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Quotient Supermanifolds BULL. AUSTRAL. MATH. SOC. 58A50 VOL. 58 (1998) [107-120] QUOTIENT SUPERMANIFOLDS CLAUDIO BARTOCCI, UGO BRUZZO, DANIEL HERNANDEZ RUIPEREZ AND VLADIMIR PESTOV A necessary and sufficient condition for the existence of a supermanifold structure on a quotient defined by an equivalence relation is established. Furthermore, we show that an equivalence relation it on a Berezin-Leites-Kostant supermanifold X determines a quotient supermanifold X/R if and only if the restriction Ro of R to the underlying smooth manifold Xo of X determines a quotient smooth manifold XQ/RQ- 1. INTRODUCTION The necessity of taking quotients of supermanifolds arises in a great variety of cases; just to mention a few examples, we recall the notion of supergrassmannian, the definition of the Teichmiiller space of super Riemann surfaces, or the procedure of super Poisson reduction. These constructions play a crucial role in superstring theory as well as in supersymmetric field theories. The first aim of this paper is to prove a necessary and sufficient condition ensuring that an equivalence relation in the category of supermanifolds gives rise to a quotient supermanifold; analogous results, in the setting of Berezin-Leites-Kostant (BLK) super- manifolds, were already stated in [6]. Secondly and rather surprisingly at that, it turns out that for Berezin-Leites-Kostant supermanifolds any obstacles to the existence of a quotient supermanifold can exist only in the even sector and are therefore purely topological. We demonstrate that an equivalence relation Ron a. BLK supermanifold X determines a quotient BLK supermanifold X/R if and only if the restriction of R to the underlying manifold Xo of X determines a quotient smooth manifold. The definition of supermanifold we adopt here encompasses both the BLK super- manifold theory [5] and, in a certain sense, the DeWitt-Rogers approach. The basic idea is to introduce an axiomatics yielding a category of supermanifolds where the usual Received 22nd December, 1997 Research partly supported by the Italian Ministry for University and Research through the research project 'Geometria delle varieta differenziabili', by the Spanish DGICYT, by the Italian GNSAGA, by the Internal Grants Committee of the Victoria University of Wellington, and by the Marsden Fund grant 'Foundations of supergeometry'. Copyright Clearance Centre, Inc. Serial-fee code: 0004-9729/98 SA2.00+0.00. 107 Downloaded from https://www.cambridge.org/core. IP address: 170.106.39.241, on 04 Oct 2021 at 17:31:52, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700032044 108 C. Bartocci, U. Bruzzo, D. Hernandez Ruiperez and V. Pestov [2] geometric constructions (products, bundles, characteristic classes, ...) work reasonably well; for a more detailed presentation the reader is referred to [1, 2, 3]. Let B — BQ © B\ be a Z2-graded-commutative associative unital K-algebra (or C- algebra); in other words, for a,/3 G Z2, we have a0 Ba B0 C Ba+0, ab = (-l) ba if a e Ba, b £ Bp . We denote by Bm>n the direct sum B™ © Bf. For the sake of simplicity we shall assume that B is finite-dimensional and that it is generated, as a unital algebra, by the odd part B\. (Equivalently: B is a graded factor-algebra of a finite-dimensional Grassmann algebra, for example, the exterior algebra A #i) The latter assumption easily implies that every element 1 6 Bis uniquely decomposed as the sum of a (real or complex) number P(x) and a nilpotent a(x). By superspace over B we mean a triple (X, A, ev), where X is a paracompact topo- logical space, A is a sheaf of Z2-graded-commutative B-algebras, and ev: A -» Cx is a morphism of sheaves of graded B-algebras (here Cx is the sheaf of 5-valued continu- ous functions on X). We shall sometimes write ip for ev(<p). A morphism of superspaces (/> /"): [X, A, evx J —> \Y, B, evy) is a pair consisting of a continuous map /: X -»• Y and a morphism of sheaves of graded 5-algebras /': B —> f*A such that ev* o /" = /* o evY. By the morphism ev: A —t Cx one can evaluate germs of superfunctions — that is, sections of A — at a point p € X. In this way, we can define the graded ideal £p of the stalk Ap formed by the germs of superfunctions vanishing at p: £p = {<p € Ap I (pip) = 0}. A supermanifold of dimension (m, n) is by definition a superspace (X, A, ev) satis- fying the following four Axioms. (The supermanifolds we characterise here were called i?°°-supermanifolds in reference [3].) AXIOM 1. The graded .4-dual of the sheaf of derivations, Ver'A, is a locally free graded .4-module of rank (m, n). Every point p € X has an open neighbourhood U with sections m n m 1 n x\...,x e A(U)0, y\...,y e A(U)l such that {dx\...,dx ,dy ,..., dy ) is a graded basis of Ver*A(U) over A(U). AXIOM 2. Given a coordinate chart (U, xl,..., xm, y1,..., yn), the assignment defines a homeomorphism of U onto an open subset in Bm'n. AXIOM 3 For every pel the ideal Sip is finitely generated. In order to state the last remaining Axiom we need to topologise the algebras of sections A(U). Let V(A) denote the sheaf of differential operators over A, that is, the Downloaded from https://www.cambridge.org/core. IP address: 170.106.39.241, on 04 Oct 2021 at 17:31:52, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700032044 [3] Quotient supermanifolds 109 graded ,4-module generated multiplicatively by Ver A over A. We define the family of prenorms PL,KM = max L(<p)(p) , P€K II IIB with L € V\A\u\ K C U compact, and || ||B a Banach norm on B. In this way we induce in A{U) the structure of a locally convex graded B-algebra. AXIOM 4. For every open subset U C X, the topological algebra A(U) is complete Hausdorff. We notice that Axiom 2 implies that X is locally Euclidean, while by Axiom 3 one can prove — via a graded version of Nakayama's lemma — the existence of local Taylor expansions. One can prove that any (TO, n) dimensional supermanifold is locally isomorphic to m the standard supermanifold (B '",Gm>n) [3]. Quite obviously, the above definition of supermanifold includes the usual notion of smooth or complex analytic manifold; in this case, the graded algebra B reduces to R or C and the sheaf A is the sheaf of germs of the appropriate (smooth or holomorphic) class of functions on X. When B is R or C, but A is a genuine sheaf of graded algebras (that is A\ ^ 0), the notion of BLK supermanifold is recovered. Finally, in the case of a Grassmann algebra B over R or C, the four Axioms we have stated are equivalent to the definition of G-supermanifold; a detailed treatment of this equivalence, together with an analysis of the axiomatics, can be found in [1]. The case of an infinite-dimensional ground algebra B is examined in [3]. In the next Section we collect some preliminary results concerning sub-supermani- folds and morphisms of supermanifolds, while in Section 3 we shall prove our main the- orem. In Section 4 we study quotients on BLK supermanifolds. 2. SUB-SUPERMANIFOLDS, TRANSVERSALITY, AND FIBRE PRODUCT We introduce the notions of immersion, submersion in the category of supermanifolds as straightforward generalisations of the corresponding properties of smooth manifolds. DEFINITION 2.1: A morphism of supermanifolds /: (Y, B) -*• (X, A) is (1) a closed immersion if /: Y —» X is a closed topological embedding, and /": A -» /»B is an epimorphism; (2) an open immersion if /: Y —> X is an open topological embedding, and B ~ pA. In local coordinates any closed immersion reduces to an immersion of open sets of model superspaces. PROPOSITION 2.2. Let f:(Y,B) -» {X,A) be a closed immersion, with dim (Y, B) = (m, n) and dim (X, B) — (r, s). For each y eY there are a neighbourhood U Downloaded from https://www.cambridge.org/core. IP address: 170.106.39.241, on 04 Oct 2021 at 17:31:52, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700032044 110 C. Bartocci, U. Bruzzo, D. Hernandez Ruiperez and V. Pestov [4] ofy and a neighbourhood V of f(y), together with isomorphisms (u,B\u) ^ (U,Gm,n) and (V,A\v) ^ (v, Gr,s), such that the induced morphism (U,Gm,n) -* (v,Gr^ is the natural immersion. PROOF: After fixing a coordinate system (x1,..., xT, £l,..., £*) in a neighbourhood of f(y), one must show that the image in T*(Y, B) of the differentials dx, d£ contains a basis. This is a direct consequence of simple algebraic facts about free graded modules. D Notice that in fact the property formulated in the above Proposition 2.2 is not characteristic of closed immersions (as simple examples at the level of smooth manifolds show), but rather of arbitrary sub-supermanifolds, see Definition 2.5 below. DEFINITION 2.3: A morphism of supermanifolds /: (Y,B) —• (X, A) is a submer- sion if for all y € Y the morphism of graded 5-modules fj{y): T*s{y) (X, A) -> T*{Y,B) is injective. The following Proposition is dual to Proposition 2.2, and is proved analogously.
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