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Symmetries of Supermanifolds Symmetries of Supermanifolds DISSERTATION zur Erlangung des Doktorgrades der Naturwissenschaften an der Fakult¨atf¨urMathematik der Ruhr-Universit¨atBochum 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 exis- tence 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 1j1 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 (1j1)-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 ' : 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 ' : W ⊂ R × R ! R, '(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 super- manifolds. The existence of a local action with a given infinitesimal action on a super- manifold 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 distribu- tion 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 2 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 ' : W ⊆ G × M ! M on M which induces the infinitesimal action λ. Moreover, any local action ' : 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¯.
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