Introduction to supergeometry
Introduction to supergeometry Introduction to supergeometry
while on the other hand, holds. Using the flow equation, we obtain X i (x(t)) = X i (x) + t n∑ j=1 ∂X i ∂x j vj v i = X i (x) and n∑ j=1 ∂X i ∂x j vj = 0, which combines into n∑ j=1 ∂X i ∂x j Xj = 0 ⇔ [X, X] = 0. So [X, X] = 0 turns out to be a necessary and sufficient condition for the integrability of X. This conclusion can be seen as a special instance of Frobenius Theorem for smooth graded manifolds, see [6]. 4 Graded symplectic geometry 4.1 Differential forms Locally, the algebra of differential forms on a graded manifold M is constructed by adding new coordinates (dx i ) n i=1 to a system of homogeneous local coordinates (xi ) n i=1 of M. Moreover, one assigns the degree |x i | + 1 to dx i . Remark 4.1. 1. If x i is a coordinate of odd degree, dx i is of even degree and consequently (dx i ) 2 ≠ 0. 2. On an ordinary smooth manifold, differential forms have two important properties: they can be differentiated – hence the name differential forms – and they also provide the right objects for an integration theory on submanifolds. It turns out that on graded manifolds, this is no longer true, since the differential forms we introduced do not come along with a nice integration theory. To solve this problem, one has to introduce new objects, called ‘integral forms’. The interested reader is referred to [14]. A global description of differential forms on M is as follows: the shifted tangent bundle T [1]M carries a natural structures of a dg manifold with a cohomological vector field Q that reads n∑ dx i ∂ ∂x i i=1 in local coordinates. The de Rham complex (Ω(M), d) of M is C ∞ (T [1]M), equipped with the differential corresponding to the cohomological vector field Q. Remark 4.2. The classical Cartan calculus extens to the graded setting: 8
1. Let X be a graded vector field on M and ω a differential form. Assume their local expressions in a coordinate system (x i ) n i=1 are X = n∑ i=1 X i ∂ ∂x i and ω = n∑ ω i1···i k dx i1 · · · dx i k , i 1 ,··· ,i k =1 respectively. The contraction ι X ω of X and ω is given locally by n∑ i 1 ,··· ,i k =1 l=1 k∑ ±ω i1···i k X i l dx i1 · · · ̂dx i l · · · dx i k . Alternatively – and to get the signs right – one uses the following rules: • ι ∂ ∂x i dx j = δ j i , • ι ∂ ∂x i is a graded derivation of the algebra of differential forms of degree |x i | − 1. 2. The Lie derivative is defined via Cartan’s magic formula L X ω := ι X dω + (−1) |X| dι X ω. If one considers both ι X and d as graded derivations, L X is their graded commutator [d, ι X ]. It follows immediately that holds. [L X , d] = 0 Let us compute the Lie derivative with respect to the graded Euler vector field E in local coordinates (x i ). By definition and consequently L E x i = |x i |x i L E dx i = dL E x i = |x i |dx i . This implies that L E acts on homogenous differential forms on M by multiplication by the difference between the total degree – i.e. the degree in C ∞ (T [1]M) – and the form degree – which is given by counting the ‘d’s, loosly speaking. We call this difference the degree of a differential form ω and denote it by deg ω. 4.2 Basic graded symplectic geometry Definition 4.3. A graded symplectic form of degree k on a graded manifold M is a two-form ω which has the following properties: • ω is homogeneous of degree k, • ω is closed with respect to the de Rham differential, 9
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1. Let X be a graded vec<strong>to</strong>r field on M and ω a differential form. Assume their local<br />
expressions in a coordinate system (x i ) n i=1 are<br />
X =<br />
n∑<br />
i=1<br />
X i<br />
∂<br />
∂x i and ω =<br />
n∑<br />
ω i1···i k<br />
dx i1 · · · dx i k<br />
,<br />
i 1 ,··· ,i k =1<br />
respectively.<br />
The contraction ι X ω of X and ω is given locally by<br />
n∑<br />
i 1 ,··· ,i k =1 l=1<br />
k∑<br />
±ω i1···i k<br />
X i l<br />
dx i1 · · · ̂dx i l · · · dx<br />
i k<br />
.<br />
Alternatively – and <strong>to</strong> get the signs right – one uses the following rules:<br />
• ι ∂<br />
∂x i dx j = δ j i ,<br />
• ι ∂<br />
∂x i is a graded derivation of the algebra of differential forms of degree |x i | − 1.<br />
2. The Lie derivative is defined via Cartan’s magic formula<br />
L X ω := ι X dω + (−1) |X| dι X ω.<br />
If one considers both ι X and d as graded derivations, L X is their graded commuta<strong>to</strong>r<br />
[d, ι X ]. It follows immediately that<br />
holds.<br />
[L X , d] = 0<br />
Let us compute the Lie derivative with respect <strong>to</strong> the graded Euler vec<strong>to</strong>r field E in local<br />
coordinates (x i ). By definition<br />
and consequently<br />
L E x i = |x i |x i<br />
L E dx i = dL E x i = |x i |dx i .<br />
This implies that L E acts on homogenous differential forms on M by multiplication by the<br />
difference between the <strong>to</strong>tal degree – i.e. the degree in C ∞ (T [1]M) – and the form degree –<br />
which is given by counting the ‘d’s, loosly speaking. We call this difference the degree of a<br />
differential form ω and denote it by deg ω.<br />
4.2 Basic graded symplectic geometry<br />
Definition 4.3. A graded symplectic form of degree k on a graded manifold M is a two-form<br />
ω which has the following properties:<br />
• ω is homogeneous of degree k,<br />
• ω is closed with respect <strong>to</strong> the de Rham differential,<br />
9
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