03e0986c0.png)
Example 2.2.
Let be a real vector bundle of fiber dimension
over the ordinary manifold
and
the associated algebra bundle of alternating multilinear forms on
. Then its sheaf of sections gives a super manifold
of dimension
, denoted by
. In the current smooth setting, Marjorie Batchelor proved in [Batchelor1979] that every super manifold is isomorphic to one of this type (this is not true for analytic super manifolds). More precisely, let
denote the category of real vector bundles over smooth manifolds, and for
, consider the vector bundle
over
with sheaf of sections
. Then the functors
![\displaystyle \Pi : \BM \to \SMan \quad \text{ and } \quad J: \SMan \to \BM](/images/math/3/2/3/3237fc6f36b18fe169b740e07c97baae.png)
come equipped with natural isomorphisms but there are only non-natural isomorphisms
, coming from a choice of a partition of unity. In other words, these functors induce a bijection on isomorphism classes of objects and inclusions on morphisms but they are not equivalences of categories because there are many more morphims in
than the linear bundle maps coming from
.
The following proposition gives two extremely useful ways of looking at morphisms between super manifolds. We shall use the notation for the algebra of (global) functions on a super manifold
.
Proposition 2.3.
For , the functor
induces natural bijections
![\displaystyle \SMan(S,M) \cong \Alg(C^\infty(M), C^\infty(S)).](/images/math/b/0/b/b0b8080d1848f8e9eccc745a0b106d16.png)
If is an open super submanifold (a
),
is in bijective correspondence with those
in
that satisfy
![\displaystyle (|f_1|(s),...,|f_p|(s)) \in |M| \subseteq \R^{p} \text{ for all } s \in |S|.](/images/math/f/4/4/f44d9e444b5457b8db8ec2648bc4c0bb.png)
![f_i](/images/math/3/8/a/38a54257d1bd19d28f5fff8954fea1e7.png)
![\eta_j](/images/math/a/8/b/a8bbd4ba517c703e75842dacf42fc56f.png)
![\phi\in \SMan(S,M)](/images/math/d/6/b/d6bd5e80f8fe0c7712e970a5fa743452.png)
![\displaystyle f_i = \phi^*(x_i) \quad \text{ and } \quad \eta_j = \phi^*(\theta_j),](/images/math/0/7/b/07b2484f5ba6bbd1eabd344265819462.png)
where are coordinates on
. Moreover, by the first part we see that
and hence
.
The proof of the first part is based on the existence of partitions of unity for super manifolds, so it is false in analytic settings. The second part always holds and is proved in [Leites1980].
3 The Functor of Points
![S](/images/math/2/0/c/20c08b85a1d0a48a17a99f4d187a66a6.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\SMan(S,M)](/images/math/0/e/f/0ef6b9ddb20854b0bf6f16f5c469319f.png)
![S](/images/math/2/0/c/20c08b85a1d0a48a17a99f4d187a66a6.png)
![S](/images/math/2/0/c/20c08b85a1d0a48a17a99f4d187a66a6.png)
![\SMan](/images/math/e/b/8/eb874ffbbcafe98b2bc469dbc0cdf12b.png)
![\SMan](/images/math/e/b/8/eb874ffbbcafe98b2bc469dbc0cdf12b.png)
Tex syntax errorby
Tex syntax error
![\SMan](/images/math/e/b/8/eb874ffbbcafe98b2bc469dbc0cdf12b.png)
![Y](/images/math/3/9/a/39abfebb66c060cd7541c76ff73c12da.png)
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
![S](/images/math/2/0/c/20c08b85a1d0a48a17a99f4d187a66a6.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
4 Super Lie Groups
These are simply group objects in . According to the functor of points approach, such a group object in
can be described by giving a functor
such that the composition with the forgetful functor
is representable.
Example 4.1.
The simplest super Lie group is the additive group structure on . It is given by the following composition law on
, obviously natural in
:
![\displaystyle (f_1,...,\eta_q) \times (h_1,...,\psi_q) \mapsto (f_1 + h_1,...,\eta_q + \psi_q) .](/images/math/8/e/9/8e9193a234621fa78a07c047769c720e.png)
The super general linear group is defined by
![\displaystyle GL(p|q)(S) := \text{Aut}_{\cO_S}(\cO_S^{p|q}) \cong \Aut_{C^\infty(S)} (C^\infty(S)^{p|q}) ,](/images/math/d/3/e/d3e13682bfd555c359d016564a76860b.png)
![A^{p|q}](/images/math/a/f/b/afb46cd5081daac1ff435bab3a76d756.png)
![A](/images/math/b/8/9/b8921ca1d75b852da96e95cda4aafeb8.png)
![p](/images/math/2/a/0/2a039ed8fdbf4ceaa9e79cdc3aecd1a2.png)
![q](/images/math/e/b/6/eb6af5b4e510c3c874d7d1f51d72393a.png)
![GL(p|q)(\_)](/images/math/0/d/3/0d355a22e098b85bac0633a16bf90e55.png)
![G \subset \R^{p^2 + q^2| 2pq}](/images/math/4/b/4/4b44319c294733beb0b8ff70bae506b2.png)
![\displaystyle |G| = \{ \> x \in \R^{p^2 + q^2} \> | \> x \in GL_p \times GL_q \> \}.](/images/math/2/0/f/20f9ef4022736908ca994d93bbc276ad.png)
This follows directly from proposition 2.3 using that a map between super algebras is invertible if and only if it is invertible modulo nilpotent elements.
5 Super Vector Bundles
A (super) vector bundle over a super manifold is a locally free sheaf
of
-modules of dimension
. The most basic example of a super vector bundle is the tangent bundle of a super manifold
. It is the sheaf of
-modules
defined by
![\displaystyle \mathcal{T} M(U) := \Der {\cO_M (U)} .](/images/math/7/5/0/750439b83645ac409ede0cbfe22c24c2.png)
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
![\displaystyle E(S)= \{ (f,g) \ \mid\ f\in\SMan(S,M), g \in f^*(\mathcal{E}^{ev}(M)) \}.](/images/math/b/9/a/b9a000803dfa9205416d325d3253dd30.png)
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
.
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\displaystyle \Pi : \Vect_M\ra \Vect_M](/images/math/7/3/6/73651532a2279128a69413d3742df4b8.png)
![E](/images/math/e/4/d/e4de1fb0b52291b425e79d56646e3c6e.png)
![\alpha](/images/math/d/a/d/dad2cde42eb00bcf43e15d68d3efabab.png)
![(E,-\alpha)](/images/math/0/d/7/0d77760ecdc9d617efed86632187a8be.png)
![\Pi](/images/math/d/3/5/d355428021402a26ac65f3b260f8485c.png)
![\Pi(E)=\epsilon_{0|1}\otimes E](/images/math/a/5/3/a532ff9a76cf9c084c3fecbd05913400.png)
![\epsilon_{0|1}](/images/math/e/9/1/e9135449197a52098e8ca83ed4171f2a.png)
![0|1](/images/math/f/1/e/f1e8e17bccddf418b572d75ccf2492d3.png)
![\cO_M](/images/math/8/c/f/8cf9b61f635637652a23517889d233a9.png)
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
:
![\displaystyle S \times G \stackrel{f \times \id}{\longrightarrow} G \times G \overset{\mu} \longrightarrow G .](/images/math/4/2/0/420a81fa4dc17e65f028947e15f19f11.png)
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.
![\Z/2](/images/math/8/0/f/80fcb476110ebdddc90b1c369907d774.png)
![\End(V)](/images/math/c/a/1/ca101a9e3551279f76ade450d1e24039.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![\Z/2](/images/math/8/0/f/80fcb476110ebdddc90b1c369907d774.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![\displaystyle \sigma=\sigma_{V,W} : V\otimes W \overset{\cong}{\longrightarrow} W \otimes V.](/images/math/b/5/2/b5225adb50a186e30cbf2c87e4307b45.png)
There are two standard choices, yielding two very different symmetric monoidal categories. For super vector spaces one has
![\displaystyle \sigma(v\otimes w) = (-1)^{|v|\cdot |w|} w\otimes v ,](/images/math/9/0/b/90b2b2d5e7ce6e31da08d2a17e358b95.png)
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
.
![a,b \in A](/images/math/b/b/0/bb073fbf397263f5084987bb9f914301.png)
![\displaystyle ab = (-1)^{|a| |b|} ba ,](/images/math/2/d/7/2d7c5aafcef828633f287a675f62ae99.png)
![\Z/2](/images/math/8/0/f/80fcb476110ebdddc90b1c369907d774.png)
![\Lambda^*(\R^q)](/images/math/e/8/1/e818262f1a90c11020608836480b5a0e.png)
![\Lambda^*(\R^q)](/images/math/e/8/1/e818262f1a90c11020608836480b5a0e.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![H^*(X;\R)](/images/math/5/5/8/55824b1d976b061f9c1f90dc95038a81.png)
Let be a commutative super algebra. The derivations of
are endomorphisms
satisfying the Leibniz rule: [1]
![\displaystyle D(a\cdot b) = Da \cdot b + (-1)^{|D| |a|} a \cdot Db.](/images/math/d/9/3/d93a69e37edea871d12e70d49818ea35.png)
![\Der A](/images/math/0/4/4/04469f8c80f959e7b5d42016524a8500.png)
![\displaystyle [D,E] := DE - (-1)^{|D| |E|}ED](/images/math/f/a/b/fabce79898f93e07c6541fe82853532a.png)
![L=\Der A](/images/math/0/a/5/0a54b497d5bb97843b2d8fb16bc96a89.png)
![L](/images/math/9/3/4/934cf4f6e8d65e941a602d24451533b6.png)
![[\cdot, \cdot]: L\otimes L\to L](/images/math/6/1/5/615ad7472d37b5dff1fb18d6b71dbc62.png)
![\displaystyle [ D,E ] + (-1)^{|D| |E|} [ E,D ] = 0](/images/math/6/c/1/6c1f534b9f85a41f1abe28df4b859fc4.png)
![\displaystyle [D, [ E,F ]] + (-1)^{|D| (|E| + |F|)} [ E, [F,D ]] + (-1)^{|F| (|D| + |E|)} [ F, [ D,E ]] = 0.](/images/math/f/5/9/f595128d405d34d21d951cb31b74b3c6.png)
Note that we cyclically permuted the 3 symbols and put down the signs according to the above sign rule.
2 Super Manifolds
We will define super manifolds as ringed spaces following [Deligne&Morgan1999]. By a morphism we will always mean a map of ringed spaces. The local model for a super manifold of dimension is
equipped with the sheaf
of commutative super
-algebras
.
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![p|q](/images/math/5/2/0/520f8c4c02bc48931b2e0c4fc1333758.png)
![(|M|,\cO_M)](/images/math/9/6/b/96b7d4143994651dfca0ed82a4557998.png)
![|M|](/images/math/b/2/8/b28417c7d7758277bf6c14675ba97e64.png)
![\R](/images/math/d/8/3/d8312bf230adacd916d047420a399944.png)
![\cO_M](/images/math/8/c/f/8cf9b61f635637652a23517889d233a9.png)
![(\R^{p},\cO_{\R^{p|q}})](/images/math/c/6/2/c625a39b9ae3c2e93b1bbdb4b9f1513e.png)
![f=(|f|,F)](/images/math/e/b/d/ebdf15dc3eafd35fec4aee8efcb3dd7b.png)
![M,N](/images/math/1/8/9/18955b9072fcd747963b5d25257dc4b9.png)
![|f|:|M|\to |N|](/images/math/7/2/2/722aef6fbf7b8799c6ebf6a4290d2309.png)
![F](/images/math/7/9/8/79851a1fc5f19464a229ccdf66c8beb2.png)
![|f|](/images/math/2/8/a/28ab75c8a710b11c4496377a5a5d8158.png)
![U\subseteq |N|](/images/math/7/b/7/7b729af206e273173e38239e205c8c56.png)
![\displaystyle F(U): \cO_N(U) \ra \cO_M(|f|^{-1}(U))](/images/math/6/a/c/6acc2644f70770f9890f42df0e4c729a.png)
![f^*](/images/math/d/4/6/d4676db1e00cb90d2833cf440d088bea.png)
![F](/images/math/7/9/8/79851a1fc5f19464a229ccdf66c8beb2.png)
![\SMan](/images/math/e/b/8/eb874ffbbcafe98b2bc469dbc0cdf12b.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\displaystyle M^{red}:=(|M|,\cO_M / \Nil)](/images/math/b/7/a/b7a4eb6d29379e02f01ae4be83b5b5f5.png)
![|M|](/images/math/b/2/8/b28417c7d7758277bf6c14675ba97e64.png)
![M^{red} \into M](/images/math/2/e/5/2e5beb54ba53e730106ee569daf34f8e.png)
![\Nil \subset \cO_M](/images/math/b/5/5/b55761b052cd5c80834c401e39fd5fc1.png)
![\R](/images/math/d/8/3/d8312bf230adacd916d047420a399944.png)
![\C](/images/math/c/2/e/c2ed9023772e6a40d82075d0095fe7bb.png)
![C^\infty](/images/math/3/a/3/3a3602d2e793dd07042cb34d8945b2af.png)
![\C](/images/math/c/2/e/c2ed9023772e6a40d82075d0095fe7bb.png)
![\cO_{\R^{p|q}}\otimes \C](/images/math/f/b/5/fb525bc6a8f0cbc061a215a03e0986c0.png)
Example 2.2.
Let be a real vector bundle of fiber dimension
over the ordinary manifold
and
the associated algebra bundle of alternating multilinear forms on
. Then its sheaf of sections gives a super manifold
of dimension
, denoted by
. In the current smooth setting, Marjorie Batchelor proved in [Batchelor1979] that every super manifold is isomorphic to one of this type (this is not true for analytic super manifolds). More precisely, let
denote the category of real vector bundles over smooth manifolds, and for
, consider the vector bundle
over
with sheaf of sections
. Then the functors
![\displaystyle \Pi : \BM \to \SMan \quad \text{ and } \quad J: \SMan \to \BM](/images/math/3/2/3/3237fc6f36b18fe169b740e07c97baae.png)
come equipped with natural isomorphisms but there are only non-natural isomorphisms
, coming from a choice of a partition of unity. In other words, these functors induce a bijection on isomorphism classes of objects and inclusions on morphisms but they are not equivalences of categories because there are many more morphims in
than the linear bundle maps coming from
.
The following proposition gives two extremely useful ways of looking at morphisms between super manifolds. We shall use the notation for the algebra of (global) functions on a super manifold
.
Proposition 2.3.
For , the functor
induces natural bijections
![\displaystyle \SMan(S,M) \cong \Alg(C^\infty(M), C^\infty(S)).](/images/math/b/0/b/b0b8080d1848f8e9eccc745a0b106d16.png)
If is an open super submanifold (a
),
is in bijective correspondence with those
in
that satisfy
![\displaystyle (|f_1|(s),...,|f_p|(s)) \in |M| \subseteq \R^{p} \text{ for all } s \in |S|.](/images/math/f/4/4/f44d9e444b5457b8db8ec2648bc4c0bb.png)
![f_i](/images/math/3/8/a/38a54257d1bd19d28f5fff8954fea1e7.png)
![\eta_j](/images/math/a/8/b/a8bbd4ba517c703e75842dacf42fc56f.png)
![\phi\in \SMan(S,M)](/images/math/d/6/b/d6bd5e80f8fe0c7712e970a5fa743452.png)
![\displaystyle f_i = \phi^*(x_i) \quad \text{ and } \quad \eta_j = \phi^*(\theta_j),](/images/math/0/7/b/07b2484f5ba6bbd1eabd344265819462.png)
where are coordinates on
. Moreover, by the first part we see that
and hence
.
The proof of the first part is based on the existence of partitions of unity for super manifolds, so it is false in analytic settings. The second part always holds and is proved in [Leites1980].
3 The Functor of Points
![S](/images/math/2/0/c/20c08b85a1d0a48a17a99f4d187a66a6.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\SMan(S,M)](/images/math/0/e/f/0ef6b9ddb20854b0bf6f16f5c469319f.png)
![S](/images/math/2/0/c/20c08b85a1d0a48a17a99f4d187a66a6.png)
![S](/images/math/2/0/c/20c08b85a1d0a48a17a99f4d187a66a6.png)
![\SMan](/images/math/e/b/8/eb874ffbbcafe98b2bc469dbc0cdf12b.png)
![\SMan](/images/math/e/b/8/eb874ffbbcafe98b2bc469dbc0cdf12b.png)
Tex syntax errorby
Tex syntax error
![\SMan](/images/math/e/b/8/eb874ffbbcafe98b2bc469dbc0cdf12b.png)
![Y](/images/math/3/9/a/39abfebb66c060cd7541c76ff73c12da.png)
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
![S](/images/math/2/0/c/20c08b85a1d0a48a17a99f4d187a66a6.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
4 Super Lie Groups
These are simply group objects in . According to the functor of points approach, such a group object in
can be described by giving a functor
such that the composition with the forgetful functor
is representable.
Example 4.1.
The simplest super Lie group is the additive group structure on . It is given by the following composition law on
, obviously natural in
:
![\displaystyle (f_1,...,\eta_q) \times (h_1,...,\psi_q) \mapsto (f_1 + h_1,...,\eta_q + \psi_q) .](/images/math/8/e/9/8e9193a234621fa78a07c047769c720e.png)
The super general linear group is defined by
![\displaystyle GL(p|q)(S) := \text{Aut}_{\cO_S}(\cO_S^{p|q}) \cong \Aut_{C^\infty(S)} (C^\infty(S)^{p|q}) ,](/images/math/d/3/e/d3e13682bfd555c359d016564a76860b.png)
![A^{p|q}](/images/math/a/f/b/afb46cd5081daac1ff435bab3a76d756.png)
![A](/images/math/b/8/9/b8921ca1d75b852da96e95cda4aafeb8.png)
![p](/images/math/2/a/0/2a039ed8fdbf4ceaa9e79cdc3aecd1a2.png)
![q](/images/math/e/b/6/eb6af5b4e510c3c874d7d1f51d72393a.png)
![GL(p|q)(\_)](/images/math/0/d/3/0d355a22e098b85bac0633a16bf90e55.png)
![G \subset \R^{p^2 + q^2| 2pq}](/images/math/4/b/4/4b44319c294733beb0b8ff70bae506b2.png)
![\displaystyle |G| = \{ \> x \in \R^{p^2 + q^2} \> | \> x \in GL_p \times GL_q \> \}.](/images/math/2/0/f/20f9ef4022736908ca994d93bbc276ad.png)
This follows directly from proposition 2.3 using that a map between super algebras is invertible if and only if it is invertible modulo nilpotent elements.
5 Super Vector Bundles
A (super) vector bundle over a super manifold is a locally free sheaf
of
-modules of dimension
. The most basic example of a super vector bundle is the tangent bundle of a super manifold
. It is the sheaf of
-modules
defined by
![\displaystyle \mathcal{T} M(U) := \Der {\cO_M (U)} .](/images/math/7/5/0/750439b83645ac409ede0cbfe22c24c2.png)
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
![\displaystyle E(S)= \{ (f,g) \ \mid\ f\in\SMan(S,M), g \in f^*(\mathcal{E}^{ev}(M)) \}.](/images/math/b/9/a/b9a000803dfa9205416d325d3253dd30.png)
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
.
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\displaystyle \Pi : \Vect_M\ra \Vect_M](/images/math/7/3/6/73651532a2279128a69413d3742df4b8.png)
![E](/images/math/e/4/d/e4de1fb0b52291b425e79d56646e3c6e.png)
![\alpha](/images/math/d/a/d/dad2cde42eb00bcf43e15d68d3efabab.png)
![(E,-\alpha)](/images/math/0/d/7/0d77760ecdc9d617efed86632187a8be.png)
![\Pi](/images/math/d/3/5/d355428021402a26ac65f3b260f8485c.png)
![\Pi(E)=\epsilon_{0|1}\otimes E](/images/math/a/5/3/a532ff9a76cf9c084c3fecbd05913400.png)
![\epsilon_{0|1}](/images/math/e/9/1/e9135449197a52098e8ca83ed4171f2a.png)
![0|1](/images/math/f/1/e/f1e8e17bccddf418b572d75ccf2492d3.png)
![\cO_M](/images/math/8/c/f/8cf9b61f635637652a23517889d233a9.png)
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
:
![\displaystyle S \times G \stackrel{f \times \id}{\longrightarrow} G \times G \overset{\mu} \longrightarrow G .](/images/math/4/2/0/420a81fa4dc17e65f028947e15f19f11.png)
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.
![\Z/2](/images/math/8/0/f/80fcb476110ebdddc90b1c369907d774.png)
![\End(V)](/images/math/c/a/1/ca101a9e3551279f76ade450d1e24039.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![\Z/2](/images/math/8/0/f/80fcb476110ebdddc90b1c369907d774.png)
![V](/images/math/e/7/7/e7702470ff62164890e1229fbdb3419a.png)
![\displaystyle \sigma=\sigma_{V,W} : V\otimes W \overset{\cong}{\longrightarrow} W \otimes V.](/images/math/b/5/2/b5225adb50a186e30cbf2c87e4307b45.png)
There are two standard choices, yielding two very different symmetric monoidal categories. For super vector spaces one has
![\displaystyle \sigma(v\otimes w) = (-1)^{|v|\cdot |w|} w\otimes v ,](/images/math/9/0/b/90b2b2d5e7ce6e31da08d2a17e358b95.png)
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
.
![a,b \in A](/images/math/b/b/0/bb073fbf397263f5084987bb9f914301.png)
![\displaystyle ab = (-1)^{|a| |b|} ba ,](/images/math/2/d/7/2d7c5aafcef828633f287a675f62ae99.png)
![\Z/2](/images/math/8/0/f/80fcb476110ebdddc90b1c369907d774.png)
![\Lambda^*(\R^q)](/images/math/e/8/1/e818262f1a90c11020608836480b5a0e.png)
![\Lambda^*(\R^q)](/images/math/e/8/1/e818262f1a90c11020608836480b5a0e.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![H^*(X;\R)](/images/math/5/5/8/55824b1d976b061f9c1f90dc95038a81.png)
Let be a commutative super algebra. The derivations of
are endomorphisms
satisfying the Leibniz rule: [1]
![\displaystyle D(a\cdot b) = Da \cdot b + (-1)^{|D| |a|} a \cdot Db.](/images/math/d/9/3/d93a69e37edea871d12e70d49818ea35.png)
![\Der A](/images/math/0/4/4/04469f8c80f959e7b5d42016524a8500.png)
![\displaystyle [D,E] := DE - (-1)^{|D| |E|}ED](/images/math/f/a/b/fabce79898f93e07c6541fe82853532a.png)
![L=\Der A](/images/math/0/a/5/0a54b497d5bb97843b2d8fb16bc96a89.png)
![L](/images/math/9/3/4/934cf4f6e8d65e941a602d24451533b6.png)
![[\cdot, \cdot]: L\otimes L\to L](/images/math/6/1/5/615ad7472d37b5dff1fb18d6b71dbc62.png)
![\displaystyle [ D,E ] + (-1)^{|D| |E|} [ E,D ] = 0](/images/math/6/c/1/6c1f534b9f85a41f1abe28df4b859fc4.png)
![\displaystyle [D, [ E,F ]] + (-1)^{|D| (|E| + |F|)} [ E, [F,D ]] + (-1)^{|F| (|D| + |E|)} [ F, [ D,E ]] = 0.](/images/math/f/5/9/f595128d405d34d21d951cb31b74b3c6.png)
Note that we cyclically permuted the 3 symbols and put down the signs according to the above sign rule.
2 Super Manifolds
We will define super manifolds as ringed spaces following [Deligne&Morgan1999]. By a morphism we will always mean a map of ringed spaces. The local model for a super manifold of dimension is
equipped with the sheaf
of commutative super
-algebras
.
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![p|q](/images/math/5/2/0/520f8c4c02bc48931b2e0c4fc1333758.png)
![(|M|,\cO_M)](/images/math/9/6/b/96b7d4143994651dfca0ed82a4557998.png)
![|M|](/images/math/b/2/8/b28417c7d7758277bf6c14675ba97e64.png)
![\R](/images/math/d/8/3/d8312bf230adacd916d047420a399944.png)
![\cO_M](/images/math/8/c/f/8cf9b61f635637652a23517889d233a9.png)
![(\R^{p},\cO_{\R^{p|q}})](/images/math/c/6/2/c625a39b9ae3c2e93b1bbdb4b9f1513e.png)
![f=(|f|,F)](/images/math/e/b/d/ebdf15dc3eafd35fec4aee8efcb3dd7b.png)
![M,N](/images/math/1/8/9/18955b9072fcd747963b5d25257dc4b9.png)
![|f|:|M|\to |N|](/images/math/7/2/2/722aef6fbf7b8799c6ebf6a4290d2309.png)
![F](/images/math/7/9/8/79851a1fc5f19464a229ccdf66c8beb2.png)
![|f|](/images/math/2/8/a/28ab75c8a710b11c4496377a5a5d8158.png)
![U\subseteq |N|](/images/math/7/b/7/7b729af206e273173e38239e205c8c56.png)
![\displaystyle F(U): \cO_N(U) \ra \cO_M(|f|^{-1}(U))](/images/math/6/a/c/6acc2644f70770f9890f42df0e4c729a.png)
![f^*](/images/math/d/4/6/d4676db1e00cb90d2833cf440d088bea.png)
![F](/images/math/7/9/8/79851a1fc5f19464a229ccdf66c8beb2.png)
![\SMan](/images/math/e/b/8/eb874ffbbcafe98b2bc469dbc0cdf12b.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\displaystyle M^{red}:=(|M|,\cO_M / \Nil)](/images/math/b/7/a/b7a4eb6d29379e02f01ae4be83b5b5f5.png)
![|M|](/images/math/b/2/8/b28417c7d7758277bf6c14675ba97e64.png)
![M^{red} \into M](/images/math/2/e/5/2e5beb54ba53e730106ee569daf34f8e.png)
![\Nil \subset \cO_M](/images/math/b/5/5/b55761b052cd5c80834c401e39fd5fc1.png)
![\R](/images/math/d/8/3/d8312bf230adacd916d047420a399944.png)
![\C](/images/math/c/2/e/c2ed9023772e6a40d82075d0095fe7bb.png)
![C^\infty](/images/math/3/a/3/3a3602d2e793dd07042cb34d8945b2af.png)
![\C](/images/math/c/2/e/c2ed9023772e6a40d82075d0095fe7bb.png)
![\cO_{\R^{p|q}}\otimes \C](/images/math/f/b/5/fb525bc6a8f0cbc061a215a03e0986c0.png)
Example 2.2.
Let be a real vector bundle of fiber dimension
over the ordinary manifold
and
the associated algebra bundle of alternating multilinear forms on
. Then its sheaf of sections gives a super manifold
of dimension
, denoted by
. In the current smooth setting, Marjorie Batchelor proved in [Batchelor1979] that every super manifold is isomorphic to one of this type (this is not true for analytic super manifolds). More precisely, let
denote the category of real vector bundles over smooth manifolds, and for
, consider the vector bundle
over
with sheaf of sections
. Then the functors
![\displaystyle \Pi : \BM \to \SMan \quad \text{ and } \quad J: \SMan \to \BM](/images/math/3/2/3/3237fc6f36b18fe169b740e07c97baae.png)
come equipped with natural isomorphisms but there are only non-natural isomorphisms
, coming from a choice of a partition of unity. In other words, these functors induce a bijection on isomorphism classes of objects and inclusions on morphisms but they are not equivalences of categories because there are many more morphims in
than the linear bundle maps coming from
.
The following proposition gives two extremely useful ways of looking at morphisms between super manifolds. We shall use the notation for the algebra of (global) functions on a super manifold
.
Proposition 2.3.
For , the functor
induces natural bijections
![\displaystyle \SMan(S,M) \cong \Alg(C^\infty(M), C^\infty(S)).](/images/math/b/0/b/b0b8080d1848f8e9eccc745a0b106d16.png)
If is an open super submanifold (a
),
is in bijective correspondence with those
in
that satisfy
![\displaystyle (|f_1|(s),...,|f_p|(s)) \in |M| \subseteq \R^{p} \text{ for all } s \in |S|.](/images/math/f/4/4/f44d9e444b5457b8db8ec2648bc4c0bb.png)
![f_i](/images/math/3/8/a/38a54257d1bd19d28f5fff8954fea1e7.png)
![\eta_j](/images/math/a/8/b/a8bbd4ba517c703e75842dacf42fc56f.png)
![\phi\in \SMan(S,M)](/images/math/d/6/b/d6bd5e80f8fe0c7712e970a5fa743452.png)
![\displaystyle f_i = \phi^*(x_i) \quad \text{ and } \quad \eta_j = \phi^*(\theta_j),](/images/math/0/7/b/07b2484f5ba6bbd1eabd344265819462.png)
where are coordinates on
. Moreover, by the first part we see that
and hence
.
The proof of the first part is based on the existence of partitions of unity for super manifolds, so it is false in analytic settings. The second part always holds and is proved in [Leites1980].
3 The Functor of Points
![S](/images/math/2/0/c/20c08b85a1d0a48a17a99f4d187a66a6.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\SMan(S,M)](/images/math/0/e/f/0ef6b9ddb20854b0bf6f16f5c469319f.png)
![S](/images/math/2/0/c/20c08b85a1d0a48a17a99f4d187a66a6.png)
![S](/images/math/2/0/c/20c08b85a1d0a48a17a99f4d187a66a6.png)
![\SMan](/images/math/e/b/8/eb874ffbbcafe98b2bc469dbc0cdf12b.png)
![\SMan](/images/math/e/b/8/eb874ffbbcafe98b2bc469dbc0cdf12b.png)
Tex syntax errorby
Tex syntax error
![\SMan](/images/math/e/b/8/eb874ffbbcafe98b2bc469dbc0cdf12b.png)
![Y](/images/math/3/9/a/39abfebb66c060cd7541c76ff73c12da.png)
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
![S](/images/math/2/0/c/20c08b85a1d0a48a17a99f4d187a66a6.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
4 Super Lie Groups
These are simply group objects in . According to the functor of points approach, such a group object in
can be described by giving a functor
such that the composition with the forgetful functor
is representable.
Example 4.1.
The simplest super Lie group is the additive group structure on . It is given by the following composition law on
, obviously natural in
:
![\displaystyle (f_1,...,\eta_q) \times (h_1,...,\psi_q) \mapsto (f_1 + h_1,...,\eta_q + \psi_q) .](/images/math/8/e/9/8e9193a234621fa78a07c047769c720e.png)
The super general linear group is defined by
![\displaystyle GL(p|q)(S) := \text{Aut}_{\cO_S}(\cO_S^{p|q}) \cong \Aut_{C^\infty(S)} (C^\infty(S)^{p|q}) ,](/images/math/d/3/e/d3e13682bfd555c359d016564a76860b.png)
![A^{p|q}](/images/math/a/f/b/afb46cd5081daac1ff435bab3a76d756.png)
![A](/images/math/b/8/9/b8921ca1d75b852da96e95cda4aafeb8.png)
![p](/images/math/2/a/0/2a039ed8fdbf4ceaa9e79cdc3aecd1a2.png)
![q](/images/math/e/b/6/eb6af5b4e510c3c874d7d1f51d72393a.png)
![GL(p|q)(\_)](/images/math/0/d/3/0d355a22e098b85bac0633a16bf90e55.png)
![G \subset \R^{p^2 + q^2| 2pq}](/images/math/4/b/4/4b44319c294733beb0b8ff70bae506b2.png)
![\displaystyle |G| = \{ \> x \in \R^{p^2 + q^2} \> | \> x \in GL_p \times GL_q \> \}.](/images/math/2/0/f/20f9ef4022736908ca994d93bbc276ad.png)
This follows directly from proposition 2.3 using that a map between super algebras is invertible if and only if it is invertible modulo nilpotent elements.
5 Super Vector Bundles
A (super) vector bundle over a super manifold is a locally free sheaf
of
-modules of dimension
. The most basic example of a super vector bundle is the tangent bundle of a super manifold
. It is the sheaf of
-modules
defined by
![\displaystyle \mathcal{T} M(U) := \Der {\cO_M (U)} .](/images/math/7/5/0/750439b83645ac409ede0cbfe22c24c2.png)
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
![\displaystyle E(S)= \{ (f,g) \ \mid\ f\in\SMan(S,M), g \in f^*(\mathcal{E}^{ev}(M)) \}.](/images/math/b/9/a/b9a000803dfa9205416d325d3253dd30.png)
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
.
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\displaystyle \Pi : \Vect_M\ra \Vect_M](/images/math/7/3/6/73651532a2279128a69413d3742df4b8.png)
![E](/images/math/e/4/d/e4de1fb0b52291b425e79d56646e3c6e.png)
![\alpha](/images/math/d/a/d/dad2cde42eb00bcf43e15d68d3efabab.png)
![(E,-\alpha)](/images/math/0/d/7/0d77760ecdc9d617efed86632187a8be.png)
![\Pi](/images/math/d/3/5/d355428021402a26ac65f3b260f8485c.png)
![\Pi(E)=\epsilon_{0|1}\otimes E](/images/math/a/5/3/a532ff9a76cf9c084c3fecbd05913400.png)
![\epsilon_{0|1}](/images/math/e/9/1/e9135449197a52098e8ca83ed4171f2a.png)
![0|1](/images/math/f/1/e/f1e8e17bccddf418b572d75ccf2492d3.png)
![\cO_M](/images/math/8/c/f/8cf9b61f635637652a23517889d233a9.png)
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
:
![\displaystyle S \times G \stackrel{f \times \id}{\longrightarrow} G \times G \overset{\mu} \longrightarrow G .](/images/math/4/2/0/420a81fa4dc17e65f028947e15f19f11.png)
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.