Integration of Vector Fields on Smooth

and Holomorphic Supermanifolds

Stéphane Garnier and Tilmann Wurzbacher

Received: September 19, 2012 Revised: December 12, 2012

Communicated by Christian Bär

Abstract. We give a new and self-contained proof of the existence and unicity of the flow for an arbitrary (not necessarily homogeneous) smooth vector field on a real supermanifold, and extend these results to the case of holomorphic vector fields on complex supermanifolds. Furthermore we discuss local actions associated to super vector fields, and give several examples and applications, as, e.g., the construction of an exponential morphism for an arbitrary finite-dimensional Lie supergroup.

2010 Mathematics Subject Classification: Primary 58A50, 37C10; Secondary 57S20, 32C11 Keywords and Phrases: Supermanifolds; Vector fields; Flows; Group actions

1 Introduction

The natural problem of integrating vector fields to obtain appropriate “flow maps” on supermanifolds is considered in many articles and monographs (compare, e.g., [17], [2], [19], [14] and [3]) but a “general answer” was to our knowledge only given in the work of J. Monterde and co-workers (see [12] and [13]). Let us consider a supermanifold M = (M, O_{M}) together with a vector field X in T_{M}(M), and an initial condition ϕ in Mor(S, M), where S = (S, O_{S}) is an arbitrary supermanifold and Mor(S, M) denotes the set of morphisms from S to M. The case of classical, ungraded, manifolds leads one to consider the following question: does there exist a “flow map” F defined on an open sub supermanifold V ⊂ ℝ^{1|1} × S and having values in M and an appropriate derivation on ℝ^{1|1}, D = ∂_{t} + ∂_{τ} + τ(a∂_{t} + b∂_{τ}), where ∂_{t} = \frac{∂}{∂t}, ∂_{τ} = \frac{∂}{∂τ} and a, b are real numbers, such that the following equations are fulfilled

\begin{aligned}D ∘ F^{∗} & = & F^{∗} ∘ X \\ F ∘ inj_{\{0\}×S}^{V} & = & ϕ . \end{aligned}  (1)

Of course, V should be a “flow domain”, i.e. an open sub supermanifold of ℝ^{1|1} × S such that {0} × S is contained in the body V of V and for x in S, the set I_{x} ⊂ ℝ defined by I_{x} × {x} = (ℝ × {x}) ∩ V is an open interval. Furthermore inj_{{0}×S}^{V} denotes the natural injection morphism of the closed sub supermanifold {0} × S of V into V. Of course, we could concentrate on the case S = M and ϕ = id_{M}, but it will be useful for our later arguments to state all results in this (formally) more general setting.

Though for homogeneous vector fields (X = X_{0} or X = X_{1}) system (1) does always have a solution, in the general case (X = X_{0}+X_{1} with X_{0} ≠ 0 and X_{1} ≠ 0) the system is overdetermined. A simple example of an inhomogeneous vector field such that (1) is not solvable is given by X = X_{0} + X_{1} = (\frac{∂}{∂x} + ξ \frac{∂}{∂ξ} ) +

(\frac{∂}{∂ξ} + ξ \frac{∂}{∂x}) on M = ℝ^{1|1}. The crucial novelty of [13] is to consider instead of (1) the following modified, weakened, problem

(inj_{ℝ}^{ℝ^{1|1}})^{∗} ∘ D ∘ F^{∗} = (inj_{ℝ}^{ℝ^{1|1}})^{∗} ∘ F^{∗} ∘ X
F ∘ inj_{{0}×S}^{V}  = ϕ , (2)

where inj_{ℝ}^{ℝ^{1|1}} = inj_{ℝ×S}^{ℝ^{1|1}×S} is again the natural injection (and where the above

more general derivation D could be replaced by ∂_{t} + ∂_{τ} since (inj_{ℝ}^{ℝ^{1|1}})^{∗} annihilates germs of superfunctions of the type τ · f, f ∈ O_{ℝ×S}).

In [13] (making indispensable use of [12]) it is shown that in the smooth case (2) has a unique maximal solution F, defined on the flow domain V = (V, O_{ℝ^{1|1}×S}|_{V}), where V ⊂ ℝ × S is the maximal flow domain for the flow of the reduced vector field X̃ = X̃_{0} on M with initial condition ϕ̃. Since the results of [12] are obtained by the use of a Batchelor model for M, i.e. a real vector bundle E → M such that M ≅ (M, Γ_{ΛE^{∗}}^{∞}), and a connection on E, we follow here another road, closer to the classical, ungraded, case and also applicable in the case of complex supermanifolds and holomorphic vector fields.

Our new method of integrating smooth vector fields on a supermanifold in Section 2 consists in first locally solving a finite hierarchy of ordinary differential equations, and is here partly inspired by the approach of [3], where the case of homogeneous super vector fields on compact supermanifolds is treated. We then show existence and unicity of solutions of (2) on smooth supermanifolds and easily deduce the results of [13] from our Lemmata 2.1 and 2.2.

A second beautiful result of [13] (more precisely, Theorem 3.6 of that reference) concerns the question if the flow F solving (2) fulfills “flow equations”, as in the ungraded case. Hereby, we mean the existence of a Lie supergroup structure on ℝ^{1|1} such that F is a local action of ℝ^{1|1} on M (in case S = M, ϕ = id_{M}). Again, the answer is a little bit unexpected: in general, given X and its flow F : ℝ^{1|1} × M ⊃ V → M, there is no Lie supergroup structure on ℝ^{1|1} such that F is a local ℝ^{1|1}-action (with regard to this structure). The condition for the existence of such a structure on ℝ^{1|1} is equivalent to the condition that (2) holds without the post-composition with (inj_{ℝ}^{ℝ^{1|1}})^{∗}, i.e. the overdetermined system (1) is solvable. Furthermore, both conditions cited are equivalent to the condition that ℝX_{0} ⊕ ℝX_{1} is a sub Lie superalgebra of T_{M}(M), the Lie superalgebra of all vector fields on M.

After discussing Lie supergroup structures and right invariant vector fields on ℝ^{1|1}, as well as local Lie group actions in the category of supermanifolds in general, we show in Section 3 the equivalence of the above three conditions, already given in [13]. We include our proof here notably in order to be able to apply it in the holomorphic case in Section 5 (see below) by simply indicating how to adapt it to this context. Let us nevertheless observe that our result is slightly more general since we do not need to ask for any normalization of the supercommutators between X_{1} and X_{0} resp. X_{1}, thus giving the criterion some extra flexibility in applications.

In Section 4, we give several examples of vector fields on supermanifolds, homogeneous and inhomogeneous, and explain their integration to flows. Notably, we construct an exponential morphism for an arbitrary finite-dimensional Lie supergroup, via a canonically defined vector field and its flow. We comment here also on the integration of what are usually called “(infinitesimal) supersymmetries” in physics, i.e., purely odd vector fields having non-vanishing self-commutators.

Finally, in Section 5 we adapt our method to obtain flows of vector fields (compare Section 2 and notably Lemma 2.1) to the case of holomorphic vector fields on holomorphic supermanifolds. To avoid monodromy problems one has, of course, to take care of the topology of the flow domains, and maximal flow domains are -as already in the ungraded holomorphic case- no more unique. Otherwise the analogues of all results in Section 2 and 3 continue to hold in the holomorphic setting.

Throughout the whole article we will work in the ringed space-approach to supermanifolds (see, e.g., [9], [10], [11] and [15] for detailed accounts of this approach). Given two supermanifolds M = (M, O_{M}) and N = (N, O_{N}), a “morphism” ϕ = (ϕ̃, ϕ^{∗}) : M → N is thus given by a continuous map ϕ̃ : M → N between the “bodies” of the two supermanifolds and a sheaf homomophism ϕ^{∗} : O_{N} → ϕ̃_{∗}O_{M}. The topological space M comes canonically with a sheaf C_{M}^{∞} = O_{M}/J, where J is the ideal sheaf generated by the germs of odd superfunctions, such that (M, C_{M}^{∞}) is a smooth real manifold. Then ϕ̃ is a smooth map from (M, C_{M}^{∞}) to (N, C_{N}^{∞}). Let us recall that a (super) vector field on M = (M, O_{M}) is, by definition, an element of the Lie superalgebra T_{M}(M) = (Der_{ℝ}(O_{M}))(M) and that X always induces a smooth vector field X̃ on (M, C_{M}^{∞}). For p in M and f + J_{p} ∈ (C_{M}^{∞})_{p} = (O_{M}/J)_{p} one defines X̃_{p}(f + J_{p}) = X_{0}(f)(p), where X_{0} is the even part of X and for g ∈ (O_{M})_{p}, g(p) ∈ ℝ is the value of g in the point p of M.

2 Flow of a vector field on a real supermanifold

In this section we give our main result on the integration of general (i.e. not necessarily homogeneous) vector fields by a new method, avoiding auxiliary choices of Batchelor models and connections, as in [12]. Our more direct approach is inspired, e.g., by [3], where the case of homogeneous vector fields on compact manifolds is treated, and it can be adapted to the holomorphic case (see Section 4).

For the sake of readability we will often use the following shorthand: if P is a supermanifold, we write inj _{ℝ}^{ℝ^{1|1}} for inj _{ℝ×P}^{ℝ^{1|1}×P}. Furthermore, the canonical coordinates of ℝ^{1|1} will be denoted by t and τ, with ensueing vector fields ∂_{t} = \frac{∂}{∂t} and ∂_{τ} = \frac{∂}{∂τ}.

Lemma 2.1. Let U ⊂ ℝ^{m|n} and W ⊂ ℝ^{p|q} be superdomains, X ∈ T_{W}(W) be a super vector field on W (not necessarily homogeneous) and ϕ ∈ Mor(U, W), and t_{0} ∈ ℝ. Let furthermore H : V → W be the maximal flow of X̃ ∈ X (W), i.e. ∂_{t} ∘ H^{∗} = H^{∗} ∘ X̃, subject to the initial condition H(t_{0}, ·) = ϕ̃ : U → W. Let now V be (V, O_{ℝ^{1|1}×U}|_{V}) and (t, τ) the canonical coordinates on ℝ^{1|1}, then there exists a unique F : V → W such that

(inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ (∂_{t} + ∂_{τ}) ∘ F^{∗} = (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ F^{∗} ∘ X and  (3)
F ∘ inj _{{t_{0}}×U}^{V} = ϕ.  (4)

Moreover, F̃ : V → W equals the underlying classical flow map H of the vector field X̃ with initial condition ϕ̃.

Proof. Let (u_{i}) = (x_{i}, ξ_{r}) and (w_{j}) = (y_{j}, η_{s}) denote the canonical coordinates on ℝ^{m|n} and ℝ^{p|q}, respectively. Then there exist smooth functions a_{J}^{j} ∈ C_{ℝ^{p}}^{∞}(W) such that

X = ∑_{j=1}^{p+q} (∑_{J} a_{J}^{j}(y)η^{J}) ∂_{w_{j}},

where J = (β_{1}, …, β_{q}) runs over the index set {0, 1}^{q} and η^{J} = ∏_{s=1}^{q} η_{s}^{β_{s}}. We

then have, of course,

X_{0} = ∑_{j} ( ∑_{|J|=|w_{j}|} a_{J}^{j}(y)η^{J} ) ∂_{w_{j}} resp. X_{1} = ∑_{j} ( ∑_{|J|=|w_{j}|+1} a_{J}^{j}(y)η^{J} ) ∂_{w_{j}}.

Here, |J| equals β_{1}+⋯+β_{q} mod 2 and |w_{j}| is the parity of the coordinate function w_{j}. The morphism F determines and is uniquely determined by functions f_{I}^{j}, g_{I}^{j} ∈ C_{ℝ×ℝ^{m}}^{∞}(V) fulfilling for each j ∈ {1, …, p + q}

F^{∗}(w_{j}) = ∑_{|I|=|w_{j}|} f_{I}^{j}(t, x)ξ^{I} + ∑_{|I|=|w_{j}|+1} g_{I}^{j}(t, x)τξ^{I}

(and f_{I}^{j} = 0 if |I| ≠ |w_{j}|, g_{I}^{j} = 0 if |I| ≠ |w_{j}| + 1) as is well-known from the standard theory of supermanifolds (compare, e.g., Thm. 4.3.1 in [20]). Here and in the sequel I = (α_{1}, …, α_{n}) is an element of the set {0, 1}^{n} and ξ^{I} stands for the product ξ_{1}^{α_{1}} · ξ_{2}^{α_{2}} ⋯ ξ_{n}^{α_{n}}. The notation |I| again denotes the parity of I, i.e. |I| = α_{1} + ⋯ + α_{n} mod 2.

Equation (3) is equivalent to the following equations:

(inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ ∂_{t} ∘ F^{∗} = (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ F^{∗} ∘ X_{0}  (5)
(inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ ∂_{τ} ∘ F^{∗} = (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ F^{∗} ∘ X_{1}  (6)

Applying (5) to the canonical coordinate functions on W, we get the following system, which is equivalent to (5):

∑_{|I|=|w_{j}|} ∂_{t}f_{I}^{j} · ξ^{I} = ∑_{|J|=|w_{j}|} F̌^{∗}(a_{J}^{j})F̌^{∗}(η^{J}) for all j in {1, …, p + q},  (7)

and (6) is equivalent to

∑_{|I|=|w_{j}|+1} g_{I}^{j} · ξ^{I} = ∑_{|J|=|w_{j}|+1} F̌^{∗}(a_{J}^{j})F̌^{∗}(η^{J}) for all j in {1, …, p + q}, (8)

where F̌ := F ∘ inj _{ℝ}^{ℝ^{1|1}} : V̌ := (V, O_{ℝ×U}|_{V}) → W. Let us immediately observe that the underlying smooth map of F̌ equals F̃, the smooth map underlying the morphism F.

Moreover the initial condition (4) is equivalent to

∑_{|I|=|w_{j}|} f_{I}^{j}(t_{0}, x)ξ^{I} = ϕ^{∗}(w_{j}) for all j in {1, …, p + q}.  (9)

We are going to show that (7) and (9) uniquely determine the functions f_{I}^{j} on V, i.e. the morphism F̌. Then the functions g_{I}^{j} are unambiguously given by (8) on V, and the morphism F is fully determined.

Let us develop Equation (7) for a fixed j:

∑_{|I|=|w_{j}|} ∂_{t}f_{I}^{j} · ξ^{I} = ∑_{\begin{matrix}|J|=|w_{j}| \\ J=(β_{1},...,β_{q}) \end{matrix}} F̌^{∗}(a_{J}^{j}) ∏_{s=1}^{q} F̌^{∗}(η_{s})^{β_{s}}  (10)

and thus ∑_{|I|=|w_{j}|} ∂_{t}f_{I}^{j} · ξ^{I} = ∑_{\begin{matrix}|J|=|w_{j}| \\ J=(β_{1},...,β_{q}) \end{matrix}} F̌^{∗}(a_{J}^{j}) ∏_{s=1}^{q} ( ∑_{|L|=1} f_{L}^{p+s}ξ^{L})^{β_{s}} . (11)

For fixed j this is an equation of Grassmann algebra-valued maps in the variables t and x that can be split in a system of scalar equations as follows. For K = (α_{1}, …, α_{n}) ∈ {0, 1}^{n}, we will denote the coefficient h_{K} in front of ξ^{K} of a superfunction h = ∑_{M} h_{M}(t, x)ξ^{M} ∈ O_{ℝ^{m+1|n}} compactly by (h|ξ^{K}) in the sequel of this proof.

Let us first describe the coefficients for F̌^{∗}(a_{J}^{j}) in (11):

(F̌^{∗}(a_{J}^{j})|ξ^{K}) = 0 if |K| = 1  (12)

and, if |K| = 0,

(F̌^{∗}(a_{J}^{j})|ξ^{K}) = a_{J}^{j} ∘ F̃ if K = (0, …, 0),

and

(F̌^{∗}(a_{J}^{j})|ξ^{K}) = ∑_{μ=1}^{p}(∂_{y_{μ}}a_{J}^{j})(F̃(t, x)) · f_{K}^{μ} + R (a_{J}^{j}, (f_{I}^{ν})_{ν,deg(I)<deg(K)}) (13)

if deg(K) > 0.

Here for I = (α_{1}, …, α_{n}), deg(I) = α_{1} + ⋯ + α_{n}, and -more importantly- R = R_{j,J,K} is a polynomial function in a_{J}^{j} and its derivatives in the y-variables up to order q included, and in the functions {f_{I}^{ν}|1 ≤ ν ≤ p+q, 0 ≤ deg(I) < deg(K)}. Equation (12) is obvious since a_{J}^{j} is an even function, whereas equation (13) can be deduced from standard analysis on superdomains. More precisely, let a be a smooth function on ℝ^{p} and ψ : ℝ^{m+1|n} → ℝ^{p|q} a morphism (of course to be applied to a = a_{J}^{j}, ψ = F̌). Then we can develop ψ^{∗}(a) as follows (compare the proof of Theorem 4.3.1 in [20]):

ψ^{∗}(a) = ∑_{γ} \frac{1}{γ!}(∂_{γ}a)(ψ̃^{∗}(y_{1}), …, ψ̃^{∗}(y_{p})) · ∏_{μ=1}^{p}(ψ^{∗}(y_{μ}) − ψ̃^{∗}(y_{μ}))^{γ_{μ}}

= a(ψ̃(t, x)) + ∑_{μ=1}^{p}(∂_{y_{μ}}a)(ψ̃(t, x)) · ( ∑_{M≠0} f_{M}^{μ} · ξ^{M} ) +

\frac{1}{2} ∑_{μ′,μ′′=1}^{p}(∂_{y_{μ′}} ∂_{y_{μ′′}} a)(ψ̃(t, x)) · ( ∑_{M′≠0} f_{M′}^{μ′} · ξ^{M′} ) · ( ∑_{M′′≠0} f_{M′′}^{μ′′} · ξ^{M′′} ) + …,

where ∑_{M≠0} f_{M}^{μ} · ξ^{M} = ψ^{∗}(y_{μ}) − ψ̃^{∗}(y_{μ}) with f_{M}^{μ} depending on t and x. We

observe that the last RHS is a finite sum since we work in the framework of finite-dimensional supermanifolds. In order to get a contribution to (ψ^{∗}(a)|ξ^{K}) we can either extract f_{K}^{μ} from the “linear term” or from products coming from the higher order terms in the above development. Thus

(ψ^{∗}(a)|ξ^{K}) = ∑_{μ=1}^{p}(∂_{y_{μ}}a)(ψ̃(t, x)) · f_{K}^{μ} + R(a, (f_{I}^{ν})_{ν,deg(I)<deg(K)}),

where R is a polynomial as described after Equation (13).

Furthermore, for an element J = (β_{1}, …, β_{q}) with |J| = 0 we have for deg(K) > 0

( ∏_{s=1}^{q} ( ∑_{|L|=1} f_{L}^{p+s}ξ^{L})^{β_{s}} | ξ^{K} ) = R ((f_{I}^{j})_{j,deg(I)<deg(K)}) .  (14)

And for an element J = (β_{1}, …, β_{q}) with |J| = 1 we get for deg(K) > 0

(∏_{s=1}^{q} ( ∑_{|L|=1} f_{L}^{p+s}ξ^{L})^{β_{s}} | ξ^{K} ) = \{ \begin{matrix}f_{K}^{p+l} + R ((f_{I}^{j})_{j,deg(I)<deg(K)}) \\ if deg(J) = 1 and l ∈ \{1, …, q\} \\ such that β_{s} = δ_{s,l} ∀s, \\ R ((f_{I}^{j})_{j,deg(I)<deg(K)}) \\ if deg(J) > 1. \end{matrix} (15)

Obviously, the coefficient of ξ^{K} of the LHS of Equation (7) is given by

( ∑_{|I|=|w_{j}|} ∂_{t}f_{I}^{j} · ξ^{I} | ξ^{K} ) = \{ \begin{matrix}∂_{t}f_{K}^{j} & if |K| = |w_{j}| \\ 0 & if |K| = |w_{j}| + 1 \end{matrix} for 1 ≤ j ≤ p + q.

Taking into account the above descriptions of the ξ^{K}-coefficients, we will show the existence (and uniqueness) of the solution functions {f_{I}^{j}|1 ≤ j ≤ p + q, I ∈ {0, 1}^{n}} for (t, x) ∈ V by induction on deg(I) and upon observing that all ordinary differential equations occuring are (inhomogeneous) linear equations for the unknown functions.

Let us start with deg(I) = 0 that is I = (0, ⋯, 0). The “0-level” of the equations (11) and (9) is ∂_{t}f_{(0,··· ,0)}^{j} = a_{(0,··· ,0)}^{j} ∘ F̃ and f_{(0,··· ,0)}^{j}(t_{0}, x) =

y_{j} ∘ ϕ̃(x) for all j such that |w_{j}| = 0. We remark that f_{(0,··· ,0)}^{j} is simply y_{j} ∘ F̃

and a_{(0,··· ,0)}^{j} is X̃(y_{j}). Thus F̃ is the flow of X̃ with initial condition ϕ̃ at

t = t_{0}, i.e., F̃ = H on V. Thus the claim is true for I = (0, ⋯, 0).

Suppose k > 0 and that the functions f_{I}^{j} are uniquely defined on V for all j and all I such that deg(I) < k. Let K be such that deg(K) = k. Let us distinguish the two possible parities of k in order to determine f_{K}^{j} for all j. Recall that f_{K}^{j} = 0 if the parities of K and j are different.

If k is even, i.e., |K| = 0, we only have to consider j such that |w_{j}| = 0. Putting (13) and (14) together, we find in this case

∂_{t}f_{K}^{j} = ( ∑_{\begin{matrix}|J|=0 \\ J=(β_{1},...,β_{q}) \end{matrix}} F̌^{∗}(a_{J}^{j}) ∏_{s=1}^{q} ( ∑_{|L|=1} f_{L}^{p+s}ξ^{L})^{β_{s}} | ξ^{K} )

= ( ∑_{\begin{matrix}deg(J)=0 \\ J=(β_{1},...,β_{q}) \end{matrix}} F̌^{∗}(a_{J}^{j}) ∏_{s=1}^{q} ( ∑_{|L|=1} f_{L}^{p+s}ξ^{L})^{β_{s}}

+ ∑_{\begin{matrix}|J|=0 \\ deg(J)>0 \\ J=(β_{1},...,β_{q}) \end{matrix}} F̌^{∗}(a_{J}^{j}) ∏_{s=1}^{q} ( ∑_{|L|=1} f_{L}^{p+s}ξ^{L})^{β_{s}} | ξ^{K} )

= (F̌^{∗}(a_{(0,··· ,0)}^{j}) + ∑_{\begin{matrix}|J|=0 \\ deg(J)>0 \\ J=(β_{1},...,β_{q}) \end{matrix}} F̌^{∗}(a_{J}^{j}) ∏_{s=1}^{q} ( ∑_{|L|=1} f_{L}^{p+s}ξ^{L})^{β_{s}} | ξ^{K} )

= ∑_{μ=1}^{p} (∂_{y_{μ}}a_{(0,··· ,0)}^{j} ∘ F̃) f_{K}^{μ} + R ((a_{J}^{j})_{J}, (f_{I}^{ν})_{ν,deg(I)<deg(K)}) .

Moreover, the initial condition gives f_{K}^{j}(t_{0}, x) = (ϕ^{∗}(y_{j})|ξ^{K}), for all j in {1, …, p}. Since the a_{J}^{j} are the (given) coefficients of the vector field X and the functions f_{I}^{ν} with deg(I) < k are known by the induction hypothesis, we have a unique local solution function f_{K}^{j}. Since the ordinary differential equation for f_{K}^{j} is linear its solution is already defined for all (t, x) ∈ V. Thus in the case that k is even f_{K}^{j} is unambiguously defined on V for all j ∈ {1, …, p + q} and for all K with deg(K) = k.

Now, if k is odd, i.e., |K| = 1, we only have to consider j such that |w_{j}| = 1. Using (13) and (15), we find in this case:

∂_{t}f_{K}^{j} = ( ∑_{\begin{matrix}|J|=1 \\ J=(β_{1},...,β_{q}) \end{matrix}} F̌^{∗}(a_{J}^{j}) ∏_{s=1}^{q} ( ∑_{|L|=1} f_{L}^{p+s}ξ^{L})^{β_{s}} | ξ^{K} )

= ( ∑_{\begin{matrix}deg(J)=1 \\ J=(β_{1},...,β_{q}) \end{matrix}} F̌^{∗}(a_{J}^{j}) ∏_{s=1}^{q} ( ∑_{|L|=1} f_{L}^{p+s}ξ^{L})^{β_{s}} +

+ ∑_{\begin{matrix}|J|=1 \\ deg(J)>1 \\ J=(β_{1},...,β_{q}) \end{matrix}} F̌^{∗}(a_{J}^{j}) ∏_{s=1}^{q} ( ∑_{|L|=1} f_{L}^{p+s}ξ^{L})^{β_{s}} | ξ^{K} )

= (∑_{s=1}^{q} F̌^{∗}(a_{(δ_{1s},··· ,δ_{qs})}^{j}) ( ∑_{|L|=1} f_{L}^{p+s}ξ^{L})

+ ∑_{\begin{matrix}|J|=1 \\ deg(J)>1 \\ J=(β_{1},...,β_{q}) \end{matrix}} F̌^{∗}(a_{J}^{j}) ∏_{s=1}^{q} ( ∑_{|L|=1} f_{L}^{p+s}ξ^{L})^{β_{s}} | ξ^{K} )

= ∑_{s=1}^{q} (a_{(δ_{1s},··· ,δ_{qs})}^{j} ∘ F̃) f_{K}^{p+s} + R ((a_{J}^{j})_{J}, (f_{I}^{ν})_{ν,deg(I)<deg(K)}) .

Moreover, the initial condition gives

f_{K}^{j}(t_{0}, x) = (ϕ^{∗}(w_{j})|ξ^{K}) for all j in {p + 1, …, p + q}.

It follows as in the case of |K| = 0, that f_{K}^{j} exists uniquely for all (t, x) ∈ V, for all j ∈ {1, …, p + q} and for all K with deg(K) = k.

We conclude that the functions {f_{I}^{j}|1 ≤ j ≤ p + q, I ∈ {0, 1}^{n}} are uniquely defined on the whole of V. Since the {g_{I}^{j}|1 ≤ j ≤ p + q, I ∈ {0, 1}^{n}} are determined by Equation (8) from the {f_{I}^{j}|1 ≤ j ≤ p + q} via comparison of coefficients, the morphism F : V → W is uniquely determined. □

We now consider the global problem of integrating a vector field on a supermanifold. In order to prove that there exists a unique maximal flow of a vector field, the following lemma will be crucial.

Lemma 2.2. Let M = (M, O_{M}) and S = (S, O_{S}) be supermanifolds, X a vector field in T_{M}(M) and ϕ in Mor(S, M). Then

(i) there exists an open sub supermanifold V = (V, O_{ℝ^{1|1}×S}|_{V}) of ℝ^{1|1}×S with V open in ℝ × S such that {0} × S ⊂ V and for all x in S, (ℝ × {x}) ∩ V is an interval, and a morphism F : V → M satisfying:

(inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ (∂_{t} + ∂_{τ}) ∘ F^{∗} = (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ F^{∗} ∘ X and  (16)
F ∘ inj _{{0}×S}^{V} = ϕ .  (17)

(ii) Let furthermore F_{1} : V_{1} → M and F_{2} : V_{2} → M be morphisms satisfying (16) and (17) where V_{i} = (V_{i}, O_{ℝ^{1|1}×S}|_{V_{i}}) with V_{i} open in ℝ × S such that {0} × S ⊂ V_{i} and for all x in S, (ℝ × {x}) ∩ V_{i} is an interval, for i = 1, 2. Then F_{1|V_{12}} = F_{2|V_{12}} on V_{12} = (V_{12}, O_{ℝ^{1|1}×S}|_{V_{12}}), where V_{12} = V_{1} ∩ V_{2}.

Proof. (i) Let ϕ̃: S → M denote the induced map of the underlying classical manifolds. Given now s in S and coordinate domains U_{s} of s and W_{s} of ϕ̃(s), isomorphic to superdomains Ǔ_{s} ⊂ ℝ^{m|n} resp. W̌_{s} ⊂ ℝ^{p|q}, by Lemma 2.1 we get solutions of (16) and (17) near s (upon reducing the size of U_{s} if

necessary): ℝ^{1|1} × S ⊃ ℝ^{1|1} × U_{s} ⊃ V_{s} ⟶^{F^{s}} W_{s} ⊂ M. If V_{s_{1}} ∩ V_{s_{2}} ≠ ∅ (compare

Figure 1) we know, again by Lemma 2.1, that F^{s_{1}} and F^{s_{2}} coincide on this intersection. Thus, by taking the union V of V_{s} for all s in S, we get a morphism F : ℝ^{1|1} × S ⊃ V → M such that F_{|V_{s}} = F^{s} for all s, and fulfilling (16) and (17).

Figure 1

(ii) We define A as the set of points (t, x) ∈ V_{12} such that there exists ϵ = ϵ_{(t,x)} > 0 and U = U_{(t,x)} an open sub supermanifold of S, such that its body U contains x and for V = V_{(t,x)} = (V_{(t,x)}, O_{ℝ^{1|1}×S}) = (] − ϵ, t + ϵ[×U, O_{ℝ^{1|1}×S}) we have F_{1|V} = F_{2|V}. Of course, if t < 0 the interval will be of the type ]t − ϵ, ϵ[ (See Figure 2). The claim of the Lemma is now equivalent to A = V_{12}. The set A is obviously open.

Figure 2

By an easy application of Lemma 2.1, A contains {0} × S. The assumptions imply that for all x ∈ S, the set I_{x} ⊂ ℝ, defined by (ℝ×{x})∩V_{12} = I_{x} ×{x}, is an open interval containing 0. The definition of A implies that the set J_{x} ⊂ I_{x} such that (ℝ × {x}) ∩ A = J_{x} × {x} is an open interval containing 0 as well. Assuming now that A ≠ V_{12}, then there exists a point (t, x_{0}) ∈ V_{12}\A such that J_{x_{0}} ≠ I_{x_{0}}. Without loss of generality we can assume that t > 0 and that for 0 ≤ t′ < t, (t′, x_{0}) ∈ A. Let U_{0} be an open coordinate neighborhood of x_{0} in S and δ > 0 such that, with V_{0} :=]t − δ, t + δ[×U_{0} ⊂ V_{12}, H(V_{0}) ⊂ W, where W = (W, O_{M}|_{W}) is a coordinate patch of M and H is the maximal flow of X̃ as in Lemma 2.1. Choose t_{0} ∈]t − δ, t[. Then (t_{0}, x_{0}) ∈ A and thus there exists ϵ > 0 and U an open sub supermanifold of U_{0} = (U_{0}, O_{S}|_{U_{0}}) containing x_{0} such that

F_{1|V} = F_{2|V}, where V =] − ϵ, t_{0} + ϵ[×ℝ^{0|1} × U ⊂ V_{12}.  (18)

On V′ =]t − δ, t + δ[×ℝ^{0|1} × U ⊂ V_{12}, F_{1} and F_{2} are defined and for i = 1, 2 the maps F_{i} ∘ inj_{{t_{0}}×U}^{V} coincide by (18) (Compare Figure 3 for the relative positions of the underlying topological spaces of these open sub supermanifolds of ℝ^{1|1} × S).

Figure 3 By Lemma 2.1 we have F_{1|V′} = F_{2|V′}. Thus F_{1} = F_{2} on V ∪ V′, and we conclude that (t, x_{0}) ∈ A. This contradiction shows that V_{12} = A. □

Remarks. (1) Obviously, Lemma 2.2 holds true for an arbitrary t_{0} ∈ ℝ replacing t_{0} = 0.

(2) Let us call a “flow domain for X with initial condition ϕ ∈ Mor(S, M) (with respect to t_{0} ∈ ℝ)” a domain V ⊂ ℝ^{1|1} × S such that {t_{0}} × S ⊂ V and for all s in S, (ℝ × {s}) ∩ V is connected, i.e. an interval (times {s}) and such that a solution F (a “flow”) of (16) and (17) exists on V. By the preceding lemma there exists such “flow domains”.

Theorem 2.3. Let M and S be supermanifolds, X be a vector field in T_{M}(M), ϕ in Mor(S, M) and t_{0} in ℝ. Then there exists a unique map F : V → M such that

(inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ (∂_{t} + ∂_{τ}) ∘ F^{∗} = (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ F^{∗} ∘ X and
F ∘ inj _{{t_{0}}×S}^{V} = ϕ ,

where V = (V, O_{ℝ^{1|1}×U}|_{V}) is the maximal flow domain for X with the given initial condition. Moreover, F̃ : V → M is the maximal flow of X̃ ∈ X (M) subject to the initial condition ϕ̃ at t = t_{0}.

Proof. The proof of the theorem follows immediately from the Lemmata 2.1 and 2.2, upon taking the union of all flow domains and flows for X as defined in the preceding remark. □

3 Supervector fields and local ℝ^{1|1}-actions

Given a vector field on a classical, ungraded, manifold, the flow map F̃ (for S = M, ϕ̃ = id_{M}) is always a local action of ℝ with its usual (and unique up to isomorphism) Lie group structure, the standard addition. The flow maps for vector fields described in the preceding section (taking here S = M, ϕ = id_{M}), do not always have the analogous property of being local actions of ℝ^{1|1} with an appropriate Lie supergroup structure. Two characterizations of those vector fields X = X_{0} +X_{1} that generate a local ℝ^{1|1}-action were found by J. Monterde and O. A. Sánchez-Valenzuela. We will give in this section a short proof of a slightly more general result, whose condition (iii) seems to be more easily verified in practice than those given in [13] (compare Thm. 3.6 and its proof there).

Let us begin by giving a useful two-parameter family of Lie supergroup structures on the supermanifold ℝ^{1|1} and their right invariant vector fields.

Lemma 3.1. Let a and b be real numbers such that a · b = 0 and μ_{a,b} = μ :

ℝ^{1|1} × ℝ^{1|1} → ℝ^{1|1} be defined by

μ̃(t_{1}, t_{2}) = t_{1} + t_{2},
μ^{∗}(t) = t_{1} + t_{2} + aτ_{1}τ_{2},
μ^{∗}(τ) = τ_{1} + e^{bt_{1}}τ_{2}  .

Then

(i) there exists a unique Lie supergroup structure on ℝ^{1|1} such that the multiplication morphism is given by μ_{a,b},

(ii) the right invariant vector fields on (ℝ^{1|1}, μ_{a,b}) are given by the graded vector space ℝD_{0} ⊕ ℝD_{1}, where

D_{0} := ∂_{t} + b · τ∂_{τ} and D_{1} := ∂_{τ} + a · τ∂_{t},

and they obey [D_{0}, D_{0}] = 0, [D_{0}, D_{1}] = −bD_{1} and [D_{1}, D_{1}] = 2aD_{0}.

Proof. Both assertions follow by straightforward verifications. □

Remarks. (1) It can easily be checked that the above family yields only three non-isomorphic Lie supergroup structures on ℝ^{1|1}, since (ℝ^{1|1}, μ_{a,0}) with a ≠ 0 is isomorphic to (ℝ^{1|1}, μ_{1,0}) and (ℝ^{1|1}, μ_{0,b}) with b ≠ 0 is isomorphic to (ℝ^{1|1}, μ_{0,1}) and the three multiplications μ_{0,0}, μ_{1,0} and μ_{0,1} correspond to nonisomorphic Lie supergroup structures on ℝ^{1|1}. Nevertheless it is very convenient to work here with the more flexible two-parameter family of multiplications.

(2) In fact, all Lie supergroup structures on ℝ^{1|1} are equivalent to μ_{0,0}, μ_{1,0} or μ_{0,1}. See, e.g., [4] for a direct approach to the classification of all Lie supergroup structures on ℝ^{1|1}.

Definition 3.2. Let G = (G, O_{G}) resp. M = (M, O_{M}) be a Lie supergroup with multiplication morphism μ and unit element e resp. a supermanifold. A “local action of G on M” is given by the following data: a collection Π of pairs of open subsets π = (U_{π}, W_{π}) of M, where U_{π} is relatively compact in W_{π}, with associated open sub supermanifolds U_{π} ⊂ W_{π} ⊂ M such that {U_{π}|π ∈ Π} is an open covering of M, and for all π in Π an open sub supermanifold G_{π} ⊂ G, containing the neutral element e and a morphism

Φ_{π} : G_{π} × U_{π} → W_{π}

fulfilling

(1) Φ_{π} ∘ (e × id_{U_{π}}) = id_{U_{π}}, where e : {pt} → G is viewed as a morphism,

(2) Φ_{π} ∘ (μ × id_{M}) = Φ_{π} ∘ (id_{G} × Φ_{π}) , where both sides are defined,

(3) if U_{π} ∩ U_{π′} ≠ ∅, Φ_{π} = Φ_{π′} on (G_{π} ∩ G_{π′}) × (U_{π} ∩ U_{π′}) .

Proposition 3.3. Let G = (G, O_{G}) resp. M = (M, O_{M}) be a Lie supergroup resp. a supermanifold. Then

(i) a local G-action on M, specified by a set Π and morphisms {(U_{π}, W_{π}, G_{π}, Φ_{π})|π ∈ Π}, gives rise to an open sub supermanifold V ⊂ G × M containing {e} × M and a morphism Φ_{V} : V → M such that

Φ_{V} ∘ (μ × id_{M}) = Φ_{V} ∘ (id_{G} × Φ_{V}) ,  (∗)

where both sides are defined and such that

Φ_{π} = Φ_{V} on (G_{π} × U_{π}) ∩ V, ∀π ∈ Π,  (∗∗)

(ii) an open sub supermanifold V ⊂ G × M containing {e} × M and a morphism Φ : V → M such that (∗) is fulfilled, where it makes sense, yields a local G-action on M such that (∗∗) holds.

Proof. As in the classical case of ungraded manifolds and Lie groups. □

Theorem 3.4. Let M be a supermanifold, X a vector field on M and V ⊂ ℝ^{1|1} × M the domain of the maximal flow F : V → M satisfying

(inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ (∂_{t} + ∂_{τ}) ∘ F^{∗} = (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ F^{∗} ∘ X and
F ∘ inj _{{0}×M}^{V} = id_{M}.

Let a and b be real numbers such that a · b = 0. Then the following assertions are equivalent:

(i) the map F fulfills

(∂_{t} + ∂_{τ} + τ(a∂_{t} + b∂_{τ})) ∘ F^{∗} = F^{∗} ∘ X,

(ii) the map F is a local (ℝ^{1|1}, μ_{a,b})-action on M,

(iii) ℝX_{0} ⊕ ℝX_{1} is a sub Lie superalgebra of T_{M}(M) with commutators [X_{0}, X_{1}] = −bX_{1} and [X_{1}, X_{1}] = 2aX_{0}.

Proof. Recall that F fulfills

(inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ ∂_{t} ∘ F^{∗} = (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ F^{∗} ∘ X_{0} and
(inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ ∂_{τ} ∘ F^{∗} = (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ F^{∗} ∘ X_{1}.

Denoting the projection from ℝ^{1|1} to ℝ by p, we have

id_{ℝ^{1|1}}^{∗} = p^{∗} ∘ (inj _{ℝ}^{ℝ^{1|1}})^{∗} + τ · p^{∗} ∘ (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ ∂_{τ}

which we will write more succinctly as

id_{ℝ^{1|1}}^{∗} = (inj _{ℝ}^{ℝ^{1|1}})^{∗} + τ · (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ ∂_{τ} .  (19)

Using relation (19) and the equations fulfilled by F^{∗} we get

F^{∗} ∘ X = (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ F^{∗} ∘ X + τ · (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ ∂_{τ} ∘ F^{∗} ∘ X
= (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ (∂_{t} + ∂_{τ}) ∘ F^{∗} + τ · (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ F^{∗} ∘ X_{1} ∘ X
= (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ (∂_{t} + ∂_{τ}) ∘ F^{∗}
+ τ · (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ F^{∗} ∘ ([X_{1}, X_{0}] + X_{0} ∘ X_{1} + \frac{1}{2}[X_{1}, X_{1}]) .

Since

(inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ F^{∗} ∘ X_{0} ∘ X_{1} = (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ ∂_{t} ∘ F^{∗} ∘ X_{1}
= ∂_{t} ∘ (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ F^{∗} ∘ X_{1}
= ∂_{t} ∘ (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ ∂_{τ} ∘ F^{∗}
= ∂_{t} ∘ ∂_{τ} ∘ F^{∗}
= ∂_{τ} ∘ ∂_{t} ∘ F^{∗},

we arrive at

F^{∗} ∘ X = (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ (∂_{t} + ∂_{τ}) ∘ F^{∗}+
τ · F^{∗} ∘ ([X_{1}, X_{0}] + \frac{1}{2}[X_{1}, X_{1}]) + τ · ∂_{τ} ∘ ∂_{t} ∘ F^{∗}. (20)

On the other hand, if a and b are real numbers, we have, again using (19)

(∂_{t} + ∂_{τ} + τ(a∂_{t} + b∂_{τ})) ∘ F^{∗}
= ((inj _{ℝ}^{ℝ^{1|1}})^{∗} + τ · (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ ∂_{τ} ) ∘ (∂_{t} + ∂_{τ}) ∘ F^{∗}
+τ · (a · (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ ∂_{t} + b · (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ ∂_{τ} ) ∘ F^{∗}
= (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ (∂_{t} + ∂_{τ}) ∘ F^{∗} + τ · ∂_{τ} ∘ ∂_{t} ∘ F^{∗}
+τ · F^{∗} ∘ (aX_{0} + bX_{1}) .

Thus we have

\begin{matrix}(∂_{t} + ∂_{τ} + τ(a∂_{t} + b∂_{τ})) ∘ F^{∗} − F^{∗} ∘ X \\ = τ · F^{∗} ∘ (aX_{0} − \frac{1}{2}[X_{1}, X_{1}] + bX_{1} − [X_{1}, X_{0}]) . \end{matrix}  (21)

Since F satisfies the initial condition (inj _{{0}×M}^{V})^{∗} ∘ F^{∗} = id_{M}, τ · F^{∗} is injective and thus Equation (21) easily implies the equivalence of (i) and (iii).

We remark that, in this case, we automatically have a · b = 0 since the Jacobi identity implies that [X_{1}, [X_{1}, X_{1}]] = [[X_{1}, X_{1}], X_{1}] + (−1)^{1·1}[X_{1}, [X_{1}, X_{1}]], i.e., 2a · b · X_{1} = [X_{1}, [X_{1}, X_{1}]] = 0.

Assume now that a and b are real numbers such that (i) satisfied, and let μ = μ_{a,b} be as in Lemma 3.1. We have to show that F is a local action of (ℝ^{1|1}, μ). Let us define

G := F ∘ (id_{ℝ^{1|1}} × F) : ℝ^{1|1} × (ℝ^{1|1} × M) → M

and

H := F ∘ (μ × id_{M}) : (ℝ^{1|1} × ℝ^{1|1}) × M ≅ ℝ^{1|1} × (ℝ^{1|1} × M) → M.

In order to prove that F is a ℝ^{1|1}-action on, we have to show that G = H. We observe that G is the integral curve of X subject to the initial condition F ∈ Mor(ℝ^{1|1} × M, M).

Let us prove that the morphism H satisfies the following conditions:

(inj _{ℝ×(ℝ^{1|1}×M)}^{ℝ^{1|1}×(ℝ^{1|1}×M)})^{∗} ∘ (∂_{t_{1}} + ∂_{τ_{1}}) ∘ H^{∗} = (inj _{ℝ×(ℝ^{1|1}×M)}^{ℝ^{1|1}×(ℝ^{1|1}×M)})^{∗} ∘ H^{∗} ∘ X (22)
H ∘ inj _{{0}×(ℝ^{1|1}×M)}^{ℝ^{1|1}×(ℝ^{1|1}×M)} = F.  (23)

Then by the unicity of integral curves we have H = G.

Equation (23) holds true since μ ∘ inj _{{0}×ℝ^{1|1}}^{ℝ^{1|1}×ℝ^{1|1}} = id_{ℝ^{1|1}}.

Defining D := D_{0} + D_{1} = ∂_{t_{1}} + ∂_{τ_{1}} + τ_{1}(a∂_{t_{1}} + b∂_{τ_{1}}) and writing inj _{|t_{1}} for

inj _{ℝ×(ℝ^{1|1}×M)}^{ℝ^{1|1}×(ℝ^{1|1}×M)} and using right invariance of D, we arrive at equation (22) as follows

(inj _{ℝ×(ℝ^{1|1}×M)}^{ℝ^{1|1}×(ℝ^{1|1}×M)})^{∗} ∘ (∂_{t_{1}} + ∂_{τ_{1}}) ∘ H^{∗}
= (inj _{|t_{1}})^{∗} ∘ D ∘ H^{∗}
= (inj _{|t_{1}})^{∗} ∘ (((D ⊗ id_{ℝ^{1|1}}^{∗}) ∘ μ^{∗}) × id_{M}^{∗}) ∘ F^{∗}
= (inj _{|t_{1}})^{∗} ∘ ((μ^{∗} ∘ D) × id_{M}^{∗}) ∘ F^{∗}
= (inj _{|t_{1}})^{∗} ∘ (μ^{∗} × id_{M}^{∗}) ∘ F^{∗} ∘ X
= (inj _{|t_{1}})^{∗} ∘ H^{∗} ∘ X.

Thus we obtain that (i) implies (ii).

Assume now that (ii) is satisfied, i.e., there exists a Lie supergroup structure on ℝ^{1|1} with multiplication μ such that

F ∘ (id_{ℝ^{1|1}} × F) = F ∘ (μ × id_{M}).  (24)

Since F is a flow for X, with initial condition ϕ = id_{M}, the LHS of the preceding equality is a flow for X with initial condition ϕ = F, (24) implies

(inj _{|t_{1}})^{∗} ∘ (∂_{t_{1}} + ∂_{τ_{1}}) ∘ (μ^{∗} × id_{M}^{∗}) ∘ F^{∗} = (inj _{|t_{1}})^{∗} ∘ (μ^{∗} × id_{M}^{∗}) ∘ F^{∗} ∘ X. (25)

Integration of Vector Fields on Supermanifolds 535

By Equation (20), the RHS gives for t_{1} = 0:

(inj _{|t_{1}=0})^{∗} ∘ (μ^{∗} × id_{M}^{∗}) ∘ F^{∗} ∘ X = F^{∗} ∘ X
= (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ (∂_{t} + ∂_{τ}) ∘ F^{∗}

+τ · [F^{∗} ∘ ([X_{1}, X_{0}] + \frac{1}{2}[X_{1}, X_{1}]) + ∂_{τ} ∘ ∂_{t} ∘ F^{∗}] .

Moreover, we have by direct comparison

(∂_{t_{1}} + ∂_{τ_{1}}) ∘ μ^{∗} = (∂_{t_{1}} + ∂_{τ_{1}})(μ^{∗}(t)) · (μ^{∗} ∘ ∂_{t})
+(∂_{t_{1}} + ∂_{τ_{1}})(μ^{∗}(τ)) · (μ^{∗} ∘ ∂_{τ}).

Thus, if μ : ℝ^{1|1} × ℝ^{1|1} → ℝ^{1|1} is given by

μ^{∗}(t) = μ̃(t_{1}, t_{2}) + α(t_{1}, t_{2})τ_{1}τ_{2}
μ^{∗}(τ) = β(t_{1}, t_{2})τ_{1} + γ(t_{1}, t_{2})τ_{2},

and upon using (inj _{{0}×ℝ^{1|1}}^{ℝ^{1|1}×ℝ^{1|1}})^{∗} ∘ μ^{∗} = id_{ℝ^{1|1}}^{∗}, we have

(inj _{{0}×ℝ^{1|1}}^{ℝ^{1|1}×ℝ^{1|1}})^{∗}(∂_{t_{1}} + ∂_{τ_{1}}) ∘ μ^{∗} = ((∂_{t_{1}} μ̃)(0, t) + α(0, t)τ) · ∂_{t}
+ (β(0, t) + (∂_{t_{1}}γ)(0, t)τ) · ∂_{τ} .

Using again (19), we have

∂_{t} ∘ F^{∗} = (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ F^{∗} ∘ X_{0} + τ · ∂_{τ} ∘ ∂_{t} ∘ F^{∗} and
∂_{τ} ∘ F^{∗} = (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ F^{∗} ∘ X_{1}.

Then the LHS of (25) at t_{1} = 0 is

(inj _{|t_{1}=0})^{∗} ∘ (∂_{t_{1}} + ∂_{τ_{1}}) ∘ (μ^{∗} × id_{M}^{∗}) ∘ F^{∗}

= (∂_{t_{1}} μ̃)(0, t) · (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ F^{∗} ∘ X_{0}
+β(0, t) · (inj _{ℝ}^{ℝ^{1|1}})^{∗} ∘ F^{∗} ∘ X_{1}
+τ · ((∂_{t_{1}} μ̃)(0, t) · ∂_{τ} ∘ ∂_{t} ∘ F^{∗}
+α(0, t) · F^{∗} ∘ X_{0}
+(∂_{t_{1}}γ)(0, t) · F^{∗} ∘ X_{1}).

Using the obtained identities for its LHS and RHS, the “τ-part” of Equation (25) at t_{1} = 0 gives us:

τ · [F^{∗} ∘ ([X_{1}, X_{0}] + \frac{1}{2}[X_{1}, X_{1}]) + ∂_{τ} ∘ ∂_{t} ∘ F^{∗}]

= τ · [(∂_{t_{1}} μ̃)(0, t) · ∂_{τ} ∘ ∂_{t} ∘ F^{∗} + α(0, t) · F^{∗} ∘ X_{0} + (∂_{t_{1}}γ)(0, t) · F^{∗} ∘ X_{1}].

Documenta Mathematica 18 (2013) 519–545

Since μ̃(t_{1}, 0) = t_{1}, we have (∂_{t_{1}} μ̃)(0, 0) = 1 and therefore the preceding equation evaluated at t = 0 yields

[X_{1}, X_{0}] + \frac{1}{2}[X_{1}, X_{1}] = (∂_{t_{1}}γ)(0, 0) · X_{1} + α(0, 0) · X_{0}

finishing the proof that (ii) implies (iii). □

4 Examples and applications

(4.1) If X = X_{0} is an even vector field, the fact that it integrates to a (local) action of ℝ = ℝ^{1|0} is almost folkloristic. The relatively recent proof of [3] - in the case of compact supermanifolds - is close to our approach. A non-trivial (local) action of ℝ^{1|0} can obviously be extended to a (local) action of (ℝ^{1|1}, μ_{a,b}) if and only if a = 0. Of course, the ensueing action of ℝ^{1|1} will not even be almost-effective, since the positive-dimensional sub Lie supergroup ℝ^{0|1} acts trivially.

(4.2) Our preferred example of an even vector field gives rise to the exponential map on Lie supergroups.

Let us first recall that an even vector field X on a supermanifold M corresponds to a section σ_{X} of the tangent bundle T M → M (see, e.g., Sections 7 and 8 of [15] for a construction of T M and a proof of this statement, and compare also the remark after Thm. 2.19 in [5]). Given an auxiliary supermanifold S and a morphism ψ : S → M, one calls for i ∈ {0, 1}

Der_{ψ}(O_{M}(M), O_{S}(S))_{i} :=

{D : O_{M}(M) → O_{S}(S)|D is ℝ-linear and ∀f, g ∈ O_{M}(M) homogeneous,

D(f · g) = D(f) · ψ^{∗}(g) + (−1)^{i·|f|}ψ^{∗}(f) · D(g)},

the “space of derivations of parity i along ψ”. In category-theoretical terms the tangent bundle T M represents then the functor from supermanifolds to sets given by S ↦ {(ψ, D)|ψ ∈ Mor(S, M) and D ∈ Der_{ψ}(O_{M}(M), O_{S}(S))_{0}} (compare, e.g., Section 3 of [6]).

Let now G = (G, O_{G}) be a Lie supergroup with multiplication μ=μ^{G} and neutral element e. We define X in Der (O_{G×T_{e}G}(G × T_{e}G)) to be the even vector field on G × T_{e}G corresponding to the following section σ_{X} of T G × T (T_{e}G) ≅ T (G × T_{e}G) → G × T_{e}G. We denote the zero-section of T G → G by σ_{0} and the canonical inclusion T_{e}G → T G by i_{e}. Then σ_{X} := (T μ ∘ (σ_{0} × i_{e}), 0), where T μ : T G × T G ≅ T (G × G) → T G is the tangential morphism associated to the multiplication morphism. (For simplicity, we write 0 for the zero-section of T (T_{e}G) → T_{e}G here and in the sequel.)

Let us recall that for S an arbitrary supermanifold, and ϕ : M → N a morphism between supermanifolds, we have an induced map ϕ(S) : M(S) = Mor(S, M) → Mor(S, N) = N (S), ϕ(S)(ψ) := ϕ ∘ ψ. Given a finitedimensional Lie superalgebra 𝔤 or a Lie supergroup G, one easily checks that for all k ≥ 0, 𝔤(ℝ^{0|k}) resp. G(ℝ^{0|k}) is a finite-dimensional classical (i.e. even) Lie algebra resp. Lie group. Furthermore, T_{e}(G(ℝ^{0|k})) is canonically isomorphic to (T_{e}G)(ℝ^{0|k}), where the first e is the obvious constant morphism from ℝ^{0|k} to G and the second e denotes the neutral element of G. (Compare, e.g., [16] for more information on the superpoint approach to Lie supergroups.)

Lemma 4.1. Let G be a Lie supergroup with multiplication μ^{G}, and the vector field X as above. Then

(i) the induced vector field X̃ on the underlying manifold G × T_{e}G is given

as

X̃_{(g,ξ)} = (ξ^{L}(g), 0) ∀(g, ξ) ∈ G × T_{e}G,

(ii) the (even) vector fields X̃ and X are complete.

Proof. (i) For k ≥ 0, let σ_{X}(ℝ^{0|k}) be the section of T (G(ℝ^{0|k})) × T (T_{e}G(ℝ^{0|k})) → G(ℝ^{0|k}) × T_{e}G(ℝ^{0|k}) induced by σ_{X}, and let X^{k} be the corresponding derivation on G(ℝ^{0|k}) × T_{e}G(ℝ^{0|k}). Since G(ℝ^{0|k}) × (T_{e}G)(ℝ^{0|k}) is an ungraded manifold,

σ_{X}(ℝ^{0|k})(g, ξ) = (T μ^{G} ∘ (σ_{0} × i_{e}) ∘ (g × ξ), 0)
= (T μ^{G} ∘ (0_{g} × ξ), 0)
= ((T μ^{G(ℝ^{0|k})})_{(g,e)}(0, ξ), 0)
= ((T l_{g}^{G(ℝ^{0|k})})_{e}(ξ), 0) ,

where μ^{G(ℝ^{0|k})} is the multiplication on G(ℝ^{0|k}) and l_{g}^{G(ℝ^{0|k})} is the leftmultiplication by the element g of the group G(ℝ^{0|k}). We conclude that σ_{X}(ℝ^{0|k})(g, ξ) (or equivalently X_{(g,ξ)}^{k}) corresponds to (ξ^{L}(g), 0), where ξ^{L} is the unique left-invariant vector field on G(ℝ^{0|k}) such that its value in e is ξ. (Observe that G(ℝ^{0|k}) is a classical Lie group and not only a group object in the category of supermanifolds, allowing us to argue “point-wise”.) (ii) The flows of X^{k} are simply given by F^{X^{k}} : ℝ × G(ℝ^{0|k}) × (T_{e}G)(ℝ^{0|k}) → G(ℝ^{0|k}) × (T_{e}G)(ℝ^{0|k}), (t, g, ξ) ↦ (g · exp^{G(ℝ^{0|k})}(tξ), ξ). All fields X^{k} are thus complete, in particular this holds for X̃ = X^{0}, the induced vector field on G = G(ℝ^{0|0}). By Theorem 2.3 the flow F^{X} : ℝ × G × T_{e}G → G × T_{e}G is then global as well, i.e. X is complete. □

Definition 4.2. Let G = (G, O_{G}) be a Lie supergroup with multiplication μ and neutral element e, and with the even vector field X and its flow morphism F = F^{X} as above. Then the “exponential morphism of G” is given by exp^{G} = proj_{1}∘F ∘inj_{{1}×{e}×T_{e}G}^{ℝ×G×T_{e}G} : T_{e}G → G, where proj_{1} : G ×T_{e}G → G is the projection on the first factor. Diagrammatically, one has

[figure: ℝ × G × T_{e}G ⊃ {1} × {e} × T_{e}G; G × T_{e}G; T_{e}G; G.; F; proj_{1}; ≅; exp^{G}]

Theorem 4.3. The exponential morphism exp^{G} : T_{e}G → G for a Lie supergroup G fulfills and is uniquely determined by the following condition: for all k ≥ 0, exp^{G}(ℝ^{0|k}) : T_{e}G(ℝ^{0|k}) → G(ℝ^{0|k}) is the exponential map exp^{G(ℝ^{0|k})} of the finite-dimensional, ungraded Lie group G(ℝ^{0|k}).

Proof. Using the notations of Lemma 4.1, a straightforward calculation shows that the flow F^{X^{k}} of X^{k} on (G × T_{e}G)(ℝ^{0|k}) is given as follows (t, (g, ξ)) ↦ F^{X} ∘ inj_{t} ∘ (g × ξ) and, notably, we have F^{X^{k}} ∘ inj_{(1,e)}(ξ) = F^{X} ∘ inj_{1} ∘ (e × ξ) = F^{X} ∘ inj_{(1,e)} ∘ ξ. Hence

exp^{G(ℝ^{0|k})}(ξ) = proj_{1} ∘ F^{X^{k}} ∘ inj_{(1,e)}(ξ)
= proj_{1} ∘ F^{X} ∘ inj_{(1,e)} ∘ ξ
= exp^{G} ∘ξ
= (exp^{G})(ℝ^{0|k})(ξ).

On the other hand, it is clear that the subcategory of superpoints with objects {ℝ^{0|k}|k ≥ 0} generates the category of supermanifolds in the following sense: given two different morphisms ϕ_{1}, ϕ_{2} : M → N between supermanifolds, there exists a k ≥ 0 and a morphism ψ : ℝ^{0|k} → M such that ϕ_{1} ∘ ψ ≠ ϕ_{2} ∘ ψ. Thus it follows that the family {exp^{G}(ℝ^{0|k})|k ≥ 0} uniquely fixes exp^{G}. □

(4.3) Part of our interest in the integration of supervector fields stemmed from the construction of a geodesic flow in [5] . Given a homogeneous (i.e., even or odd) Riemannian metric on a supermanifold M, the associated geodesic flow is, in fact, defined as the flow of an appropriate Hamiltonian vector field on its (co-)tangent bundle.

(4.4) An odd vector field X_{1} on a supermanifold M is called “homological” if X_{1} ∘ X_{1} = \frac{1}{2}[X_{1}, X_{1}] = 0. Its flow is given by the following ℝ^{0|1}-action Φ : ℝ^{0|1} × M → M, Φ^{∗}(f) = f + τ · X_{1}(f), ∀f ∈ O_{M}(M). This action can, of course, be extended to a (not almost-effective) (ℝ^{1|1}, μ_{a,b})-action if and only if b = 0. Typical examples arise as follows: let E → M be a vector bundle over a classical manifold and T be an “appropriate” ℝ-linear operator on sections of ∧E^{∗}, then T yields a vector field on ΠE := (M, Γ_{∧E^{∗}}^{∞}), the supermanifold associated to E → M by the Batchelor construction. If E = T M → M, we have Γ_{∧E^{∗}}^{∞} = Ω_{M}^{•}(M), the sheaf of differential forms on M, with its natural ℤ/2ℤ-grading. Taking T = d, we get an odd vector field that is obviously homological. Taking T = ι_{ξ}, the contraction of differential forms with a vector field ξ on the (here classical) base manifold M, we again get a homological vector field on ΠT M. Since ι_{ξ} ∘ ι_{η} + ι_{η} ∘ ι_{ξ} = 0, the vector space of all vector fields on M is realized as a commutative, purely odd sub Lie superalgebra of all vector fields on ΠT M. More generally, a section s of E → M always gives rise to a contraction ι_{s} : Γ_{∧E^{∗}}^{∞}(M) → Γ_{∧E^{∗}}^{∞}(M) that is an odd derivation (i.e. an anti-derivation of degree -1 in more classical language). Furthermore, given two sections s and t of E, the associated odd vector fields commute. In the article [1] this construction is studied in the special case that E is the spinor bundle over a classical spin manifold M.

(4.5) If G is a Lie group acting on a classical manifold M, then the action can of course be lifted to an action on the total space of the tangent and the cotangent bundle of M. The induced vector fields on ΠT M are even and ξ in 𝔤 = Lie(G), the Lie algebra of G, acts on O_{ΠT M} by L_{ξ}, the Lie derivative with respect to the fundamental vector field on M associated to ξ. Putting together these fields and the contractions constructed in Example (4.4), we get a Lie superalgebra with underlying vector space 𝔤 ⊕ 𝔤, the first resp. second summand being the even resp. odd part. The commutators in 𝔤 ⊕ 𝔤 are given as follows: [L_{ξ}, L_{η}] = L_{[ξ,η]}, [ι_{ξ}, ι_{η}] = 0 and [L_{ξ}, ι_{η}] = ι_{[ξ,η]} for all ξ, η ∈ 𝔤. In fact, the above can be interpreted as an action of the Lie supergroup ΠT G on ΠT M. The Lie algebra 𝔤 ⊕ 𝔤 can be extended by a one-dimensional odd direction generated by the exterior derivative d. The extended algebra 𝔤⊕(𝔤⊕ℝ·d), still a sub Lie superalgebra of T_{ΠT M}(M), has the following additional commutators:

[L_{ξ}, d] = 0 and [d, ι_{ξ}] = L_{ξ} for all ξ ∈ 𝔤.

(4.6) If M = ℝ^{1|1} with coordinates (x, ξ), the vector field X_{1} = ∂_{ξ} + ξ∂_{x} is obviously odd and non-homological since X_{1} ∘ X_{1} = ∂_{x}. Direct inspection shows that the map Φ : ℝ^{0|1} × M → M, Φ^{∗}(f) = f + τ · X_{1}(f) (compare Example (4.1)) does not fulfill ∂_{τ} ∘ Φ^{∗} = Φ^{∗} ∘ X_{1}. Nevertheless, the trivial extension of Φ to a morphism F : ℝ^{1|1} × M → M is the flow of X_{1} in the sense of Theorem 2.3, fulfilling the initial condition ϕ = id_{M}. We underline that this map is not an action of ℝ^{1|1}. Upon extending X_{1} to X := X_{0} + X_{1} with X_{0} := \frac{1}{2}[X_{1}, X_{1}] = X_{1} ∘ X_{1}, we obtain by Theorem 3.4 an action of (ℝ^{1|1}, μ_{1,0}) as the flow map of X. Let us observe that the above vector field X_{1} (and not X) is the prototype of what is called a “supersymmetry” in the physics literature (compare, e.g., [21] and the other relevant texts in these volumes.). More recently, the associated Lie supergroup structure on ℝ^{1|1} (and an analogous structure on ℝ^{2|1}) were introduced by S. Stolz and P. Teichner into their program to geometrize the cocycles of elliptic cohomology (compare [7] and [18]).

Obviously, one can generalize this construction to ℝ^{m|n} (m, n ≥ 1) with coordinates (x_{1}, …, x_{m}, ξ_{1}, …, ξ_{n}) by setting for 1 ≤ k ≤ m, 1 ≤ α ≤ n

D_{α,k} := ∂_{ξ_{α}} + ξ_{α} · ∂_{x_{k}}.

We then have [D_{α,k}, D_{β,l}] = δ_{α,β} · (∂_{x_{k}} + ∂_{x_{l}}) and [D_{α,k}, ∂_{x_{l}}] = 0. Taking X_{1} = D_{α,k} and X_{0} = ∂_{x_{k}} we reproduce a copy of the preceding situation.

(4.7) The vector field X = X_{0} + X_{1} on M = ℝ^{1|1} with X_{0} = ∂_{x} + ξ · ∂_{ξ} and X_{1} = ∂_{ξ}+ξ·∂_{x}, already mentioned in the introduction, is a very simple example of an inhomogeneous vector field not generating any local ℝ^{1|1}-action, since, e.g., condition (iii) in Theorem 3.4 is violated. Thus integration of X is only possible in the sense of Theorem 2.3, i.e. upon using the evaluation map. Let us observe that the sub Lie superalgebra 𝔤 of T_{M}(M) generated by X, i.e. by X_{0} and X_{1} since sub Lie superalgebras are by definition graded sub vector spaces, is four-dimensional with two even generators Z, W and two odd generators D, Q such that: Z is central, [W, D] = Q, [W, Q] = D, D^{2} = −Q^{2} = Z and [D, Q] = 0. (This amounts in physical interpretation to the presence of two commuting supersymmetries D and Q, generating the same supersymmetric Hamiltonian Z plus an even symmetry commuting with the Hamiltonian and exchanging the supersymmetries D and Q.)

5 Flow of a holomorphic vector field on a holomorphic supermanifold

In this section we extend our results to the holomorphic case. We will always denote the canonical coordinates on ℂ^{1|1} by z and ζ and write ∂_{z} resp. ∂_{ζ} for \frac{∂}{∂z} resp. \frac{∂}{∂ζ}. All “auxiliary” supermanifolds S and morphisms having these as sources will be assumed to be holomorphic in this section.

Definition 5.1. Let M = (M, O_{M}) be a holomorphic supermanifold and X a holomorphic vector field on M and S a supermanifold with a morphism ϕ ∈ Mor(S, M) and z_{0} in ℂ.

(1) A “flow for X (with initial condition ϕ and with respect to z_{0})” is an open sub supermanifold V ⊂ ℂ^{1|1} × S, such that {z_{0}} × S ⊂ V (S and V the bodies of S and V) and such that for all s in S, (ℂ × {s}) ∩ V is connected, together with a morphism of holomorphic supermanifolds F : V → M such that

(inj _{ℂ}^{ℂ^{1|1}})^{∗} ∘ (∂_{z} + ∂_{ζ}) ∘ F^{∗} = (inj _{ℂ}^{ℂ^{1|1}})^{∗} ∘ F^{∗} ∘ X and
F ∘ inj _{{z_{0}}×S}^{V} = ϕ.

Sometimes we call the supermanifold V (or abusively its body V) the “flow domain” of X.

(2) A flow domain V of a flow (V, F) for X is called “fibrewise 1-connected (relative to the projection V → S)” (or “fibrewise 1-connected over S”) if for all s in S, (ℂ × {s}) ∩ V is connected and simply connected.

Remark. We avoid the term “complex supermanifold” here, since it is often used to describe supermanifolds that are, as ringed spaces, locally isomorphic to open sets D ⊂ ℝ^{k} with structure sheaf C_{D}^{∞} ⊗_{ℝ} ∧ℂ^{l}. “Holomorphic supermanifolds” are of course locally isomorphic to open sets D ⊂ ℂ^{k} with structure sheaf O_{D} ⊗_{ℂ} ∧ℂ^{l}, where O_{D} denotes the sheaf of holomorphic functions on D.

Let us first give the holomorphic analogue of Lemma 2.1.

Lemma 5.2. Let U ⊂ ℂ^{m|n} and W ⊂ ℂ^{p|q} be superdomains, X a holomorphic vector field on W (not necessarily homogeneous), ϕ in Mor(U, W) and z_{0} in ℂ. Then

(i) it exists a holomorphic flow (V, F̃) for the reduced holomorphic vector field X̃ on U with initial condition ϕ̃ and with respect to z_{0} such that the flow domain V ⊂ ℂ × U is fibrewise 1-connected over U. Furthermore on every flow domain in the sense of Definition 5.1 the holomorphic flow is unique.

(ii) Let now (V, F̃) be a fibrewise 1-connected flow domain for X̃ over U. Then there exists a unique holomorphic flow F : V → W for X, with V the open sub supermanifold of ℂ^{1|1} × U with body equal to V.

Remark. The example of the holomorphic vector field X = (w^{2} + w^{3}ξ_{1}ξ_{2})\frac{∂}{∂w} on W = ℂ^{1|2} with coordinates (w, ξ_{1}, ξ_{2}) shows that the condition of fibrewise 1-connectivity of V is not only a technical assumption to our proof. The underlying vector field X̃ = w^{2} \frac{∂}{∂w} on ℂ, with initial condition ϕ̃ = id : ℂ → ℂ with respect to z_{0} = 0, can be integrated to the flow F̃ : V = ℂ^{2}\{z · w = 1} → ℂ, F̃(z, w) = \frac{1}{1/w−z} for w ≠ 0 and F̃(z, 0) = 0. Obviously, for w ≠ 0, (ℂ×\{w\})∩V is connected, but not simply connected. Direct inspection now shows that the flow F of X with initial condition ϕ = id and with respect to z_{0} = 0 cannot be defined on the whole of V = (V, O_{ℂ^{1|1}×ℂ^{1|2}}|_{V}).

Proof of Lemma 5.2. (i) The existence (and the stated unicity property) of a flow (V̌ , F̃) for X̃, with {z_{0}} × U ⊂ V̌ ⊂ ℂ × U fulfilling the initial condition ϕ̃ with respect to z_{0} ∈ ℂ is of course a classical application of the existence of solutions of holomorphic ordinary differential equations (see, e.g., [8]). Upon reducing the size of V̌ we always find flow domains that are fibrewise 1-connected.

(ii) The induction procedure of the proof of Lemma 2.1 can be applied here upon recalling the following standard facts from the theory of holomorphic linear ordinary differential equations (compare, e.g., [8]):

Fact 1. Let Ω ⊂ ℂ be open and 1-connected (i.e. connected and simply connected), and z_{0} ∈ Ω. If A : Ω → Mat(N × N, ℂ) and b : Ω → ℂ^{N} are holomorphic and ψ_{0} ∈ ℂ^{N}, then there exists a unique holomorphic map ψ : Ω → ℂ^{N} fulfilling

\frac{∂}{∂z} ψ(z) = A(z)ψ(z) + b(z)

such that ψ(z_{0}) = ψ_{0}.

Fact 2. Let Ω and z_{0} be as in Fact 1, and let P be a holomorphic manifold (“a parameter space”), and let A : Ω × P → Mat(N × N, ℂ), b : Ω × P → ℂ^{N}, as well as ψ_{0} : P → ℂ^{N} be holomorphic maps. Then there exists a unique holomorphic map ψ : Ω × P → ℂ^{N} fulfilling

\frac{∂}{∂z} ψ(z, x) = A(z, x)ψ(z, x) + b(z, x)

such that ψ(z_{0}, x) = ψ_{0}(x), ∀x ∈ P.

Obviously, to apply these facts in our context, we need the fibrewise 1connectivity of the “underlying flow domain” V for X̃. □

Before stating and proving our central result in the holomorphic case, we give the following useful shorthand.

Definition 5.3. Let S be a supermanifold, z_{0} in ℂ and N ⊂ ℂ^{1|1} × S be an open sub supermanifold containing {z_{0}} × S. Then N^{z_{0}} is defined as the open sub supermanifold of N = (N, O_{N}) whose body equals ⨿_{s∈S} ((ℂ × {s}) ∩ N)^{(z_{0},s)},

where ((ℂ × {s}) ∩ N)^{(z_{0},s)} is the connected component of (ℂ × {s}) ∩ N containing (z_{0}, s).

Remark. A flow domain V in the sense of Definition 5.1(1) is always open and contains {z_{0}} × S. Furthermore for all s in S the section (ℂ × {s}) ∩ V is connected. The preceding definition will in fact be useful for discussing intersections of flow domains in the next theorem.

Theorem 5.4. Let M be a holomorphic supermanifold and X a holomorphic vector field on M, and let S be a holomorphic supermanifold with a holomorphic morphism ψ : S → M, and z_{0} ∈ ℂ. Then

(i) there exists a flow (V, F̃) for the reduced vector field X̃ with initial condition ϕ̃ with respect to z_{0} such that the flow domain V ⊂ ℂ × S is fibrewise 1-connected over S,

(ii) if (V, F̃) is as in (i), then there exists a unique flow for X with initial condition ϕ with respect to z_{0}, F : V → M, where V ⊂ ℂ^{1|1} × S is the open sub supermanifold with body V, (iii) if (V_{1}, F_{1}) and (V_{2}, F_{2}) are two flows for X, both with initial condition ϕ

with respect to z_{0}, then F_{1} = F_{2} on the flow domain (V_{1} ∩ V_{2})^{z_{0}},

(iv) there exists maximal flow domains for X and the germs of their flows coincide on {z_{0}} × S.

Proof. (i) and (ii). It easily follows from Lemma 5.2 that S can be covered by open sub supermanifolds {U^{α}|α ∈ A} such that X_{|U^{α}} has a holomorphic flow with initial condition ϕ_{|U_{α}} with respect to z_{0}, F^{α} : V^{α} = Δ_{r_{α}}(z_{0}) × ℂ^{0|1} × U^{α} → M, where r_{α} > 0, and for r > 0, Δ_{r}(z_{0}) is the open disc of radius r centred in z_{0}. Since F^{α} = F^{β} on V^{α} ∩ V^{β} by the unicity part of Lemma 5.2, we can glue these flows to obtain ℂ^{1|1} × S ⊃ V := ∪_{α∈A}V^{α} ⟶^{F} M, a flow for X on M with initial condition ϕ

with respect to z_{0}. Obviously, the “fibres” (ℂ × {s}) ∩ V are 1-connected for all s in S, i.e., the flow domain V is fibrewise 1-connected over S. Note that if we have a flow F̃ for the reduced vector field X̃ on a flow domain V that is fibrewise 1-connected over S, then part (ii) of Lemma 5.2 yields a flow for X defined on V = (V, O_{ℂ^{1|1}×S}|_{V}).

(iii) The body of (V_{1} ∩ V_{2})^{z_{0}} has as a strong deformation retract the body of {z_{0}}×S. Without loss of generality we can assume that S and thus (V_{1} ∩V_{2})^{z_{0}} are connected. The local unicity in Lemma 5.2 together with the identity principle for holomorphic morphisms of holomorphic supermanifolds imply that F_{1} = F_{2} on (V_{1} ∩ V_{2})^{z_{0}}.

(iv) By Zorn’s lemma we get maximal flow domains and by part (iii) the corresponding flows coincide near {z_{0}} × S. □

Remarks. (1) The non-unicity of maximal flow domains for holomorphic vector fields is a well-known phenomenon already in the ungraded case. A simple example for this is the vector field X on ℂ^{∗} such that X(w) = \frac{1}{w} \frac{∂}{∂w} for all w in ℂ^{∗}.

(2) Given the above theorem, the analogues of Lemma 3.1, Proposition 3.3 and Theorem 3.4 can now without difficulty be proven to hold for holomorphic supermanifolds.

Acknowledgements. We gratefully acknowledge the support of the SFB/TR 12 Symmetries and Universality in Mesoscopic Systems program of the Deutsche Forschungsgemeinschaft. Furthermore, we thank the referee for very helpful comments.

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Stéphane Garnier Fakultät für Mathematik Ruhr-Universität Bochum D-44780 Bochum, Germany Stephane.Garnier@rub.de Tilmann Wurzbacher LMAM-UMR 7122 Université de Lorraine et C.N.R.S. F-57045 Metz, France and Fakultät für Mathematik Ruhr-Universität Bochum D-44780 Bochum, Germany Tilmann.Wurzbacher@rub.de 546

Documenta Mathematica 18 (2013)