Symmetries of Supermanifolds

Home , Supermanifold

DISSERTATION

zur Erlangung des Doktorgrades der Naturwissenschaften an der FakultätfürMathematik der Ruhr-UniversitätBochum

vorgelegt von

Hannah Bergner aus Remscheid

im Juni 2015

Contents

Introduction 5

1 Preliminaries 13 1.1 Supermanifolds ...... 13 1.2 Morphisms and vector ﬁelds on supermanifolds ...... 14

2 Lie supergroups and their actions 17 2.1 Basic definitions ...... 17 2.2 Local and infinitesimal actions of Lie supergroups ...... 18 2.3 Flows of vector fields ...... 19 2.4 Harish-Chandra pairs ...... 24 2.5 Effective actions of Lie supergroups ...... 26

3 Distributions on supermanifolds and Frobenius’ theorem 29 3.1 Commuting vector ﬁelds ...... 30 3.2 Involutive distributions and Frobenius’ theorem ...... 31

4 Globalizations of infinitesimal actions on supermanifolds 37 4.1 Distributions associated with infinitesimal actions ...... 38 4.2 Existence of local actions ...... 42 4.2.1 Uniqueness of local actions ...... 43 4.2.2 Construction of local actions ...... 43 4.3 Globalizations of infinitesimal actions ...... 47 4.3.1 Univalent leaves and holonomy ...... 49 4.3.2 The action on G × M from the right and invariant functions . . 54 4.3.3 Characterization of globalizable infinitesimal actions ...... 56 4.4 Examples ...... 61 4.5 Actions of simply-connected Lie supergroups ...... 64

5 Automorphism groups of compact complex supermanifolds 67 5.1 The group of automorphisms Aut0¯(M)...... 68 5.1.1 The topology on Aut0¯(M)...... 69 5.1.2 Non-existence of small subgroups of Aut0¯(M) ...... 73 5.1.3 One-parameter subgroups of Aut0¯(M) ...... 76 5.2 The Lie superalgebra of vector ﬁelds ...... 80 5.3 The automorphism group ...... 81 5.4 The functor of points of the automorphism group ...... 84 5.5 The case of a superdomain with bounded underlying domain ...... 88 5.6 Examples ...... 88 4 Introduction

Supermanifolds are generalizations of classical manifolds, whose definition is motivated by concepts stemming from supersymmetry. A supermanifold is a real or complex manifold with an enriched structure sheaf, meaning that locally, in addition to the usual coordinates, there are “odd” coordinates θ1, . . . θn which anticommute, i.e. θiθj = −θjθi. In this monograph, we study symmetries of supermanifolds. We will be concerned with two main questions: The first is about the relation of infinitesimal symmetries and local or global symmetries on supermanifolds. The second question is about the structure of the global symmetries of a compact complex supermanifold and the existence of Lie supergroup of automorphisms. Our motivation are the following classical results. A basic result in the study of symmetries on manifolds is Palais’ theory (see [Pal57]) on the relation of finite-dimensional Lie algebras of vector fields on manifolds and actions of the corresponding Lie groups. Palais proves that any finite-dimensional Lie algebra of vector fields on a manifold is induced by a local action of the corresponding Lie group and provides necessary and sufficient conditions for the Lie algebra of vector fields to be induced by a global action of the Lie group. In complex geometry, an important result is the fact that the automorphism group of a compact complex manifold carries the structure of a complex Lie group such that its action on the compact complex manifold is holomorphic (see [BM47]). One fundamental fact is that the Lie algebra Vec(M) of holomorphic vector fields on a compact complex manifold M is finite-dimensional. The results of Palais’ imply that the simply-connected Lie group with Lie algebra Vec(M) acts on M. It turns out that this Lie group, is up to ineffectivity, the connected component of the automorphism group of M. A vector field(1) on a supermanifold M, can be written as a sum of an even and an odd vector field on M. As proven in [MSV93] and [GW13], the flow of the vector field 1|1 X defines a local R -action on M if and only if the even part and and the odd part of the vector field X are contained in a (1|1)-dimensional Lie subsuperalgebra of the Lie superalgebra of vector fields on M. This generalizes the well known result that the flow of a vector field X on a classical manifold M is a local R-action; note that in this case RX is automatically a Lie subalgebra of Vec(M). Just as a (local) R-action on a manifold M induces a vector field, any (local) action of a Lie group G on the manifold M induces an infinitesimal action on M, i.e. a homomorphism g → Vec(M) of Lie algebras, where g is the Lie algebra of right-invariant vector fields on G and Vec(M)

(1)To be more precise, a vector field on a supermanifold should be called a super vector field. But to avoid excessive use of the word “super”, we will simply talk about vector fields on supermanifolds and always mean super vector fields.

5 denotes the set of vector fields on M. Similarly, any (local) action of a Lie supergroup G on a supermanifold M induces an infinitesimal action g → Vec(M), where g is now the Lie superalgebra of right-invariant vector fields on the Lie supergroup G; cf. Section 2.2. It is a natural question to ask in which cases it is possible to go the other way: Let M be a supermanifold and g a finite-dimensional Lie superalgebra. We ask which homomorphisms λ : g → Vec(M) of Lie superalgebras are induced by a local or global action of a Lie supergroup G with Lie superalgebra g. In the setting of classical manifolds and Lie groups, Palais studied these questions in [Pal57]. Palais proved that, given a Lie group G with Lie algebra of right-invariant vector fields g and an infinitesimal action λ : g → Vec(M) on a manifold M, there always exists a local action ϕ : W ⊆ G × M → M of the Lie group G which induces λ. It is easy to construct examples where the infinitesimal action is not induced by a global action of G on M: If M 0 is a manifold with G-action and we take an open subset M ⊂ M 0 which is not G-invariant, then the restriction of the infinitesimal action to M is not induced by a G-action on M. Another example is e.g. given by the vector field 2 ∂ x −x ∂x on R. Its flow ϕ : W ⊂ R × R → R, ϕ(t, x) = 1+tx is not global. But if we embed R into the real projective space, R ,→ P1(R), x 7→ [1 : x], we have the R-action given by (t, [x0 : x1]) 7→ [x0 + tx1 : x1]. In the coordinate chart R → U0 ⊂ P1(R), x 7→ [x : 1], the action is simply given by addition (t, x) 7→ x + t. The R-action on 2 ∂ P1(R) extends the flow on R and induces the vector field −x ∂x on R. Note that in both cases the infinitesimal action on M is still induced by a G- action on the larger manifold M 0. We call infinitesimal actions which arise in this way globalizable. Necessary and sufficient conditions for an infinitesimal action to be globalizable are found in [Pal57]. We generalize the results of [Pal57] to the case of infinitesimal actions on supermanifolds. The existence of a local action with a given infinitesimal action on a supermanifold is proven and conditions for the globalizability of an infinitesimal actions are found. As in the classical case, a key point in the proof is the study of a certain distribution Dλ on the product G × M associated with an infinitesimal action λ : g → Vec(M) on the supermanifold M, where G is again a Lie supergroup with Lie superalgebra of right-invariant vector fields g; see Section 4.1. The distribution Dλ is spanned by vector fields of the form X + λ(X) for X ∈ g, considering X and λ(X) as vector fields on the product G × M, so more formally we ∗ ∗ should write X ⊗ idM + idG ⊗ λ(X) instead of X + λ(X). Regarding the equivalence of infinitesimal and local actions up to shrinking, we have the full analogue of the classical result: Theorem 1 (see Section 4.2). Let G be a Lie supergroup with Lie superalgebra of right-invariant vector fields g, and let λ : g → Vec(M) be an infinitesimal action on a supermanifold M. Then there exists a local G-action ϕ : W ⊆ G × M → M on M which induces the infinitesimal action λ. Moreover, any local action ϕ : W → M is uniquely determined by its induced infinitesimal action and domain of definition. Using the notation of the preceding theorem, we now ask when the infinitesimal action λ is globalizable. The notion of univalence of an infinitesimal action λ, which is related to properties of the distribution Dλ, is extended to the case of supermanifolds.

6 A first guess could be that λ is globalizable if and only if the underlying infinitesimal action on the underlying manifold M is globalizable to an action of the Lie group G which underlies G. This guess is actually false. It is necessary that the underlying infinitesimal action is globalizable, but this is in general not sufficient. In the case where the underlying infinitesimal action of λ is globalizable, the obstruction for λ to be globalizable is a holonomy phenomenon of the distribution Dλ, which only appears in the setting of supermanifolds. Let g0¯ be the even part of the Lie superalgebra g, and g1¯ the odd, g = g0¯ + g1¯. The Lie algebra of the Lie group G, which is the underlying group of G, can be identified with g0¯. The main result for the conditions of globalizability is the following: Theorem 2 (see Section 4.3). The infinitesimal action λ : g → Vec(M) is globalizable if and only if one of the following equivalent conditions is satisfied:

(i) The restricted inﬁnitesimal action λ|g0¯ : g0¯ → Vec(M) is globalizable to an action of G on a supermanifold M0 which contains M as an open subsupermanifold and the G-action induces λ on M.

(ii) The inﬁnitesimal action λ is univalent.

(iii) The underlying inﬁnitesimal action is globalizable, and all leaves Σ ⊂ G × M of the distribution Dλ are holonomy free. Condition (iii) of this theorem together with the appropriate classical results of Palais formulated in [Pal57] then yield the following corollaries stated in Section 4.5.

Corollary 1. Let G be a simply-connected Lie supergroup, i.e. a Lie supergroup whose underlying Lie group is simply connected, and λ : g → Vec(M) an inﬁnitesimal action whose support is relatively compact in M. Then there exists a global G-action on M which induces λ. In particular, there is a one-to-one correspondence between inﬁnitesimal and global actions of a simply-connected Lie supergroup on a supermanifold with compact underlying manifold.

Corollary 2. Let λ : g → Vec(M) be an infinitesimal action of a simply-connected Lie supergroup G and let {Xi}i∈I , Xi ∈ g0¯, be a set of generators of g0¯ such that the underlying vector fields of λ(Xi), i ∈ I, on M have global flows. Then there exists a global G-action on M which induces λ.

If M is a compact complex manifold, the Lie algebra Vec(M) of holomorphic vector fields is finite-dimensional. Therefore, the simply-connected complex Lie group G with Lie algebra Vec(M) acts holomorphically on M by the version of Corollary 1 for classical manifolds. The Lie group G thus is a candidate for the connected component of the automorphism group Aut(M) of M. Since Aut(M) is a complex Lie group which acts holomorphically on M by [BM47], the simply-connected Lie group G is, after taking a quotient by the ineffectivity, indeed the connected component of Aut(M). We want to obtain similar results on the automorphism group of a compact complex supermanifold, i.e. a complex supermanifold whose underlying manifold is compact. In Section 5.2, we prove that the Lie superalgebra Vec(M) of holomorphic vector fields on a compact complex supermanifold M is finite-dimensional. The infinitesimal action

7 λ : Vec(M) → Vec(M) given by the identity map has necessarily compact support since M is compact. By applying Corollary 1 to this special situation, we get that the simply-connected Lie supergroup G with Lie superalgebra Vec(M) acts on M such that the induced infinitesimal action is the identity Vec(M) → Vec(M). This gives a first idea what the automorphism group of a compact complex supermanifold should be: Up to ineffectivity, the Lie supergroup G with Lie superalgebra Vec(M) should be the connected component of the automorphism Lie supergroup of M. Furthermore, we would like the underlying Lie group to consist of automorphisms of M. An approach to the automorphism group of a compact complex supermanifold M is to consider the set of automorphism of M, which we denote by Aut0¯(M). Since an automorphism ϕ of the supermanifold M (with structure sheaf OM) is “even” in the ∗ sense that its pullback ϕ : OM → ϕ˜∗(OM) is a parity-preserving morphism, we can (at most) expect the set Aut0¯(M) to carry the structure of a classical Lie group if we require its action on M to be smooth or holomorphic. Thus Aut0¯(M) cannot be a Lie supergroup of positive odd dimension. In order to obtain a Lie supergroup as automorphism group we also have to consider the Lie superalgebra of holomorphic vector fields on M. This is a candidate for the Lie superalgebra of the automorphism group. It is import to establish a connection between the Lie superalgebra of vector fields on M and the set of automorphisms Aut0¯(M). For this, we need to prove that Aut0¯(M) carries the structure of a complex Lie group that acts holomorphically on M, which then implies that its Lie algebra is isomorphic to the Lie algebra of even holomorphic vector fields on M. First, an analogue of the compact-open topology is introduced on the set Aut0¯(M). The group Aut0¯(M) endowed with this topology is a topological group. There are two main steps in the proof that Aut0¯(M) carries the structure of a complex Lie group with Lie algebra isomorphic to the Lie algebra of even holomorphic vector fields on M; The first is to prove that the topological group Aut0¯(M) carries the structure of a Lie group. The crucial step here is the following proposition on the topological properties of Aut0¯(M).

Propostion 1 (see Section 5.1). The topological group Aut0¯(M) does not contain small subgroups, i.e. there is a neighbourhood of the identity such that the trivial subgroup is the only subgroup contained in this neighbourhood. A result on the existence of Lie group structures on locally compact topological groups without small subgroups (see [Yam53]) then implies that Aut0¯(M) carries the structure of real Lie group. The second step is the proof of a regularity statement for actions of continuous one- parameter subgroups of Aut0¯(M). This is needed to prove that the action of Aut0¯(M) on the supermanifold M is holomorphic and to relate even holomorphic vector ﬁelds and one-parameter subgroups of Aut0¯(M). We have the following:

Propostion 2 (see Section 5.1). Let ϕ : R → Aut0¯(M) be a continuous one-parameter subgroup of Aut0¯(M). Then the natural action of ϕ on M is analytic. A consequence of the two preceding propositions is the desired result on the structure of Aut0¯(M):

Theorem 3 (see Section 5.1). The topological group Aut0¯(M) carries the structure of a complex Lie group such that its Lie algebra is isomorphic to the Lie algebra

8 of even holomorphic vector ﬁeld on M and the natural action of Aut0¯(M) on the supermanifold M is holomorphic.

We have a representation α of Aut0¯(M) on Vec(M) induced by the action of (2) Aut0¯(M) on M. Using the equivalence of Lie supergroups and Harish-Chandra pairs we can now give the following deﬁnition.

Definition (Automorphism Lie supergroup). The automorphism group Aut(M) of a compact complex supermanifold is the unique complex Lie supergroup associated with the Harish-Chandra pair (Aut0¯(M), Vec(M)) with representation α. Remark that the simply-connected Lie supergroup G with Lie superalgebra Vec(M) is, up to ineffectivity, really the connected component of the automorphism Lie supergroup of M. This was a priori not obvious. Before knowing that Aut0¯(M) is a complex Lie group, which acts holomorphically on M, we did not know that the connected component of Aut0¯(M) is generated by the flows of even holomorphic vector fields on M. The natural action of the automorphism Lie supergroup Aut(M) on M is holomorphic, i.e. we have a morphism Ψ : Aut(M) × M → M of complex supermanifolds. The automorphism Lie supergroup Aut(M) satisfies the following universal property (see Section 5.3):

Theorem 4. If G is a complex Lie supergroup with a holomorphic action ΨG : G×M → M on M, then there is a unique morphism σ : G → Aut(M) of Lie supergroups such that the diagram

ΨG G × M / M 8

σ×id Ψ M ' Aut(M) × M is commutative.

Using the “functor of points” approach to supermanifolds, an alternative deﬁnition of the automorphism group as a functor in analogy to [SW11] and [Ost15] is possible. We can deﬁne this functor from the category of superpoints, i.e. supermanifolds over a point, to the category of sets by the assignment

0|k 0|k 0|k 7→ {ϕ : × M → × M | ϕ is invertible, and pr 0|k ◦ ϕ = pr 0|k }, C C C C C 0|k 0|k where pr 0|k : × M → denotes the projection onto the ﬁrst component. In C C C Section 5.4 we prove that the two approaches to the automorphism group are equivalent and that the constructed automorphism group Aut(M) represents the just deﬁned functor. m In the classical case, automorphism groups of bounded domains in C are also known to carry the structure of a Lie group (see e.g. [Car79]). This result cannot be generalized to the category of supermanifolds as will be shown by an example in Section 5.5.

(2)As in the case of vector ﬁelds on supermanifolds, it would be more precise to talk about super Harish-Chandra pairs.

9 Outline First, we review some basic definitions related to supermanifolds and introduce some notation. This is followed by a chapter on Lie supergroups and their actions which mostly aims at collecting facts which are used later on. After recalling some basic definitions, we explain how (local) actions of Lie supergroups induce infinitesimal actions and we collect important facts about flows. Then we describe the equivalence of Lie supergroups and Harish-Chandra pairs. At the end of the chapter, the notion of an effective action of a Lie supergroup is introduced and an equivalent formulation in terms of the corresponding Harish-Chandra pair is found. In Chapter 3, distributions on supermanifolds are studied. The local Frobenius’ theorem for distributions on supermanifolds is discussed. We also provide an example which shows that a global Frobenius’ theorem does not hold in the context of supermanifold since integral manifolds do not give enough information about a distribution. This is due to the fact that we do “point evaluation” transverse to integral manifolds when going from the distribution to the integral manifolds. The relation between infinitesimal, local, and global actions on supermanifolds is analyzed in Chapter 4. In the last chapter, the automorphism group of a compact complex manifold is studied.

The main results of this thesis are published [Ber14] and [BK15].

10 Acknowledgements

First and foremost, I would like to thank my supervisor Prof. Dr. Peter Heinzner for introducing me to the field of research of this thesis, for mathematical discussions, for continuous support and encouragement during the last years, and for motivating me to study mathematics. I am very grateful to Prof. Dr. Dr. h.c. mult. Alan T. Huckleberry for mathematical advice and for accepting to be a referee for this thesis. I am also indebted to Prof. Dr. Martin R. Zirnbauer for accepting to be a referee for this thesis. My thanks also go to Prof. Dr. Tilmann Wurzbacher who first introduced me to supermanifolds in a lecture course. Moreover, I am very grateful to Dr. Matthias Kalus for mathematical discussions and fruitful collaboration. I would like to thank all colleagues and friends at “Arbeitsgruppe Transformation- sgruppen” and “Lehrstuhl fürKomplexe Geometrie” in Bochum for creating a fertile research environment as well as for the pleasant and friendly working atmosphere, which I appreciated very much. The financial support of SFB/TR 12 “Symmetries and Universality in Mesocopic Systems” is gratefully acknowledged. I am deeply grateful to my family, friends, and Christian for love, encouragement, and practical and emotional support.

11 12 Chapter 1

Preliminaries

In this chapter, we recall import deﬁnitions and facts in the context of supermanifolds. Throughout, we work with the “Berezin-Leites-Kostant”-approach to supermanifolds (see e.g. [Ber87], [Le˘ı80],or [Kos77], and also [Var04]).

1.1 Supermanifolds

The notion of a supermanifold extends the notion of a usual, in the following often called classical, manifold. Heuristically, a supermanifold is a manifold where we also have anticommuting variables θ1, . . . , θn with θiθj = −θjθi in addition to the usual V n local coordinates x1, . . . , xm. Let R denote the exterior algebra of the vector space n R , which is isomorphic to R[θ1, . . . , θn] with relations θiθj = −θjθi. The model space m|n can be deﬁned as the pair ( m, C∞ ⊗ V n), i.e. the underlying space is m and R R Rm R R m|n m ∞ V n ∼ “functions” of on open subsets of U ⊆ are elements of C m (U) ⊗ = R R R R C∞ (U) ⊗ [θ , . . . , θ ]. Thus, any element f ∈ C∞ (U) ⊗ V n can be written as Rm R 1 n Rm R X ν f = fνθ n ν∈(Z2)

ν ν1 νn for smooth functions fν on U, where we write θ for θ1 . . . θn if ν = (ν1, . . . , ν2) ∈ n ¯ ¯ n 0¯ 1¯ m|n V n (Z2) = {0, 1} , and set θj = 1, θj = θj. Similarly, we deﬁne C by replacing R V n by C and smooth by holomorphic functions. Before giving the deﬁnition of a supermanifold, we make a few remarks about the algebraic structure of the structure sheaf C∞ ⊗ V n of m|n, which also apply to the Rm R R structure sheaf of m|n. We have a natural -grading on C∞ ⊗ V n induced by the C Z2 Rm R V n V n grading of R . On the exterior algebra R we have the grading

^ n ^ n ^ n M ^2k n M ^2k+1 n R = R ⊕ R = (R ) ⊕ (R ) , 0¯ 1¯ k k V n where the ﬁrst summand is the even part, i.e. 0-part,¯ of R , and the second is the odd part, i.e. 1-part,¯ Accordingly, a function f = P f θν ∈ C∞ (U) ⊗ V n is even ν ν Rm R if fν = 0 for all ν with |ν| = 1¯ and odd if fν = 0 for all ν with |ν| = 0,¯ where ¯ ¯ |ν| = |(ν1, . . . , νn)| = ν1 + ... + νn ∈ Z2 = {0, 1}. This gives a grading

∞ ^ n ∞ ^ n ∞ ^ n C m ⊗ R = C m ⊗ R ⊕ C m ⊗ R R R 0¯ R 1¯

13 on the structure sheaf and the multiplication of two elements respects this grading. Con- sequently, the structure sheaf C∞ ⊗ V n is a sheaf of -graded algebras. We call an Rm R Z2 element f ∈ C∞ (U) ⊗ V n homogeneous of parity |f| ∈ if f ∈ (C∞ (U) ⊗ V n) . Rm R Z2 Rm R |f| m|n m|n m|n m A superdomain U in R (or C ) consists of an open subset U ⊆ R (or m ∞ V n V n U ⊆ ) and the restriction of the structure sheaf C ⊗ (or O m ⊗ ) to U. C Rm R C C A supermanifold can now be deﬁned to be a space that is locally isomorphic to m|n m|n superdomains in R (or C ). More precisely:

Definition 1.1.1. A real supermanifold M is a locally ringed space (M, OM), consisting of a second-countable Hausdorff topological space M and a sheaf OM of Z2-graded m|n algebras on M, which is locally isomorphic to superdomains in R . Similarly, M = (M, OM) is a complex supermanifold if it is locally isomorphic to m|n superdomains in C . The pair (m|n) is defined to be the dimension of the supermanifold M, and we also call m the even and n the odd dimension of M.

The underlying space M of a real or complex supermanifold M = (M, OM) carries a unique structure of a real or complex manifold with sheaf of smooth or holomorphic functions OM such that the evaluation ev = evM : OM → OM , locally given by P ν ˜ f = ν fνθ 7→ f0, is a morphism of sheaves. We also write f for ev(f). In the following, if we consider the underlying space M of a supermanifold M as a manifold, we always implicitly mean the just described structure. Usually, we denote a supermanifold by a calligraphic letter, e.g. M, and then the underlying manifold by the corresponding standard uppercase letters, e.g. M. The notation for the structure sheaf of a supermanifold M is OM.

Example 1.1.2. We have the following examples of supermanifolds:

m|n m|n m|n m m (1) R , C , U for U ⊆ R or U ⊆ C open.

(2) (M, ΩM ) for a manifold M with sheaf of diﬀerential forms ΩM .

(3) (M, V E), where E is the sheaf of sections of a vector bundle E → M on M.

Supermanifolds which are isomorphic to a supermanifold associated with a vector bundle as in the third example are called split supermanifold. All real supermanifolds are split, given any real supermanifold M = (M, OM) there exists a smooth vector bundle E → M on M with sheaf of sections E such that M is isomorphic to (M, V E) (see [Bat79]). This result is however not true for complex supermanifolds, there are complex supermanifolds which are not split; see e.g. [Gre82].

1.2 Morphisms and vector ﬁelds on supermanifolds

In this section, the definitions of a morphism between supermanifolds and (super) vector fields on supermanifolds are recalled. Moreover, a useful fact about the relation of invertible morphisms of a supermanifold satisfying a nilpotency condition and nilpotent (super) vector fields on the supermanifold is stated.

14 Deﬁnition 1.2.1. A morphism between supermanifolds M and N is a morphism ∗ ϕ = (ϕ, ˜ ϕ ):(M, OM) → (N, ON ) of ringed spaces, i.e.ϕ ˜ : M → N is a continuous ∗ (1) map, ϕ : ON → ϕ˜∗(OM) a morphism of sheaves of Z2-graded algebras, andϕ ˜ and ∗ ∗ ϕ are compatible in the sense that evM(ϕ (f)) = evN (f) ◦ ϕ˜ for any f ∈ ON (U), U ⊆ N open.

For any morphism ϕ : M → N of supermanifolds we denote the underlying map M → N byϕ ˜, and the morphism ON → ϕ˜∗(ON ) on the level of sheaves is denotes by ϕ∗ and referred to as the pullback of ϕ. The underlying mapϕ ˜ : M → N of a morphism M → N is smooth if M and N are real supermanifolds and holomorphic if M and N are complex.

Deﬁnition 1.2.2. An immersed subsupermanifold of a supermanifold M is a supermanifold N together with an immersion ϕ : N → M with injective underlying mapϕ ˜ (cf. [Le˘ı80]). We say that N is an open subsupermanifold of M if there is an open subset U ⊆ M such that N = (U, OM|U ). In this case we also write N ⊆ M.

m|n m|n Superdomains are for instance examples of open subsupermanifolds of R or C . Deﬁnition 1.2.3. A vector ﬁeld (or derivation) X of parity |X| on a supermanifold M is a derivation of graded sheaves X : OM → OM, i.e. X is linear and for any f, g ∈ OM(U), U ⊆ M open, f homogeneous of parity |f|, we have

X(fg) = X(f)g + (−1)|X||f|fX(g).

The vector field X of parity |X| is called even if |X| = 0¯ and odd if |X| = 1.¯ A vector field (or derivation) on M is then a morphism OM → OM which is the sum of an even vector field X0¯ and an odd vector field X1¯, X = X0¯ + X1¯. We denote by Der(OM) = TM the sheaf of derivations or tangent sheaf of M. Moreover, we set Vec(M) = TM(M) for the space of vector fields. This space Vec(M) is a Lie superalgebra, possibly of infinite dimension, with commutator [X,Y ] = XY − (−1)|X||Y |YX for vector fields X and Y of parity |X| and |Y |.

It would be more precise to talk about “super vector ﬁelds” on supermanifolds and their “super commutators”. For the convenience of notation, we will however stick to the words “vector ﬁeld” and “commutator“, and always implicitly mean the corresponding objects on supermanifolds.

Example 1.2.4. Let x , . . . , x , θ , . . . , θ denote coordinates on m|n. Then ∂ 1 m 1 n R ∂xj defined by ∂ ∂ ∂ (xj) = 1, (xk) = 0 for j 6= k, and (θk) = 0 ∂xj ∂xj ∂xj is an even vector field on m|n. Similarly, ∂ defined by R ∂θj ∂ ∂ ∂ (xk) = 0, (θj) = 1, and (θk) = 0 for j 6= k ∂θj ∂θj ∂θj

(1) −1 The sheafϕ ˜∗(OM) denotes the direct image sheaf, i.e.ϕ ˜∗(OM)(U) = OM(ϕ ˜ (U)) for any U ⊆ N open.

15 is an odd vector field. m|n Moreover, any vector field X on R has the form m n X ∂ X ∂ X = aj + bk ∂xj ∂θk j=1 k=1 for suitable a , b ∈ C∞ ( m) ⊗ V n. j k Rm R R We have the following relation between nilpotent even vector fields on a supermanifold and morphisms of this supermanifold satisfying a certain nilpotency condition:

Lemma 1.2.5. Let ϕ : M → M be a morphism of supermanifolds with underlying ∗ ∗ map ϕ˜ = idM and such that ϕ − idM : OM → OM is nilpotent, i.e. there is N ∈ N ∗ ∗ N with (ϕ − idM) = 0. Then

N X (−1)n+1 X = log(ϕ∗) = (ϕ∗ − id∗ )n n M n=1 is a nilpotent even vector ﬁeld on M and we have

X 1 ϕ∗ = exp(X) = Xn. n! n≥0

Moreover, for any nilpotent even vector field X on M, the (finite) sum exp(X) defines a map OM → OM which is the pullback of an invertible morphism M → M with the identity as underlying map, and the pullback of the inverse is exp(−X).

This lemma follows directly from the next technical result on the relation of algebra homomorphisms and derivations (cf. [vdE00], Proposition 2.1.3 and Lemma 2.1.4):

Lemma 1.2.6. Let A be an algebra (containing Q), and χ : A → A an algebra homomorphism such that (χ − idA) is nilpotent. Then the map D : A → A deﬁned by

X (−1)n+1 D(a) = log(χ)(a) = (χ − id )n(a) for a ∈ A n A n≥1

P 1 n is a derivation and χ = exp(D) = n≥0 n! D . Moreover, exp(D): A → A is an automorphism with inverse exp(−D) for any nilpotent derivation D.

16 Chapter 2

Lie supergroups and their actions

Lie supergroups and their actions on supermanifolds are import for the study of symmetries of supermanifolds. They generalize the notion of Lie groups and their actions on classical manifolds. In this chapter, Lie supergroups, their actions and related concepts are discussed. First, we recall the basic definitions of a Lie supergroup and actions on supermanifolds. Then we give the definition of local and infinitesimal actions on supermanifolds and explain how local actions induce infinitesimal actions. In the next section, we describe how to ”go from infinitesimal to local“ in the case of one vector field on a supermanifold. For this we collect important facts about flows of vector fields on supermanifolds (cf. [MSV93] and [GW13]), and we study a few properties of the flows. A Lie supergroup is uniquely determined by its underlying classical Lie group, its Lie superalgebra, and a representation of the classical Lie group on the Lie superalgebra, which is formalized by the concept of a Harish-Chandra pair. In Section 2.4, we recall the related definitions and state the equivalence between Lie supergroups and Harish- Chandra pairs (cf. [Kos83], and [Vis11] for the case of complex Lie supergroups). In the last section of this chapter, the notion of an effective action of a Lie supergroup on a supermanifold is introduced, generalizing the classical definition of an effective action. We also prove an equivalent characterization of the effectiveness of an action in terms of the corresponding action of the associated Harish-Chandra pair.

2.1 Basic deﬁnitions

In this section, the basic deﬁnitions in the context of Lie supergroups are recalled.

Deﬁnition 2.1.1. A Lie supergroup is a supermanifold G together with morphism µ : G × G → G, ι : G → G, and eG : {e} ,→ G for multiplication, inversion and the neutral element, so that the usual group axioms are satisﬁed.

The underlying manifold G of a Lie supergroup G = (G, OG) is a classical Lie group. A vector ﬁeld X on a Lie supergroup G is called right-invariant if

∗ ∗ ∗(1) µ ◦ X = (X ⊗ idG) ◦ µ .

17 We define the Lie superalgebra g of G to be the set of right-invariant vector fields on G. Its even part g0¯ can be identified with the Lie algebra (of right-invariant vector fields) of G. Similarly, one can define a complex Lie supergroup as complex supermanifold with holomorphic morphisms for multiplication, inversion, and the neutral element. The underlying manifold then carries the structure of a complex Lie group and the set of right-invariant holomorphic vector fields forms a complex Lie superalgebra.

1|1 Example 2.1.2. Let K = R or C. Then the supermanifold K carries three non- isomorphic structures of a Lie supergroup (cf. [GW13], Lemma 3.1). Let t and τ 1|1 1|1 1|1 1|1 denote coordinates on K . Then the multiplication µ = µa,b : K × K → K ∗ ∗ t b given by µ (t) = t1 + t2 + aτ1τ2, µ (τ) = τ1 + e 1 τ2 for any a, b ∈ K with ab = 0 is the 1|1 multiplication of a Lie supergroup structure on K , and any Lie supergroup structure 1|1 1|1 on the supermanifold K is isomorphic to one of the three Lie supergroups (K , µ0,0), 1|1 1|1 (K , µ1,0) and (K , µ0,1). 1|1 The Lie superalgebra of right-invariant vector fields on (K , µa,b) is isomorphic to KX0¯ ⊕ KX1¯ with relations [X0¯,X0¯] = 0, [X0¯,X1¯] = −bX1¯ and [X1¯,X1¯] = 2aX0¯. As in the classical case, an action of a Lie supergroup on a supermanifold can be defined. Definition 2.1.3. An action of a Lie supergroup G on a supermanifold M is a morphism Ψ : G × M → M satisfying the usual action properties, i.e.

(i) Ψ ◦ ιe = idM, where ιe : M → {e} × M ⊂ G × M is the canonical inclusion, and

(ii) Ψ ◦ (µ × idM) = Ψ ◦ (idG × Ψ). In the same manner, an action of a complex Lie supergroup on a complex supermanifold is deﬁned.

2.2 Local and inﬁnitesimal actions of Lie supergroups

A local action of a Lie supergroup on a supermanifold can be defined in complete analogy to the classical case. We give the precise definition here in order to recall the requirements on the domain of definition of a local action.

Deﬁnition 2.2.1. A local action of a Lie supergroup G = (G, OG) on a supermanifold M = (M, OM) is a morphism ϕ : W → M, W = (W, OG×M|W ), where W is an open neighbourhood of {e} × M in G × M such that Wp = {g ∈ G| (g, p) ∈ W } is connected for each p ∈ M, satisfying the action properties:

(i) ϕ ◦ ιe = idM (ii) The equation ϕ ◦ (µ × idM) = ϕ ◦ (idG × ϕ) holds on the open subsupermanifold of G × G × M where both sides are deﬁned.

(1) ∗ The vector field (X ⊗idG ) is the extension of the vector field X on G to a vector field on the product ∗ ∗ ∗ ∗ G × G such that X = (X ⊗ idG ) ◦ π1 and 0 = (X ⊗ idG ) ◦ π2 if πi : G × G → G, i = 1, 2, is the projection onto the i-th component.

18 Let G be a classical Lie group with Lie algebra g of right-invariant vector fields. Any (local) G-action ψ : G × M → M on a manifold M induces an ”infinitesimal action“ by mapping a right-invariant vector field X ∈ g on G to the induced vector ∂ field ∂t 0 ψ(exp(tX), −) on M and the map

∂ g → Vec(M),X 7→ ψ(exp(tX), −) ∂t 0 is a homomorphism of Lie algebras. The same statement can be extended to the case of action of Lie supergroups on supermanifolds. First, we define the notion of an infinitesimal action. Definition 2.2.2. Let G be a Lie supergroup with Lie superalgebra g, and M a supermanifold. An infinitesimal action of G on M is a homomorphism λ : g → Vec(M) of Lie superalgebras. If G and M are complex, we require λ to be a homomorphism of complex Lie superalgebras. As in the classical case, (local) actions of Lie supergroups induce infinitesimal actions: Proposition 2.2.3. Let Ψ be a local action of a Lie supergroup G, with Lie superalgebra ∗ ∗ g, on a supermanifold M. Then (X(e)⊗idM)◦Ψ is a vector field on M for any X ∈ g, where X(e) is the evaluation of the vector field in e. The map

∗ ∗ λΨ : g → Vec(M) = Der OM(M),X 7→ (X(e) ⊗ idM) ◦ Ψ is a homomorphism of Lie superalgebras.

∗ Proof. Let X,Y ∈ g be homogeneous. The vector ﬁeld X ⊗ idM has the parity |X| of X. Let f, g ∈ OM(M) and let f be homogeneous with parity |f|. Then, using ϕ ◦ ιe = idM, we get

∗ ∗ ∗ ∗ ∗ ∗ ((X(e) ⊗ idM) ◦ Ψ )(fg) = ιe(X ⊗ idM)(Ψ (f)Ψ (g)) ∗ ∗ |X||f| ∗ ∗ = ((X(e) ⊗ idM) ◦ Ψ )(f) g + (−1) f ((X(e) ⊗ idM) ◦ Ψ )(g) .

Since Ψ is an action, we have Ψ ◦ (µ × idM) = Ψ ◦ (idG × Ψ). A calculation using this ∗ ∗ identity results in λΨ(X)λΨ(Y ) = (XY (e) ⊗ idM) ◦ Ψ . Thus

|X||Y | ∗ ∗ [λΨ(X), λΨ(Y )] = XY (e) − (−1) YX(e) ⊗ idM ◦ Ψ = λΨ([X,Y ]).

Deﬁnition 2.2.4. Let M be a supermanifold and G a Lie supergroup with Lie superalgebra g. The induced inﬁnitesimal action of a G-action Ψ on M is the homomorphism

∗ ∗ λΨ : g → Vec(M), λΨ(X) = (X(e) ⊗ idM) ◦ Ψ .

2.3 Flows of vector ﬁelds

After explaining how local actions induce infinitesimal actions on supermanifolds, we now describe how an infinitesimal symmetry on a supermanifold, given by one vector field, can be integrated to a local symmetry, given by the flow of the vector field.

19 In analogy to the classical case, we can define the flow of a vector field on a supermanifold. In the case of an odd vector field or a non-homogeneous vector field, i.e. a vector field X = X0¯ + X1¯, where X0¯ 6= 0 is an even vector field and X1¯ 6= 0 is an odd vector field, some subtleties appear in the definition of a flow or the existence of flows (see Remark 2.3.9 or e.g. [GW13]). For our purposes, the definition of the flow of an even vector field suffices.

Definition 2.3.1. Let X be an even vector field on a supermanifold M. A flow of X (with respect to the initial condition idM : M → M and t0 = 0 ∈ R) is a X morphism ϕ = ϕ : W → M, where W = (W, OR×M|W ) and W ⊆ R × M is an open neighbourhood of {0} × M such that Wp = {t ∈ R|(t, p) ∈ W } ⊆ R is connected for each p ∈ M, with

(i) ϕ ◦ ι0 = idM, where ι0 : M ,→ {0} × M ⊂ W is the canonical inclusion, and

∂ ∗ ∗ (ii) ∂t ◦ ϕ = ϕ ◦ X. 0 0 0 The ﬂow ϕ : W → M is called maximal if for any ﬂow ϕ : W = (W , OR×M|W 0 ) → 0 0 M of X we have W ⊆ W and ϕ = ϕ|W 0 .

Remark 2.3.2. Any vector field X on a supermanifold M induces a vector field X˜ on the underlying manifold M. The reduced vector field X˜ can be defined by ˜ ˜ ∞ X(f) = ev(X(F )), if F is a function on M with ev(F ) = F = f, where ev : OM → CM denotes the evaluation map.

As in the classical case, there exists a unique maximal ﬂow of an even vector ﬁeld on a real supermanifold.

Theorem 2.3.3 (see [MSV93], Theorem 3.5/3.6, or [GW13], Theorem 2.3 and Theorem 3.4). Let X be an even vector field on a real supermanifold M. Then there exists a unique maximal flow ϕ of X. The reduced map ϕ˜ : W → M then is the unique maximal flow of the reduced vector field X˜ on M. Moreover, the flow ϕ defines a local R-action on M.

Remark 2.3.4. For an even holomorphic vector field X on a complex supermanifold M, there also exists a flow map ϕ : W ⊂ C×M → M (see [GW13], Theorem 5.4). But in contrast to the real case there is in general no unique maximal domain of definition, as already in the classical case, since two flow maps ϕ1 : W1 = (W1, OC×M|W1 ) → M and ϕ2 : W2 = (W2, OC×M|W2 ) → M of the same holomorphic vector field X do not necessarily agree on their common domain of definition since for p ∈ M the set (W1 ∩ W2)p = {z ∈ C| (z, p) ∈ (W1 ∩ W2)} might not be connected. Nevertheless, the flow maps ϕ1 and ϕ2 coincide on the open subsupermanifold of C × M with underlying set [ ◦ ((W1 ∩ W2)p) × {p} p∈M ◦ where ((W1 ∩ W2)p) is the connected component of 0 in (W1 ∩ W2)p. Moreover, it is not always possible to define the flow map of X on the open supermanifold whose underlying set is the domain of definition for a given flow map of the reduced vector field.

20 As in the classical case, a ﬂow map ϕ : W → M of a holomorphic vector ﬁeld

X on M may not satisfy the equation ϕ ◦ (idC × ϕ) = ϕ ◦ (µC × idM) on the open subsupermanifold of C × C × M on which both sides of the equation are defined, where µC denotes the multiplication on C. Therefore, in the complex case not every flow is a local C-action. A flow does however yield a local C-action after shrinking its domain of definition.

Lemma 2.3.5. Let ϕ : W = (W, OC×M|W ) → M be the flow of a holomorphic vector field X on a supermanifold M. Then the maps ψ1 = ϕ ◦ (idC × ϕ): Y1 → M and ψ2 = ϕ ◦ (µC × idM): Y2 → M, where Yi = (Yi, OC×C×M|Yi ), i = 1, 2, are open subsupermanifolds of C × C × M, both satisfy C×M (a) ψi ◦ ι0 = ϕ, and ∂ ∗ ∗ (b) ∂t ◦ ψi = ψi ◦ X, C×M for i = 1, 2, where ι0 is the inclusion C×M ι0 : C × M ,→ {0} × C × M ⊂ C × C × M and t denotes the coordinate on the first component of the product C × C × M. N Proof. Let ι0 : N ,→ {0} × N ⊂ C × N be the canonical inclusion for N = C, M, or C × M. We have C×M C×M M ψ1 ◦ ι0 = (ϕ ◦ (idC × ϕ)) ◦ ι0 = ϕ ◦ ι0 ◦ ϕ = idM ◦ ϕ = ϕ and

C×M C×M C ψ2 ◦ ι0 = (ϕ ◦ (µC × idM)) ◦ ι0 = ϕ ◦ (µC × idM) ◦ (ι0 × idM) C = ϕ ◦ ((µC ◦ ι0 ) × idM) = ϕ ◦ (idC × idM) = ϕ. ∂ ∗ ∗ Using ∂t ◦ ϕ = ϕ ◦ X, we calculate ∂ ∂ ∂ ◦ ψ ∗ = ◦ (id∗ ⊗ ϕ∗) ◦ ϕ∗ = (id∗ ⊗ ϕ∗) ◦ ( ◦ ϕ∗) ∂t 1 ∂t C C ∂t ∗ ∗ ∗ ∗ = (idC ⊗ ϕ ) ◦ (ϕ ◦ X) = ψ1 ◦ X and ∂ ∂ ∂ ◦ ψ ∗ = ◦ (µ∗ ⊗ id∗ ) ◦ ϕ∗ = ◦ µ∗ ⊗ id∗ ◦ ϕ∗ ∂t 2 ∂t C M ∂t C M ∂ ∂ = µ∗ ◦ ⊗ id∗ ◦ ϕ∗ = (µ∗ ⊗ id∗ ) ◦ ◦ ϕ∗ C ∂t M C M ∂t ∗ ∗ ∗ ∗ = (µC ⊗ idM) ◦ (ϕ ◦ X) = ψ2 ◦ X.

Corollary 2.3.6. Let ϕ : W = (W, OC×M|W ) → M be the flow of a holomorphic vector field X on a supermanifold M as before. Then ψ1 = ϕ ◦ (idC × ϕ): Y1 → M and ψ2 = ϕ ◦ (µC × idM): Y2 → M are both flow maps of X, with respect to the initial condition ϕ and z0 = 0, on an open neighbourhood of (0, 0, p) for any p ∈ M. Therefore, ψ1 and ψ2 agree on their common domain of definition if (Y1 ∩ Y2) ∩ (C × C × Mα) is connected for each connected component Mα of M. In particular, the underlying set W of W ⊆ C × M can be shrunk such that ϕ : W → M is a local C-action.

21 Proof. We have (0, 0, p) ∈ Y1 and (0, 0, p) ∈ Y2 for any p ∈ M. The preceding lemma implies now that ψ1 and ψ2 are flow maps of X with respect to the initial condition ϕ and z0 = 0 on some open neighbourhood of (0, 0, p). Hence, the local uniqueness of such flow maps (cf. [GW13], Theorem 5.4) yields ψ1 = ψ2 near (0, 0, p). Thus, if (Y1 ∩Y2)∩(C×C×Mα) is connected for each connected component Mα of M, ψ1 and ψ2 agree on there common domain of definition (Y1 ∩

Y2, OC×C×M|Y1∩Y2 ) by the identity principle for holomorphic maps. The set W ⊆ C×M can be shrunk such that the intersection (Y1 ∩Y2)∩(C×C×Mα) of underlying set Y1 ∩ Y2 of the common domain of deﬁnition ψ1 = ϕ ◦ (idC × ϕ) and ψ2 = ϕ ◦ (µC × idM) and an arbitrary connected component Mα of M is connected. For (z, p) ∈ W , we have

(Y1)(z,p) = {t ∈ C| (t, z, p) ∈ Y1} = {z ∈ C| (t, ϕ˜(z, p)) ∈ W } = Wϕ˜(z,p) and

(Y2)(z,p) = {t ∈ C| (t + z, p) ∈ W } = {t − z ∈ C| t ∈ Wp} = Wp − z, and (Y1)(z,p) and (Y2)(z,p) are both non-empty open connected neighbourhoods of 0 ∈ C. Therefore, (Y1 ∩ Y2) ∩ (C × C × Mα) is connected if (Y1 ∩ Y2)(z,p) = (Y1)(z,p) ∩ (Y2)(z,p) is connected for each (z, p) ∈ W . For example, this can be accomplished by shrinking W such that Wp = {t ∈ C| (t, p)} ⊆ C is convex for each p ∈ M. Lemma 2.3.7. Let X and Y be vector ﬁelds on M, assume that Y is even and let ϕY be the ﬂow of Y . If ιt : M ,→ {t} × M ⊂ R × M is the canonical inclusion, denote by X X ϕt the composition ϕ ◦ ιt. Then

∂ Y [X,Y ] = (ϕt )∗X ∂t 0

Y Y ∗ Y ∗ ∂ for (ϕt )∗X = (ϕ−t) ◦ X ◦ (ϕt ) , where ∂t is again considered as a vector ﬁeld on the ∂ ∗ ∂ product R × M and ∂t s = ιs ◦ ∂t for s ∈ R. Proof. Since vector ﬁelds are derivations of sheaves, it is enough to show the statement Y ∗ locally. For f ∈ OM(V ), V ⊆ M open, the pullback (ϕt ) (f) is a function on some subset of R × V . In local coordinates (t, x1, . . . , xk, ξ1, . . . ξl) on R × M, we have

Y ∗ X ν X ν1 νl (ϕt ) (f) = gν(t, x)ξ = gν(t, x1, . . . , xk)ξ1 . . . ξl l l ν∈(Z2) ν∈(Z2) for some smooth functions gν on a subset of R×V . For each gν, Taylor expansion yields R 1 ∂ P ν gν(t, x) = gν(0, x) + tgˆν(t, x) forg ˆν(t, x) = 0 ( ∂r r=st gν(r, x))ds. If g = ν gν(0, x)ξ P ν Y ∗ Y ∗ ∗ andg ˆt = ν gˆν(t, x)ξ , then (ϕt ) (f) = g + tgˆt. Since (ϕ0 ) = idM, we have g = Y ∗ (ϕ0 ) (f) = f and consequently

Y ∗ (ϕt ) (f) = f + tgˆt.

Furthermore,

∂ ∂ Y ∗ gˆ0 = (g + tgˆt) = (ϕt ) (f) = Y (f). ∂t 0 ∂t 0 This now yields

22 ∂ Y ∂ Y ∗ Y ∗ ∂ Y ∗ (ϕt )∗X (f) = (ϕ−t) (X((ϕt ) (f))) = (ϕ−t) (X(f + tgˆt)) ∂t 0 ∂t 0 ∂t 0

∂ Y ∗ ∂ Y ∗ = (ϕ−t) (X(f)) + (ϕ−t) (X(tgˆt)) = −YX(f) + X(ˆg0) = [X,Y ](f). ∂t 0 ∂t 0 Corollary 2.3.8. Under the assumptions of the above lemma, we have:

∂ Y Y (i) ∂t s (ϕt )∗X = (ϕs )∗[X,Y ] Y Y ∗ Y ∗ (ii) If [X,Y ] = 0, then (ϕt )∗X = X, i.e. (ϕt ) ◦ X = X ◦ (ϕt ) , for all t. (iii) If [X,Y ] = 0 and X is also even, then the flows of X and Y locally commute, i.e. X Y Y X ϕt ◦ ϕs = ϕs ◦ ϕt for small s, t. Proof. The statements of the corollary can be concluded from the above lemma in the same manner as in the classical case. Remark first that for any diffeomorphism ψ : M → M the map

Y Y −1 ψ∗(ϕt ) = ψ ◦ ϕt ◦ ψ is the ﬂow of the even vector ﬁeld ψ∗Y because ∂ ∂ ∂ (ψ ◦ ϕY ◦ ψ−1)∗(f) = (ψ−1)∗((ϕY )∗(ψ∗(f))) = (ψ−1)∗ (ϕY )∗(ψ∗(f)) ∂t t ∂t t ∂t t −1 ∗ Y ∗ ∗ Y −1 ∗ −1 ∗ ∗ = (ψ ) ((ϕt ) (Y (ψ (f)))) = (ψ ◦ ϕt ◦ ψ ) ((ψ ) ◦ Y ◦ ψ )(f) Y ∗ = (ψ∗(ϕt )) (ψ∗Y )(f)

Y Y −1 −1 for all f ∈ OM(M) and ψ∗(ϕt ) ◦ ι0 = ψ ◦ ϕ0 ◦ ψ = ψ ◦ idM ◦ ψ = idM. Y Y Applying this to ψ = ϕs , we get (ϕs )∗Y = Y since the ﬂows of the two vector ﬁelds Y Y Y Y Y Y Y coincide because (ϕs )∗(ϕt ) = ϕs ◦ ϕt ◦ ϕ−s = ϕs+t−s = ϕt for all t. Therefore, using preceding lemma we get

∂ Y ∂ Y ∂ Y Y Y (ϕt )∗X = (ϕt+s)∗X = (ϕt )∗((ϕs )∗(X)) = [(ϕs )∗X,Y ] ∂t s ∂t 0 ∂t 0 Y Y Y = [(ϕs )∗X, (ϕs )∗Y ] = (ϕs )∗[X,Y ], which is part (i). Assume now [X,Y ] = 0. Part (i) then yields

∂ Y Y (ϕt )∗X = (ϕs )∗[X,Y ] = 0 ∂t s

Y Y for small s, and thus (ϕt )∗X = (ϕ0 )∗X = idM∗X = X for arbitrary t. X Y To prove (iii), let X be even and denote its ﬂow by ϕ . The ﬂow of X = (ϕs )∗X Y X Y X Y is then also given by (ϕs )∗(ϕt ) = ϕs ◦ ϕt ◦ ϕ−s. Consequently,

X Y X Y ϕt = ϕs ◦ ϕt ◦ ϕ−s and the ﬂows of X and Y commute.

23 Remark 2.3.9. It is also possible to define the flow of a not necessarily homogeneous vector field X on a supermanifold M; cf. [MSV93], [GW13]. Then the flow map is a 1|1 1|1 morphism W ⊆ R × M → M. But it does not always define a local R -action on M. In contrast to the classical case, the span RX of the vector field X is in general not a Lie subsuperalgebra of Vec(M). Let X0¯ be the even part of X, and X1¯ the odd, X = X0¯ + X1¯. Even the span RX0¯ ⊕ RX1¯ is in general not a Lie superalgebra since we might have [X0¯,X1¯] ∈/ RX1¯ or [X1¯,X1¯] = 2X1¯X1¯ ∈/ RX0¯. 1|1 It turns out that the flow of X defines a local R -action precisely if X0¯ and X1¯ are contained in a (1|1)-dimensional Lie subsuperalgebra g of Vec(M). Note that the 1|1 Lie supergroup structure on R is not unique (see Example 2.1.2), and the flow is 1|1 1|1 an R -action with respect to structure of a Lie supergroup on R induced by the structure of the Lie superalgebra g. In the special case of an odd vector field X which commutes with itself, i.e. [X,X] = 0|1 0|1 2XX = 0, there is an R -action inducing this vector field. The R -action is given 0|1 ∗ by ψ : R × M → M, ψ (f) = f + τX(f), where τ denotes the coordinate on 0|1 0|1 0|1 R ⊂ R × M and f and X(f) are considered as functions on the product R × M on the right-hand side of the equation.

2.4 Harish-Chandra pairs

An important tool for the study of Lie supergroups is the equivalence of Lie supergroups and Harish-Chandra pairs. We briefly recall some important definitions and how the isomorphism of the category of Lie supergroups and Harish-Chandra pairs can be realized. Definition 2.4.1. A pair (G, g) consisting of a (real or complex) Lie group G and a (real or complex) Lie superalgebra g = g0¯ ⊕ g1¯ together with a parity-preserving representation σ of G on g, i.e. a (smooth or holomorphic) morphism σ : G → GL(g0¯)× GL(g1¯), is called a (real or complex) Harish-Chandra pair if the following holds:

(i) The Lie algebra of G is g0¯ and the restriction of σ to g0¯ coincides with the adjoint action of G on its Lie algebra.

(ii) The diﬀerential of σ at the identity e ∈ G coincides with the adjoint action of g0¯ on g. Let (G, g) with representation σ and (H, h) with representation τ be two Harish- Chandra pairs. A morphism between those consists of a homomorphism ϕ : G → H of Lie groups and a homomorphism λ : g → h of Lie superalgebras such that the diﬀerential of ϕ at the identity e ∈ G coincides with λ|g0¯ and such that τ(ϕ(g)) = λ ◦ σ(g) for all g ∈ G. If G is a real or complex Lie supergroup, then we have an associated Harish-Chandra pair: The Lie group G is the underlying Lie group of G, g the Lie superalgebra of G, and σ is the adjoint representation of G on g. Furthermore, any morphism of Lie supergroups G → H induces a morphism of the corresponding Harish-Chandra pairs. Theorem 2.4.2. The categories of Lie supergroups and Harish-Chandra pairs are isomorphic.

24 For a proof see [Kos83], and [Vis11] in the complex case. For convenience, we brieﬂy describe how a Lie supergroup can be constructed from a Harish-Chandra pair. Let (G, g) together with the representation σ be a Harish-Chandra pair. Let U(g) denote the universal enveloping algebra of g and U(g0¯) the universal enveloping algebra of g0¯. Furthermore, let FG denote the sheaf of smooth functions on G if G is a real Lie group and the sheaf of holomorphic functions if G is a complex Lie group. Since g0¯ is the Lie algebra of G, we have a natural action of U(g0¯) on the sheaf FG. Furthermore, U(g0¯) acts on U(g) by multiplication from the left if U(g0¯) is considered as a subset of U(g). We deﬁne a sheaf O on the G by setting

O(U) = Hom (U(g), F (U)) U(g0¯) G for open subsets U ⊆ G. The elements of Hom (U(g), F (U)) are U(g¯)-linear U(g0¯) G 0 morphisms U(g) → FG(U). We consider FG(U) as an “even” object, and define a morphism U(g) → FG(U) to be even if it is parity-preserving and odd if it is parity- interchanging. In this way, O is a sheaf of Z2-graded vector spaces. Using the Hopf superalgebra structure on the universal enveloping algebra U(g), it it possible to define the multiplication of two elements in O(U), and G = (G, O) carries the structure of a (real or complex) supermanifold. Moreover, using the explicit realization of the sheaf O, the pullbacks of morphisms for multiplication, inversion, and the neutral element on G can be defined and thus G is a Lie supergroup. For details of this construction see e.g. [Vis11]. Let Ψ : G × M → M be the action of a Lie supergroup on the supermanifold M. Then by composing with the canonical inclusion G,→ G we get an action ψ : G × M → M of the underlying Lie group G on the supermanifold M. Moreover, we ∗ ∗ have the induced infinitesimal action λΨ : g → Vec(M), X 7→ (X(e) ⊗ idM) ◦ Ψ as in Proposition 2.2.3. Let (G, g) be the Harish-Chandra pair associated with G with adjoint representation σ. The maps ψ and λΨ satisfy

∗ ∗ λΨ(X) = (X(e) ⊗ idM) ◦ ψ for all X ∈ g0¯, and (2.1) −1 ∗ ∗ λΨ(σ(g)(Y )) = (ψg ) λΨ(Y )(ψg) for all g ∈ G, Y ∈ g, (2.2) where ψg : M → M is the automorphism of M given by the composition of ψ and the canonical inclusion {g} × M ,→ G × M.

Deﬁnition 2.4.3. An action of a Harish-Chandra pair (G, g) with representation σ on a supermanifold M is an action ψ : G×M → M and a homomorphism λ : g → Vec(M) of Lie superalgebras satisfying the compatibility conditions as in (2.1) and (2.2).

There is a one-to-one correspondence between actions of Lie supergroups and actions of Harish-Chandra pairs (see e.g. [BCC09]). Above, we described how we can associate an action of a Harish-Chandra pair with an action of the corresponding Lie supergroup. Conversely, if (G, g) is a Harish-Chandra pair and an action of (G, g) on a supermanifold M is given by ψ : G × M → M and λ : g → Vec(M), we can reconstruct the action of the corresponding Lie supergroup G using the explicit realization of the structure sheaf of G by O (U) = Hom (U(g), F (U)) for U ⊆ G open. The structure sheaf on the G U(g0¯) G product manifold G × M is then given by

O ⊗ˆ O ∼ Hom (U(g), F ⊗ˆ O ), G M = U(g0¯) G M

25 where ⊗ˆ denotes the completed tensor product in the sense of [Gro55], and the action of G has the following form: Proposition 2.4.4. Let Ψ be the action of the Lie supergroup G on M, corresponding to the action of the corresponding Harish-Chandra pair given by ψ and λ. Then the pullback of Ψ has the form

O (V ) → Hom (U(g), (F ⊗ˆ O )(ψ˜−1(V ))), M U(g0¯) G M f 7→ X 7→ (−1)|X||f|(ψ∗ ◦ λ(X))(f) for any f ∈ OM(V ), V ⊆ M open, where we extend λ : g → Vec(M) to a map λ : U(g) → U(Vec(M)) by setting

∗ ∗ λ(X1 ...Xr) = ((X1 ...Xr)(e) ⊗ idM) ◦ Ψ = λ(X1) . . . λ(Xr) for X1,...,Xr ∈ g. In [BCC09], Proposition 4.3, the corresponding formula is given if working with left-invariant vector ﬁelds.

2.5 Eﬀective actions of Lie supergroups

In this section, the notion of an effective action of a Lie supergroup on a supermanifold is defined, and an equivalent characterization of an effective action in terms of the Harish-Chandra pair is given. As a generalization of an effective action of a classical Lie group, we define: Definition 2.5.1. Let Ψ : G × M → M be the action of a Lie supergroup G on a supermanifold M. The action Ψ is called effective if it satisfies the following property: Let N be a supermanifold and χ1, χ2 : N → G be arbitrary morphisms. Then the equality Ψ ◦ (χ1 × idM) = Ψ ◦ (χ2 × idM) implies χ1 = χ2. We can characterize the effectiveness of an action of a Lie supergroup by properties of the action of the corresponding Harish-Chandra pair: Proposition 2.5.2. Let G be a Lie supergroup and (G, g) with representation σ the associated Harish-Chandra pair. Let Ψ: G×M → M be an action on the supermanifold M, and suppose the corresponding action of the Harish-Chandra pair (G, g) is given by ψ : G × M → M and λ : g → Vec(M). The action Ψ is effective if and only if the G-action ψ is effective and λ is injective. Remark 2.5.3. The action ψ of the classical Lie group G on the supermanifold M is effective if and only if

0 0 ψg = ψg0 for g, g ∈ G implies g = g , where ψh : M → M denotes again the automorphism of M given by the composition of ψ and the inclusion M ,→ {h} × M for h ∈ G.

26 Proof of Proposition 2.5.2. First, suppose that Ψ is effective. Let g, g0 ∈ G such that ψg = ψg0 . Define N = {0} and let χ1, χ2 : N → G be given by the inclusions {0} ,→ 0 {g} ⊂ G and {0} ,→ {g } ⊂ G. Identifying N ×M with M, we have Ψ◦(χ1 ×idM) = ψg and Ψ ◦ (χ2 × idM) = ψg0 . Therefore, Ψ ◦ (χ1 × idM) = Ψ ◦ (χ2 × idM) and it follows 0 χ1 = χ2 since Ψ is supposed to be effective. Consequently, we have g = g . Now, let X ∈ g such that λ(X) = (X(e) ⊗ id∗) ◦ Ψ∗ = 0, where X(e) denotes again the evaluation of the right-invariant vector field X in the identity e of G. We first consider the case where X is even, i.e. X is contained in g0¯, which is the Lie algebra of G. Let exp(tX) denote the corresponding one-parameter subgroup of G, and define

χ1 : K → G by the pullback OG → OK, f 7→ f(exp(tX)), and χ2 : K → G as the composition C → {e} ,→ G for the identity element e of G, where K = R if G is a real Lie supergroup and K = C if G is a complex Lie supergroup. For any element f ∈ OG(U), U ⊆ G open, we have

∂ ∗ ∗ ∗ ∂ ∗ ∂ ∗ (χ1 ⊗ id ) ◦ Ψ (f) = exp(tX) (f) = exp((t + s)X) (f) ∂t s ∂t s ∂t 0

∂ ∗ ∗ ∗ = exp(tX) (exp(sX) (f)) = λ(X)(exp(sX) (f)) = 0, ∂t 0 where t denotes the coordinate on K and s is contained in some neighbourhood Ω of ∂ ∗ ∗ ∗ ∂ ∗ ∗ ∗ 0 in K. Thus we have ∂t s (χ1 ⊗ id ) ◦ Ψ = 0 = ∂t s (χ2 ⊗ id ) ◦ Ψ for s ∈ Ω. ∗ ∗ ∗ ∗ ∗ ∗ Since we also have ev|t=0 (χ1 ⊗ id )(Ψ (f)) = f = ev|t=0 (χ2 ⊗ id )(Ψ (f)) for any f ∈ OG(U) if ev|t=0 denotes the evaluation of in the ﬁrst component in t = 0, we deduce Ψ ◦ (χ1|Ω × id) = Ψ ◦ (χ2|Ω × id). This implies χ1|Ω = χ2|Ω due to the eﬀectiveness of

Ψ and thus exp(tX) = 0 for all t ∈ Ω, i.e. X = 0. Consequently, λ|g0¯ is injective. ∗ ∗ Consider now the case X ∈ g1¯ and suppose again λ(X) = (X(e) ⊗ id ) ◦ Ψ = 0. Since 2 X is an odd vector field on G, we have [X,X] = 2X ∈ g0¯. As λ|g0¯ is injective and λ([X,X]) = [λ(X), λ(X)] = 0, the vector field X commutes with itself, i.e. [X,X] = 0. 0|1 ∗ Therefore, the morphism γ : K × M → M given by the pullback γ (f) = f + τX(f), 0|1 ∗ ∂ ∗ if τ is the coordinate on K , defines a flow of X with ev|τ=0 γ (f) = f and ∂τ γ (f) = 0|1 ∗ X(f); see also [GW13], § 4. Define χ1 : K → G by χ1(f) = f(e) + τX(f)(e), where 0|1 f(e) and X(f)(e) are the evaluations of f and X(f) in e. Moreover, let χ2 : K → G ∗ be given by χ2(f) = f(e). We have χ1 = χ2 precisely if X = 0. A calculation yields ∂ ∗ ∂τ 0 χ1 = X(e). Thus we have ∂ ∗ ∗ ∗ ∗ ∗ ∗ ⊗ id (χ1 ⊗ id ) ◦ Ψ = (X(e) ⊗ id ) ◦ Ψ = λ(X) = 0 ∂τ 0 ∂ ∗ ∗ ∗ ∗ = ⊗ id (χ2 ⊗ id ) ◦ Ψ . ∂τ 0 Since also

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ (evτ=0 ⊗ id )(χ1 ⊗ id ) ◦ Ψ = id = (evτ=0 ⊗ id )(χ2 ⊗ id ) ◦ Ψ , we have Ψ ◦ (χ1 × id) = Ψ ◦ (χ2 × id) and thus by assumption χ1 = χ2, which implies X = 0. Suppose now that the action ψ of the classical Lie group G on the supermanifold M is eﬀective and that λ is injective. Let χj : N → G, j = 1, 2, be morphisms with Ψ ◦ (χ1 × id) = Ψ ◦ (χ2 × id). We need to show χ1 = χ2.

27 0|k First, we consider the case where N = K andχ ˜1(0) = e =χ ˜2(0). Let τ1, . . . , τk 0|k denote coordinates on K . Then there are morphisms αν, βν : OG → C such that

∗ X ν ∗ X ν χ1(f) = τ αv(f) and χ2(f) = τ βv(f) k k ν∈(Z2) ν∈(Z2) for all f, and αν, βν are of parity |ν| = |(ν1, . . . , νk)| = ν1 + ... + νk ∈ Z2. Since χ˜j(0) = e, we have α0(f) = f(e) = β0(f), where f(e) denotes the evaluation of f in the k identity e. Assume now χ1 6= χ2, and let ν0 ∈ (Z2) such that αν0 6= βν0 and such that there is no µ with αµ 6= βµ and ||µ|| < ||ν0|| for ||µ|| = ||(µ1, . . . , µk)|| = µ1+...+µk ∈ Z. ∼ (In the deﬁnition of ||µ|| for µ ∈ Z2, each µj is considered as an element of {0, 1}(= Z2) and the sum µ1 + ... + µk is understood as a sum in Z). Deﬁne cν : OG → C as cν = αν − βν. By the choice of ν0, we have cν = 0 if ||ν|| < ||ν0||. For any homogeneous f, g we have

X ν ∗ ∗ ∗ ∗ ∗ ∗ τ cν(fg) = (χ1 − χ2)(fg) = χ1(f)χ1(g) − χ2(f)χ2(g) ν,||ν||≥||ν0|| ∗ ∗ ∗ ∗ ∗ ∗ = χ1(f)(χ1 − χ2)(g) + (χ1 − χ2)(f)χ2(g) !     ! X ν X µ X µ X ν = τ αν(f)  τ cµ(g) −  τ cµ(f) τ βν(f) . ν µ,||µ||≥||ν0|| µ,||µ||≥||ν0|| ν

A comparison of the coeﬃcient of τ ν0 on both sides of this equation yields

|f||ν0| |f||ν0| cν0 (fg) = cν0 (f)β0(g) + (−1) α0(f)cν0 (g) = cν0 (f)g(e) + (−1) f(e)cν0 (g).

Thus, cν0 deﬁnes a point-derivation OG,e → C of parity |ν0| on G in the identity e. Let ∗ ∗ Y denote the corresponding right-invariant vector ﬁeld on G, i.e. Y = (cν0 ⊗ id ) ◦ µ if

µ denotes the multiplication of G, and Y (e) = cν0 . Since Ψ◦(χ1 ×id) = Ψ◦(χ2 ×id), we ∗ ∗ ∗ ∗ P ν ∗ ∗ ∗ have 0 = ((χ1 − χ2) ⊗ id ) ◦ Ψ = (( ν τ cν) ⊗ id ) ◦ Ψ and thus 0 = (cν0 ⊗ id ) ◦ Ψ = (Y (e) ⊗ id∗) ◦ Ψ∗. This implies Y = 0 since we supposed that the inﬁnitesimal action

λ is injective. Consequently, we have cν0 = Y (e) = 0 which is a contradiction to the choice of ν0 with cν0 = αν0 − βν0 6= 0. Therefore, the morphisms χ1 and χ2 have to be identical. The general case can be reduced to the just described special case. In order to do so, consider now arbitrary N and χj : N → G, j = 1, 2, with Ψ ◦ (χ1 × id) = Ψ ◦ (χ2 × id). 0|k If χ1 6= χ2 then there is a morphism χ : K → N such that χ1 ◦ χ 6= χ2 ◦ χ. Therefore, 0|k 0 we may assume N = K . Let g =χ ˜1(0) and g =χ ˜2(0). We have ψg = ψg0 since 0 Ψ◦(χ1 ×id) = Ψ◦(χ2 ×id). This implies g = g by assumption. Let µg : G → G denote ∗ ∗ ∗ 0 the left-multiplication with g, i.e. µg = (evg ⊗ id ) ◦ µ . Deﬁne χj = µg ◦ χj, j = 1, 2. 0 0 Since µg is invertible we have χ1 = χ2 if and only if χ1 = χ2. By construction we have 0 0 0 0 χ˜1(0) = e =χ ˜2(0) and Ψ ◦ (χ1 × id) = Ψ ◦ (χ2 × id). By the preceding considerations 0 0 we get χ1 = χ2 and thus χ1 = χ2. Hence, the action Ψ is eﬀective.

28 Chapter 3

Distributions on supermanifolds and Frobenius’ theorem

In this chapter, we study distributions, and in particular involutive distributions, on supermanifolds. Distributions play an important role when studying how infinitesimal symmetries can be integrated to local or global symmetries. Thus, the results of this chapter are an important tool in the next chapter where we construct local actions from infinitesimal actions and find conditions for the existence of global actions. We start by recalling the definition of a distribution, which is a straightforward generalization of the classical definition in a sheaf-theoretic formulation. As a technical preparation for the study of the local structure of an involutive distribution, commuting vector fields on supermanifolds and their flows are studied in Section 3.1. Then the structure of involutive distributions on supermanifolds is examined. A local version of Frobenius’ theorem for involutive distributions is proven; a different proof of this theorem can also be found in [Var04]. Contrary to the classical case however, there is no global version of Frobenius’ theorem on supermanifolds since integral subsupermanifold do not contain enough information about the distribution. We give an example of a non-involutive distribution on a supermanifold which has integral subsupermanifolds through each point. The results of this chapter are included in [Ber14].

Deﬁnition 3.0.4. A (smooth or holomorphic) distribution D on a (real or complex) supermanifold M is a graded OM-subsheaf of the tangent sheaf TM = Der OM of M which is locally a direct factor, i.e. for each point p ∈ M there exists an open neighbourhood U of p in M and a subsheaf E of Der OM|U on U such that D|U ⊕ E = Der OM|U .

Remark 3.0.5 (cf. [Var04], Section 4.7). Locally there exist independent vector fields X1,...,Xr,Y1,...,Ys spanning a given distribution D, where the Xi are even and the Yj are odd. Moreover, (r|s) = dim D(p) for any p ∈ M (if M is connected) and (r|s) is called the rank of the distribution D. If ψ : M → N is a diffeomorphism and D is a distribution on M, then there is a distribution ψ∗(D) on N which is spanned by vector fields of the form ψ∗(X) = (ψ−1)∗ ◦ X ◦ ψ∗ for vector fields X on M belonging to D.

29 Remark 3.0.6. A distribution D of rank (r|s) on a supermanifold M induces a distribution De of rank r on the underlying classical manifold M by defining De(p) = {X˜(p)| X ∈ D} ⊆ TpM for p ∈ M. A vector field on M belongs to the reduced distribution De if and only if it is the reduced vector field X˜ of some vector field X on M belonging to D.

3.1 Commuting vector ﬁelds

In this section, commuting vector ﬁelds on a supermanifold M are studied. The results are used in the next section for a local characterization of involutive distributions. m|n m|n m|n m|n Let µ : R × R → R denote the addition on R which is given by ((s, σ), (t, τ)) 7→ (s + t, σ + τ) in coordinates. The goal of this section is the proof of the following theorem.

Theorem 3.1.1. Let M = (M, OM) be a supermanifold, let X1,...,Xm be even and let Y1,...,Yn be odd vector ﬁelds on M which all commute. Then, for any p ∈ M m there exists an open connected neighbourhood U of 0 in R , an open neighbourhood V ∗ of p in M and a morphism ϕ = (ϕ, ˜ ϕ ): U × V → M, where U = (U, O m|n | ) and R U V = (V, OM|V ), such that:

(i) The map ϕ has the action property, i.e. we have ϕ ◦ ι0 = idM if ι0 : M ,→ m|n {0} × M ⊂ R × M denotes the canonical inclusion, and the equality

ϕ ◦ (id m|n × ϕ) = ϕ ◦ (µ × id ) R M m|n m|n holds on the open subsupermanifold of R × R × M on which both sides are deﬁned.

m|n (ii) If (t, τ) are coordinates on R , then for all i = 1, . . . , m, j = 1 . . . , n we have

∂ ∗ ∗ ∂ ∗ ∗ (1) ◦ ϕ = ϕ ◦ Xi and ◦ ϕ = ϕ ◦ Yj. ∂ti ∂τj

m|n m|n By replacing R by C an analogous result also holds true for a complex supermanifold M and holomorphic vector fields X1,...,Xm and Y1,...,Yn. m|n Remark 3.1.2. Note that ϕ locally defines a local action of R on M in the sense that ϕ would be a local action if the assumption that {0} × M is contained in the domain of definition was dropped.

Proof of Theorem 3.1.1. Let X1,...,Xm be even and Y1,...,Yn be odd vector fields X on M which all commute. Furthermore, let ϕ i : Wi → M denote the flow of Xi for any i. By Corollary 2.3.8 (iii) these flows all commute. Given p ∈ M, there exist an m open neighbourhood V ⊆ M of p and an open connected neighbourhood U ⊆ R of 0 such that β : U × V → M, β = ϕX1 ◦ ... ◦ ϕXm , t t1 tm m is defined, where V = (V, OM|V ), t = (t1, . . . , tm) coordinates on U ⊆ R , and βt is the evaluation of β in the first component in t. Since the flows ϕXi commute and each

(1)Considering ∂ and ∂ as vector ﬁelds on the product m|n × M. ∂ti ∂τj R

30 flow defines a local R-action on M, the map β has the action property, i.e. β ◦ι0 = idM and βs ◦ βs = βs+t for all s, t for which both sides of the equation are defined. 0|n Let τ1, . . . , τn be coordinates on R and define n 0|n ∗ X (2) α : R × M → M by α (f) = exp τjYj (f). j=1

The underlying map isα ˜ = idM . The sum

n ∞ n X X 1 X exp( τ Y )(f) = ( τ Y )k(f) j j k! j j j=1 k=0 j=1

Pn n+1 is ﬁnite because ( j=1 τjYj) = 0. Since the odd vector ﬁelds Yj all commute, i.e.

[Yi,Yj] = YiYj + YjYi = 0, P P P we get (σiYi)(τjYj) = (τjYj)(σiYi), and thus exp( i σiYi) exp( j τjYj) = exp( k(σk + 0|n 0|n τk)Yk) if (σ1, . . . , σn, τ1, . . . , τn) are coordinates on R × R . Consequently, the map 0|n α deﬁnes an R -action on M. Now, let W = (U × V, O m|n | ) and deﬁne R ×M U×V

ϕ : W → M, ϕ = β ◦ (idU × α) on W.

Then ϕ∗(f) = exp(P τ Y )(ϕX1 )∗ ◦ ... ◦ (ϕXm )∗(f). The map ϕ satisﬁes ϕ ◦ ι = id j j j t1 tm 0 M and ∂ ◦ ϕ∗ = X ◦ ϕ∗, ∂ ◦ ϕ∗ = Y ◦ ϕ∗ for all i, j, making use of the commutativity of ∂ti i ∂τj j the vector ﬁelds and Corollary 2.3.8. A calculation using the action properties of α and

β and again Corollary 2.3.8 shows that ϕ ◦ (id m|n × ϕ) = ϕ ◦ (µ × id ) on the open R M m|n m|n subsupermanifold of × ×M on which both ϕ◦(id m|n ×ϕ) and ϕ◦(µ×id ) R R R M are deﬁned.

Remark 3.1.3. The complex version of the result can be proven along the lines of the real case using holomorphic ﬂows and an analogue of Corollary 2.3.8 for holomorphic vector ﬁelds.

3.2 Involutive distributions and Frobenius’ theorem

As in the classical case, the notion of an involutive distribution can be deﬁned.

Deﬁnition 3.2.1. A distribution D is called involutive if Dp is a Lie subalgebra of (Der OM)p for each p ∈ M, i.e. if for any two vector ﬁelds X and Y on M belonging to D their commutator [X,Y ] also belongs to D.

In the following involutive distributions on supermanifolds will be studied further. The goal is the proof of the local version of Frobenius’ theorem.

(2) Pn P∞ 1 Pn k The sum exp( j=1 τj Yj )(f) = k=0 k! ( j=1 τj Yj ) (f) has to be understood in the following way: The vector fields Yj , which are a priori vector fields on M, are considered as vector fields on the product 0|n R ×M and similarly f is considered as a function on this product. Hence, τj Yj (f) here in fact means ∗ ∗ 0|n 0|n τj ·(id 0|n ⊗Yj )(πM(f)), where τj is now considered as a coordinate on ×M, πM : ×M → M R R R ∗ 0|n is the canonical projection onto M, and id 0|n ⊗ Yj is the extension of Yj to a vector field on × M. R R

31 Theorem 3.2.2 (Local version of Frobenius’ Theorem, cf. [Var04] Theorem 4.7.1). A distribution D on a real ( complex) supermanifold M is involutive if and only if there are local coordinates (x, θ) = (x1, . . . , xm, θ1, . . . , θn) around each point p ∈ M such that D is locally spanned by ∂ ,..., ∂ , ∂ ,..., ∂ , i.e. there exist supermanifolds U and ∂x1 ∂xr ∂θ1 ∂θs S and a diffeormorphism (resp. biholomorphism) onto its image ψ : U × S → M such that ψ∗(DU ) = D for the distribution DU on U × S in U-direction, i.e. the distribution ∗ spanned by vector fields of the form X ⊗ idS on U × S where X is a vector field on U.

If the distribution D is locally spanned by ∂ ,..., ∂ , ∂ ,..., ∂ , it follows di- ∂x1 ∂xr ∂θ1 ∂θs rectly that D is involutive. Consequently, only the existence of such coordinates for an involutive distribution remains to be proven. The proof in [Var04] discusses the even and odd part of the distribution D separately. Here, ﬁrst the existence of r even and s odd commuting vector ﬁelds that locally span D will be shown, similarly as in [Var04]. Then Theorem 3.1.1 is used to obtain the local coordinates (x, θ) with the required property.

Lemma 3.2.3. Let D be an involutive distribution of rank (r|s) on a real (resp. complex) supermanifold M. Then, there locally exist smooth (resp. holomorphic) even vector ﬁelds X1,...,Xr and odd vector ﬁelds Y1,...,Ys which locally span D and which all commute.

The proof of this lemma can be carried out as in the classical case.

0 0 0 0 Proof. Let X1,...,Xr,Y1,...,Ys be r even and s odd independent vector ﬁelds on some open subset U of M that span the distribution D on U. If x1, . . . , xm, θ1, . . . , θn are local coordinates on U, then

m n m n 0 X 0 ∂ X 0 ∂ 0 X 0 ∂ X 0 ∂ Xi = aik + bil and Yj = cjk + bjl ∂xk ∂θl ∂xk ∂θl k=1 l=1 k=1 l=1

0 0 0 0 0 for some functions aik, djl ∈ OM(U)0¯ and bil, cjk ∈ OM(U)1¯. Deﬁning the matrices A , 0 0 0 0 0 B , C and D by Aij = aij etc., we get

 0   ∂  X1 ∂x1  .   .   .   .   0  0 0  ∂  X  A B    r =  ∂xm  . Y 0  C0 D0  ∂  1  ∂θ1   .   .   .   .     .  0 ∂ Ys ∂θn

A0 B0 The matrix is an (r + s) × (m + n)-matrix and its rank is maximal since C0 D0 0 0 0 0 the vector ﬁelds X1,...,Xr,Y1,...,Ys were assumed to be independent. Hence, after interchanging coordinates, the matrices A0, B0, C0 and D0 may be written as A0 = 0 0 0 A1 B1 (A1 A2), B = (B1 B2), C = (C1 C2) and D = (D1 D2) so that is an C1 D1

32 invertible (r + s) × (r + s)-matrix. Consequently, by deﬁning

   0  X1 X1  .   .   .   .    −1  0  Xr A1 B1 Xr   =  0  , Y1  C1 D1 Y     1   .   .   .   .  0 Ys Ys we get independent vector ﬁelds X1,...,Xr,Y1,...Ys locally spanning the distribution D such that m n ∂ X ∂ X ∂ Xi = + aik + bil ∂xi ∂xk ∂θl k=r+1 l=s+1 and similarly m n ∂ X ∂ X ∂ Yj = + cjk + djl ∂θj ∂xk ∂θl k=r+1 l=s+1 for some aik, djl ∈ OM(U)0¯ and bil, cjk ∈ OM(U)1¯. Since D is involutive, the commutator

m m h ∂ ∂ i P h ∂ ∂ i P h ∂ ∂ i [Xi,Yj] = , + , cjk + aik , + ... ∂xi ∂θj ∂xi ∂xk ∂xk ∂θj k=r+1 k=r+1 Pm ∂ Pn ∂ = λk + µl , k=r+1 ∂xk l=s+1 ∂θl also belongs to D, where λk ∈ OM(U)1¯ and µl ∈ OM(U)0¯. Hence, this commutator may be written as a combination of the vector fields X1,...,Xr,Y1,...Ys. The special form of those vector fields now yields [Xi,Yj] = 0. A similar argument shows [Xi,Xj] = 0 and [Yi,Yj] = 0 for all i, j. Therefore, the vector fields X1,...,Xr,Y1,...Ys all commute.

Proof of Theorem 3.2.2. Let X1,...,Xr,Y1,...,Ys be r even and s odd independent and commuting vector ﬁelds, as constructed in the above lemma, which locally span the distribution D. r r By Theorem 3.1.1 there exist an open connected neighbourhood U ⊆ R (resp. C ) r of 0 ∈ R and an open neighbourhood V ⊆ M of a given point p ∈ M such that there is a map ϕ : U ×V → M, where U = (U, O r|s | ) (resp. (U, O r|s | )) and V = (V, O | ), R U C U M V satisfying the properties mentioned in Theorem 3.1.1. Consider now the restricted map ϕ|U×{p} : U × {p} → M. For the derivative ∗ dϕ|U×{p}(v) = v ◦ (ϕ|U×{p}) for v ∈ T(0,p)(U × {p}) we have ∂ ∂ ∗ dϕ|U×{p} = ◦ (ϕ|U×{p}) = Xi(p) ∂ti 0 ∂ti 0 and ∂ ∂ ∗ dϕ|U×{p} = ◦ (ϕ|U×{p}) = Yj(p), ∂τj 0 ∂τj 0 r|s r|s where (t, τ) are coordinates on R (resp. C ).

33 Since the vector ﬁelds X1,...,Xr,Y1,...,Ys are independent, the restricted map ϕ|U×{p} is an immersion in (0, p). Hence, after possibly shrinking U, ϕ|U×{p} is an immersion and U × {p} =∼ ϕ(U × {p}) is a subsupermanifold of M. Let S be a subsupermanifold of M which is transverse to ϕ(U × {p}) inϕ ˜(0, p) = p. Moreover, deﬁne

ψ : U × S → M by ψ = ϕ|U×S .

For the derivative of ψ in (0, p) we have dψ(T(0,p)(U × {p})) = T(0,p)(ϕ(U × {p})) and dψ(T(0,p)({0} × S)) = d(ϕ|{0}×S )({0} × TpS) = TpS. Therefore, making use of the inverse function theorem on supermanifolds (see e.g. Theorem 2.3.1 in [Le˘ı80]), the map ψ is a local diffeomorphism (resp. biholomorphism) in (0, p) as S was chosen to be transversal to ϕ(U ×{p}) and the dimensions of U ×S and M coincide. After shrinking the domain of definition, the map ψ is a diffeomorphism (resp. biholomorphism) onto its image. Furthermore, ∂ −1 ∗ ∂ ∗ −1 ∗ ∗ ψ∗ = (ψ ) ◦ ◦ ψ = (ψ ) ◦ ψ ◦ Xi = Xi ∂ti ∂ti and similarly ψ ∂ = Y , which now yields the existence of appropriate local coor- ∗ ∂τj j dinates. Therefore, the distribution D can be written in the required form with respect to these coordinates.

As in the classical case, the notion of an integral manifold of a distribution can be deﬁned.

Deﬁnition 3.2.4. Let D be a distribution of rank (r|s) on a supermanifold M = (M, OM). An (r|s)-dimensional subsupermanifold j : N → M, N = (N, ON ), of M is called an integral manifold of D through p ∈ M if

(i) the point p is contained in ˜(N), and

(ii) for any vector field X on M belonging to D there exists a vector field X¯ on N such that X¯ ◦ j∗ = j∗ ◦ X, or equivalently X(ker j∗) ⊆ ker j∗, and all vector fields on N are of the form X¯ for some vector field X on M which belongs to D.

Remark 3.2.5. In the case of supermanifolds, integral manifolds of a distribution do not provide as much information about the distribution as in the classical case. This is illustrated in the subsequent example. Contrary to the classical case, there is no global version of Frobenius’ theorem for the above deﬁned notion of an integral manifold. The existence of integral manifolds through each point does not imply that the distribution is involutive. Nevertheless, the local version of Frobenius’ theorem still guarantees the existence of integral manifolds through each point for an involutive distribution. There exist non-involutive distributions which have integral manifolds through each point.

0|2 Example 3.2.6 (A non-involutive distribution with integral manifolds). Let M = R , with coordinates θ1 and θ2, and let D be the distribution on M spanned by the odd vector ﬁeld X = ∂ + θ θ ∂ . Since the distribution D is a sheaf on 0 = {0}, it is ∂θ1 1 2 ∂θ2 R

34 uniquely determined by its stalk D0 in 0 and thus D will be identiﬁed with D0 in this example. The distribution D is not involutive since ∂ ∂ ∂ ∂ ∂ [X,X] = 2XX = 2 + θ1θ2 + θ1θ2 = 2θ2 ∈/ D. ∂θ1 ∂θ2 ∂θ1 ∂θ2 ∂θ2

0|1 0|1 0|1 0|1 0|1 ∼ Let N = R , with coordinate θ, and j : N = R ,→ R × {0} ⊂ R × R = 0|2 ∗ ∗ R = M be the natural inclusion with pullback j (θ1, θ2) = (θ, 0) and kernel ker j = ∗ O(M) · θ2. The map j is surjective, hence N is a closed submanifold of M. The vector field X = ∂ + θ θ ∂ is tangent to N because X(θ ) = θ θ ∈ ker j∗. ∂θ1 1 2 ∂θ2 2 1 2 Therefore, we have Y (ker j∗) ⊆ ker j∗ for every element Y ∈ D and consequently Y defines a vector field Y¯ : O(M)/ ker j∗ =∼ O(N ) → O(M)/ ker j∗ =∼ O(N ) on N . The ¯ ∂ ¯ ¯ ∗ ∗ vector field X on N induced by X equals ∂θ since X(θ) = X(j (θ1)) = j (X(θ1)) = j∗(1) = 1. Hence, the distribution D is not involutive, but there exist integral manifolds of D through each point (which is only 0). Moreover, remark that the involutive distribution spanned by ∂ has the same ∂θ1 integral manifolds as D.

35 36 Chapter 4

Globalizations of inﬁnitesimal actions on supermanifolds

Let g be a finite-dimensional Lie superalgebra, G a Lie supergroup with g as its Lie superalgebra of right-invariant vector fields, M a supermanifold and denote the Lie superalgebra of vector fields on M by Vec(M). Any action Ψ : G × M → M, or local action, of the Lie supergroup G on M induces an infinitesimal action g → Vec(M) on M as described in Proposition 2.2.3. Starting with an infinitesimal action λ : g → Vec(M) of G on M, it is a natural question to ask in which cases this infinitesimal action is induced by a local or global action of G on M, or some larger supermanifold M∗ containing M as an open subsupermanifold. In the case of a smooth manifold M and a Lie group G, Palais studied these questions in detail ([Pal57]). Concerning the existence of local actions, he proved that every infinitesimal action λ is induced by a local action of G on M. This generalizes the fact that the flow of any vector field on M defines a local R-action on M. In [Pal57], Palais also found necessary and sufficient conditions for the existence of a globalization, i.e. a (possibly non-Hausdorff) manifold M ∗, containing M as an open submanifold, with a G-action ϕ : G × M ∗ → M ∗ on M ∗ that induces the infinitesimal action λ on M and satisfies G · M := ϕ(G × M) = M ∗. In this chapter, we extend these results to the case of (real or complex) supermanifolds and (real or complex) Lie supergroups. The existence of a local action with a given infinitesimal action and conditions for the existence of a globalization are proven. A key point in the proof is, as in the classical case in [Pal57], the study of the distribution D = Dλ on the product G × M spanned by vector fields of the form

X + λ(X) for X ∈ g, considering X and λ(X) as vector fields on the product G × M. Also, the fact that the flow of one even vector field on a supermanifold defines a local R-action, or C-action in the complex case, is used (see [MSV93] and [GW13]). The results of this chapter are also published in [Ber14].

37 4.1 Distributions associated with inﬁnitesimal actions

Given an infinitesimal action λ : g → Vec(M), define the distribution D = Dλ on G×M as the distribution spanned by vector fields of the form X + λ(X)(1) for X ∈ g (cf. [Pal57], Chapter II, Definition 7). The rank of the distribution D equals the dimension of the Lie superalgebra g. If the homomorphism λ : g → Vec(M) is given, we can take G to be any Lie supergroup with Lie superalgebra g. One choice is for example the unique Lie supergroup with simply-connected underlying Lie group and Lie superalgebra g (for the existence of such G see e.g. [Kos83], and in the complex case [Vis11]). In the following, properties of the distribution associated with an infinitesimal action are studied.

Lemma 4.1.1. The distribution D = Dλ associated with an inﬁnitesimal action λ is involutive.

Proof. Since D is spanned by vector ﬁelds of the form X +λ(X) for X ∈ g, it is enough to check that the bracket of two such vector ﬁelds belongs again to D. Using that λ is a homomorphism of Lie superalgebras we get

[X + λ(X),Y + λ(Y )] = [X,Y ] + [λ(X), λ(Y )] = [X,Y ] + λ([X,Y ]) for any X,Y ∈ g and thus [X + λ(X),Y + λ(Y )] = [X,Y ] + λ([X,Y ]) also belongs to the distribution D.

The local version of Frobenius’ theorem now yields that the distribution D associated with an inﬁnitesimal action locally looks like a standard distribution on a product. In the following, local charts for the distribution D which locally transform D to a standard distribution and satisfy a few more properties with respect to the product structure of G × M are of special interest.

Definition 4.1.2 (flat chart). Let G be a Lie supergroup, M a supermanifold and D the distribution on G × M associated with an infinitesimal action on M. Let U ⊆ G be an open connected neighbourhood of a point g ∈ G, U = (U, OG|U ), and denote by ιg : M ,→ {g} × M ⊂ G × M the canonical inclusion. Moreover, let V ⊆ M be open, V = (V, OM|V ), and let ρ : V → M be a diffeomorphism onto its image. Denote by DG the standard distribution on G × M in G-direction, which is spanned by vector fields ∗ X ⊗ idM on G × M, where X is an arbitrary vector field on G. A diffeomorphism onto its image ψ : U × V → G × M is called a flat chart with respect to (D, U, V, g, ρ), or simply a flat chart (in g), if the following conditions are satisfied:

(i) ψ∗(DG|W ) = D|ψ˜(W ) for each open subset W ⊆ U × V ,

(ii) πG|U×V = πG ◦ ψ for the projection πG : G × M → G,

(iii) ψ ◦ ιg|V = ιg|ρ˜(V ) ◦ ρ.

(1)The vector ﬁeld X + λ(X) is here considered as a vector ﬁeld on G × M, so more formally one ∗ ∗ should write X ⊗ idM + idG ⊗ λ(X) for X + λ(X).

38 Remark 4.1.3. If ψ : U × V1 → G × M is a flat chart with respect to (D, U, V1, g, ρ1) and ρ2 : V2 → M is a diffeomorphism onto its image withρ ˜2(V2) ⊆ V1, then the map 0 ψ = ψ ◦ (idU × ρ2) is a flat chart with respect to (D, U, V2, g, ρ1 ◦ ρ2). Lemma 4.1.4. Let D be the distribution on G × M associated with the infinitesimal action λ : g → Vec(M), λ0 = λ|g0¯ : g0¯ → Vec(M) the restriction of λ to the even part g0¯ of g, and D0 the distribution on G × M associated with λ0. Let ψ : U × V → G × M be a flat chart with respect to (D, U, V, g, ρ) and define ψ0 : U × V → G × M by ψ0 = (idG × ϕ0) ◦ (diag × idV ), where diag : U → U × U denotes the diagonal map and ϕ0 is the composition of πM ◦ ψ and the canonical inclusion U × V ,→ U × V. Then ψ0 is a flat chart with respect to (D0, U, V, g, ρ).

Proof. It can be checked by direct calculations that ψ0 is a flat chart, using that the even right-invariant vector fields on G can be identified with the right-invariant vector fields ∼ on G, i.e. Lie(G) = g0¯ if g0¯ denotes the even part of the Lie superalgebra g = g0¯ + g1¯ of G.

Proposition 4.1.5 (Local existence of flat charts). Let D be the distribution associated with the infinitesimal action λ : g → Vec(M) on the supermanifold M and let G be a Lie supergroup with g as its Lie superalgebra of right-invariant vector fields. For any point (g, p) ∈ G × M there are an open connected neighbourhood U of g in G and an open neighbourhood V of p in M such that there exists a flat chart ψ : U × V → G × M with respect to (D, U, V, g, ρ = idV ). By Remark 4.1.3 this implies moreover the existence of U and V and a flat chart with respect to (D, U, V, g, ρ) for arbitrary ρ.

Proof. Let X1,...,Xk,Y1,...,Yl be a basis of g such that X1,...,Xk are even and Y1,...,Yl odd vector ﬁelds. Then the tangent vectors

0 0 0 0 0 X1(g ),...,Xk(g ),Y1(g ),...,Yl(g ) ∈ Tg0 G = {X(g )| X ∈ g} are linearly independent for all g0 ∈ G. Since the distribution D is spanned by vector fields of the form X + λ(X) for X ∈ g, there exist local coordinates (t, τ) for G on an open connected neighbourhood U ⊆ G of g and local coordinates (x, θ) for M on an open neighbourhood V ⊆ M of p so that D is locally spanned by the commuting vector fields m n m n ∂ X ∂ X ∂ ∂ X ∂ X ∂ A = + a + b and B = + c + d i ∂t iu ∂x iv ∂θ j ∂τ ju ∂x jv ∂θ i u=1 u v=1 v j u=1 u v=1 v for i = 1 . . . , k and j = 1, . . . , l, where aiu, biv, cju, djv ∈ OG×M(U × V ). After shrinking U and V , U = (U, OG|U ) can be assumed to be an open subsuper- k|l manifold of R with g = 0 and there is a morphism ϕ : U ×(U ×V) → G×M associated with the above defined commuting vector fields, satisfying the properties ϕ ◦ ι0 = id and ϕ ◦ (id k|l × ϕ) = ϕ ◦ (µ k|l × id ) (cf. Theorem 3.1.1). Since R R M

∂ ∗ ∗ ∂ ∗ ∗ ◦ ϕ = ϕ ◦ Ai and ◦ ϕ = ϕ ◦ Bj, ∂ti ∂τj the subsupermanifold V =∼ {0}×V ⊂ U ×V is transverse to ϕ(U ×{(0, p)}) inϕ ˜(0, 0, p) = (0, p). The map ψ : U ×V → G×M, ψ = ϕ|U×{0}×V , identiﬁes the standard distribution

39 DU on U × V with D, i.e. ψ∗(DG|W ) = ψ∗(DU |W ) = D|ψ˜(W ) for all open subsets W ⊆ U × V . The action property of the map ϕ moreover implies that

ψ ◦ ιg|V = ψ ◦ ι0|V = ϕ|U×{0}×V ◦ ι0|V = idU×V |{0}×V = ι0|V = ιg|V .

Hence, it only remains to show that πG|U×V = πG ◦ ψ in order to prove that ψ ∗ ∗ is a ﬂat chart. This is equivalent to showing ψ (ti) = ti and ψ (τj) = τj for all i, j, where ti and τj are now considered as local coordinate functions on G × M. Since ∂ (ψ∗(t )) = ψ∗(ψ ( ∂ )(t )) = ψ∗(0) = 0 for all r 6= i, ∂ (ψ∗(t )) = 1 and ∂tr i ∗ ∂tr i ∂ti i ∂ (ψ∗(t )) = ψ∗(ψ ( ∂ )(t )) = ψ∗(0) = 0 for all s, the function ψ∗(t ) ∈ O (U ×V ) ∂τs i ∗ ∂τs i i G×M is of the form ∗ X ν ψ (ti) = (ti + ci(x)) + cν(x)θ ν6=0 for some smooth functions ci, cν(ν 6= 0) on V . The property ψ ◦ ι0 = ι0 now implies ∗ ∗ ∗ P ν 0 = ι0(ti) = ι0 ◦ ψ (ti) = (0 + ci(x)) + ν6=0 cν(x)θ and therefore ci = cν = 0. Hence ∗ ∗ ψ (ti) = ti as required. A similar argument yields ψ (τj) = τj.

Lemma 4.1.6. Let ψ : U × V → G × M be a diﬀeomorphism onto its image such that

(i) ψ∗(DG|W ) = D|ψ˜(W ) for a distribution D associated with an inﬁnitesimal action and any open subset W ⊆ U × V , and

(ii) πG|U×V = πG ◦ ψ.

Then given any element g ∈ U there exists a diﬀeomorphism onto its image ρ : V → M such that ψ ◦ ιg|V = ιg|ρ˜(V ) ◦ ρ. Hence ψ is a ﬂat chart with respect to (D, U, V, g, ρ).

˜ Proof. The property πG ◦ ψ = πG|U×V implies (ψ ◦ ˜ιg)(V ) ⊂ {g} × M. To show that ∼ ψ ◦ ιg|V : V → G × M induces a map ρ : V → M = {g} × M, it is enough to check that ∗ ∗ (ψ ◦ ιg|V ) ◦ πG = evg, where evg : OG(G) → R denotes the evaluation in g. ∗ ∗ ∗ The fact πG ◦ ψ = πG|U×V implies (ψ ◦ ιg|V ) ◦ πG = (πG ◦ ψ ◦ ιg|V ) = (πG|U×V ◦ ∗ ιg|V ) = evg since (πG ◦ ιg) is the unique map M → {g} ⊂ G. The map ρ satisfies ψ ◦ ιg|V = ιg|ρ˜(V ) ◦ ρ by definition and is a diffeomorphism onto its image since ψ is a diffeomorphism onto its image and V and M have the same dimension.

Proposition 4.1.7 (Uniqueness of flat charts). If D is a distribution on G × M associated to an infinitesimal action λ : g → Vec(M), then a flat chart ψ : U × V → G × M with respect to (D, U, V, g, ρ) is unique.

The proof of this proposition is carried out in two steps:

(i) First, it is shown that two ﬂat charts ψ1 and ψ2 with respect to (D, U, V, g, ρ) coincide on an open neighbourhood of {g} × V in U × V .

(ii) Second, the local statement and the connectedness of U are used to globally get ψ1 = ψ2.

40 Proof. (i) Consider first the case where ρ = id and D = DG, i.e. the infinitesimal action λ is the zero map. Now, let ψ be a flat chart with respect to (DG, U, V, g, id). Since the identity idG×M|U×V is also a flat chart with respect to (DG, U, V, g, id), the equality of ψ and the identity in a neighbourhood of any point (g, p) for p ∈ V needs to be shown. Let (t, τ) be local coordinates on a connected neighbourhood U 0 of g ∈ G such that g = 0 in these coordinates and let (x, θ) be local coordinates on a neighbourhood V 0 of p ∈ V . Then (t, τ, x, θ) are local coordinates for G × M in a neighbourhood of (0, p) = ψ˜(0, p) and therefore on a neighbourhood of ψ˜(U 00 × V 00) ⊆ U 0 × V 0 for 00 0 00 0 appropriate subsets U ⊆ U and V ⊆ V . Since πG|U×V = πG ◦ ψ, we have

∗ ∗ ∗ ∗ ψ (t, τ) = ψ (πG(t, τ)) = πG(t, τ) = (t, τ).

Moreover, ψ∗(DG) = DG implies

∂ ∂ ∂ ∂ ∂ ∂ ψ∗ , ψ∗ ∈ span ,..., , ,..., ∂tr ∂τs ∂t1 ∂tk ∂τ1 ∂τl for all r, s. Hence

∂ ∂ ψ∗ (x, θ) = ψ∗ (x, θ) = (0, 0), ∂tr ∂τs and then ∂ (ψ∗(x, θ)) = ψ∗(ψ ( ∂ )(x, θ)) = ψ∗(0, 0) = (0, 0), and similarly we get ∂tr ∗ ∂tr ∂ (ψ∗(x, θ)) = (0, 0). Consequently, we have ∂τs ∗ ∗ ∗ ∗ ψ (x, θ) = ι0 ◦ ψ (x, θ) = ι0(x, θ) = (x, θ) and thus ψ = id on U 00 × V 00. Let ψ1 and ψ2 now be flat charts with respect to (D, U, V, g, ρ) for a distribution D associated with an arbitrary infinitesimal action and arbitrary ρ. For any p ∈ V , we have ψ˜1(g, p) = ψ˜1 ◦ ˜ιg(p) = ˜ιg(˜ρ(p)) = (g, ρ˜(p)) = ψ˜2 ◦ ˜ιg(p) = ψ˜2(g, p). Now let U 0 and U 00 be open connected neighbourhoods of g and V 0 and V 00 open neighbourhoods of p with U 00 ⊆ U 0 and V 00 ⊆ V 0 such that the associated open subsu- 0 0 00 00 permanifolds U of G and V of M are isomorphic to superdomains and ψ˜1(U × V ) is ˜ 0 0 −1 00 00 contained in ψ2(U × V ). The composition ψ2 ◦ ψ1 is defined on U × V and a flat 00 00 −1 chart with respect to (DG,U ,V , g, id). By the above argumentation, ψ2 ◦ ψ1 = id 00 00 and thus ψ1 = ψ2 on U × V .

(ii) Let ψ1 and ψ2 be two flat charts with respect to (D, U, V, g, ρ). For each p ∈ V define the subset Wp ⊆ U containing the points t ∈ U such that ψ1 = ψ2 on an open neighbourhood of (t, p) in U × V . The sets Wp are open by definition and contain g as a consequence of (i). To prove ψ1 = ψ2, i.e. Wp = U for each p, it is therefore enough to show that each Wp is also closed in U due to the connectedness of U. If Wp is not closed, then there is a point t0 ∈ U \ Wp such that Wp ∩ Ω 6= ∅ for every open neighbourhood Ω of t0. The continuity of the underlying maps implies 0 0 ψ˜1(t0, p) = ψ˜2(t0, p). Let U be an open neighbourhood of t0 in U and V an open neighbourhood of p in V such that the associated open subsupermanifolds U 0 and V0 are isomorphic to superdomains. Now let U 00 ⊆ U 0 be an open connected neighbourhood

41 00 0 00 00 0 0 of t0 and V ⊆ V an open neighbourhood of p such that ψ˜1(U × V ) ⊆ ψ˜2(U × V ). 00 Moreover, let s0 be an element of U ∩Wp which exists by the choice of t0. After shrink- 0 00 0 ing V and V , the maps ψ1 and ψ2 coincide on an open neighbourhood of {s0} × V . 0 By Lemma 4.1.6 there exists a diﬀeomorphism onto its image ρ0 : V → M such that 0 0 0 0 the restrictions of ψ1 and ψ2 to U ×V are ﬂat charts with respect to (D,U ,V , s0, ρ0). 00 00 The same argument as given in (i) then shows that ψ1 and ψ2 coincide on U × V which is a contradiction to the assumption t0 ∈/ Wp.

0 0 Remark 4.1.8. Let ψ and ψ be flat charts and denote by ψ0 and ψ0 the associated 0 0 flat charts for the even part (cf. Lemma 4.1.4). Then ψ = ψ if and only if ψ0 = ψ0. Moreover, if ψ0 : U × V → G × M is a flat chart with respect to (D0, U, V, g, ρ) then the uniqueness and local existence of flat charts imply that there is a flat chart ψ : U × V → G × M with respect to (D, U, V, g, ρ).

4.2 Existence of local actions

X In the classical case, the flow ϕ :Ω ⊆ R × M → M of a vector field X on a manifold M defines a local R-action on M. More generally, as proven in [Pal57], Chapter II, if we have a Lie algebra homomorphism λ : g → Vec(M) of a finite dimensional Lie algebra g into the Lie algebra of vector fields on M, there is a local action ϕ :Ω ⊆ G×M → M of a Lie group G, with Lie algebra of right-invariant vector fields g, such that its induced infinitesimal action

∂ ∗ ∗ g → Vec(M),X 7→ ϕ(exp(tX), −) = (X(e) ⊗ idM ) ◦ ϕ , ∂t 0 coincides with λ. A typical example is the case where X1,...,Xk are vector ﬁelds on

M whose R-span g = spanR{X1,...,Xk} is a Lie subalgebra of Vec(M) and λ is the inclusion g ,→ Vec(M). The goal in this section is the proof of an analogous theorem on supermanifolds. Important for the proof is the study of the distribution associated with an infinitesimal action, as in the classical case in [Pal57]. This also generalizes the result in [MSV93] and [GW13] saying that the flow of one 1|1 vector field X on a supermanifold M is a local R -action if and only if X is contained in a (1|1)-dimensional Lie subsuperalgebra g ⊂ Vec(M). The results in this section are formulated for the real case, but are equally true in the complex case, i.e. for complex supermanifolds M, complex Lie supergroups G and morphisms λ : g → Vec(M) of complex Lie superalgebras. As in the classical case, there is an equivalence of infinitesimal actions and local actions up to shrinking. This is the content of the following theorem whose proof is carried out in the remainder of this section.

Theorem 4.2.1. Let λ : g → Vec(M) be an infinitesimal action. Then there exists a local G-action ϕ : W ⊆ G × M → M on M such that its induced infinitesimal action λϕ equals λ. Moreover, any local action ϕ : W → M is uniquely determined by its induced infinitesimal action and domain of definition.

42 First, the uniqueness of a local action up to shrinking is proven. Then we describe how a local action can be constructed from an inﬁnitesimal action using the ﬂat charts of the distribution studied in the preceding section.

4.2.1 Uniqueness of local actions We need the following lemma to prove the uniqueness of local actions with a prescribed inﬁnitesimal action up to shrinking of the domain of deﬁnition.

Lemma 4.2.2. Let ϕ : W ⊆ G ×M → M be the local action of the Lie supergroup G on a supermanifold M with induced inﬁnitesimal action λϕ. Let D denote the distribution associated with λϕ. Deﬁne

ψ = (idG × ϕ) ◦ (diag × idM): W → G × M where diag : G → G × G denotes the diagonal map. Let p ∈ M and U ⊂ Wp = {g ∈ G| (g, p) ∈ W } be a relatively compact connected open neighbourhood of e ∈ Wp ⊂ G. Then there exists an open neighbourhood V of p ∈ M with U × V ⊂ W . The restriction ψ|U×V is a ﬂat chart with respect to (D, U, V, e, id).

Proof. Since U × {p} ⊂ W is compact, we can ﬁnd an open neighbourhood V of p with U × V ⊆ W . By deﬁnition of ψ, we have πG ◦ ψ = πG. Moreover, ψ ◦ ιe = ιe since ϕ is a local action. ∗ ∗ ∗ ∗ Let X ∈ g, then X ◦ diag = diag ◦ (X ⊗ idG + idG ⊗ X) since X is a derivation. If G G ιe : G ,→ {e} × G ⊂ G × G is the inclusion, then µ ◦ ιe = idG for the multiplication µ on ∗ ∗ G. Since X is right-invariant, we have X = (X(e) ⊗ idG) ◦ µ . By a calculation, using these facts and that ϕ is a local action, we obtain

∗ ∗ ∗ ψ∗(X ⊗ id ) = X ⊗ id + id ⊗ λϕ(X), which yields ψ∗(DG) = D.

The lemma implies that every local action of a Lie supergroup is uniquely determined by its domain of deﬁnition and its induced inﬁnitesimal action:

Corollary 4.2.3. The domain of definition of a local action and the induced infinitesimal action uniquely determine the local action, i.e. if ϕ1 : W → G × M and ϕ2 : W → G × M are local actions of the Lie supergroup G on the supermanifold M with the same induced infinitesimal action λ = λϕ1 = λϕ2 , then ϕ1 = ϕ2.

Proof. By the preceding lemma and the uniqueness of ﬂat charts we have ψ1 = (idG × ϕ1) ◦ (diag × idM) and ψ2 = (idG × ϕ2) ◦ (diag × idM) and hence ϕ1 = πM ◦ ψ1 = πM ◦ ψ2 = ϕ2.

4.2.2 Construction of local actions In the following, let λ : g → Vec(M) be a fixed infinitesimal action and G a Lie supergroup with multiplication µ : G × G → G and Lie superalgebra of right-invariant vector fields g.

43 The goal is to find a local G-action on M with induced infinitesimal action λ. Such a local action of G on M is constructed using flat charts for the distribution D associated with λ. The domain of definition of the constructed action depends, in general, on some choices. After restricting two local actions with the same infinitesimal action to a neighbourhood of {e} × M in G × M, the local actions coincide as proven in the previous paragraph. Nevertheless, in general there is, as in the classical case (cf. [Pal57] Chapter III.4), no unique maximal domain of definition on which the local action can be defined.

Deﬁnition of a local action: Choose a neighbourhood basis {Uα}α∈A of the identity e ∈ G such that (cf. [Pal57], Chapter II, §7):

−1 −1 (i) Each Uα is connected and Uα = (Uα) = {g ∈ G| g ∈ Uα}.

(ii) For α, β ∈ A either Uα ⊆ Uβ or Uβ ⊆ Uα holds.

Note that the two conditions guarantee the connectedness of Uα ∩ Uβ for arbitrary α, β ∈ A. For each p ∈ M choose α(p) ∈ A and a neighbourhood Vp ⊆ M of p such that there is a ﬂat chart 2 ψp : Uα(p) × Vp → G × M 2 2 with respect to (D,Uα(p),Vp, e, id), where Uα(p) and Vp denote again the open subsuper- 2 manifolds of G and M with underlying sets Uα(p) = {gh| g, h ∈ Uα} and Vp. For two elements p, q ∈ M, we may assume Uα(p) ⊆ Uα(q). Therefore, if the intersection

2 2 2 (Uα(p) × Vp) ∩ (Uα(q) × Vq) = Uα(p) × (Vp ∩ Vq) is non-empty, then the restrictions of ψp and ψq to their common domain of deﬁnition 2 are both ﬂat charts with respect to (D,Uα(p), (Vp ∩ Vq), e, id) and hence coincide. Let [ W = (Uα(p) × Vp) ⊆ G × M. p∈M

The set W is open by deﬁnition and contains {e} × M. Furthermore, for each p ∈ M the subset Wp = {g ∈ G| (g, p) ∈ W } ⊆ G is connected since all Uα(q) are connected.

Let W = (W, OG×M|W ) and define a morphism ψ : W → G × M by ψ|Uα(p)×Vp = ψp for each p ∈ M. Let ϕ : W → M, ϕ = πM ◦ ψ. We now show that ϕ defines a local group action with induced infinitesimal action λ.

Proposition 4.2.4. The map ϕ deﬁnes a local action of G on the supermanifold M.

Proof. Since the map ψ deﬁning ϕ = πM◦ψ is locally given by ﬂat charts with respect to (D,Uα(p),Vp, e, id), we have ψ◦ιe = ιe and therefore ϕ◦ιe = πM ◦ψ◦ιe = πM ◦ιe = idM. Thus it remains to show that

ϕ ◦ (µ × idM) = ϕ ◦ (idG × ϕ)(?)

44 on the open subsupermanifold of G × G × M on which both sides are deﬁned. To prove (?) the special form of the distribution associated with an inﬁnitesimal action with respect to the group structure of G is used. The commutativity of the following diagram will be shown:

χ×idM G × G × M / G × G × M

−1 idG ×ψ (χ ×idM)◦(idG ×ψ)

G × G × M / G × G × M (τ×idM)◦(idG ×ψ)◦(τ×idM)

In the above diagram all maps are only deﬁned on appropriate open subsupermanifolds of G × G × M. The map τ : G × G → G × G denotes the map which interchanges the two components. Moreover, the map χ : G × G → G × G is deﬁned by

χ = (idG × µ) ◦ (diag × idG), such that µ = π2 ◦ χ and π1 = π1 ◦ χ, where πi, i = 1, 2, is the projection onto the i-th factor. The underlying map is given byχ ˜(g, h) = (g, gh). Note that if G = M and the infinitesimal action λ is the canonical inclusion g ,→ Vec(G) of the right-invariant vector fields, then χ is a flat chart for the distribution D with respect to (D, G, M, e, id), χ and ψ coincide (on their common domain of definition) and (?) is equivalent to the associativity of the multiplication µ. Let Ψ1 = (τ × idM) ◦ (idG × ψ) ◦ (τ × idM) ◦ (idG × ψ) and −1 Ψ2 = (χ × idM) ◦ (idG × ψ) ◦ (χ × idM).

The underlying maps are given by Ψ˜ 1(g, h, p) = (g, h, ϕ˜(g, ϕ˜(h, p))) and Ψ˜ 2(g, h, p) = (g, h, ϕ˜(gh, p)). The open subsupermanifold of G ×G ×M on which both morphisms Ψ1 and Ψ2 are defined is exactly the open subsupermanifold which is the common domain of definition of ϕ◦(µ×idM) and ϕ◦(idG ×ϕ). A calculation shows πM◦Ψ1 = ϕ◦(idG ×ϕ) and πM ◦ Ψ2 = ϕ ◦ (µ × idM) and the commutativity of the above diagram directly implies (?). To show the equality of Ψ1 and Ψ2, we consider the distribution D1⊗λ on G×(G×M) 1 ∗ associated with the infinitesimal action ⊗ λ : g → Vec(G × M), X 7→ (idG ⊗ λ(X)), of G on G × M and show that Ψ1 and Ψ2 are both locally flat charts for this distribution. Let DG be the distribution on G × G × M which is spanned by vector fields of the ∗ ∗ form X ⊗ idG ⊗ idM for X ∈ g. The distribution D1⊗λ is spanned by vector fields of ∗ ∗ ∗ ∗ 1 ∗ ∗ the form X ⊗ idG ⊗ idM + idG ⊗ idG ⊗ λ(X). For X ∈ g and writing for idG or idM, we have

(Ψ1)∗(X ⊗ 1 ⊗ 1) = (τ × idM)∗(idG × ψ)∗(τ × idM)∗(idG × ψ)∗(X ⊗ 1 ⊗ 1)

= (τ × idM)∗(idG × ψ)∗(1 ⊗ X ⊗ 1) = (τ × idM)∗(1 ⊗ X ⊗ 1 + 1 ⊗ 1 ⊗ λ(X)) = X ⊗ 1 ⊗ 1 + 1 ⊗ 1 ⊗ λ(X),

45 where the fact that ψ is a ﬂat chart for the distribution D associated with λ is used. Similarly, using the fact that χ∗(X ⊗ 1) = (X ⊗ 1 + 1 ⊗ X), we get (Ψ2)∗(X ⊗ 1 ⊗ 1) = X ⊗ 1 ⊗ 1 + 1 ⊗ 1 ⊗ λ(X). Hence, Ψ1 and Ψ2 both transform DG into D1⊗λ, i.e. (Ψi)∗(DG) = D1⊗λ. Remark that πG ◦ Ψ1 = πG = πG ◦ Ψ2 if πG : G × G × M → G denotes the projection G×M onto the ﬁrst component. Now, let ιe : G × M ,→ {e}×G×M⊂G×G×M and M ιe : M ,→ {e} × M ⊂ G × M denote the inclusions. Then a calculation shows

G×M G×M G×M G×M Ψ1 ◦ ιe = ιe ◦ ψ and Ψ2 ◦ ιe = ιe ◦ ψ

M M G×M G×M since ψ ◦ ιe = ιe and (χ × idM) ◦ ιe = ιe using that e is the identity element of G and the definition of χ. This shows that Ψ1 and Ψ2 are both locally flat charts in e. In order to check that Ψ1 and Ψ2 coincide everywhere, the special form of the 0 sets in {Uα}α∈A is important. Let (g, g , p) ∈ G × G × M such that Ψ1 and Ψ2 are defined on a neighbourhood of (g, g0, p), i.e. such that (g0, p), (gg0, p), (g, ϕ˜(g, p)) ∈ W . Then by definition of ψ, there exists Uα ∈ {Uγ}γ∈A containing g, a neighbourhood 0 2 Vα of q =ϕ ˜(g , p) in M and a flat chart ψα : Uα × Vα → G × M with respect to 2 (D,Uα,Vα, e, id) and with ψ|Uα×Vα = ψα|Uα×Vα . Furthermore, choose Uβ ∈ {Uγ}γ∈A 0 0 containing g and gg and a neighbourhood Vβ of p in M such that there exists a flat 2 2 chart ψβ : Uβ × Vβ → G × M with respect to (D,Uβ ,Vβ, e, id) and with ψ|Uβ ×Vβ =

ψβ|Uβ ×Vβ . 0 Shrink Vβ and choose a neighbourhood U of g in G with U ⊆ Uβ such thatϕ ˜(U × Vβ) ⊆ Vα. By the special choice of the neighbourhood basis {Uγ}γ∈A of e in G either Uα ⊆ Uβ or Uβ ⊆ Uα is true. First, suppose that Uα ⊆ Uβ. The map Ψ1 is defined on Uα × U × Vβ and the restriction of Ψ1 to Uα × U × Vβ is a flat chart with respect to (D1⊗λ,Uα,U × Vβ, e, ψ). 2 −1 Moreover, we haveµ ˜(Uα ×U) = Uα ·U ⊆ Uβ and thus Ψ2,β = (χ ×idM)◦(idG ×ψβ)◦ (χ×idM) is also defined on Uα ×U ×Vβ and a flat chart with respect to (D1⊗λ,Uα,U × Vβ, e, ψ). By the uniqueness of flat charts, Ψ1 and Ψ2,β coincide on Uα × U × Vβ so 0 that Ψ1 = Ψ2 near (g, g , p) ∈ Uα × U × Vβ. Consider now the case Uβ ⊆ Uα. The map Ψ1,α = (τ ×idM)◦(idG ×ψα)◦(τ ×idM)◦ 2 2 (idG × ψ) is defined on Uα × U × Vβ and is a flat chart with respect to (D1⊗λ,Uα,U × 0 −1 0 −1 −1 2 2 Vβ, e, ψ). The set Uβ(g ) contains e and g. Since Uβ(g ) ⊂ Uβ · Uβ = Uβ ⊆ Uα, 0 −1 the map Ψ1,α may be restricted to Uβ(g ) × U × Vβ and is then a flat chart with 0 −1 respect to (D1⊗λ,Uβ(g ) ,U × Vβ, e, ψ). After possibly shrinking U, we may assume 0 −1 0 −1 2 0 −1 µ˜(Uβ(g ) × U) = Uβ(g ) · U ⊆ Uβ . Then Ψ2 is defined on Uβ(g ) × U × Vβ and 0 −1 is a flat chart with respect to (D1⊗λ,Uβ(g ) ,U × Vβ, e, ψ). Again, the uniqueness of 0 0 −1 flat charts implies Ψ1 = Ψ2 near (g, g , p) ∈ Uβ(g ) × U × Vβ.

Proposition 4.2.5. The inﬁnitesimal action

∗ ∗ λϕ : g → Vec(M),X 7→ (X(e) ⊗ idM) ◦ ϕ , induced by the local G-action ϕ is λ.

Proof. Since the map ψ is locally a flat chart, ψ is a local diffeormorphism, πG ◦ ψ = πG and ψ ◦ ιe = ιe. Moreover, we have locally ψ∗(DG) = D. Therefore, the vector field

46 ∗ ψ∗(X ⊗ idM) on G × M belongs to the distribution D for each vector ﬁeld X ∈ g. We have ∗ ∗ −1 ∗ ∗ ∗ ∗ −1 ∗ ∗ ∗ ψ∗(X ⊗ idM) ◦ πG = (ψ ) ◦ (X ⊗ idM) ◦ ψ ◦ πG = (ψ ) ◦ (X ⊗ idM) ◦ πG −1 ∗ ∗ ∗ ∗ ∗ = (ψ ) ◦ πG ◦ X = πG ◦ X = (X ⊗ idM) ◦ πG.

Let X1,...,Xk+l be a basis of g and a1, . . . , ak+l local functions on G × M such that k+l ∗ X ∗ ∗ ψ∗(X ⊗ idM) = ai(Xi ⊗ idM + idG ⊗ λ(Xi)), i=1 ∗ which is possible since ψ∗(X ⊗idM) belongs to D. Then, combining the above, we have k+l ! ∗ ∗ X ∗ ∗ ∗ (X ⊗ idM) ◦ πG = ai(Xi ⊗ idM + idG ⊗ λ(Xi)) ◦ πG i=1 k+l ! ! X ∗ ∗ = aiXi ⊗ idM ◦ πG, i=1 Pk+l which implies X = i=1 aiXi. Since X ∈ g and X1,...,Xk+l is a basis of g, the ai’s are all constants and hence ∗ ∗ ∗ ψ∗(X ⊗ idM) = X ⊗ idM + idG ⊗ λ(X). Therefore, ∗ ∗ ∗ ∗ ∗ ∗ λϕ(X) = (X(e) ⊗ idM) ◦ ϕ = ιe ◦ (X ⊗ idM) ◦ ψ ◦ πM ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ = ιe ◦ ψ ◦ ψ∗(X ⊗ idM) ◦ πM = (ψ ◦ ιe) ◦ (X ⊗ idM + idG ⊗ λ(X)) ◦ πM ∗ ∗ = ιe ◦ (0 + πM ◦ λ(X)) = λ(X).

4.3 Globalizations of inﬁnitesimal actions

After we proved the existence of a local action of a Lie supergroup G on a supermanifold M with a given induced infinitesimal action λ : g → Vec(M), it is natural to ask in which cases this extends to a global G-action on M. A simple way to obtain examples of a local action which is not global is to start with an action on a supermanifold M0. This action then induces a local action, which is not global, on every non-invariant open subsupermanifold M ⊂ M0 The aim of this section is to characterize all infinitesimal action which “arise” in the just described way from a global action. These infinitesimal actions are called globalizable. In the classical case (see [Pal57], Chapter III), Palais found necessary and sufficient conditions for an infinitesimal action to be globalizable, allowing the larger manifold M 0, a globalization of the infinitesimal action, to be a possibly non-Hausdorff manifold. In this section, similar conditions for the existence of globalizations of infinitesimal actions of Lie supergroups on supermanifolds are proven, and differences to the classical case are pointed out. It is also shown by an example that an infinitesimal action on a supermanifold may not be globalizable even if its underlying infinitesimal action is. In analogy to the classical case (cf. [Pal57], Chapter III, Definition II), we define the notion of a globalization of an infinitesimal action.

47 Deﬁnition 4.3.1. A globalization of an inﬁnitesimal action λ : g → Vec(M) of a Lie supergroup G on a supermanifold M is a pair (M0, ϕ0) with the following properties:

(i) M0 is a supermanifold, whose underlying manifold M 0 is allowed to be a non- Hausdorﬀ manifold, and M is an open subsupermanifold of M0, and

(ii) ϕ0 : G × M0 → M0 is an action of the Lie supergroup G on M0 such that its inﬁnitesimal action restricted to M coincides with λ, and

(iii)ϕ ˜0(G × M) = M 0.

If there is no chance of confusion the supermanifold M0 is also called a globalization. The inﬁnitesimal action λ is called globalizable if there exists a globalization (M0, ϕ0) of λ.

Deﬁnition 4.3.2. The domain of deﬁnition W = (W, OW ) of a local G-action ϕ : W → M is called maximally balanced if Wp = Wϕ˜(h,p)h = {gh| g ∈ Wϕ˜(h,p)} for all (h, p) ∈ W , where again Wq = {g ∈ G| (g, q) ∈ W } for any q ∈ M.

Remark 4.3.3. In [Pal57] a maximally balanced domain of definition of a local action is called maximum (see Definition VII in Chapter III, [Pal57]). Here, the term maximally balanced is used instead of maximum in order to avoid any confusion with maximal domains of definition.

In the classical case, the following theorem states necessary and suﬃcient conditions for the existence of a globalization of an inﬁnitesimal action.

Theorem 4.3.4 (see [Pal57], Chapter III, Theorem X). Let λ : g → Vec(M) be an inﬁnitesimal action of a Lie group G on a manifold M. Then the following statements are equivalent:

(i) The inﬁnitesimal action λ is globalizable.

(ii) There exists a local action ϕ : W → M of G on M with infinitesimal action λ whose domain of definition W is maximally balanced; this local action with a maximally balanced domain of definition is then unique and any other local action with the same infinitesimal action λ is a restriction of ϕ.

(iii) Let D be the distribution on G × M associated with the inﬁnitesimal action λ and Σ ⊂ G × M any leaf of D, i.e. a maximal connected integral manifold. Then the map πG|Σ :Σ → M is injective, where πG : G × M → G denotes the projection onto G.

Remark 4.3.5. If ϕ0 : G×M0 → M0 is a globalization of the infinitesimal action λ, then the underlying actionϕ ˜0 : G × M 0 → M 0 is a globalization of the reduced infinitesimal ˜ action λ : g0¯ → Vec(M). Therefore, a necessary condition for an infinitesimal action to be globalizable is the existence of a globalization M 0 of the reduced infinitesimal action.

48 4.3.1 Univalent leaves and holonomy In this paragraph, the notion of univalent leaves of the distribution associated with an infinitesimal action is introduced and holonomy phenomena are discussed. These no- tions play an important role in the characterization of globalizable infinitesimal actions on supermanifolds. Throughout, let λ : g → Vec(M) be a fixed infinitesimal action of a Lie supergroup G and let D be the associated distribution on G × M. Trying to generalize the classical result (Theorem 4.3.4) to supermanifolds, the question of an appropriate formulation of condition (iii) arises since integral manifolds do not uniquely determine a distribution on a supermanifold (cf. Example 3.2.6). For that purpose the notion of a univalent leaf Σ ⊂ G × M, extending the classical notion, is introduced. Then holonomy phenomena of the distribution D are studied and a connection between the absence of such phenomena and the notion of univalent leaves is established.

Remark 4.3.6. The infinitesimal action λ : g → Vec(M) induces an infinitesimal ˜ action λ : g0¯ → Vec(M) of the classical Lie group G on the manifold M, where the Lie algebra of G is identified with the even part g0¯ of g. For any vector field X ∈ g0¯(⊆ g) we define λ˜(X) to be the reduced vector field Y˜ of Y = λ(X).

˜ Lemma 4.3.7. Let λ : g0¯ → Vec(M) denote the induced inﬁnitesimal action of G on M and Dλ˜ the associated distribution on G × M. Then we have De = Dλ˜, where De denotes the distribution on G × M induced by D (cf. Remark 3.0.6).

Proof. For any odd vector field Y on a supermanifold we have Y˜ = 0. Since X + λ(X) has the same parity as X if X is homogeneous, the reduced distribution De is spanned ˜ ˜ by vector fields of the form X + λ(X) for X ∈ g0¯. These vector fields also generate Dλ˜ and thus De = Dλ˜.

As a corollary of the identity De = Dλ˜, we get the following relation between integral manifolds of the involutive distributions D on G × M and Dλ˜ on G × M.

Corollary 4.3.8. Every integral manifold N ⊂ G × M of the distribution Dλ˜ is the underlying manifold of some integral manifold N ⊂ G × M of the distribution D and conversely the underlying manifold of every integral manifold of D is an integral manifold of Dλ˜.

In the following, by a leaf Σ ⊂ G × M a leaf, i.e. a maximal connected integral manifold, of the distribution Dλ˜ = De is meant. By the preceding corollary every leaf Σ is as well the underlying manifold of an integral manifold of the distribution D.

The fact that the distribution Dλ˜ is involutive guarantees the existence of a leaf through each point (g, p) ∈ G × M. This leaf is denoted by Σ(g,p).

Definition 4.3.9. A leaf Σ ⊂ G×M is called univalent (with respect to D) if for every path γ : [0, 1] → Σ ⊂ G×M there exists a flat chart ψ : U ×V → G ×M with respect to ˜ (D, U, V, γG(0), id), for γG := πG ◦ γ : [0, 1] → G, such that ψ(U × V ) contains γ([0, 1]). The infinitesimal action λ is called univalent if all leaves Σ ⊂ G × M are univalent.

49 Remark 4.3.10. By Remark 4.1.3, a leaf Σ is univalent if for every path γ : [0, 1] → Σ there exists a ﬂat chart ψ with respect to (D, U, V, γG(0), ρ) for some ρ or equivalently, after shrinking, for any ρ, with γ([0, 1]) ⊂ ψ˜(U × V ).

Remark 4.3.11. In the definition of a univalent leaf Σ it is enough for the defining property to hold true for closed paths γ : [0, 1] → Σ because for any path γ0 the composition (γ0)−1 · γ0 of γ0 and (γ0)−1,(γ0)−1(t) = γ0(1 − t), is a closed path with γ0([0, 1]) = ((γ0)−1 · γ0)([0, 1]). ˜ Remark 4.3.12. If λ is univalent, then so is the induced infinitesimal action λ : g0¯ → Vec(M).

In [Pal57], an infinitesimal action (in the classical case) is called univalent if the restriction of the projection πG : G × M → G to an arbitrary leaf Σ ⊂ G × M is injective. The above defined notion of univalent infinitesimal actions on supermanifolds extends this definition:

Proposition 4.3.13. In the case of classical manifolds G = G and M = M, an inﬁnitesimal action is univalent if and only if the projection πG|Σ :Σ → G is injective for each leaf Σ ⊂ G × M.

Proof. Let Σ ⊂ G × M be any leaf, x = (g, p), y = (g, q) ∈ Σ, and γ : [0, 1] → Σ a path from x to y. Since λ is univalent, there is a flat chart ψ : U × V → G × M with respect to (D, U, V, g, id) with γ([0, 1]) ⊂ ψ(U × {p}). We have (g, q) = γ(1) = ψ(g, p) = (g, p) because πG ◦ ψ = πG. Assume now that πG|Σ is injective for each leaf Σ ⊂ G × M. Let Σ ⊂ G × M be a leaf and γ : [0, 1] → Σ a path. Using the compactness of γ([0, 1]) there are 0 = t0 < . . . < tk = 1 and flat charts ψi : Ui × Vi → G × M, i = 0, . . . , k − 1, with respect to (D,Ui,Vi, γG(ti), id) such that the intersection Ui ∩ Uj is connected for all i, j and γ([ti, ti+1]) ⊂ ψ(Ui × Vi). 0 0 Set ψ0 = ψ0 and, after possibly shrinking V0, inductively define flat charts ψi : Ui × V0 → G × M for i ≥ 1 by composing ψi and a local diffeomorphism of the form 0 0 (id × ρi) such that ψi and ψi+1 coincide on (Ui ∩ Ui+1) × V0 for i = 0, . . . , k − 2 and 0 0 γ([ti, ti+1]) ⊂ ψi(Ui × V0) holds. We have ψ0(U0 × {p}) ⊂ Σ(g,p) for g := γG(0) and 0 0 0 any p ∈ V0, and then by induction ψi(Ui × {p}) ⊂ Σ(g,p) for any i. Therefore, ψi = ψj on (Ui ∩ Uj) × V0 since πG|Σ(g,p) is injective and πG ◦ ψ = πG for any flat chart ψ. Sk−1 Consequently, we can define a flat chart ψ : U × V → G × M, U := i=0 Ui, V := V0, 0 with respect to (D, U, V, g = γG(0), id) by setting ψ|Ui×V0 = ψi. The map ψ satisfies γ([0, 1]) ⊂ ψ(U × V ) by construction.

Proposition 4.3.14. The infinitesimal action λ is univalent if and only if for any two flat charts ψi : Ui × Vi → G × M with respect to (D,Ui,Vi, g, ρi), i = 1, 2, and ρ1 = ρ2 on V1 ∩ V2 we have ψ1 = ψ2 on their common domain of definition. Remark that by Proposition 4.1.7 any two flat charts coincide on their common domain of definition (U1 ∩ U2) × (V1 ∩ V2) if U1 ∩ U2 is connected.

Proof. Let λ be univalent and ψi ﬂat charts with respect to (D,Ui,Vi, g, ρi) and ρ1 = ρ2 on V1 ∩ V2. Let (h, p) ∈ (U1 ∩ U2) × (V1 ∩ V2). Since ψ1 and ψ2 are ﬂat charts, ˜ ˜ ψ1(U1 ×{p}) and ψ2(U2 ×{p}) are both contained in the leaf Σ = Σ(g,ρ˜1(p)) = Σ(g,ρ˜2(p)).

50 ˜ The univalence of the reduced infinitesimal action λ implies that πG|Σ :Σ → G is injective and thus ψ˜1(h, p) = ψ˜2(h, p). Let γ : [0, 1] → U1 ∪ U2 be a closed path with 1 1 1 γ(0) = γ(1) = g, γ([0, 2 ]) ⊂ U1, γ([ 2 , 1]) ⊂ U2 and γ( 2 ) = h. Then ( ψ˜ (γ(t), p), t ≤ 1 γ0 : [0, 1] → Σ, γ0(t) = 1 2 ˜ 1 ψ2(γ(t), p), t > 2 is a closed path. As λ is univalent, there is a flat chart ψ : U×V → G×M with respect to 0 ˜ 1 ˜ 1 ˜ (D, U, V, g, id) with γ ([0, 1]) = ψ1(γ([0, 2 ])×{p})∪ψ2(γ([ 2 , 1])×{p}) ⊂ ψ(U×V ). Then, after possibly shrinking V1, ψ ◦ (id × ρ1) is a flat chart with respect to (D, U, V1, g, ρ1). By Proposition 4.1.7 the flat charts ψ ◦ (id × ρ1) and ψ1 coincide on a neighbourhood 1 of γ([0, 2 ]) × {p}. Moreover, ψ ◦ (id × ρ1) and ψ2 coincide on a neighbourhood of 1 γ([ 2 , 1]) × {p} since ρ1 = ρ2 on V1 ∩ V2. In particular, we get ψ1 = ψ ◦ (id × ρ1) = ψ2 1 near (h, p) = (γ( 2 ), p). Suppose now that any two flat charts ψi with respect to (D,Ui,Vi, g, ρi), i = 1, 2, with ρ1 = ρ2 already coincide on their common domain of definition. Assume that there is a leaf Σ ⊂ G × M which is not univalent and let γ : [0, 1] → Σ be a path for which there is no flat chart ψ : U × V → G × M with γ([0, 1]) ⊂ ψ˜(U × V ). Define I to be the set of points t ∈ [0, 1] such that there exists a flat chart ψ : U × V → G × M with γ([0, t]) ⊂ ψ˜(U × V ). The set I is open and I 6= [0, 1] by assumption. Let s be the minimum of [0, 1] \ I. There is a flat chart ψ1 : U1 × V1 → G × M with respect to ˜ (D,U1,V1, πG(γ(s)), id) with γ(s) ∈ ψ1(U1 × V1). By the choice of s there is t ∈ [0, s) with γ([t, s]) ⊂ ψ˜1(U1 × V1) such that there is a flat chart ψ2 : U2 × V2 → G × M with ˜ respect to (D,U2,V2, πG(γ(0)), id) with γ([0, t]) ∈ ψ2(U2 × V2). Then h = πG(γ(t)) ∈ U1 ∩ U2 and after possibly shrinking V2 there exists a diffeomorphism ρ such that 0 ψ1 ◦ (id × ρ) is a flat chart with respect to (D,U1,V2, h, ρ ) and ψ2 with respect to 0 0 (D,U2,V2, h, ρ ) for some ρ : V2 → M. Therefore, ψ1 ◦ (id × ρ) and ψ2 agree on their common domain of definition and they define a flat chart ψ :(U1 ∪ U2) × V2 → G × M with γ([0, s]) ⊂ ψ˜((U1 ∪ U2) × V2), contradicting the definition of s.

The preceding proposition allows us to glue together ﬂat charts in the case of a univalent inﬁnitesimal action. This also implies the next corollary.

Corollary 4.3.15. The inﬁnitesimal action λ is univalent if and only if for any compact subset Σ0 of a leaf Σ ⊂ G × M there exists a ﬂat chart ψ : U × V → G × M with Σ0 ⊂ ψ˜(U × V ).

In the following the structure of the distribution D associated with λ is investigated further. Let Σ ⊂ G × M be a leaf, (g, p) ∈ Σ and γ : [0, 1] → Σ a closed path with γ(0) = γ(1) = (g, p) and let γG = πG ◦ γ. We now want to associate a germ of a local diffeormorphism of M around p measuring the holonomy along the path γ. To do so, let 0 = t0 < . . . < tk = 1 be a partition of [0, 1] such that there are flat charts ψi : Ui×V → G×M, i = 0, . . . , k−1, with γ([ti, ti+1]) ⊂ ψ˜i(Ui×{p}), such that ψi and ψi+1 coincide on their common domain of definition and such that ψ0 is a flat chart with respect to (D,U0, V, g, id). We have ψ˜i(Ui × {p}) ⊂ Σ for any i. By Lemma 4.1.6 there is a diffeomorphism ρ : V → M onto its image such that ψk−1 is a flat chart with ˜ respect to (D,Uk−1, V, g = γG(1), ρ). Since (g, p) = γ(1) ∈ ψk−1(Uk−1 × {p}), we have ˜ (g, ρ˜(p)) = ψk−1(g, p) = (g, p) and thusρ ˜(p) = p.

51 Define Φ(γ) to be the germ of the local diffeomorphism ρ in p. The local uniqueness of flat chart implies that Φ(γ) does not depend on the actual choice of the flat charts ψi : Ui × Vi → G × M. Let Diffp(M) denote the set of germs of local diffeomorphisms χ : V1 → V2 in p ∈ M, where Vi = (Vi, OM), i = 1, 2, are open subsupermanifolds of M with p ∈ Vi. Then Φ(γ) is an element of Diffp(M) for each closed path γ : [0, 1] → Σ with γ(0) = γ(1) = (g, p). In the case of complex supermanifolds and holomorphic maps Diffp(M) should be replaced by Holp(M), the set of germs of local biholomorphisms χ : V1 → V2. Proposition 4.3.16. The germ Φ(γ) only depends on the homotopy class [γ] of the closed path γ with γ(0) = γ(1) = (g, p). Therefore, the assignment [γ] 7→ Φ([γ]) = Φ(γ) defines a map Φ = ΦΣ = ΦΣ,(g,p) : π1(Σ, (g, p)) → Diffp(M).

Proof. Let γs : [0, 1] → Σ, s ∈ [0, 1], be a continuous family of closed paths with γs(0) = γs(1) = (g, p) and γ0 = γ. Let s0 ∈ [0, 1], 0 = t0 < . . . < tk = 1 and ˜ ψi : Ui × Vi → G × M, i = 0, . . . , k − 1, flat charts with γs0 ([ti, ti+1]) ⊂ ψi(Ui × Vi) and such that ψ0 is a flat chart with respect to (D,U0,V0, g, id) and ψi and ψi+1 coincide on their common domain of definition. Then ψk−1 is a flat chart with respect to

(D,Uk−1,Vk−1, g, ρ) for some local diﬀeomorphism ρ around p and Φ(γs0 ) is the germ of ρ in p. Since all intervals [ti, ti+1] are compact there exists an open neighbourhood J ⊆ [0, 1] of s0 such that γs([ti, ti+1]) ⊂ ψ˜i(Ui × Vi) for i = 0, . . . , k − 1 and all s ∈ J.

Therefore, Φ(γs0 ) = Φ(γs) for all s ∈ J and the set {s ∈ [0, 1]| Φ(γs) = Φ(γ)} is open and closed. Since γ = γ0 we get Φ(γs) = Φ(γ) for all s ∈ [0, 1]. Remark 4.3.17. The deﬁnition of Φ implies that it is a group homomorphism: If γ is a constant path, then Φ([γ]) is the germ of the identity id : M → M and we have Φ([γ1] · [γ2]) = Φ([γ1]) ◦ Φ([γ2]) for the composition γ1 · γ2 of two closed paths γ1 and γ2. Remark 4.3.18. By Lemma 4.1.4 and Remark 4.1.8 the morphism Φ does not depend on whether its construction is done with respect to the distribution D on G × M associated with λ or to the distribution D0 on G × M associated with the inﬁnitesimal action λ0 = λ|g0¯ .

Remark 4.3.19. Due to the connectedness of the leaves Σ the map ΦΣ,(g,p) is trivial for some (g, p) ∈ Σ if and only if ΦΣ,(h,q) is trivial for all (h, p) ∈ Σ. The triviality of the map Φ = ΦΣ = ΦΣ,(g,p) can be viewed as an absence of holonomy for the leaf Σ.

1 0|2 Example 4.3.20. Let G = S , with coordinate φ, and M = R , with coordinates θ , θ . Let X = θ ∂ and consider the infinitesimal S1-action λ : Lie(S1) =∼ → 1 2 1 ∂θ2 R Vec(M), λ(t) = tX. The unique leaf of the distribution D on S1 × M, spanned by ∂ 1 1 1 ∂φ + X, is Σ = S × {0} = S × M. Let φ0 ∈ S and r :Ω → R be a local inverse 1 1 it around 1 ∈ Ω ⊂ S of R → S , t 7→ e . Then ∗ ∗ ∗ −1 ∗ ψ (φ, θ1, θ2) = (φ, ρ (θ1), ρ (θ2)) + (0, 0, r(φφ0 )ρ (θ1)) defines the pullback of a flat chart ψ with respect to (D, U, V, φ0, ρ) for V = {0} = M, 1 −1 0|2 0|2 U ⊂ S with φ0 ∈ U and Uφ0 ⊆ Ω and a diffeormorphism ρ : R → R . For any

52 0 0 0 0 0|2 0|2 φ0 ∈ U the map ψ is also a ﬂat chart with respect to (D, U, V, φ0, ρ ) for ρ : R → R with pullback

0 ∗ ∗ 0 −1 ∗ (ρ ) (θ1, θ2) = ρ (θ1, θ2) + (0, r(φ0φ0 )ρ (θ1)) 0 ! Z φ0 ∗ ∗ = ρ (θ1, θ2) + 0, 1dφ ρ (θ1) , φ0

0 ∗ ∗ where the integral might be taken along any path in Ω. In particular (ρ ) (θ1) = ρ (θ1). 0|2 0|2 A calculation shows that the map Φ : π1(Σ, (φ0, 0)) → Diff0(R ) = Diff(R ) is given by Z Z ∗ ∗ Φ([γ]) (θ1, θ2) = id (θ1, θ2) + 0, 1dφ θ1 = θ1, θ2 + 1dφ θ1 γ γ 1 ∼ for any γ : [0, 1] → S = Σ, γ(0) = γ(1) = φ0. Thus, identifying π1(Σ, (φ0, 0)) and Z, we get 0|2 ∗ Φ: Z → Diff(R ), Φ(k) (θ1, θ2) = (θ1, θ2 + 2πkθ1). We now establish an equivalence between univalent infinitesimal actions and infinitesimal actions with univalent reduced infinitesimal action and leaves without holonomy. Proposition 4.3.21. The infinitesimal action λ is univalent if and only if the reduced ˜ infinitesimal action λ : g0¯ → Vec(M) is univalent and for all leaves Σ the map ΦΣ is trivial. Proof. If the infinitesimal action λ is univalent, then the reduced infinitesimal action λ˜ is univalent as a direct consequence. Moreover, for any closed path γ : [0, 1] → Σ there is a flat chart ψ : U × V → G × M with γ([0, 1]) ⊂ ψ˜(U × V ) and thus Φ([γ]) = id. ˜ Now, suppose that λ is univalent and ΦΣ is trivial for all leaves Σ ⊂ G × M. We need to show that any two flat charts ψi : Ui × Vi → G × M, i = 1, 2, with respect to (D,Ui,Vi, g, ρi) with ρ1 = ρ2 coincide on their common domain of definition (cf. −1 Proposition 4.3.14). By replacing ψi by ψi ◦ (id × ρi ) it is enough to show ψ1 = ψ2 in the case of ρ1 = ρ2 = id. Let (h, p) ∈ (U1 × V1) ∩ (U1 × V1) = (U1 ∩ U2) × (V1 ∩ V2) be arbitrary. We ˜ ˜ ˜ ˜ have ψi(h, p) ∈ ψi(Ui × {p}) ⊂ Σ(g,p) and thus ψ1(h, p) = ψ2(h, p) since πG|Σ(g,p) is injective. After shrinking V2 there is a diffeomorphism ρ : V2 → M such that ψ2 ◦ (id × ρ) is defined and coincides with ψ1 near (h, p). Note thatρ ˜(p) = p since ψ˜1(h, p) = ψ˜2(h, p). The composition ψ2 ◦ (id × ρ) is a flat chart with respect to (D,U2,V2, g, ρ). Let α : [0, 1] → Σ(g,p) be a closed path with α(0) = α(1) = (g, p), 1 ˜ ˜ ˜ 1 ˜ 1 α( 2 ) = ψ1(h, p) = ψ2(h, ρ˜(p)) = ψ2(h, p) and α([0, 2 ]) ⊂ ψ1(U1 × {p}) and α([ 2 , 1]) ⊂ ˜ ˜ ψ2(U2 × {p}) = ψ2((id × ρ˜)(U2 × {p})). By the definition of Φ = ΦΣ(g,p) we have Φ([α]) = ρ. The triviality of Φ then gives ρ = id so that ψ1 and ψ2 agree near (h, p).

Corollary 4.3.22. The inﬁnitesimal action λ is univalent if and only if its restriction

λ0 = λ|g0¯ is univalent. Proof. The statement follows from the preceding proposition and the observation formulated in Remark 4.3.18.

53 4.3.2 The action on G × M from the right and invariant functions In this paragraph, the action R : G × (G × M) → G × M of the Lie supergroup G on the product G × M from the right, which is in the classical case given by (g, (h, x)) 7→ (hg−1, x), is introduced. Its behaviour with respect to the distribution D associated with λ : g → Vec(M) and in particular D-invariant functions is studied. D Definition 4.3.23. Let OG×M be the sheaf of D-invariant functions on G × M, i.e. D OG×M(Ω) = {f ∈ OG×M(Ω)| D · f = 0} for any open subset Ω ⊂ G × M. Remark 4.3.24. For any D-invariant function f on G × M, the underlying function f˜ is constant along the leaves, and in the classical case every such function is D-invariant. Moreover, if ψ : U × V → G × M is a flat chart and f is D-invariant, then ψ∗(f) is DG-invariant since ψ∗(DG) = D. The DG-invariant functions on U × V ⊆ G × M are of ∗ the form fM = 1 ⊗ fM for fM ∈ OM(V ) so that ψ (f) = 1 ⊗ fM for an appropriate fM. We have the following identity principle for D-invariant functions on G × M: Lemma 4.3.25. Let W ⊆ G × M be open, Σ ⊂ G × M be a leaf with Σ ⊂ W and let D f1, f2 ∈ OG×M(W ). If f1 = f2 on an open neighbourhood of some x ∈ Σ, then f1 and f2 coincide on an open neighbourhood of the leaf Σ. 0 Proof. Define Σ ⊆ Σ to be the subset of points y ∈ Σ such f1 and f2 are equal on some open neighbourhood of y. The set Σ0 is open and contains x by assumption. ˜ ∗ For any flat chart ψ : U ×V → G ×M with ψ(U ×V ) ⊂ W we have ψ (fi) = 1⊗fM,i ˜ for appropriate fM,i by Remark 4.3.24. Therefore, f1 = f2 near ψ(U × {q}) if and only 0 if fM,1 = fM,2 near q ∈ M. It follows that the set Σ is also closed and thus equal to Σ.

Definition 4.3.26. Let µ : G × G → G denote the multiplication of G, ι : G → G the inversion and τ : G ×G → G ×G the morphism which interchanges the two components. Then define the action of G on itself from the right as r : G × G → G, r = µ ◦ τ ◦ (ι × id). The underlying action is given by (g, h) 7→ hg−1. Define now a G-action on G × M by

R : G × (G × M) → G × M,R = r × id.

Lemma 4.3.27. For every right-invariant vector ﬁeld X on G we have

(id∗ ⊗ (X ⊗ id∗ + id∗ ⊗ λ(X))) ◦ R∗ = R∗ ◦ (X ⊗ id∗ + id∗ ⊗ λ(X)).

Proof. The right-invariance of X is equivalent to µ∗ ◦ X = (X ⊗ id∗) ◦ µ∗ and a short calculation yields (id∗ ⊗X)◦r∗ = r∗ ◦X, which directly implies the desired equality.

Corollary 4.3.28. We have ∗ D ˜ 1⊗D R (OG×M) ⊆ R∗(OG×G×M), where 1⊗D is the distribution on G×(G×M) spanned by vector ﬁelds of the form id∗⊗Y 1⊗D 1 for vector ﬁelds Y belonging to D and OG×G×M denotes the sheaf of ⊗ D-invariant functions.

54 Proof. Using the preceding lemma, we have

(id∗ ⊗ (X ⊗ id∗ + id∗ ⊗ λ(X)))(R∗(f)) = R∗ ◦ (X ⊗ id∗ + id∗ ⊗ λ(X))(f) = R∗(0) = 0 for any D-invariant function f on G×M and X ∈ g. Thus, R∗(f) is 1⊗D-invariant.

Deﬁnition 4.3.29. For g ∈ G, let rg : G → G denote the composition of the action r and the inclusion G ,→ {g} × G ⊂ G × G, and deﬁne

Rg : G × M → G × M,Rg = (rg × id).

−1 −1 Since r is an action, rg and Rg are diﬀeomorphisms, (rg) = rg−1 and (Rg) = Rg−1 .

Lemma 4.3.30. For each g ∈ G the map Rg : G × M → G × M satisﬁes

(i) (Rg)∗(DG) = DG, and

(ii) (Rg)∗(D) = D.

Proof. Property (i) can be directly obtained from the deﬁnition of Rg. Property (ii) follows from the fact that (rg)∗(X) = X for every right-invariant vector ﬁeld X, and therefore (Rg)∗(X + λ(X)) = (rg)∗(X) + λ(X) = X + λ(X) so that (Rg)∗(D) = D.

The composition of ﬂat charts and maps of the form Rg exhibits a special behaviour as speciﬁed in the following lemma.

Lemma 4.3.31. Let g ∈ G and let ψ : U × V → G × M be a ﬂat chart with respect to (D, U, V, h, ρ). Then the composition

0 ψ = Rg−1 ◦ ψ ◦ (Rg|Ug×V ): Ug × V → G × M is a ﬂat chart with respect to (D, Ug, V, hg, ρ) where Ug = {ug| u ∈ U} and Ug = (Ug, OG|Ug).

0 Proof. We have (ψ )∗(DG) = D using (Rg)∗(DG) = DG and (Rg)∗(D) = D. Moreover, 0 0 direct calculations show πG ◦ ψ = πG and ψ ◦ ιhg = ιhg ◦ ρ.

Corollary 4.3.32. The underlying classical Lie group G acts on the space of leaves ˜ ˜ Σ ⊂ G × M by (g, Σ) 7→ Rg(Σ). For (h, p) ∈ G × M we have Rg(Σ(h,p)) = Σ(hg−1,p).

Proof. Since Rg preserves the distribution D, R˜g maps leaves diﬀeomorphically onto ˜ −1 ˜ ˜ leaves. We have Rg(Σ(h,p)) = Σ(hg−1,p) because (hg , p) = Rg(h, p) ∈ Rg(Σ(h,p)).

Remark 4.3.33. If there exists an action ϕ : G × M → M with infinitesimal action λ, ˜ then Rg(Σ(e,p)) = Σ(g−1,p) = Σ(e,ϕ˜(g,p)) for any p ∈ M: Let ψ = (id × ϕ) ◦ (diag × id) : G × M → G × M. The map ψ is a flat chart (cf. Lemma ˜ ˜ ˜ ˜ 4.2.2) and ψ(G × {p}) = Σ(e,p). We have (e, ϕ˜(g, p)) = Rg(g, ϕ˜(g, p) = Rg(ψ(g, p)) ∈ ˜ ˜ ˜ Rg(ψ(G × {p})) and thus Rg(Σ(e,p)) = Σ(g−1,p) = Σ(e,ϕ˜(g,p)). Proposition 4.3.34. The infinitesimal action λ : g → Vec(M) is univalent if and only if every leaf of the form Σ(e,p) for p ∈ M is univalent.

55 Proof. If λ is univalent, then all leaves are univalent, in particular each leaf of the form Σ(e,p). Assume now that all leaves Σ(e,p) are univalent. Let Σ be an arbitrary leaf and ˜ ˜ let (g, p) ∈ Σ. We have Σ(e,p) = Rg(Σ(g,p)) = Rg(Σ). If Ω ⊂ Σ = Σ(g,p) is a relatively ˜ compact subset, then Rg(Ω) ⊂ Σ(e,p) is relatively compact and the univalence of Σ(e,p) yields the existence of a ﬂat chart ψ : U × V → G × M with R˜g(Ω) ⊂ ψ˜(U × V ). By Lemma 4.3.31 the map

Rg−1 ◦ ψ ◦ Rg : Ug × V → G × M is a ﬂat chart and ˜ ˜ ˜ ˜ ˜ ˜ ˜ Ω = Rg−1 (Rg(Ω)) ⊂ Rg−1 (ψ(U × V )) = (Rg−1 ◦ ψ ◦ Rg)(Ug × V ).

4.3.3 Characterization of globalizable infinitesimal actions We now study conditions for the existence of globalizations. The main result is the following: Theorem 4.3.35. Let λ : g → Vec(M) be an infinitesimal action. Then the following statements are equivalent: (i) The infinitesimal action λ is globalizable.

(ii) The restricted infinitesimal action λ0 = λ|g0¯ is globalizable. (iii) The infinitesimal action λ is univalent. ˜ (iv) The reduced infinitesimal action λ is univalent, i.e. πG|Σ is injective for each leaf Σ, and all leaves Σ ⊂ G × M are holonomy free, i.e. the morphism ΦΣ is trivial. (v) There exists a local action ϕ : W → M with induced infinitesimal action λ whose domain of definition is maximally balanced. The equivalence of (iii) and (iv) is the content of Proposition 4.3.21 and once the equivalence of (i) and (iii) is established the equivalence of (i) and (ii) is a consequence of Corollary 4.3.22. The other equivalences are proven in the following: The implication (iii) ⇒ (i) is the content of Proposition 4.3.44, (i) ⇒ (v) follows from Proposition 4.3.47, and (v) ⇒ (iii) is proven in Proposition 4.3.48. Remark 4.3.36 (see [Pal57], Chapter III, Theorem IV and Theorem V). In the classical case, Palais shows that for a univalent infinitesimal action the space of leaves Σ ⊂ G × M of the distribution D, which is denoted by M ∗ = (G × M)/∼, carries in a natural way the structure of a possibly non-Hausdorff manifold, the action of G on G × M by g · (h, p) = (hg−1, p) induces an action on the quotient space M ∗ and ∗ M → M , p 7→ Σ(e,p) is an injective embedding. A similar construction is also important in the case of supermanifolds. Definition 4.3.37. Let λ be an infinitesimal action of G on M and D the associated distribution on G × M. Define

M ∗ = (G × M)/∼

56 ∗ to be the space of leaves Σ ⊂ G × M and denote byπ ˜ : G × M → M ,π ˜(g, p) = Σ(g,p), ∗ the projection. Now endow M with the quotient topology and deﬁne the sheaf OM∗ ∗ of Z2-graded algebras on M by setting

D OM∗ =π ˜∗(OG×M),

D −1 ∗ D i.e. OM∗ (Ω) = OG×M(˜π (Ω)) for any open subset Ω ⊂ M , where OG×M denotes again the sheaf of D-invariant functions on G × M. Deﬁne the ringed space

∗ ∗ M = (M , OM∗ )

∗ ∗ ∗ and let π = (π , π˜): G × M → M , where π : OM∗ → π˜∗(OG×M) is given by the D −1 −1 canonical inclusion OM∗ (Ω) = OG×M(˜π (Ω)) ,→ OG×M(˜π (Ω)) for any open subset Ω ⊆ M ∗.

Deﬁnition 4.3.38. For any open subset V ⊆ M, V = (V, OM|V ), and g ∈ G we deﬁne a morphism of ringed spaces

∗ ιV,g : V → M by ιV,g = π ◦ ιg, where ιg : V ,→ {g} × V ⊂ G × M denotes again the inclusion. The reduced map of ιV,g is given by p 7→ Σ(g,p). For g = e and V = M, let

∗ ιM = ιe : M → M , ιM = π ◦ ιe.

Remark 4.3.39. Let ψ : U × V → G × M be a ﬂat chart with respect to (D, U, V, g, id) and let πM = πM|U×V : U × V → V denote the projection onto the second component. As ιg : V → U × V is a section of πM and ψ ◦ ιg = ιg, the diagram

ψ U × V / G × M

πM π V / M∗ ιV,g is commutative.

∗ We will see later on that M carries the structure of a supermanifold and ιM : M → M∗ is an open embedding if and only if the inﬁnitesimal action λ is globalizable. In this case M∗ is itself a globalization of λ and the G-action on M∗ is induced by the G-action R on G × M.

Remark 4.3.40. The topological space M ∗ only depends on the underlying inﬁnites- ˜ imal action λ : g0¯ → Vec(M) since De = Dλ˜ (cf. Lemma 4.3.7). Hence, the mapπ ˜ is an open map as in the classical case (cf. [Pal57], Chapter I, Theorem III, or [HI97], § 2, Proposition 2). The topological space M ∗ fulﬁlls the second axiom of countability becauseπ ˜ is an open quotient map and G × M a manifold.

∗ ∗ Lemma 4.3.41. The space M = (M , OM∗ ) is a locally ringed space.

57 D −1 ∗ Proof. For f ∈ OM∗ (Ω) = OG×M(˜π (Ω)) with Σ ∈ Ω ⊆ M denote by [f]Σ the germ of f in the stalk (OM∗ )Σ. We can deﬁne the ideal ˜ mΣ = {[f]Σ| f(Σ) = 0} C (OM∗ )Σ ∗ for any leaf Σ ∈ M . Assume that mΣ is not a maximal ideal in the stalk (OM∗ )Σ. Then there is a proper ideal IC(OM∗ )Σ which is not contained in mΣ. Let [f]Σ ∈ I\mΣ. Then we have f˜(Σ) 6= 0 and the continuity of f˜ gives the existence of an open neighbourhood Ω of Σ ∈ M ∗ such that f is deﬁned onπ ˜−1(Ω), i.e.

D −1 −1 f ∈ OM∗ (Ω) = OG×M(˜π (Ω)) ⊂ OG×M(˜π (Ω)), ˜ −1 −1 and f(x) 6= 0 for all x ∈ π˜ (Ω). Consequently, there exists g ∈ OG×M(˜π (Ω)) with gf = fg = 1. The function g is also D-invariant since

0 = Y (1) = Y (gf) = Y (g)f + (−1)|g||Y |gY (f) = Y (g)f + 0 = Y (g)f for all vector ﬁelds Y belonging to D and thus Y (g) = 0. Therefore, g deﬁnes an element in OM∗ (Ω) and [g]Σ[f]Σ = [gf]Σ = 1 ∈ I which contradicts the assumption I 6= (OM∗ )Σ. Lemma 4.3.42. The action of G on G × M from the right R : G × (G × M) → G × M induces an action χ of G on the space M∗, i.e. a morphism of ringed spaces satisfying the usual action properties, such that

R G × (G × M) / G × M

id×π π

∗ ∗ G × M χ / M commutes. In particular, if M∗ is a supermanifold, then χ is an action of the Lie supergroup G on M∗. Proof. The underlying actionχ ˜ of G on M ∗ is given by the formula ∗ ∗ ˜ χ˜ : G × M → M , (g, Σ(h,p)) 7→ Rg(Σ(h,p)) = Σ(hg−1,p) and continuous sinceπ ˜ ◦ R˜ =χ ˜ ◦ (idG × π˜). If f is any D-invariant function on G × M, then R∗(f) is (1 ⊗ D)-invariant (see Corollary 4.3.28). Therefore, ∗ 1⊗D ˜ −1 ∼ ˆ D ˜ −1 (π ◦ R) (f) ∈ (OG×G×M)((˜π ◦ R) (Ω)) = (OG⊗OG×M)((˜π ◦ R) (Ω))

D −1 ∗ ∗ for any f ∈ OM∗ (Ω) = OG×M(˜π (Ω)), Ω ⊆ M , and (π ◦ R) induces a morphism ∗ ∗ ∗ ∗ ∗ χ : OM∗ → χ˜∗(OG×M∗ ) with R ◦ π = (id × π) ◦ χ . The map χ deﬁnes an action of G on M∗ since R is an action and χ inherits the respective properties.

The inﬁnitesimal action

∗ ∗ ∗ λχ : g → Vec(M ), λχ(X) = (X(e) ⊗ id ) ◦ χ on the ringed space M∗ extends the inﬁnitesimal action λ on M in the following sense:

58 Lemma 4.3.43. For any X ∈ g we have

∗ ∗ λ(X) ◦ ιM = ιM ◦ λχ(X).

∗ ∗ −1 Proof. Let Ω ⊆ M be open and f ∈ OM∗ (Ω). Then π (f) is D-invariant onπ ˜ (Ω) ⊂ G × M and thus (id∗ ⊗ λ(X))(π∗(f)) = −(X ⊗ id∗)(π∗(f)) for X ∈ g. Consequently, we have

∗ ∗ ∗ ∗ ∗ ∗ (λ(X) ◦ ιM )(f) = (λ(X) ◦ ιe )(π (f)) = ιe ((id ⊗ λ(X))(π (f))) = −((X(e) ⊗ id∗)(π∗(f)).

A calculation using the identities π ◦ R = χ ◦ (id × π) and R ◦ (id × ιe) = (ι × id) gives ∗ ∗ ∗ ∗ ∗ ιM ◦ λχ(X) = ((−X(e)) ⊗ id ) ◦ π so that λ(X) ◦ ιM = ιM ◦ λχ(X). Proposition 4.3.44. Let λ be univalent. Then M∗ is a supermanifold, with a possibly non-Hausdorff underlying manifold M ∗. ∗ Moreover, the morphism ιM : M → M is an open embedding, the infinitesimal ac- ∼ ∗ tion λχ induced by χ extends the infinitesimal action λ on M = (˜ιM(M), OM |˜ιM(M)), ∗ and we have G · ˜ιM(M) =χ ˜(G × ˜ιM(M)) = M . Thus M∗ is a globalization of λ and the infinitesimal action λ is globalizable. The proof of the proposition makes use of the next lemma. Lemma 4.3.45. Let λ be univalent and W ⊆ G × M an open connected subset. For D any f ∈ OG×M(W ) there exists a unique extension ˆ D −1 ˆ f ∈ OG×M(˜π (˜π(W ))) with f|W = f. Proof. Note that the open set [ π˜−1(˜π(W )) = Σ Σ leaf Σ∩W 6=∅ is again connected and by Lemma 4.3.25 a D-invariant extension fˆ of f is unique. Let Σ be a leaf with Σ∩W 6= ∅, i.e. Σ ∈ π˜(W ), and let Σ0 ⊂ Σ be relatively compact. Since λ is univalent, there exists a flat chart ψ : U × V → G × M with Σ0 ⊂ ψ˜(U × V ) ˜ 0 0 ∗ and ψ(U × V ) ⊂ W for some open subset U ⊆ U. The function (ψ|ψ˜−1(W )) (f) is ∗ 0 ˜−1 DG-invariant and thus of the form (ψ|ψ˜−1(W )) (f) = 1 ⊗ fM on U × V ⊆ ψ (W ) for ˆ some fM ∈ OM(V ). Now, 1 ⊗ fM is already defined on U × V and we define f on ψ˜(U × V ) by ˆ −1 ∗ f|ψ˜(U×V ) = (ψ ) (1 ⊗ fM). This yields a well-defined function fˆ onπ ˜−1(˜π(W )) due to the uniqueness of flat charts following from the univalence of λ (see Proposition 4.3.14). Moreover, fˆ is D-invariant ˆ by construction and f|W = f. Proof of Proposition 4.3.44. We prove that in the case of a univalent infinitesimal ∗ ∗ action the morphism ιV,g : V → M defines a chart for M if there is a flat chart ψ : U × V → G × M with respect to (D, U, V, g, id). Due to the local existence of flat charts this implies that M∗ is a supermanifold.

59 By Proposition 4.3.21, the restrictionπ ˜G|Σ :Σ → G of the projectionπ ˜G : G×M → G is injective for all leaves Σ ⊂ G × M. Therefore, we have ˜ιV,g(p) = Σ(g,p) = Σ(g,q) = 0 ˜ιV,g(q) if and only if p = q. The map ˜ιV,g is open because for any open subset V ⊆ V 0 0 ˜ ∗ the set ˜ιV,g(V ) = ˜ιV,g(˜πM(U × V )) =π ˜(ψ(U × V )) is open in M using that ψ is a local diﬀeomorphism andπ ˜ an open map. Consequently, ˜ιV,g is a homeomorphism onto its image. ∗ ∗ To show that ιV,g is injective, let Ω ⊆ M be open and f1, f2 ∈ OM∗ (Ω). If ∗ ∗ ιV,g (f1) = ιV,g (f2), then also

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ψ (π (f1)) = πM (ιV,g (f1)) = πM (ιV,g (f2)) = ψ (π (f2)).

∗ D ∗ Since π : OM∗ =π ˜∗(OG×M) ,→ π˜∗(OG×M) is the canonical inclusion, π (fi) and ∗ ∗ −1 −1 fi can be identiﬁed. If ψ (f1) = ψ (f2), then f1 = f2 on ψ˜(ψ˜ (˜π (Ω))) and thus f1 = f2 by Lemma 4.3.25. −1 For any fV ∈ OM|V (˜ιV,g(Ω)) we have

∗ DG −1 DG ˜−1 −1 1 ⊗ fV = πM (fV ) ∈ OG×M(U × ˜ιV,g(Ω)) = OG×M(ψ (˜π (Ω))).

−1 ∗ ˜ −1 ˜ ˜−1 −1 Thus (ψ ) (1⊗fV ) is a D-invariant function on ψ(U ×V )∩π˜ (Ω) = ψ(ψ (˜π (Ω))). ˆ D −1 By Lemma 4.3.45 there is a D-invariant extension f ∈ OG×M(˜π (Ω)) = OM∗ (Ω) of −1 ∗ ∗ ˆ ∗ ∗ ˆ ∗ −1 ∗ (ψ ) (1⊗fV ) and we have ιV,g (f) = fV since ψ (π (f)) = ψ ((ψ ) (1⊗fV )) = 1⊗fV . ∗ ∗ Consequently, ιV,g is also surjective and thus ιV,g is a chart for M . The morphism ιM = ιe is an open embedding since the univalence of λ implies that ˜ιM is injective with the same argument as for ˜ιV,g and locally ιM is of the form ιV,e such that there exists a flat chart ψ with respect to (D, U, V, e, id). By Lemma 4.3.43, the infinitesimal action λχ induced by χ extends the infinitesimal ∼ ∗ action of λ on M = (˜ιM(M), OM |˜ιM(M)). Since ˜ g · Σ(e,p) =χ ˜(g, Σ(e,p)) = Rg(Σ(e,p)) = Σ(g−1,p) for any p ∈ M and ˜ιM(M) consists exactly of those leaves which intersect {e} × M, we ∗ have G · ˜ιM(M) =χ ˜(G × ˜ιM(M)) = M .

Lemma 4.3.46. Let ϕ : W → M be a local action with induced infinitesimal action λ ˜ and maximally balanced domain of definition. Then we have ψ(Wp × {p}) = Σ(e,p) for any p ∈ M and ψ = (id × ϕ) ◦ (diag × id) : W → G × M, which is locally a flat chart.

Proof. The lemma follows from the analogous classical result (see [Pal57], Chapter II, Theorem VI) since ϕ : W → M has a maximally balanced domain of deﬁnition if and only if the domain of deﬁnition W of the reduced local action is maximally balanced.

Proposition 4.3.47. Let ϕ0 : G × M0 → M0 be a globalization of λ. Then there is a local action ϕ : W → M with maximally balanced domain of definition and infinitesimal action λ. Moreover, ϕ : W → M is the unique maximal local action with infinitesimal action λ. Any two local actions ϕi : Wi → M, i = 1, 2, with infinitesimal action λ coincide on their common domain of definition and define a local action χ : W1 ∪W2 → M with

χ|W1 = ϕ1 and χ|W2 = ϕ2.

60 0 −1 Proof. The set (ϕ ˜ ) (M) ∩ (G × M) is open in G × M and contains {e} × M. Let Wp be the connected component of e in {g ∈ G| ϕ˜0(g, p) ∈ M} for p ∈ M and define [ W = Wp × {p}. p∈M The set W ⊆ G × M is open and the largest domain of definition of a local action 0 −1 included in (ϕ ˜ ) (M)∩(G×M) (cf. [Pal57], Chapter II, Theorem I). Let W = (W, OW ) 0 and define ϕ = ϕ |W : W → M. The map ϕ is a local action of G on M. A direct calculation shows that the domain of definition W of ϕ is maximally balanced. By the ˜ preceding lemma we have ψ(Wp × {p}) = Σ(e,p) for any p ∈ M, where ψ = (id × ϕ) ◦ (diag × id) : W → G × M. Let χ : Wχ → M be any local action with induced infinitesimal action λ and set ψχ = (id × χ) ◦ (diag × id) : Wχ → G × M. For any p ∈ M, ψ˜χ(Wχ,p × {p}) is contained ˜ in the leaf Σ(e,p). Since Σ(e,p) = ψ(Wp × {p}), we get ˜ ˜ Wχ,p = πG(ψχ(Wχ,p × {p})) ⊆ πG(Σ(e,p)) = πG(ψ(Wp × {p})) = Wp, which implies Wχ ⊆ W . The uniqueness of local actions with a given domain of definition (see Corollary 4.2.3) yields ϕ = χ on Wχ. Proposition 4.3.48. Let ϕ : W → M be a local action with maximally balanced domain of definition. Then its induced infinitesimal action is univalent. Proof. Let ψ = (id × ϕ) ◦ (diag × id) : W → G × M be the locally flat chart associated ˜ with the local action ϕ. By Lemma 4.3.46 we have ψ(Wp ×{p}) = Σ(e,p) for any p ∈ M. Let Ω ⊂ Σ(e,p) be a relatively compact connected subset. By Lemma 4.2.2 there are subsets U ⊂ G and V ⊂ M, p ∈ V , such that ψ|U×V is a flat chart and Ω ⊂ ˜ ψ(U × V ). Consequently, Σ(e,p) is univalent for any p ∈ M and hence λ is univalent by Proposition 4.3.34

4.4 Examples

In this section, a family of examples of inﬁnitesimal actions on the supermanifold 0|2 M = (C \{0}) × C is provided. In all cases the underlying action is globalizable, but the inﬁnitesimal action on the supermanifold is in general not globalizable.

0|2 Example 4.4.1. Let M = (C \{0}) × C , with coordinates z, θ1, θ2, and let α : C \{0} → C be a holomorphic function. Consider the even holomorphic vector field ∂ X = (1 + α(z)θ θ ) α 1 2 ∂z on M. We now examine for which α the infinitesimal C-action λα on M, generated by Xα, is globalizable. First, note that for any α the leaves Σ ⊂ C × M of the distribution Dα on C × M are of the form Σ = Σ(t,z) = {(t + s, z + s)| s ∈ C \ {−z}} ∂ for some (t, z) ∈ C × (C \{0}), where Dα is spanned by the vector field ∂t + Xα if t denotes the coordinate on C. Each leaf Σ is therefore biholomorphic to C \{0}.

61 ˜ ∂ The reduced vector ﬁeld Xα = ∂z always generates a globalizable inﬁnitesimal ac- ∗ tion and a globalization of M = C\{0} is M = C with the usual addition as C-action. If λ is globalizable, then the globalization M∗ = (M ∗, π˜ ODα ) is a complex super- α α ∗ C×M manifold of dimension (1|2). Every complex supermanifold N with underlying manifold C is split since C is Stein (see [Oni98], Theorem 3.4). Moreover, N is isomorphic to 1|n C for some n ∈ N since all holomorphic vector bundles on C are trivial. This implies ∗ 1|2 that Mα is isomorphic to C if it is a supermanifold. Let f ∈ OC×M(C×M). There are holomorphic functions f0, f1, f2, f12 on C×M = C × (C \{0}) with f = f0 + f1θ1 + f2θ2 + f12θ1θ2. Then

∂ ∂ ∂ ∂ ∂ ∂ + X (f) = + (f + f θ + f θ ) + (α f + + f )θ θ . ∂t α ∂t ∂z 0 1 2 2 2 ∂z 0 ∂t ∂z 12 1 2

∂ Therefore, if f is Dα-invariant, i.e. ( ∂t + Xα)(f) = 0, then the functions f0, f1 and f2 are constant along the leaves Σ(t,z) = {(t + s, z + s)| s ∈ C \ {−z}} and thus there exist holomorphic functions gi : C → C, i = 1, 2, 3, with

fi(t, z) = gi(z − t).

∂ ∂ ∂ Moreover, the Dα-invariance of f implies (α ∂z f0 + ∂t + ∂z f12) = 0 and consequently f12 is locally of the form

∂ f (t, z) = −A(z) f (t, z) + g (z − t), 12 ∂z 0 12

0 where A is a (local) primitive of α, i.e. A (z) = α(z), and g12 a holomorphic function on (some open subset of) C. There are two diﬀerent cases:

(i) If α has a global primitive A : C \{0} → C we have

∗ Dα O ∗ (M ) = O ( × M) Mα C×M C = g0(z − t) + g1(z − t)θ1 + g2(z − t)θ2 ∂ + g12(z − t) − A(z) g0(z − t) θ1θ2 g0, g1, g2, g12 holomorphic ∂z

∼ O 1|2 ( ). = C C

It turns out that the inﬁnitesimal action λα is globalizable in this case with glob- ∗ ∼ 1|2 1|2 alization Mα = C and C acts on C by the usual addition on C extended to 1|2 1|2 1|2 C , i.e. the action χ : C × C → C is given by

∗ χ (z, θ1, θ2) = (z + t, θ1, θ2).

∗ Moreover, the open embedding ιM : M → Mα can be realized as

0|2 1|2 ∗ ιM : C \{0} × C ,→ C , ιM (z, θ1, θ2) = (z − A(z)θ1θ2, θ1, θ2).

62 (ii) In the case where α does not have a global primitive A on C \{0}, we get

∗ Dα O ∗ (M ) = O ( × M) Mα C×M C

= λ + g1(z − t)θ1 + g2(z − t)θ2 + g12(z − t)θ1θ2

λ ∈ C, g1, g2, g12 holomorphic ,

∗ which is not isomorphic to O 1|2 ( ). Therefore, the ringed space M is not a C C α supermanifold and λα cannot be globalizable.

A calculation shows that for arbitrary α a ﬂat chart ψ with respect to (Dα, U, V, t0, ρ) is given by the pullback ∗ ψ (t, z, θ1, θ2) ∗ ∗ = t, ρ (z, θ1, θ2) + 0, (t − t0) + A(˜ρ(z) + (t − t0)) − A(˜ρ(z)) ρ (θ1θ2), 0, 0

∗ ∗ ∗ ∗ = t, ρ (z) + (t − t0) + A(˜ρ(z) + (t − t0)) − A(˜ρ(z)) ρ (θ1θ2), ρ (θ1), ρ (θ2) , where U ⊆ C and V ⊆ C\{0} need to be open subsets such that there exists a primitive A of α on U +ρ ˜(V ) = {t + z| t ∈ U, z ∈ ρ˜(V )} ⊂ C \{0}. Let Σ ⊂ C × M be a leaf and (s, z) ∈ Σ. We shall now explicitly describe the morphism 0|2 Φ = ΦΣ,(s,z) : π1(Σ, (s, z)) → Holz(C \{0} × C ). 0 First remark that if ψ is a flat chart with respect to (Dα, U, V, t0, ρ) and t0 ∈ U, then 0 0 ψ is also a flat chart with respect to (Dα, U, V, t0, ρ ) for 0 ∗ (ρ ) (z, θ1, θ2) ∗ 0 0 ∗ = ρ (z, θ1, θ2) + (t0 − t0) + A(˜ρ(z) + (t0 − t0)) − A(˜ρ(z)) ρ (θ1θ2), 0, 0 0 0 ! Z t0 Z t0 ∗ ∗ = ρ (z, θ1, θ2) + 1dt + α(˜ρ(z) + (t − t0))dt ρ (θ1θ2), 0, 0 t0 t0 where the integrals do not depend on the path, as long as it is contained in U, since α 0 0 0 0 0 has a primitive on U +ρ ˜(V ). If ψ is another flat chart with respect to (D,U ,V , t0, ρ ) 0 00 0 0 0 0 00 00 with V ∩V 6= ∅ and t0 ∈ U , then ψ is also a flat chart with respect to (D,U ,V , t0, ρ ) for 00 ∗ (ρ ) (z, θ1, θ2) 00 00 ! Z t0 Z t0 0 ∗ 0 0 0 ∗ = (ρ ) (z, θ1, θ2) + 1dt + α(˜ρ (z) + (t − t0))dt (ρ ) (θ1θ2), 0, 0 0 0 t0 t0 0 0 ! Z t0 Z t0 ∗ ∗ = ρ (z, θ1, θ2) + 1dt + α(˜ρ(z) + (t − t0))dt ρ (θ1θ2), 0, 0 t0 t0 00 00 ! Z t0 Z t0 0 0 ∗ + 1dt + α(˜ρ(z) + (t0 − t0) + (t − t0))dt ρ (θ1θ2), 0, 0 0 0 t0 t0 00 00 ! Z t0 Z t0 ∗ ∗ = ρ (z, θ1, θ2) + 1dt + α(˜ρ(z) + (t − t0))dt ρ (θ1θ2), 0, 0 t0 t0

63 0 0 0 ∗ ∗ sinceρ ˜ (z) =ρ ˜(z) + (t0 − t0) and (ρ ) (θi) = ρ (θi), i = 1, 2, where the integrals are taken along appropriate paths. For any closed path γ : [0, 1] → Σ ⊂ C × M, γ(r) = (γ1(r), γ2(r)) ∈ Σ ⊂ C × M, γ(0) = γ(1) = (s, z), we consequently get that the pullback of the local biholomorphism Φ([γ]) = ΦΣ,(s,z)([γ]) is given by Z Z ∗ ∗ Φ([γ]) (z, θ1, θ2) = id (z, θ1, θ2) + 1dt + αs,zdt θ1θ2, 0, 0 γ1 γ1 Z = (z, θ1, θ2) + αs,zdt θ1θ2, 0, 0 γ1 for αs,z(t) = α((z −s)+t). Thus, the morphism Φ is trivial if and only if α has a global primitive on C \{0}. Using Theorem 4.3.35 this shows again that the inﬁnitesimal action λα is globalizable if and only if α has a global primitive. In the special case of α(z) = z−1, we have for example

0|2 ∗ Φ: Z → Holp(C \{0} × C ), Φ(k) (z, θ1, θ2) = (z + 2πikθ1θ2, θ1, θ2) ∼ for any leaf Σ, identifying the fundamental group of Σ = C \{0} with Z.

4.5 Actions of simply-connected Lie supergroups

In this section, a few consequences of the characterization of globalizable infinitesimal actions of a simply-connected Lie supergroup, i.e. a Lie supergroup G whose underlying Lie group G is simply-connected, are given. If G is simply-connected and acts on the classical manifold M, the leaves Σ ⊂ G×M of the distribution associated with the infinitesimal action are all isomorphic to G and thus simply-connected. This yields consequences for the existence of globalizations of infinitesimal actions of simply-connected Lie supergroups since there are no holonomy phenomena, i.e. the morphisms ΦΣ : π1(Σ) → Diffp(M) are all trivial, if the underlying infinitesimal action is global.

Deﬁnition 4.5.1. An inﬁnitesimal action λ : g → Vec(M) is called global if there is a G-action on M which induces λ.

Remark 4.5.2 (cf. [Pal57], Chapter II, Section 4). Let G be a Lie group acting on a manifold M. Then the leaves Σ ⊂ G × M of the distribution Dλ˜ associated with the inﬁnitesimal action λ˜ induced by the G-action are all isomorphic to G.

As a consequence we obtain the following theorem.

Theorem 4.5.3. Let G be a simply-connected Lie supergroup and λ : g → Vec(M) an ˜ inﬁnitesimal action of G such that its reduced action λ : g0¯ → Vec(M) is global. Then the inﬁnitesimal action λ is globalizable, and M is the unique globalization.

Proof. Let D denote again the distribution on G×M associated with λ. By Lemma 4.3.7 we have De = Dλ˜ and by deﬁnition the leaves of D are the leaves of Dλ˜. By the preceding remark all leaves Σ ⊂ G × M are isomorphic to G and consequently simply-connected. ∼ Therefore, the morphisms Φ : π1(Σ, (g, p)) = {1} → Diﬀp(M) are all trivial. Since

64 λ˜ is global and thus in particular globalizable, λ˜ is univalent. This implies that λ is globalizable using the equivalent characterizations of globalizability formulated in 0 0 Theorem 4.3.35. Let M be a globalization of λ and ιM : M → M the open embedding. Then M 0 is a globalization of λ˜ and because M 0 is the unique globalization of λ˜ (cf. 0 [Pal57], Chapter III, Theorem XII,(4)) we have M = M and ˜ιM = idM . Since ιM is 0 an open embedding, this implies ιM = idM and M = M. If the assumption on the simply-connectedness of G is dropped in the above theorem, there exist counterexamples to the statement, see e.g. Example 4.3.20. Also, as illustrated in Example 4.4.1, it is not enough for λ˜ to be globalizable. We really need that λ˜ is global. Definition 4.5.4. Let λ : g → Vec(M) be an infinitesimal action. The set of points p ∈ M such that there exists an even vector field X ∈ g0¯ with λ(X)(p) 6= 0 is called the support of λ. Remark 4.5.5. The definition of the support of an infinitesimal action λ implies that the support of λ coincides with the support of the underlying infinitesimal action λ˜. In the classical case, we have the following two theorems on actions of simply- connected Lie groups. Theorem 4.5.6 (see [Pal57], Chapter III, Theorem XVIII). Let G denote a simply- ˜ connected Lie group with Lie algebra g0¯ and let λ : g0¯ → Vec(M) be an infinitesimal action of G on a manifold M. If the support of λ˜ is relatively compact in M, then λ˜ is global. In particular, any infinitesimal action of a simply-connected Lie group G on a compact manifold M is global. ˜ Theorem 4.5.7 (see [Pal57], Chapter IV, Theorem III). Let λ : g0¯ → Vec(M) be an infinitesimal action of a simply-connected Lie group G on M. Suppose there exists a set of generators {Xi}i∈I , Xi ∈ g0¯, of the Lie algebra g0¯ such that the flow of each vector field λ˜(Xi) is global. Then the infinitesimal action λ˜ is global. Applying Theorem 4.5.3, these results in the classical case can be directly carried over to the case of infinitesimal actions of simply-connected Lie supergroups on supermanifolds. Corollary 4.5.8. Let G be a simply-connected Lie supergroup and λ : g → Vec(M) an infinitesimal action whose support is relatively compact in M. Then the infinitesimal action λ is global. In particular, any infinitesimal action of a simply-connected Lie supergroup on a supermanifold with compact underlying manifold is global. Corollary 4.5.9. Let λ : g → Vec(M) be an infinitesimal action of a simply-connected Lie supergroup G such that there exists a set of generators {Xi}i∈I , Xi ∈ g0¯, of g0¯ such that each vector field λ˜(Xi) has a global flow. Then the infinitesimal action λ is global. A slightly weaker version of this corollary, in a formulation for DeWitt supermanifolds, has been proven, in a different way, in [Tuy13]. The assumption there is that all even vector fields have global flows, and not only a set of generators.

65 66 Chapter 5

Automorphism groups of compact complex supermanifolds

The automorphism group of a compact complex manifold M carries the structure of a complex Lie group which acts holomorphically on M and whose Lie algebra consists of the holomorphic vector fields on M (see [BM47]). In this chapter, we investigate how this result can be extended to the category of compact complex supermanifolds. Let M be a compact complex supermanifold, i.e. a complex supermanifold whose underlying manifold is compact. An automorphism of M is a biholomorphic morphism M → M. A first candidate for the automorphism group of such a supermanifold is the set of automorphisms, which we denote by Aut0¯(M). However, every automorphism ϕ of a supermanifold M (with structure sheaf OM) is “even” in the sense that its ∗ pullback ϕ : OM → ϕ˜∗(OM) is a parity-preserving morphism. Therefore, we can (at most) expect this set of automorphisms of M to carry the structure of a classical Lie group if we require its action on M to be smooth or holomorphic. This way we do not obtain a Lie supergroup of positive odd dimension. We prove that the group Aut0¯(M), endowed with an analogue of the compact-open topology, carries the structure of a complex Lie group such that the action on M is holomorphic and its Lie algebra is the Lie algebra of even holomorphic vector fields on M. It should be noted that the group Aut0¯(M) is in general different from the group Aut(M) of automorphisms of the underlying manifold M. There is a group homomorphism Aut0¯(M) → Aut(M) given by assigning the underlying map to an automorphism of the supermanifold; this group homomorphism is in general neither injective nor surjective. We will define the automorphism group of a compact complex supermanifold M to be a complex Lie supergroup which acts holomorphically on M and satisfies a universal property. In analogy to the classical case, its Lie superalgebra is the Lie superalgebra of holomorphic vector fields on M, and the underlying Lie group is Aut0¯(M), the group of automorphisms of M. Using the equivalence of complex Harish-Chandra pairs and complex Lie supergroups (see [Vis11]), we construct the appropriate automorphism Lie supergroup of M. Taking a different approach to supermanifolds via functors of points, an alternative definition of the automorphism group as a functor in analogy to [SW11] could be made. In Section 5.4 it is proven that the constructed automorphism group Aut(M) represents this functor.

67 In the classical case, another class of complex manifolds where the automorphism m group carries the structure of a Lie group is given by the bounded domains in C (see [Car79]). An analogue statement is false in the case of supermanifolds as shown by an example in Section 5.5. The results of this chapter, except those formulated in Section 5.4 related to the approach via the functor of points, are contained in [BK15].

5.1 The group of automorphisms Aut0¯(M)

Let M be a compact complex supermanifold. An automorphism of M is a biholomorphic morphism ϕ : M → M. Denote by Aut0¯(M) the set of automorphisms of M. The map Aut0¯(M) → Aut(M) which assigns to an automorphism ϕ : M → M the automorphismϕ ˜ : M → M of the underlying complex manifold is a group homomorphism. Since M is compact, Aut(M) carries the structure of a complex Lie group (see [BM47]). Note that this group homomorphism Aut0¯(M) → Aut(M) is in general neither surjective nor injective as illustrated in the next two examples.

0|1 Example 5.1.1. Let M = C . Then the underlying manifold M is simply a point 0|1 and we have Aut(M) = Aut({0}) = {idM }. Let ξ denote the coordinate on M = C . ∗ ∗ For each automorphism ϕ of M there exists c ∈ C such that ϕ (ξ) = c · ξ. Thus we ∼ ∗ have Aut0¯(M) = C and the homomorphism Aut0¯(M) → Aut(M) corresponds to the ∗ unique map C → {idM }. Example 5.1.2. If M is a split complex supermanifold associated with a holomorphic vector bundle E → M, then any automorphism ϕ : M → M induces an automorphism E → E of the vector bundle E over the underlying automorphismϕ ˜ : M → M. In particular, if the homomorphism Aut0¯(M) → Aut(M), ϕ 7→ ϕ˜, is surjective, each automorphism of M necessarily lifts to an automorphism of the bundle E (over the given automorphism of M). Therefore, in order to give an example of a complex supermanifold M such that Aut0¯(M) → Aut(M) is not surjective, it suffices to find a compact complex manifold M with holomorphic vector bundle E with the property that not all automorphisms of M lift to automorphisms of the vector bundle E. If the underlying manifold M is allowed to be disconnected, we could take M to be the disjoint union of two copies of the same compact complex manifold, i.e M = N∪˙ N for a compact complex manifold N. Then we can for example define a line bundle L on M by taking two non-isomorphic line bundles on the two copies of N. In this case the automorphism of M which interchanges the two copies of N does not lift to an automorphism of the line bundle L. For an example where M is connected, consider a connected compact complex manifold N, and let Lj → N, j = 1, 2, be two non-isomorphic line bundles. Define M to be the product manifold M = N × N, and let let E = L1 × L2 the product vector bundle of rank 2. Let p ∈ N. The restriction of the vector bundle E to the submanifold N × {p} is isomorphic to the Whitney sum of L1 and the trivial line ∼ ∼ bundle L = N × C, i.e. E|N×{p} = L1 ⊕ L. Similarly we have E|{p}×N = L ⊕ L2. The automorphism χ : M = N × N → M = N × N, χ(p1, p2) = (p2, p1) for pj ∈ N, which interchanges the two factors of N does not lift to an automorphism of the bundle E

68 since χ(N × {p}) = {p} × N and the restriction of E to N × {p} is not isomorphic to the restriction of E to {p} × N.

5.1.1 The topology on Aut0¯(M)

In this paragraph, a topology on Aut0¯(M) is introduced, which generalizes the compact- open topology and topology of compact convergence of the classical case. We prove that Aut0¯(M) is a locally compact topological group with respect to this topology. Let K ⊆ M be a compact subset such that there are local odd coordinates θ1, . . . , θn for M on an open neighbourhood of K. Moreover, let U ⊆ M be open and f ∈ OM(U), n and let Uν be open subsets of C for ν ∈ (Z2) . Let ϕ : M → M be an automorphism withϕ ˜(K) ⊆ U. Then there are holomorphic functions ϕf,ν on a neighbourhood of K such that ∗ X ν ϕ (f) = ϕf,νθ . n ν∈(Z2) Let ∆(K, U, f, θj,Uν) = {ϕ ∈ Aut0¯(M)| ϕ˜(K) ⊆ U, ϕf,ν(K) ⊆ Uν}, and endow Aut0¯(M) with the topology generated by sets of this form, i.e. the sets of the form ∆(K, U, f, θj,Uν) form a subbase of the topology. Remark 5.1.3. In particular, the subsets of the form

∆(K,U) = {ϕ ∈ Aut0¯(M)| ϕ˜(K) ⊆ U} are open for K ⊆ M compact and U ⊆ M open. Hence the map Aut0¯(M) → Aut(M), associating to an automorphism ϕ of M the underlying automorphismϕ ˜ of M, is continuous.

Remark 5.1.4. The group Aut0¯(M) endowed with the above topology is a second- countable Hausdorﬀ space since M is second-countable.

Let U ⊆ M be open. Then we can define a topology on OM(U) as follows: If K ⊆ U is compact such that there exist odd coordinates θ1, . . . , θn on a neighbourhood of K, P ν write f ∈ OM(U) on K as f = ν fνθ . Let Uν ⊆ C be open subsets. Then define a topology on OM(U) by requiring that the sets of the form {f ∈ OM(U)| fν(K) ⊆ Uν} are a subbase of the topology. A sequence of functions fk converges to f if and only P ν if in all local coordinate domains with odd coordinates θ1, . . . , θn and fk = ν fk,νθ , P ν f = ν fνθ , the coefficient functions fk,ν converge uniformly to fν on compact subsets. Note that for any open subsets U1,U2 ⊆ M with U1 ⊂ U2 the restriction map

OM(U2) → OM(U1), f 7→ f|U1 , is continuous. Using Taylor expansion (in local coordinates) of automorphisms of M we can deduce the following lemma:

Lemma 5.1.5. A sequence of automorphisms ϕk : M → M converges to an automorphism ϕ : M → M with respect to the topology of Aut0¯(M) if and only if the following condition is satisﬁed: For all U, V ⊆ M open subsets such that V contains the closure of ϕ˜(U), there is an N ∈ N such that ϕ˜k(U) ⊆ V for all k ≥ N. Furthermore, for any ∗ ∗ f ∈ OM(V ) the sequence (ϕk) (f) converges to ϕ (f) on U in the topology of OM(U).

69 Lemma 5.1.6. If U, V ⊆ M are open subsets, K ⊆ M is compact with V ⊆ K, then the map ∗ ∆(K,U) × OM(U) → OM(V ), (ϕ, f) 7→ ϕ (f), is continuous.

Proof. Let ϕk ∈ ∆(K,U) be a sequence of automorphisms of M converging to ϕ ∈ ∆(K,U), and fl ∈ OM(U) a sequence converging to f ∈ OM(U). Choosing appropriate ∗ local coordinates and using Taylor expansion of the pullbacks (ϕk) (fl), it can be shown ∗ ∗ that (ϕk) (fl) converges to ϕ (f) as k, l → ∞. For this, it is used that the derivatives of a sequence of uniformly converging holomorphic functions also uniformly converge.

Lemma 5.1.7. The topological space Aut0¯(M) is locally compact.

Proof. Let ψ ∈ Aut0¯(M). For each ﬁxed x ∈ M there are open neighbourhoods Vx and Ux of x and ψ˜(x) respectively such that ψ˜(Kx) ⊆ Ux for Kx := V x. We may additionally assume that there are local odd coordinates ξ1, . . . , ξn for M on Ux, and θ1, . . . , θn local odd coordinates on an open neighbourhood of Kx. For any automorphism ϕ : M → M withϕ ˜(Kx) ⊆ Ux, let ϕj,k, ϕj,ν (for ||ν|| = ||(ν1, . . . , νn)|| = ν1 + ... + νn ≥ 3, the sum again understood as a sum in Z) be local holomorphic functions such that n ∗ X X ν ϕ (ξj) = ϕj,kθk + ϕj,νθ . k=1 ||ν||≥3

Choose bounded open subsets Uj,k,Uj,ν ⊂ C, such that ψj,k(x) ∈ Uj,k and ψj,ν(x) ∈ Uj,ν. Since ψ is an automorphism, we have

det ((ψj,k(y))1≤j,k≤n) 6= 0 for all y ∈ Kx. For later considerations shrink Uj,k such that det(C) 6= 0 for all C = (cj,k)1≤j,k≤n with cj,k ∈ Uj,k. After shrinking Vx we may assume ψj,k(Kx) ⊆ Uj,k and ψj,ν(Kx) ⊆ Uj,ν. Hence ψ is contained in the set Θ(x) = {ϕ ∈ Aut0¯(M) | ϕ˜(Kx) ⊆ U x, ϕj,k(Kx) ⊆ U j,k, ϕj,ν(Kx) ⊆ U j,ν}, which contains an open neighbourhood of ψ.

Since M is compact, M is covered by ﬁnitely many of the sets Vx, say Vx1 ,...,Vxl . Then ψ is contained in Θ = Θ(x1) ∩ ... ∩ Θ(xl). We will now prove that Θ is sequentially compact:

Let ϕk be any sequence of automorphisms contained in Θ. Then, using Montel’s theorem and passing to a subsequence, the sequence ϕk converges to a morphism ϕ : M → M. It remains to show that ϕ is an automorphism of M. The underlying mapϕ ˜ : M → M is surjective since if p∈ / ϕ˜(M), then ϕ ∈ ∆(M,M \ {p}) and therefore ϕk ∈ ∆(M,M \{p}) for k large enough which contradicts the assumption that ϕk is an automorphism. This also implies that there is an x ∈ M such that the diﬀerential Dϕ˜(x) is invertible. Using Hurwitz’s theorem (see e.g. [Nar71], p. 80) it follows det(Dϕ˜(x)) 6= 0 for all x ∈ M. Thusϕ ˜ is locally biholomorphic. Moreover, ϕ is locally invertible due to the special form of the sets Θ(xi).

In order to check thatϕ ˜ is injective, let p1, p2 ∈ M, p1 6= p2, such that q =ϕ ˜(p1) = ϕ˜(p2). Let Ωj, j = 1, 2, be open neighbourhoods of pj with Ω1 ∩ Ω2 = ∅. By [Nar71], p. 79, Proposition 5, there exists k0 with the property that q ∈ ϕ˜k(Ω1) and q ∈ ϕ˜k(Ω2) for all k ≥ k0. The fact that the ϕk’s are bijective yields a contradiction to Ω1 ∩Ω2 = ∅.

70 Proposition 5.1.8. The set Aut0¯(M) is a topological group with composition of automorphisms as multiplication and inversion of automorphisms as the inverse.

Proof. Let ϕk and ψl be two sequences of automorphisms of M converging to ϕ and ˜ ˜ ψ respectively. By the classical theory,ϕ ˜k ◦ ψl converges toϕ ˜ ◦ ψ. Let U, V, W ⊆ M ˜ ˜ be open subsets withϕ ˜(V ) ⊆ W ,ϕ ˜k(V ) ⊆ W , ψ(U) ⊆ V , ψl(U) ⊆ V , for k and l ∗ sufficiently large and let f ∈ OM(W ). Then the sequence (ϕk) (f) ∈ OM(V ) converges ∗ ∗ ∗ ∗ to ϕ (f) on V , and by Lemma 5.1.6 (ϕk ◦ ψl) (f) = (ψl) ((ϕk) (f)) converges to ψ∗(ϕ∗(f)) = (ϕ ◦ ψ)∗(f) on U as k, l → ∞ , which shows that the multiplication is continuous. −1 Consider now the inversion map Aut0¯(M) → Aut0¯(M), ϕ 7→ ϕ . Let ϕk be a sequence −1 −1 in Aut0¯(M) converging to ϕ ∈ Aut0¯(M). We will verify that ϕk converges to ϕ by considerations in local coordinates. In a first step, we make a reduction to the case where the underlying maps of ϕk and ϕ are the identity in this local setting. Secondly, we indicate how we can further reduce to the case where the maps induced by ϕk and ϕ on the vector bundle are the identity in this local setting. Since the automorphism group Aut(M) of the underlying manifold M is a topolog- −1 ical group, the inversion map Aut(M) → Aut(M) is continuous andϕ ˜k converges to ϕ˜−1. Let U, V ⊂ M be open subsets such thatϕ ˜(U) ⊂ V and such that there are coordinates z1, . . . , zm, θ1, . . . , θn on U and w1, . . . , wm, ξ1, . . . , ξn on V for M. Let 0 00 00 0 V and V be open relatively compact subsets ofϕ ˜(U) such that V ⊂ V . Sinceϕ ˜k −1 −1 converges toϕ ˜ andϕ ˜k converges toϕ ˜ we can find an open relatively compact subset 0 00 0 0 00 0 0 U ⊂ U such that V ⊂ ϕ˜(U ) ⊂ V ⊂ ϕ˜(U) and also V ⊂ ϕ˜k(U ) ⊂ V ⊂ ϕ˜k(U) ⊂ V for k sufficiently large. −1 ∗ −1 ∗ 00 We have to prove that (ϕk ) (f) converges to (ϕ ) (f) in OM(V ) for any f ∈ 0 ∗ OM(U ). Define morphisms ψ, ψk :(U, OM|U ) → (V, OM|V ) by setting ψ (w) =ϕ ˜(z) ∗ ∗ ∗ and ψ (ξj) = θj and similarly ψk(w) =ϕ ˜k(z) and ψk(ξj) = θj. These morphisms −1 −1 0 0 have inverses ψ , ψk :(V , OM|V 0 ) → (U, OM|U ) on V . Due to the convergence of −1 −1 −1 ∗ ϕ˜k toϕ ˜ andϕ ˜k toϕ ˜ it is enough to prove the convergence of (ϕk ◦ ψk) (f) = ∗ −1 ∗ −1 ∗ 0 ψk((ϕk ) (f)) to (ϕ ◦ ψ) (f) in OM(U ) for any f ∈ OM(U) since this implies that −1 ∗ −1 ∗ ∗ −1 ∗ −1 ∗ ∗ (ϕk ) (f) = (ψk ) ((ϕk ◦ ψk) (f)) converges to (ϕ ) (f) = (ψ ) ((ϕ ◦ ψ) (f)) in 00 OM(V ). Therefore, we may assumeϕ ˜ = id,ϕ ˜k = id on U without loss of generality. In local coordinates z1 . . . , zm, θ1, . . . , θn we then have n ∗ X ν ∗ X X ν ϕ (zj) = zj + aj,ν(z)θ and ϕ (θj) = bj,l(z)θl + bj,ν(z)θ |ν|=0¯,||ν||>0 l=1 |ν|=1¯,||ν||>1 for appropriate holomorphic functions aj,ν, bj,l, bj,ν on U, where again ||ν|| = ν1 + ... + νn ∈ Z considering each νl ∈ Z2 as 0 or 1 and taking the sum in Z. We define B(z) to be the matrix with entries bj,l(z) at position (j, l) and set bν = (b1,ν, . . . , bn,ν), ∗ P ν ∗ Pn and shortly write ϕ (θ) = B(z)θ + |ν|=1¯,||ν||>1 bν(z)θ for ϕ (θj) = l=1 bj,l(z)θl + P ν |ν|=1¯,||ν||>1 bj,ν(z)θ , j = 1, . . . , n. Since ϕ is invertible, B(z) is an invertible n × n- matrix for each z ∈ U, and thus B defines a holomorphic map B : U → GLn(C). Similarly, we have

∗ X ν ∗ X ν ϕk(zj) = zj + aj,ν;k(z)θ and ϕk(θ) = Bk(z)θ + bν;k(z)θ |ν|=0¯,||ν||>0 |ν|=1¯,||ν||>1

71 for appropriate holomorphic functions aj,ν;k, bj,ν;k on U and holomorphic maps Bk : U → GLn(C). Deﬁne morphisms χ, χk :(U, OM|U ) → (U, OM|U ) by ∗ ∗ χ (zj) = zj, χ (θ) = B(z)θ and similarly ∗ ∗ χk(zj) = zj, χk(θ) = Bk(z)θ. −1 ∗ −1 ∗ −1 These morphisms are invertible and we have (χ ) (zj) = zj,(χ ) (θ) = (B(z)) θ −1 ∗ −1 ∗ −1 and (χk ) (zj) = zj,(χk ) (θ) = (Bk(z)) θ. Since ϕk converges to ϕ, Bk uniformly converges to B on compact subset of U and thus χk converges to χ. Moreover, since −1 −1 the inversion in GLn(C) is continuous, (Bk) uniformly converges to B on compact −1 −1 −1 subsets of U which implies that χk converges to χ . Consequently, ϕk converges to −1 −1 −1 ϕ precisely if ϕk ◦ χk converges to ϕ ◦ χ. It is hence enough to consider the case where Bk(z) = B(z) = En, En the identity matrix of size n, because   ∗ ∗ X ν −1 X ν (ϕ ◦ χ) (θ) = χ B(z)θ + bν(z)θ  = B(z)(B(z)) θ + cν(z)θ |ν|=1¯,||ν||>1 |ν|=1¯,||ν||>1 X ν = θ + cν(z)θ |ν|=1¯,||ν||>1 for appropriate holomorphic functions cj,ν on U, cν = (c1,ν, . . . cn,ν), and similarly

∗ X ν (ϕk ◦ χk) (θ) = θ + cν;k(z)θ |ν|=1¯,||ν||>1 for appropriate holomorphic functions cj,ν;k on U. ∗ Suppose now B(z) = Bk(z) = En. For the pullback ϕ we have

∗ ∗ X ν ∗ ∗ X ν (ϕ − id )(zj) = aj,ν(z)θ and (ϕ − id )(θ) = cν(z)θ , |ν|=0¯,||ν||>0 |ν|=1¯,||ν||>1

∗ ∗ ∗ ∗ and we conclude that ϕ −id is nilpotent. Moreover, by an analogous argument ϕk −id is nilpotent for each k. Therefore, by Lemma 1.2.5

X (−1)l+1 Y = log(ϕ∗) = (ϕ∗ − id∗)l l l≥1 and X (−1)l+1 Y = log(ϕ∗) = (ϕ∗ − id∗)l k k l k l≥1 ∗ P 1 l ∗ define nilpotent even vector fields on U and ϕ = exp(Y ) = l≥0 l! Y and ϕk = exp(Yk). Here, the sums for exp and log are finite sums due to the nilpotency of Y , ∗ ∗ ∗ ∗ Yk and (ϕ − id ), (ϕk − id ). Therefore, the sequence of vector fields Yk converges to Y in the sense that for any f ∈ OM(U) the sequence Yk(f) converges to Y (f) in P ν OM(U), i.e. if fν, fν;k are holomorphic functions on U with Y (f) = ν fν(z)θ and P ν Yk(f) = ν fν;k(z)θ , then fν;k uniformly converges to fν on compact subsets of U. −1 −1 Therefore, −Yk converges to −Y and this implies that ϕk converges to ϕ since the −1 ∗ −1 ∗ pullbacks are given by (ϕ ) = exp(−Y ) and (ϕk ) = exp(−Yk).

72 Remark 5.1.9. Let M be a split supermanifold and let E → M be a vector bundle with V associated sheaf of sections E such that the structure sheaf OM is isomorphic to E. By [Mor58] the group of automorphisms Aut(E) of the vector bundle E is a complex Lie group. Each automorphism ϕ of the supermanifold M induces an automorphism ϕE of the vector bundle E over the underlying mapϕ ˜ of ϕ, and the map π : Aut0¯(M) → ∼ V Aut(E), ϕ 7→ ϕE, is continuous. The splitting OM = E yields a section σ of π. If χ : E → E is an automorphism of the bundle E with pullback χ∗, we deﬁne the ∗ ∗ automorphism σ(χ): M → M by the pullback f1 ∧ ... ∧ fk 7→ χ (f1) ∧ ... ∧ χ (fk) for Vk f1 ∧ ... ∧ fk ∈ E. In particular, π is surjective and we have an exact sequence

0 → ker π → Aut0¯(M) → Aut(E) → 0, which splits. Consequently, the topological group Aut0¯(M) is a semidirect product ∼ Aut0¯(M) = ker π o Aut(E). The kernel of π consists of those automorphisms of M which induce the identity on the vector bundle E. If ϕ : M → M is an automorphism, we have π(ϕ) = idE precisely if the underlying mapϕ ˜ is the identity on M and the pullback of ϕ satisfies M ^ (ϕ∗ − id∗)(E) ⊆ k E . k>1 In this case (ϕ∗ − id∗) is nilpotent and there is an even vector field X on M with exp(X) = ϕ∗ by Lemma 1.2.5. The vector field X is nilpotent and fulfills ^ M ^ X k E ⊆ l E l≥k+2 for all k. More generally, let  

(2)  ^k M ^l  Vec (M) = X ∈ Vec¯(M) X E ⊆ E for all k 0  l≥k+2  denote the set of even nilpotent vector fields on M which are increasing with respect to ∼ V (2) the Z-grading on OM = E. In particular, the reduction of vector fields in Vec (M) to vector fields on M is 0. The exponential map Vec(2)(M) → ker π assigning to a vector field X the automorphism with pullback exp(X) is bijective. In Section 5.2, we will prove that the Lie superalgebra Vec(M) of vector fields on M is finite-dimensional, (2) ∼ which implies that Vec (M) is also finite-dimensional. This implies that Aut0¯(M) = ker π o Aut(E) carries the structure of a complex Lie group. In the general case of a not necessarily split supermanifold M the proof that Aut0¯(M) can be endowed with the structure of a complex Lie group requires more work, which is carried out in the next paragraphs.

5.1.2 Non-existence of small subgroups of Aut0¯(M)

In this section, we prove that Aut0¯(M) does not contain small subgroups, which means that there exists an open neighbourhood of the identity in Aut0¯(M) such that each

73 subgroup contained in this neighbourhood consists only of the identity. As a consequence, the topological group Aut0¯(M) carries the structure of a real Lie group by a result of Yamabe (cf. [Yam53]). Before proving the non-existence of small subgroups, a few technical preparations m|n m|n are needed. Consider C and let z1, . . . , zm, ξ1, . . . , ξn denote coordinates on C . m P ν Let U ⊆ be an open subset. For f = f ξ ∈ O m|n (U) deﬁne C ν ν C

X X ν : ||f||U = fνξ = ||fν||U , ν U ν where ||fν||U denotes the supremum norm of the holomorphic function fν on U. For m|n any morphism ϕ : U = (U, O m|n | ) → deﬁne C U C m n X ∗ X ∗ ||ϕ||U := ||ϕ (zi)||U + ||ϕ (ξj)||U . i=1 j=1

m|n Lemma 5.1.10. Let U = (U, O m|n | ) be a superdomain in . For any relatively C U C compact open subset U 0 of U there exists ε > 0 such that any morphism ψ : U → m|n 0 C with the property ||ψ − id||U < ε is biholomorphic as a morphism from U = 0 (U , O m|n | 0 ) onto its image. C U Proof. Let r > 0 be such that the closure of the polydisc

n ∆r (z) = {(w1, . . . , wm)| |wj − zj| < r} 0 m is contained in U for any z = (z1, . . . , zm) ∈ U . Let v ∈ C be any non-zero vector. 0 Then we have z + ζv ∈ U for any z ∈ U and ζ in the closure of ∆ r (0) = {t ∈ | |t| < ||v|| C r ˜ ||v|| }. If ||ψ − id||U < ε for a given ε > 0, then we have in particular ||ψ − id||U < ε for m the supremum norm of the underlying maps ψ,˜ id : U → C . Then, for the diﬀerential m 0 Dψ˜ of ψ˜ and any non-zero vector v ∈ C and any z ∈ U we have

d Dψ˜(z)(v) − v = ψ˜(z + tv) − (z + tv) dt

Z 1 ψ˜(z + ζv) − (z + ζv) = 2 dζ 2π ∂∆ r (0) ζ ||v||

1 Z ψ˜(z + ζv) − (z + ζv) ≤ dζ 2 2π ∂∆ r (0) ζ ||v|| ε||v|| < . r ˜ ε 0 This implies ||Dψ(z) − id|| < r with respect to the operator norm, for any z ∈ U . Thus ψ˜ is locally biholomorphic on U 0 if ε is small enough. Moreover, ε may be chosen such that ψ˜ is injective (see e.g. [Hir76], Chapter 2, Lemma 1.3).

Let ψj,k, ψj,ν be holomorphic functions on U such that n ∗ X X ν ψ (ξj) = ψj,kξk + ψj,νξ . k=1 ||ν||≥3

74 It is now enough to show det((ψj,k)1≤j,k≤n(z)) 6= 0 for all z ∈ U 0 and ε small enough in order to prove that ψ is a biholomorphism form 0 U onto its image. This follows from the fact that we assumed ||ψ − id||U < ε which implies ||ψj,k||U < ε if j 6= k and ||ψj,j − 1||U < ε.

This lemma now allows us to prove that Aut0¯(M) contains no small subgroups; for a similar result in the classical case see [BM46], Theorem 1.

Proposition 5.1.11. The topological group Aut0¯(M) has no small subgroups, i.e. there is a neighbourhood of the identity which contains no non-trivial subgroup. Proof. Let U ⊂ V ⊂ W be open subsets of M such that U is relatively compact in V and V is relatively compact in W . Moreover, suppose that W = (W, OM|W ) is isomor- m|n phic to a superdomain in C and let z1, . . . , zm, ξ1, . . . , ξn be local coordinates on W. By definition ∆(V,W ) = {ϕ ∈ Aut0¯(M)| ϕ˜(V ) ⊆ W } and ∆(U,V ) are open neighbourhoods of the identity in Aut0¯(M). Choose ε > 0 as in the preceding lemma such m|n that any morphism χ : V → C with ||χ − id||V < ε is biholomorphic as a morphism from U onto its image. Let Ω ⊆ ∆(V,W ) ∩ ∆(U,V ) be the subset whose elements ϕ satisfy ||ϕ − id||V < ε. The set Ω is open and contains the identity. Since Aut0¯(M) is locally compact by Lemma 5.1.7, it is enough to show that each compact subgroup Q ⊆ Ω is trivial. Otherwise for non-compact Q, let Ω0 be an open neighbourhood of 0 the identity with compact closure Ω which is contained in Ω, and suppose Q ⊆ Ω0. 0 Then Q ⊆ Ω ⊂ Ω is a compact subgroup and Q is trivial if Q is trivial. m|n Define a morphism ψ : V → C by setting Z Z ∗ ∗ ∗ ∗ ψ (zi) = q (zi) dq and ψ (ξj) = q (ξj) dq, Q Q where the integral is taken with respect to the normalized Haar measure on Q. This m|n yields a holomorphic morphism ψ : V → C since each q ∈ Q defines a holomorphic m|n ˜ R morphism V → W ⊆ C . Its underlying map is ψ(z) = Q q˜(z) dq. The morphism ψ satisfies Z Z ∗ ∗ ∗ ||ψ (zi) − zi||V = (q (zi) − zi) dq ≤ ||q (zi) − zi||V dq Q V Q and similarly Z ∗ ∗ ||ψ (ξj) − ξj||V ≤ ||q (ξj) − ξj||V dq. Q Consequently, we have m n X ∗ X ∗ ||ψ − id||V = ||ψ (zi) − zi||V + ||ψ (ξj) − ξj||V i=1 j=1   Z m n X ∗ X ∗ ≤  ||q (zi) − zi||V + ||q (ξj) − ξj||V  dq Q i=1 j=1 Z = ||q − id||V dq < ε. Q

75 Thus by the preceding lemma, ψ|U is a biholomorphic morphism onto its image. Furthermore, on U we have ψ ◦ q0 = ψ for any q0 ∈ Q since Z Z 0 ∗ 0 ∗ ∗ 0 ∗ ∗ 0 ∗ ∗ (ψ ◦ q ) (zi) = (q ) (ψ (zi)) = (q ) q (zi) dq = (q ) (q (zi)) dq Q Q Z Z 0 ∗ ∗ ∗ = (q ◦ q ) (zi) dq = q (zi) dq = ψ (zi) Q Q due to the invariance of the Haar measure, and also

0 ∗ ∗ (ψ ◦ q ) (ξj) = ψ (ξj).

0 0 The equality ψ ◦ q = ψ on U implies q |U = idU because of the invertibility of ψ. By 0 the identity principle it follows that q = idM if M is connected, and hence Q = {idM}. In general, M has only ﬁnitely many connected components since M is compact. Therefore, a repetition of the preceding argument yields the existence of a neighbourhood of the identity of Aut0¯(M) without non-trivial subgroups. By Theorem 3 in [Yam53], the preceding proposition implies the following:

Corollary 5.1.12. The topological group Aut0¯(M) can be endowed with the structure of a real Lie group.

5.1.3 One-parameter subgroups of Aut0¯(M)

In order to obtain results on the regularity of the action of Aut0¯(M) on the compact complex supermanifold M and to characterize the Lie algebra of Aut0¯(M), we study continuous one-parameter subgroups of Aut0¯(M). Each continuous one-parameter subgroup R → Aut0¯(M) is an analytic map between the Lie groups R and Aut0¯(M). We prove that the action of each continuous one-parameter subgroup of Aut0¯(M) on M is analytic and induces an even holomorphic vector field on M. Consequently, the Lie algebra of Aut0¯(M) may be identified with the Lie algebra Vec0¯(M) of even holomorphic vector fields on M, and Aut0¯(M) carries the structure of a complex Lie group whose action on the supermanifold M is holomorphic. Definition 5.1.13. A continuous one-parameter subgroup ϕ of automorphisms of M is a family of automorphisms ϕt : M → M, t ∈ R, such that the map ϕ : R → Aut0¯(M), t 7→ ϕt, is a continuous group homomorphism.

Remark 5.1.14. Let ϕt : M → M, t ∈ R, be a family of automorphisms satisfying ϕs+t = ϕs ◦ ϕt for all s, t ∈ R, and such thatϕ ˜ : R × M → M,ϕ ˜(t, p) =ϕ ˜t(p) is continuous. Then ϕt is a continuous one-parameter subgroup if and only if the following condition is satisﬁed: Let U, V ⊂ M be open subsets, and [a, b] ⊂ R such thatϕ ˜([a, b]× U) ⊆ V . Assume moreover that there are local coordinates z1, . . . , zm, ξ1, . . . , ξn for M on U. Then for any f ∈ OM(V ) there are continuous functions fν :[a, b] × U → C with (fν)t = fν(t, ·) ∈ OM(U) for ﬁxed t ∈ [a, b] such that

∗ X ν (ϕt) (f) = fν(t, z)ξ . ν We say that the action of the one-parameter subgroup ϕ on M is analytic if each fν(t, z) is analytic in both components.

76 Proposition 5.1.15. Let ϕ be a continuous one-parameter subgroup of automorphisms of M. Then the action of ϕ on M is analytic. Remark 5.1.16. Deﬁning a continuous one-parameter subgroup as in Remark 5.1.14, the statement of Proposition 5.1.15 also holds true for complex supermanifolds M with non-compact underlying manifold M since the compactness of M is not needed for the proof. For the proof of the proposition the following technical lemma is needed: m Lemma 5.1.17. Let U ⊆ V ⊆ C be open subsets, p ∈ U, Ω ⊆ R an open connected neighbourhood of 0, and let α :Ω × U → V be a continuous map satisfying α(t, z) = α(t + s, z) − f(t, s, z) for (t, s, z) in a neighbourhood of (0, 0, p) and for some continuous function f which is analytic in (t, z). If α is holomorphic in the second component, then it is analytic on a neighbourhood of (0, p). Proof. For small t, h > 0, z near p, we have Z h Z h h · α(t, z) = α(t + s, z)ds − f(t, s, z)ds 0 0 Z h+t Z h Z h = α(s, z)ds − α(s, z)ds − (f(t, s, z) − α(s, z))ds t 0 0 Z h+t Z t Z h = α(s, z)ds − α(s, z)ds − (f(t, s, z) − α(s, z))ds h 0 0 Z t Z h = (α(s + h, z) − α(s, z))ds − (f(t, s, z) − α(s, z))ds 0 0 Z t Z h = f(s, h, z)ds − (f(t, s, z) − α(s, z))ds. 0 0 The assumption that f is a continuous function which is analytic in the ﬁrst and third component therefore implies that α is analytic.

Proof of Proposition 5.1.15. Due to the action property ϕs+t = ϕs ◦ ϕt it is enough to show the statement for the restriction of ϕ to (−ε, ε) × M for some ε > 0. Let U, V ⊆ M be open subsets such that U is relatively compact in V , and such that there are local coordinates z1, . . . , zm, ξ1, . . . , ξn on V for M. Choose ε > 0 such that ϕ˜t(U) ⊆ V for any t ∈ (−ε, ε). Let αi,ν, βj,ν be continuous functions on (−ε, ε) × U ∗ P ν ∗ P ν with (ϕt) (zi) = |ν|=0¯ αi,ν(t, z)ξ and (ϕt) (ξj) = |ν|=1¯ βj,ν(t, z)ξ , where |ν| = |(ν1, . . . , νn)| = ν1 + ... + νn ∈ Z2. We have to show that α and β are analytic in (t, z). 0 n n The induced map ψ :(−ε, ε) × U × C → V × C on the underlying vector bundle is given by      z1 α1,0(t, z)   .   .    .   .        zm  αm,0(t, z)  t,   7→ Pn  ,   v1   β1,k(t, z)vk      k=1    .   .    .   .  Pn vn k=1 βn,k(t, z)vk

77 ¯ ¯ ¯ ¯ ¯ 0 where βj,k = βj,ek if ek = (0,..., 0, 1, 0,..., 0) denotes the k-th unit vector. The map ψ n is a local continuous one-parameter subgroup on U ×C because ϕ is a continuous one- parameter subgroup. By a result of Bochner and Montgomery the map ψ0 is analytic in (t, z, v) (see [BM45], Theorem 4). Hence, the map ψ :(−ε, ε) × U → V given by ∗ ∗ Pn (ψt) (zi) = αi(t, z), (ψt) (ξj) = k=1 βj,k(t, z)ξk is analytic. Let X be the local vector ﬁeld on U induced by ψ, i.e.

∂ ∗ X(f) = (ψt) (f). ∂t 0 We may assume that X is non-degenerate, i.e. the evaluation X(p) of X in p does not vanish for all p ∈ U. Otherwise, consider instead of ϕ the diagonal action on C × M acting by addition of t in the ﬁrst component and ϕt in the second, and note that this action is analytic precisely if ϕ is analytic. For the diﬀerential dψ of ψ in (0, p) we have ! ∂ ∂ ∗ dψ = ◦ ψ = X(p) 6= 0. ∂t (0,p) ∂t (0,p)

Therefore, the restricted map ψ|(−ε,ε)×{p} is an immersion and its image ψ((−ε, ε)×{p}) is a subsupermanifold of V. Let S be a subsupermanifold of U transverse to ψ((−ε, ε)× {p}) in p. The map ψ|(−ε,ε)×S is a submersion in (0, p) since dψ(T(0,p)(−ε, ε) × {p})) = Tpψ((−ε, ε) × {p}) and dψ(T(0,p){0} × S) = TpS because ψ|{0}×U = id. Hence χ := ψ|(−ε,ε)×S is locally invertible around (0, p), and thus invertible as a map onto its image after possibly shrinking U and ε, and satisﬁes ∂ ∂ χ = (χ−1)∗ ◦ ◦ χ∗ = (χ−1)∗ ◦ χ∗ ◦ X = X. ∗ ∂t ∂t

Therefore, after deﬁning new coordinates w1, . . . , wm, θ1, . . . , θn for M on U via χ, we have X = ∂ and (ϕ )∗ is of the form ∂w1 t ∗ X ν (ϕt) (w1) = w1 + t + α1,ν(t, w)θ , |ν|=0¯,||ν||>0 ∗ X ν (ϕt) (wi) = wi + αi,ν(t, w)θ for i 6= 1, |ν|=0¯,||ν||>0 ∗ X ν (ϕt) (θj) = θj + βj,ν(t, w)θ , |ν|=1¯,||ν||>1 for appropriate αi,ν, βj,ν, where again ||ν|| = ||(ν1, . . . , νn)|| ∈ Z. For small s and t we have   ∗ ∗ ∗ X ν ϕt (ϕs(wi)) = ϕt wi + δ1,is + αi,ν(s, w)θ  |ν|=0¯,||ν||>0 X ν X ∗ ν = wi + δi,1(t + s) + αi,ν(t, w)θ + ϕt (αi,ν(s, w)θ ). (5.1) |ν|=0¯,||ν||>0 |ν|=0¯,||ν||>0

Let fi,ν(t, s, w) be such that

X ∗ ν X ν ϕt (αi,ν(s, w)θ ) = fi,ν(t, s, w)θ . (5.2) |ν|=0¯,||ν||>0 |ν|=0¯,||ν||>0

78 ν0 For fixed ν0 the coefficient fi,ν0 (t, s, w) of θ depends only on αi,ν0 (s, w + te1) and βj,µ(t, w) for µ with ||µ|| ≤ ||ν0|| − 1 as well as αj,ν(t, w) and its partial derivatives in the second component for ν with ||ν|| ≤ ||ν0|| − 2. This can be shown by a calculation ∗ ∗ using the special form of ϕt (wj) and ϕt (θj) and general properties of the pullback of a morphism of supermanifolds. Assume now that the analyticity near (0, p) of αi,ν, βj,µ is shown for ||ν||, ||µ|| < 2k and all i, j. Let ν0 be such that ||ν0|| = 2k. Then fi,ν0 (t, s, w) is a continuous function which is analytic in (t, w) near (0, p) for fixed s. ∗ ∗ ∗ Since ϕt (ϕs(wi)) = ϕt+s(wi), using (1) and (2) we get

αi,ν0 (t, w) + fi,ν0 (t, s, w) = αi,ν0 (t + s, w), and thus αi,ν0 (t, w) is analytic near (0, p) by Lemma 5.1.17. Similarly, it can be proven that βj,µ0 is analytic for ||µ0|| = 2k + 1 if αi,ν and βj,µ are analytic for ||ν||, ||µ|| < 2k + 1.

Corollary 5.1.18. The Lie algebra of Aut0¯(M) is isomorphic to the Lie algebra Vec0¯(M) of even vector ﬁelds on M, and Aut0¯(M) is a complex Lie group.

Proof. If ϕ : R → Aut0¯(M), t 7→ ϕt, is a continuous one-parameter subgroup, then by Proposition 5.1.15 the action of ϕ on M is analytic. Therefore, ϕ induces an even holomorphic vector ﬁeld X(ϕ) on M by setting

∂ ∗ X(ϕ) = (ϕt) , ∂t 0 and ϕ is the flow map of X(ϕ). On the other hand, each X ∈ Vec0¯(M) is globally integrable since M is compact (cf. [GW13], Theorem 5.4). Its flow ϕX defines a one- parameter subgroup of Aut0¯(M) which is continuous. This yields an isomorphism of Lie algebras Lie(Aut0¯(M)) → Vec0¯(M). ∼ Consequently, we have Lie(Aut0¯(M)) = Vec0¯(M) and since Vec0¯(M) is a complex Lie algebra, Aut0¯(M) carries the structure of a complex Lie group.

The Lie group Aut0¯(M) naturally acts on M; this action ψ : Aut0¯(M) × M → M ∗ ∗ ∗ is given by (evg ⊗ idM) ◦ ψ = g where evg denotes the evaluation in g ∈ Aut0¯(M).

Corollary 5.1.19. The natural action of Aut0¯(M) on M deﬁnes a holomorphic morphism of supermanifolds ψ : Aut0¯(M) × M → M.

Proof. Since the action of each continuous one-parameter subgroup of Aut0¯(M) on M is holomorphic by the preceding considerations and each g ∈ Aut0¯(M) is an automorphism g : M → M, the action ψ is a holomorphic.

If a Lie supergroup G (with Lie superalgebra g of right-invariant vector fields) acts on a supermanifold M via ψ : G × M → M, this action ψ induces an infinitesimal ∗ ∗ action dψ : g → Vec(M) defined by dψ(X) = (X(e) ⊗ idM) ◦ ψ for any X ∈ g, where ∗ X ⊗ idM denotes the canonical extension of the vector field X on G to a vector field ∗ on G × M, and (X(e) ⊗ idM) is its evaluation in the neutral element e of G.

Corollary 5.1.20. Identifying the Lie algebra of Aut0¯(M) with Vec0¯(M) as in Corol- lary 5.1.18, the induced inﬁnitesimal action of the action ψ : Aut0¯(M) × M → M in Corollary 5.1.19 is the inclusion Vec0¯(M) ,→ Vec(M).

79 5.2 The Lie superalgebra of vector ﬁelds

In this section, we prove that the Lie superalgebra Vec(M) of holomorphic super vector fields on a compact complex supermanifold M is finite-dimensional. First, we prove that Vec(M) is finite-dimensional if M is a split supermanifold using that its tangent sheaf TM is a coherent sheaf of OM -modules, where OM denotes again the sheaf of holomorphic functions on the underlying manifold M. Then the statement in the general case is deduced using a filtration of the tangent sheaf. Remark that since Aut0¯(M) is a complex Lie group with Lie algebra isomorphic to the Lie algebra Vec0¯(M) of even holomorphic vector fields on M (see Corollary 5.1.18), we already know that the even part of Vec(M) = Vec0¯(M) ⊕ Vec1¯(M) is finite- dimensional. Let M be a complex supermanifold of dimension (m|n), and let IM be the ideal subsheaf generated by the odd elements in the structure sheaf OM. As described in [Oni98], we have the filtration 0 1 2 n+1 OM = (IM) ⊃ (IM) ⊃ (IM) ⊃ ... ⊃ (IM) = 0 of the structure sheaf OM by the powers of IM. Define the quotient sheaves grk(OM) = k k+1 L (IM) /(IM) . This gives rise to the Z-graded sheaf gr OM = kgrk(OM). Further- more, gr M = (M, gr OM) is a split complex supermanifold of the same dimension as M.

Lemma 5.2.1. Let M be a split complex supermanifold. Then its tangent sheaf TM is a coherent sheaf of OM -modules. V Proof. Since M is split, its structure sheaf OM is isomorphic to E as an OM -module, where E is the sheaf of sections of a holomorphic vector bundle on the underlying manifold M. Thus, the structure sheaf OM, and hence also the tangent sheaf TM, carry the structure of a sheaf of OM -modules. Let U ⊂ M be an open subset such that there exist even coordinates z1, . . . , zm and odd coordinates ξ1, . . . , ξn on U. Any derivation D ∈ TM(U) on U can uniquely be written as

 m n  X X ∂ X ∂ D = f (z)ξν + g (z)ξν  i,ν ∂z j,ν ∂ξ  n i j ν∈(Z2) i=1 j=1 where fi,ν, gj,ν are holomorphic functions on U. Therefore, the restricted sheaf TM|U 2n(m+n) is isomorphic to (OM |U ) and TM is coherent over OM . Proposition 5.2.2. The Lie superalgebra Vec(M) of holomorphic vector ﬁelds on a compact complex supermanifold M is ﬁnite-dimensional.

Proof. First, assume that M is split. Then the tangent sheaf TM is a coherent sheaf of OM -modules. Thus, the space of global sections of TM, Vec(M) = TM(M), is finite-dimensional since M is compact (cf. [CS53]). Now, let M be an arbitrary compact complex supermanifold. We associate the split complex supermanifold gr M = (M, gr OM). As before, let IM denote the ideal subsheaf in OM generated by the odd elements. Define the filtration of sheaves of Lie superalgebras

TM =: (TM)(−1) ⊃ (TM)(0) ⊃ (TM)(1) ⊃ ... ⊃ (TM)(n+1) = 0

80 of the tangent sheaf TM by setting

k k+1 (TM)(k) = {D ∈ TM| D(OM) ⊂ (IM) ,D(IM) ⊂ (IM) } for k ≥ 0. Moreover, deﬁne grk(TM) = (TM)(k)/(TM)(k+1) and set M gr(TM) = grk(TM). k≥−1

By Proposition 1 in [Oni98] the sheaf gr(TM) is isomorphic to the tangent sheaf of the associated split supermanifold gr M. By the preceding considerations, the space of holomorphic vector ﬁelds on gr M, M Vec(gr M) = gr(TM)(M) = grk(TM)(M), k≥−1 is of ﬁnite dimension. The projection onto the quotient yields

dim(TM)(k)(M) − dim(TM)(k+1)(M) ≤ dim(grk(TM)(M)) and dim(TM)(n)(M) = dim(grn(TM)(M)) and hence by induction X dim(TM)(k)(M) ≤ dim(grj(TM)(M)), j≥k which gives dim(TM(M)) = dim (TM)(−1)(M) ≤ dim (gr(TM)(M)) .

In particular, dim(TM(M)) is ﬁnite. Remark 5.2.3. The proof of the preceding proposition also proves the following in- equality: dim(Vec(M)) ≤ dim(Vec(gr M))

5.3 The automorphism group

In this section, the automorphism group of a compact complex supermanifold is defined. This is done via the formalism of Harish-Chandra pairs for complex Lie supergroups. The underlying classical Lie group is Aut0¯(M) and the Lie superalgebra is Vec(M), the Lie superalgebra of vector fields on M. Moreover, we prove that the automorphism group satisfies a universal property, which justifies our terminology.

Consider the representation α of Aut0¯(M) on Vec(M) given by

−1 ∗ ∗ α(g)(X) = g∗(X) = (g ) ◦ X ◦ g for g ∈ Aut0¯(M),X ∈ Vec(M).

This representation α preserves the parity on Vec(M), and its restriction to Vec0¯(M) ∼ coincides with the adjoint action of Aut0¯(M) on its Lie algebra Lie(Aut0¯(M)) = Vec0¯(M). Moreover, the diﬀerential (dα)id at the identity id ∈ Aut0¯(M) is the adjoint representation of Vec0¯(M) on Vec(M):

81 Let X and Y be vector ﬁelds on M. Assume that X is even and let ϕX denote the corresponding one-parameter subgroup. Then we have

∂ X (dα)id(X)(Y ) = (ϕt )∗(Y ) = [X,Y ] ∂t 0 by Corollary 2.3.8. Therefore, the pair (Aut0¯(M), Vec(M)) together with the representation α is a complex Harish-Chandra pair, and using the equivalence between the category of complex Harish-Chandra pairs and complex Lie supergroups (cf. [Vis11], § 2), we can deﬁne the automorphism group of a compact complex supermanifold M as follows:

Deﬁnition 5.3.1. The automorphism group Aut(M) of a compact complex supermanifold is the unique complex Lie supergroup whose associated Harish-Chandra pair is (Aut0¯(M), Vec(M)) with α as the adjoint representation.

Since the action ψ : Aut0¯(M)×M → M induces the inclusion Vec0¯(M) ,→ Vec(M) as infinitesimal action (see Corollary 5.1.20), there exists a Lie supergroup action Ψ : Aut(M) × M → M with the identity Vec(M) → Vec(M) as induced infinitesimal action and we have Ψ| = ψ (cf. Theorem 4.3.35). Aut0¯(M)×M The automorphism group together with Ψ satisfies a universal property.

Theorem 5.3.2. Let G be a complex Lie supergroup with a holomorphic action ΨG : G × M → M. Then there is a unique morphism σ : G → Aut(M) of Lie supergroups such that the diagram

ΨG G × M / M 8

σ×id Ψ M ' Aut(M) × M is commutative.

Proof. Let G be the underlying Lie group of G. For each g ∈ G, we have a morphism ∗ ∗ ∗ ΨG(g): M → M by setting (ΨG(g)) = (evg ⊗ idM) ◦ (ΨG) . This morphism ΨG(g) is −1 an automorphism of M with inverse ΨG(g ) and gives rise to a group homomorphism σ˜ : G → Aut0¯(M), g 7→ ΨG(g). Let g denote the Lie superalgebra (of right-invariant super vector fields) of G, and dΨG : g → Vec(M) the infinitesimal action induced by ΨG. The restriction of dΨG to the even part g0¯ = Lie(G) of g coincides with the differential (dσ˜)e ofσ ˜ at the identity e ∈ G. Moreover, if αG denotes the adjoint action of G on g, and α denotes, as before, the adjoint action of Aut0¯(M) on Vec(M), we have

−1 ∗ ∗ −1 ∗ ∗ dΨG(αG(g)(X)) = (ΨG(g )) ◦ dΨG(X) ◦ (ΨG(g)) = (˜σ(g )) ◦ dΨG(X) ◦ (˜σ(g))

= α(˜σ(g))(dΨG(X))

82 for any g ∈ G, X ∈ g. Using the correspondence between Lie supergroups and Harish- Chandra pairs, it follows that there is a unique morphism σ : G → Aut(M) of Lie supergroups with underlying mapσ ˜ and derivative dΨG : g → Vec(M) (see e.g. [Vis11], § 2), and σ satisﬁes Ψ ◦ (σ × idM) = ΨG. The uniqueness of σ follows from the fact that each morphism τ : G → Aut(M) of Lie supergroups fulﬁlling the same properties as σ necessarily induces the map dΨG : g → Vec(M) on the level of Lie superalgebras and its underlying mapτ ˜ has to satisfy τ˜(g) = ΨG(g) =σ ˜(g).

Remark 5.3.3. Since the morphism σ in Theorem 5.3.2 is unique, the automorphism group of a compact complex supermanifold M is the unique Lie supergroup satisfying the universal property formulated in Theorem 5.3.2.

Remark 5.3.4. We say that a real Lie supergroup G acts on M by holomorphic transformations if the underlying Lie group G acts on the complex manifold M by holomorphic transformations and if there is a homomorphism of Lie superalgebras g → Vec(M) which is compatible with the action of G on M. Using the theory of Harish- Chandra pairs, we also have the Lie supergroup GC, the universal complexiﬁcation of G; see [Kal15]. The underlying Lie group of GC is the universal complexiﬁcation GC of the Lie group G. Let g = g0¯ ⊕ g1¯ denote the Lie superalgebra of G, g0¯ the Lie algebra C C of G. Then the Lie algebra g0¯ of G is a quotient of g0¯ ⊗ C, and the Lie superalgebra C C of G can be realized as g0¯ ⊕(g1¯ ⊗C). The action of G on M extends to a holomorphic GC-action on M, and the homomorphism g → Vec(M) extends to a homomorphism C g0¯ ⊕ (g1¯ ⊗ C) → Vec(M) of complex Lie superalgebras, which is compatible with the GC-action on M. Thus, we have a holomorphic GC-action on M extending the G- action. Moreover, there is a morphism σ : GC → Aut(M) of Lie supergroups as in Theorem 5.3.2.

0|1 0|1 Example 5.3.5. Let M = C . Denoting the odd coordinate on C by ξ, each vector 0|1 ∂ ∂ field on C is of the form X = aξ ∂ξ + b ∂ξ for a, b ∈ C. The flow ϕ : C × M → M ∂ ∗ at 0|1 ∂ of aξ ∂ξ is given by (ϕt) (ξ) = e ξ, and the flow ψ : C × M → M of b ∂ξ by ∗ ∂ ∂ 0|1 1|1 ψ (ξ) = bτ + ξ. Let X0¯ = ξ ∂ξ and X1¯ = ∂ξ . Then Vec(C ) = CX0¯ ⊕ CX1¯ = C , 1|1 where the Lie algebra structure on C is given by [X0¯,X1¯] = −X1¯ and [X1¯,X1¯] = 0. Note that this Lie superalgebra is isomorphic to the Lie superalgebra of right-invariant 1|1 vector fields on the Lie supergroup (C , µ0,1), where the multiplication µ = µ0,1 is ∗ ∗ t given by µ (t) = t1 + t2 and µ (τ) = τ1 + e 1 τ2; for the Lie supergroup structures on 1|1 0|1 C see Example 2.1.2. In particular, the Lie superalgebra Vec(C ) is not abelian. 0|1 ∗ Since each automorphism ϕ of C is given by ϕ (ξ) = c · ξ for some c ∈ C, c 6= 0, 0|1 ∼ ∗ we have Aut0¯(C ) = C .

Proposition 5.3.6. The action of the automorphism group Aut(M) on M is eﬀective.

Proof. Using Proposition 2.5.2, the eﬀectiveness of the Aut(M)-action follows directly from the fact that the underlying Lie group Aut0¯(M) consists of the set of automorphisms and that the induced inﬁnitesimal action is the identity map Vec(M) → Vec(M).

83 5.4 The functor of points of the automorphism group

The diffeomorphism supergroup of a real supermanifold is studied in [SW11] and is proven to be a Fréchet Lie supergroup in the case where the supermanifold is compact. In [SW11], the “functor of points” approach to supermanifolds is used, i.e. a supermanifold is a representable contravariant functor from the category of superpoints to the category of sets. Similarly, a Lie supergroup is a representable functor from the category of superpoints to the category of groups. A superpoint is a supermanifold whose underlying manifold consists of only one point, i.e. in the case of real supermanifolds these are 0|k precisely the supermanifolds R for k ∈ N. Starting with a supermanifold M we define 0|k 0|k the corresponding functor Hom(−, M) by the assignment R 7→ Hom(R , M), where 0|k 0|k Hom(R , M) denotes the set of morphisms of supermanifolds R → M, and for mor- 0|k 0|l 0|l 0|k phisms α : R → R we define Hom(−, M)(α) : Hom(R , M) → Hom(R , M) by ϕ 7→ ϕ ◦ α. In analogy to the definition in [SW11] for the diffeomorphism supergroup of a real supermanifold, we define a functor Aut(M) associated with a complex supermanifold M. The goal of this section is to prove that in the case of a compact complex supermanifold M, the automorphism Lie supergroup as defined in Section 5.3 represents the functor Aut(M), i.e. the functors Aut(M) and Hom(−, Aut(M)) are isomorphic.

Definition 5.4.1. Let M be a complex supermanifold. We define the functor Aut(M) from the category of superpoints to the category of groups as follows: On objects, we define the functor Aut(M) by the assignment

0|k 0|k 0|k 7→ {ϕ : × M → × M | ϕ is invertible, and pr 0|k ◦ ϕ = pr 0|k }, C C C C C

0|k 0|k 0|k 0|l where pr 0|k : × M → is the projection. For morphisms α : → , we C C C C C 0|l 0|k set Aut(M)(α): Aut(M)(C ) → Aut(M)(C ),

ϕ 7→ (id 0|k × (pr ◦ ϕ ◦ (α × id ))) ◦ (diag × id ), C M M M

0|k Let χ : C → Aut(M) be an arbitrary morphism of complex supermanifolds. Let 0|k 0|k 0|k diag : C → C × C denote the diagonal map and let Ψ : Aut(M) × M → M denote the natural action of Aut(M) on M. Then the composition

ϕ = (id 0|k × (Ψ ◦ (χ × id ))) ◦ (diag × id ) χ C M M

0|k 0|k is an invertible map × M → × M with pr 0|k = pr 0|k ◦ ϕ . C C C C χ 0|k 0|k Lemma 5.4.2. The assignments Hom(C , Aut(M)) → Aut(M)(C ), χ 7→ ϕχ, de- ﬁne a natural transformation Hom(−, Aut(M)) → Aut(M).

84 0|k 0|l Proof. Let α : C → C be an arbitrary morphism. We will show that

ϕ = (id 0|k × (pr ◦ ϕ ◦ (α × id ))) ◦ (diag × id ). χ◦α C M χ M M To prove this equation it is enough to check that the composition of both sides of the equation with pr coincide since the composition of both sides with pr 0|k equals M C pr 0|k . We have C

prM ◦ ϕχ◦α = Ψ ◦ ((χ ◦ α) × idM) = prM ◦ ϕχ ◦ (α × idM)

= pr ◦ (id 0|k × (pr ◦ ϕ ◦ (α × id ))) ◦ (diag × id ). M C M χ M M

0|k 0|k Each map Hom(C , Aut(M)) → Aut(M)(C ), χ 7→ ϕχ, is a homomorphism of groups, as can be veriﬁed by direct calculations. For example, for the multiplication 0|k χ1 · χ2 of two morphisms χ1, χ2 : C → Aut(M) we have

prM ◦ ϕ(χ1·χ2) = Ψ ◦ ((χ1 · χ2) × idM) = Ψ ◦ ((µ ◦ (χ1 × χ2) ◦ diag) × idM)

= Ψ ◦ (idAut(M) × Ψ) ◦ (((χ1 × χ2) ◦ diag) × idM)

= Ψ ◦ (χ × id ) ◦ (id 0|k × Ψ) ◦ (id 0|k × χ × id ) ◦ (diag × id ) 1 M C C 2 M M

= prM ◦ ϕχ1 ◦ ϕχ2 , where µ : Aut(M) × Aut(M) denotes the multiplication on Aut(M) and we used the action property Ψ ◦ (µ × idM) = Ψ ◦ (idAut(M) × Ψ) of the action Ψ on M.

The just defined natural transformation is actually an isomorphism of functors. The proof of this fact is the content of the remainder of this section. The injectivity of the assignment χ 7→ ϕχ follows from the effectivity of the Aut(M)-action on M. In the proof of the surjectivity a “normal form” of the pullback of automorphisms 0|k 0|k ϕ : × M → × M with pr 0|k ◦ ϕ = pr 0|k is used. Thus, we start with some C C C C technical preparations. 0|k 0|k Let M be a complex supermanifold and ϕ : C × M → C × M be an invertible 0|k morphism with pr 0|k ◦ ϕ = pr 0|k . Let ι : M ,→ × M denote the canonical C C C inclusion. The composition ϕ0 = prM ◦ ϕ ◦ ι is an automorphism of M. Then ϕ is uniquely determined by ϕ0 and a set of vector fields on M:

0|k 0|k Lemma 5.4.3. Let ϕ : ×M → ×M be an invertible morphism with pr 0|k ◦ϕ = C C C k pr 0|k . Then there are vector ﬁelds X of parity |ν| for ν ∈ ( ) , ν 6= 0, on M such C ν Z2 that   ∗ ∗ X ν ϕ = (id × ϕ0) exp  τ Xν , ν6=0

0|k 0|k ν if τ1, . . . , τk denote coordinates on C ⊂ C × M. By τ Xν we mean the vector field 0|k on C × M which is induced by the extension of the vector field Xν on M to a vector 0|k ν ν1 νk field on the product C × M followed by the multiplication with τ = τ1 . . . τk . In ν other words for U ⊆ M open we have τ X (f) = 0 for f ∈ O 0|k ({0}) ⊂ O 0|k ({0}× ν C C ×M ν ν U) and (τ X )(g) = τ X (g) for g ∈ O (U) ⊂ O 0|k ({0} × U) considering X (g) ν ν M C ×M ν as a function on the product.

85 Moreover,    n X X 1 X exp τ νX = τ νX  ν n!  ν ν6=0 n≥0 ν6=0

k+1 P ν is a ﬁnite sum since ν6=0 τ Xν = 0.

A version of this lemma is also proven in [SW11], Theorem 5.1. Here, we give a diﬀerent proof which makes use of the technical result formulated in Lemma 1.2.5.

Proof. It is enough to prove the statement for those morphisms ϕ with ϕ0 = idM since −1 we may otherwise consider the morphism ϕ ◦ (id 0|k × ϕ ) . C 0 Assume now ϕ = id . Since pr 0|k × ϕ = pr 0|k ,ϕ ˜ = id , and O 0|k ∼ 0 M C C {0}×M C ×M = V k k C ⊗ OM, there are morphisms aν : OM → OM for ν ∈ (Z2) , ν 6= 0, with aν(1) = 0 such that ∗ ∗ X ν ϕ = id + τ aν, ν6=0 V k 0|k i.e. for f ∈ ∼ O 0|k ({0}) considered as a function f ⊗ 1 on the product × M C = C C ∗ we have ϕ (f) = f, and for g ∈ OM(U), U ⊆ M open, considered as a function 1 ⊗ g 0|k ∗ P ν on the product C × M we have ϕ (g) = g + ν6=0 τ aν(g). ∗ ∗ P ν The diﬀerence ϕ −id = τ a is a nilpotent morphism O 0|k → O 0|k . ν6=0 ν C ×M C ×M By Lemma 1.2.5  n X (−1)n+1 X X = τ νa n  ν n≥1 ν6=0

0|k ∗ defines an even vector field on C × M, and ϕ = exp(X). Let Xν : OM → OM be P ν morphisms such that X = ν6=0 τ Xν. Since X is an even vector field we have

X ν X ν X ν τ Xν(fg) = X(fg) = X(f)g + fX(g) = τ Xν(f)g + f τ Xν(g) ν6=0 ν6=0 ν6=0 X ν |ν||f| = τ Xν(f)g + (−1) fXν(g) ν6=0 for any f and g with f homogeneous of parity |f|. Therefore, we have

|ν||f| Xν(fg) = Xν(f)g + (−1) fXν(g) , which proves that Xν is a vector ﬁeld of parity |ν| on M.

0|k 0|k Proposition 5.4.4. If ϕ : C × M → C × M is an invertible morphism with 0|k pr 0|k = pr 0|k ◦ ϕ, then there is a unique morphism χ : → Aut(M) with ϕ = ϕ . C C C χ 0|k Proof. If χ1, χ2 : C → Aut(M) are morphisms with ϕ = ϕχ1 = ϕχ2 , then we necessarily have χ1 = χ2 since the Aut(M)-action on M is effective (cf. Proposition 5.3.6). k Let Xν be vector fields on M of parity |ν|, ν ∈ (Z2) , ν 6= 0, and ϕ0 : M → M ∗ ∗ P ν an automorphism such that ϕ = (id × ϕ0) exp ν6=0 τ Xν as in Lemma 5.4.3. Since ϕ0 is an automorphism of M, it is an element of Aut0¯(M) by definition. Let

86 evϕ0 denote the evaluation in ϕ0, i.e. evϕ0 is the pullback of the canonical inclusion 0|k {ϕ0} ,→ Aut(M), and let prAut(M) : C × Aut(M) → Aut(M) be the projection. We 0|k define χ : C → Aut(M) as the morphism given by the pullback   ∗ ∗ X ν ∗ χ = (id ⊗ evϕ0 ) ◦ exp  τ (Xν)R ◦ prAut(M), ν6=0 where (Xν)R denotes the right-invariant vector field on Aut(M) corresponding to the vector field Xν on M which is an element of the Lie superalgebra Vec(M) of Aut(M). The vector fields (Xν)R on Aut(M) and Xν on M fulfill the relation

∗ ∗ ∗ ((Xν)R ⊗ id ) ◦ Ψ = Ψ ◦ Xν since (Xν)R is right-invariant, Ψ is an action, and the induced infinitesimal action of Ψ maps (Xν)R, considered as an element of the Lie superalgebra of Aut(M), to P ν 0|k Xν. The sum Y = ν6=0 τ (Xν)R defines an even vector field on C × Aut(M), and exp(Y ) is a finite sum since Y is nilpotent. Furthermore, using Lemma 1.2.6 we get exp(Y )(fg) = exp(Y )(f) exp(Y )(g) and thus exp(Y ) is the pullback of an automor- 0|k 0|k phism C × Aut(M) → C × Aut(M) with the identity as the underlying map. This implies that χ∗ is indeed the pullback of a morphism. We still need to verify that the morphism χ satisfies ϕχ = ϕ. For this it is enough to check pr ◦ ϕ = pr ◦ ϕ since we already know that pr 0|k ◦ ϕ = pr 0|k = pr 0|k ◦ ϕ. M χ M C χ C C ∗ ∗ ∗ We have prM ◦ ϕχ = Ψ ◦ (χ × idM). Using the relation ((Xν)R ⊗ id ) ◦ Ψ = Ψ ◦ Xν we get

n n X ν ∗ ∗ ∗ ∗ ∗ X ν τ (Xν)R ⊗ id ◦ (id ⊗ Ψ ) = (id ⊗ Ψ ) ◦ τ Xν for any n ≥ 0, and thus

∗ ∗ ∗ X ν ∗ ∗ ∗ (exp(Y ) ⊗ id ) ◦ (id ⊗ Ψ ) = exp τ (Xν)R ⊗ id ◦ (id ⊗ Ψ )

∗ ∗ X ν = (id ⊗ Ψ ) ◦ exp τ Xν .

This implies

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ (prM ◦ ϕχ) = (χ ⊗ id ) ◦ Ψ = (id ⊗ evϕ0 ⊗ id ) ◦ (exp(Y ) ⊗ id ) ◦ (id ⊗ Ψ ) ◦ prM ∗ ∗ ∗ ∗ X ν ∗ = (id ⊗ evϕ0 ⊗ id ) ◦ (id ⊗ Ψ ) ◦ exp τ Xν ◦ prM

∗ ∗ X ν ∗ ∗ ∗ = (id ⊗ ϕ0) ◦ exp τ Xν ◦ prM = ϕ ◦ prM,

∗ ∗ ∗ ∗ where we used the special form of ϕ and the fact that (evϕ0 ⊗ id ) ◦ Ψ = ϕ0 by deﬁnition of the action Ψ of Aut(M) on M; cf. Corollary 5.1.19.

As a direct consequence of this proposition and Lemma 5.4.2, we get:

Corollary 5.4.5. The functors Aut(M) and Hom(−, Aut(M)) are isomorphic. This isomorphism is realized by the natural transformation introduced in Lemma 5.4.2.

87 5.5 The case of a superdomain with bounded underlying domain

m In the classical case, the automorphism group of a bounded domain U ⊂ C is a (real) Lie group (see Theorem 13 in “Sur les groupes de transformations analytiques” m|n in [Car79]). If U ⊂ C is a superdomain whose underlying set U is a bounded m domain in C , it is in general not possible to endow its set of automorphisms with the structure of a Lie group such that the action on U is smooth, as will be illustrated in an example. In particular, there is no Lie supergroup satisfying the universal property as the automorphism group of a compact complex supermanifold M does as formulated in Theorem 5.3.2.

Example 5.5.1. Let U be a superdomain of dimension (1|2) whose underlying set is an arbitrary bounded domain U ⊂ C. Let z, θ1, θ2 denote coordinates for M. For ∂ any holomorphic function f on U, define the even vector field Xf = f(z)θ1θ2 ∂z . The reduced vector field X˜f = 0 is completely integrable and thus the flow of Xf can be ∗ defined on C×U (cf. [GW13] Lemma 5.2). The flow is given by (ϕt) (z) = z+t·f(z)θ1θ2 ∗ and (ϕt) (θj) = θj. For any holomorphic functions f and g we have [Xf ,Xg] = 0, and ∼ thus their flows commute by Corollary 2.3.8. Therefore, {Xf | f ∈ O(U)} = O(U) is an uncountably infinite-dimensional abelian Lie algebra. If the set of automorphisms of U carried the structure of a Lie group such that its action on U was smooth, its Lie ∼ algebra would necessarily contain {Xf | f ∈ O(U)} = O(U) as a Lie subalgebra, which is not possible.

5.6 Examples

In this section, we determine the automorphism group Aut(M) for some complex supermanifolds M with underlying manifold M = P1C. Let L1 denote the hyperplane bundle on M = P1C with sheaf of sections O(1), and ⊗k Lk = (L1) the line bundle of degree k, k ∈ Z, on P1C, and sheaf of sections O(k). Each holomorphic vector bundle on P1C is isomorphic to a direct sum of line bundles

Lk1 ⊕ ... ⊕ Lkn (see [Gro57]). Therefore, if M is a split supermanifold with M = P1C and dim M = (1|n), there exist k1, . . . , kn ∈ Z such that the structure sheaf OM of M is isomorphic to ^ (O(k1) ⊕ ... ⊕ O(kn)).

Let Uj = {[z0 : z1] ∈ P1C | zj 6= 0}, j = 1, 2, and Uj = (Uj, OM|Uj ). Moreover, deﬁne ∗ ∗ ∗ ∗ ∗ U0 = U0 \{[1 : 0]} and U1 = U1 \{[0 : 1]}, and let Uj = (Uj , OM|Uj ). We can now choose local coordinates z, θ1, . . . , θn for M on U0, and local coordinates w, η1, . . . , ηn ∗ ∗ on U1 so that the transition map χ : U0 → U1 , which determines the supermanifold structure of M, is given by 1 χ∗(w) = and χ∗(η ) = zkj θ . z j j

Example 5.6.1. Let M = (P1C, OM) be a complex supermanifold with dim M = (1|1). Since the odd dimension is 1, the supermanifold M has to be split. Let −k ∈ Z be the degree of the associated line bundle. Choose local coordinates z, θ for M on U0

88 ∗ ∗ ∗ 1 and w, η on U1 as above so that the transition map χ : U0 → U1 is given by χ (w) = z ∗ 1 and χ (η) = zk θ. We first want to determine the Lie superalgebra Vec(M) of vector fields on M.A calculation in local coordinates verifying the compatibility condition with the transition map χ yields that the restriction to U0 of any vector field on M is of the form ∂ ∂ ∂ ∂ (α + α z + α z2) + (β + kα z)θ + p(z) + q(z)θ , 0 1 2 ∂z 2 ∂θ ∂θ ∂z where α0, α1, α2, β ∈ C, p is a polynomial of degree at most k, and q is a polynomial of degree at most 2 − k. If k < 0 (resp. 2 − k < 0), the polynomial p (resp. q) is 0. The Lie algebra Vec0¯(M) of even vector fields is isomorphic to sl2(C) ⊕ C, where an isomorphism sl2(C) ⊕ C → Vec0¯(M) is given by

a b ∂ ∂ , d 7→ (−b − 2az + cz2) + ((d − ka) + kcz)θ . c −a ∂z ∂θ

Note that since the odd dimension of M is one, each automorphism ϕ : M → M gives rise to an automorphism of the line bundle L−k and vice versa. Hence, the automorphism group Aut(L−k) of the line bundle L−k and Aut0¯(M) coincide. A calculation yields that the group Aut0¯(M) of automorphisms M → M can be ∗ ∗ identified with PSL2(C) × C if k is even and with SL2(C) × C if k is odd. Consider a b ∗ a b the element c d , s , where s ∈ C and c d is either an element of SL2(C) or the representative of the corresponding class in PSL2(C). The action of the corresponding element ϕ ∈ Aut0¯(M) on M is then given by c + dz 1 ϕ∗(z) = and ϕ∗(θ) = + s θ a + bz (a + bz)k as a morphism over appropriate subsets of U0 and by aw + b 1 ϕ∗(w) = and ϕ∗(η) = + s η cw + d (cw + d)k over appropriate subsets of U1. The Lie supergroup structure on Aut(M) is now uniquely determined by Aut0¯(M), Vec(M), and the adjoint action of Aut0¯(M) on Vec(M). Since Aut0¯(M) is connected ∼ it is enough to calculate the adjoint action of Vec0¯(M) = sl2C ⊕ C on Vec1¯(M). Let Pl denote the space of polynomials of degree at most l, and set Pl = {0} for l < 0. The space of odd vector fields Vec1¯(M) is isomorphic to Pk ⊕ P2−k via ∂ ∂ p(z) ∂θ + q(z)θ ∂z 7→ (p(z), q(z)). 1 0 ∼ ∂ The element H = 0 −1 ∈ sl2(C) ⊂ sl2(C)⊕C = Vec0¯(M) corresponds to −2z ∂z − ∂ kθ ∂θ . The adjoint action of this vector field on the first factor Pk of Vec1¯(M) is given ∂ ∂ by −2z ∂z + k · Id, and on the second factor P2−k by −2z ∂z + (2 − k) · Id. Calculating the weights of the sl2(C)-representation on Pk and P2−k, we get that Pk is the unique irreducible (k + 1)-dimensional representation and P2−k the unique irreducible (3 − k)- dimensional representation. Moreover, a calculation yields that d ∈ C corresponding to ∂ d · θ ∂θ ∈ Vec0¯(M) acts on Pk by multiplication with −d and on P2−k by multiplication with d.

89 If k < 0 or k > 2, we have

[Vec1¯(M), Vec1¯(M)] = 0. ∼ ∂ ∂ ∂ In the case k = 0, we have Pk = C. Since [ ∂θ , q(z)θ ∂z ] = q(z) ∂z for any q ∈ P2, we get ∂ ∼ [Vec1¯(M), Vec1¯(M)] = a(z) a ∈ P2 = sl2(C), ∂z and the map P0 ×P2 → Vec0¯(M), (X,Y ) 7→ [X,Y ], corresponds to C×P2 → Vec0¯(M), ∂ (p, q(z)) 7→ p · q(z) ∂z . ∼ Similarly, if k = 2, we have P2−k = C, and 2 ∂ ∂ [Vec1¯(M), Vec1¯(M)] = (α0 + α1z + α2z ) + (α1 + 2α2z)θ α0, α1, α2 ∈ C ∂z ∂θ ∼ = sl2(C)

∂ ∂ ∂ 0 ∂ since [p(z) ∂θ , θ ∂z ] = p(z) ∂z +p (z)θ ∂θ . The map P2×P0 → Vec0¯(M), (X,Y ) 7→ [X,Y ], ∂ 0 ∂ corresponds to P2 × C → Vec0¯(M), (p(z), q) 7→ q · p(z) ∂z + q · p (z)θ ∂θ . ∼ 2 2 If k = 1, then Pk ⊕ P2−k = C ⊕ C . We have

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂θ , θ ∂z = ∂z , z ∂θ , θ ∂z = z ∂z + θ ∂θ ,

∂ ∂ ∂ ∂ ∂ 2 ∂ ∂ ∂θ , zθ ∂z = z ∂z , z ∂θ , zθ ∂z = z ∂z + zθ ∂θ , and consequently [Vec1¯(M), Vec1¯(M)] = Vec0¯(M). Note that Aut(M) carries the structure of a split Lie supergroup if and only if k < 0 or k > 2. For the deﬁnition of a split Lie supergroup see e.g. Proposition 4 in [Vis11].

Example 5.6.2. Let M = (P1C, OM) be a split complex supermanifold of dimension dim M = (1|2) associated with O(−k1) ⊕ O(−k2), k1, k2 ∈ Z. We determine the group Aut0¯(M) of automorphisms M → M. We choose coordinates z, θ1, θ2 for U0 and w, η1, η2 for U1 as described above so that ∗ −1 ∗ −k the transition map χ is given by χ (w) = z and χ (ηj) = z j θj. The action of PSL2(C) on P1C by M¨obiustransformations lifts to an action of a b SL2(C) on M by letting A = c d ∈ SL2(C) act by the automorphism ϕA : M → M with pullback c + dz ϕ∗ (z) = and ϕ∗ (θ ) = (a + bz)−kj θ A a + bz A j j as a morphism over appropriate subsets of U0, and aw + b ϕ∗ (w) = and ϕ∗ (η ) = (cw + d)−kj η A cw + d A j j over appropriate subsets of U1. Using the transition map χ one might also calculate the representation of ϕ in coordinates as a morphism over subsets U0 → U1 and U1 → U0. −1 0 If k1 and k2 are both even, we have ϕA = IdM for A = 0 −1 and thus we get an action of PSL2(C) on M. Consider the homomorphism of Lie groups Ψ : Aut0¯(M) → Aut(P1C) assigning to each automorphism ϕ : M → M the underlying biholomorphic mapϕ ˜ : P1C → P1C.

90 ∼ This homomorphism Ψ is surjective since Aut(P1C) = PSL2(C) and since the PSL2(C)- action on P1C lifts to an action (of SL2(C)) on the supermanifold M. The kernel ker Ψ consists of those automorphisms ϕ : M → M whose underlying mapϕ ˜ is the identity P1C → P1C. This kernel is a normal subgroup, SL2(C) acts on ker Ψ, and we obtain: ( ∼ ker Ψ o PSL2(C) if k1 and k2 are both even Aut0¯(M) = ker Ψ o SL2(C) otherwise Thus, it remains to determine ker Ψ. Let ϕ : M → M be an automorphism withϕ ˜ = Id. Let f and bjk, j, k = 1, 2, be ∼ holomorphic functions on U0 = C such that the pullback of ϕ over U0 is given by ∗ ∗ ϕ (z) = z + f(z)θ1θ2 and ϕ (θ) = B(z)θ,

b11(z) b12(z) ∗ ∗ where B(z) = and ϕ (θ) = B(z)θ is an abbreviation for ϕ (θj) = b21(z) b22(z) bj1(z)θ1 + bj2(z)θ2 for j = 1, 2. Similarly, let g and cjk be holomorphic functions ∼ on U1 = C such that the pullback of ϕ over U1 is given by ∗ ∗ ϕ (w) = w + g(w)η1η2 and ϕ (η) = C(z)η, where C(z) = c11(z) c12(z) . The compatibility condition with the transition map χ c21(z) c22(z) now gives the relation 1 f(z) = −z2−(k1+k2)g for z ∈ ∗. z C

Therefore, f and g are both polynomials of degree at most 2 − (k1 + k2), and they are zero in the case k1 + k2 > 2. For the matrices B and C we get the relation zk1 0 1 z−k1 0 B(z) = C for z ∈ ∗. 0 zk2 z 0 z−k2 C

1 ∗ If k1 = k2, this implies B(z) = C z for all z ∈ C . Thus, B(z) = B and C(w) = C are constant matrices, and B = C ∈ GL2(C) since ϕ was assumed to be invertible. Consequently, we have ∼ ker Ψ = P2−(k1+k2) o GL2(C) in the case k1 = k2, where Pl denotes again the space of polynomials of degree at most l if l ≥ 0 and Pl = {0} otherwise. The group structure on the semidirect product is given by (f1(z),B1) · (f2(z),B2) = (det B1f1(z) + f2(z),B1B2).

Let now k1 6= k2. After possibly changing coordinates we may assume k1 > k2. We have

k1 −k1 1 k1−k2 1 z 0 1 z 0 c11 z z c12 z B(z) = k2 C −k2 = k2−k1 1 1 0 z z 0 z z c21 z c22 z ∗ for all z ∈ C . This implies that b11 = c11 and b22 = c22 are constants. Since we assume k1 > k2, we also get b21 = c21 = 0 and b12 and c12 are polynomials of degree at most k1 − k2. Therefore, ∼ λ p(z) ∗ ker Ψ = P2−(k1+k2) o λ, µ ∈ C , p ∈ Pk1−k2 , 0 µ

91 and the group structure is again given by (f1(z),B1) · (f2(z),B2) = (det B1f1(z) + n o f (z),B B ) for f , f ∈ P , B ,B ∈ λ p(z) λ, µ ∈ ∗, p ∈ P . 2 1 2 1 2 2−(k1+k2) 1 2 0 µ C k1−k2 The semidirect product ker Ψ o SL2(C) (or ker Ψ o PSL2(C)) is a direct product if and only if k1 = k2 and k1 + k2 ≥ 2.

Example 5.6.3. Let M = (P1C, OM) be the complex supermanifold of dimension ∗ ∗ dim M = (1|2) given by the transition map χ : U0 → U1 with pullback

1 1 1 χ∗(w) = + θ θ and χ∗(η ) = θ . z z3 1 2 j z2 j The supermanifold M is not split and the associated split supermanifold corresponds to O(−2) ⊕ O(−2); see e.g. [BO96]. As in the previous example, the action of PSL2(C) on P1C by M¨obiustransforma- a b tions lifts to an action of PSL2(C) on M. Let A denote the class of c d ∈ SL2(C) in PSL2(C). Then A acts by the morphism ϕA : M → M whose pullback as a morphism over appropriate subsets of U0 is given by

c + dz b 1 ϕ∗ (z) = − θ θ and ϕ∗ (θ ) = θ . A a + bz (a + bz)3 1 2 A j (a + bz)2 j

∼ Let Ψ : Aut0¯(M) → Aut(P1C) = PSL2(C) denote again the Lie group homomorphism which assigns to an automorphism of M the underlying automorphism of P1C. The assignment A 7→ ϕA ∈ Aut0¯(M) deﬁnes a section PSL2(C) → Aut0¯(M) of Ψ, and we have ∼ Aut0¯(M) = ker Ψ o PSL2(C).

The section PSL2(C) → Aut0¯(M) induces on the level of Lie algebras the morphism a b σ : sl2(C) ,→ Vec0¯(M), which maps an element c −a ∈ sl2(C) to the vector ﬁeld on M whose restriction to U0 is 2 ∂ ∂ ∂ c − 2az − bz − bθ1θ2 − 2(a + bz) θ1 + θ2 . ∂z ∂θ1 ∂θ2

We now calculate the kernel ker Ψ. Let ϕ ∈ ker Ψ. Its underlying mapϕ ˜ is the identity and we thus have

∗ ∗ ϕ (z) = z + a0(z)θ1θ2 and ϕ (θ) = A0(z)θ on U0 and ∗ ∗ ϕ (w) = w + a1(w)η1η2 and ϕ (η) = A1(w)η on U1 for holomorphic functions a0 and a1 and invertible matrices A0 and A1 whose ∗ ∗ entries are holomorphic functions. The notation ϕ (θ) = A0(z)θ (and similarly ϕ (η) = ∗ A1(w)η) is again an abbreviation for ϕ (θj) = (A0(z))j1θ1 +(A0(z))j2θ2, where A0(z) = ((A0(z))jk)1≤j,k≤2. A calculation with the transition map χ then yields the relations

1 1 1 1 1 A (w) = A and a (w) = det A − 1 − a 1 0 w 1 w 0 w w 0 w

92 ∗ for any w ∈ C . Since a0, a1, A0, and A1 are holomorphic on C, we get that A0 = A1 ∼ are constant matrices, det A0 = 1, and a0 = a1 = 0. Therefore, ker Ψ = SL2(C), and its Lie algebra is ∂ ∂ a11 a12 Lie(ker Ψ) = (a11θ1 + a12θ2) + (a21θ1 + a22θ2) ∈ sl2(C) . ∂θ1 ∂θ2 a21 a22

Since Lie(ker Ψ) and σ(Lie(PSL2(C)) commute, the semidirect product ker ΨoPSL2(C) is direct and we have ∼ Aut0¯(M) = SL2(C) × PSL2(C). Remark in particular that this group is diﬀerent from the automorphism group of the corresponding split supermanifold N which is associated with O(−2) ⊕ O(−2) and for ∼ which we calculated that Aut0¯(N ) = GL2(C) × PSL2(C).

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