Super manifolds: an incomplete survey
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We present an incomplete survey on some basic notions of super manifolds which may serve as a short introduction to this subject. Almost all the material is taken from the beautiful survey article on super manifolds [Deligne&Morgan1999]. Standard references also include [Leites1980], [Berezin1987], [Manin1988] or [Voronov1991]. The material below is a prerequisite to our papers [Hohnhold&Kreck&Stolz&Teichner2010] and [Stolz&Teichner2008].
Contents |
1 Super Algebra
Let us begin by explaining briefly what super means in an algebraic context, working with the ground field of real numbers. The monoidal category of super vector spaces, with tensor products, is by definition the same as the monoidal category of -graded vector spaces, with the graded tensor product. As a consequence, a super algebra is simply a monoidal object in this category and is hence the same thing as a
-graded algebra. For example, the endomorphism ring
of a super vector space
inherits a natural
-grading from that of
. The distinction between these notions only arises from the choice of symmetry operators
![\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.