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Integration and Bordism on Supermanifolds

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Integration and Bordism on Supermanifolds NEW ZEALAND JOURNAL OF MATHEMATICS Volume 28 (1999), 125-140 INTEGRATION AND BORDISM ON SUPERMANIFOLDS C. V ic t o r ia (Received July 1996) Abstract. An integration scheme for Z2-forms on Berezin-Leites-Kostant su­ permanifolds together with a bordism theory is introduced. Its relationship with the berezinian integration is stated. Stokes formula is proved for this integration scheme, thus supplying a pairing between the cohomology of Z2- forms and bordism cycles. It is proved that the bordism groups of a given supermanifold depend only on its split (Batchelor) structure. Introduction The de Rham complex of forms on manifolds was generalized in [28] to complexes of Z2-forms on real or complex differentiable Berezin-Leites-Kostant supermani­ folds, and a cohomology ring for them was obtained. In this paper we define an integration scheme for these Z2-forms (Section 3) and we prove that it verifies Stokes theorem (Theorem 4.2). Thus, the integration of cohomology classes of Z2-forms over bordism cycles of Berezin-Leites-Kostant supermanifolds - a no­ tion introduced in Section 5 - is well defined (Proposition 5.1). Whence, a pairing between cohomology and bordism is established. We also prove (Theorem 5.2) that the bordism groups, 0 (V)(X), of a supermanifold X are isomorphic to the bordism groups fi(.,.) (Gr(X)), where Gr is the functor which associates with any supermanifold its underlying linear supermanifold (also called its split (Batchelor) supermanifold). In order to better locate this paper in the existing literature, a few words should be said about the types of supermanifolds and the various relationships among them. Nowadays, the name supermanifold has basically two meanings: G-supermanifolds, and Berezin-Leites-Kostant supermanifolds (also called graded manifolds). G-supermanifolds are the closest to Rogers’ original theory of super­ manifolds [18]; they satisfy Rothstein’s axiomatics [21], and they contain several subcategories of supermanifolds (for instance, DeWitt, GH°°, H°° supermanifolds, etc.), which were sometimes considered as separate types of supermanifolds. The specific genealogy to which the general theory of G-supermanifolds gives rise to, together with some in-depth development of the various topics within this cate­ gory can be found in the monograph [3]. Berezin-Leites-Kostant supermanifolds, on the other hand, were introduced by Berezin and Leites in the East [4] [5] [13], and by Kostant in the West [12] (see also the book by Manin [15] for a thorough introduction to this theory). Even though there is a ono-to-one correspondence 1991 AMS Mathematics Subject Classification: 14F40, 57Q20, 58A10, 58A50, 58C35, 58C50. Key words and phrases: de Rham cohomology, Cobordism, differential forms, supermanifolds and graded manifolds, integration on manifolds; measures on manifolds, analysis on supermanifolds or garded manifolds. 126 C. VICTORIA between isomorphism classes of DeWitt supermanifolds and isomorphism classes of Berezin-Leites-Kostant supermanifolds, it is well-known that these categories are not equivalent because there are much more morphisms in the former than there are in the latter (see [3]). The framework for the integration and bordism schemes developed in this paper is that of Berezin-Leites-Kostant supermanifolds. A major problem in supermani­ fold theory is precisely that of defining a “suitable” integration theory (see problem (11) in [14]). As far as we know, there does not yet exist any general integration theory on G-supermanifolds. Nevertheless, there are some partial results. For in­ stance, the presentation of the berezinian sheaf made in [11] has been successfully used in [8] to construct a berezinian-like integration on DeWitt supermanifolds. Also, a path integration on Rogers supermanifolds has been developed in [19] [20]. On the other hand, some integration schemes have been proposed in the lit­ erature for Berezin-Leites-Kostant supermanifolds. The most popular one is the integration of compactly supported berezinian sections [4] [5] [13], along with its generalization for the non-compactly supported case [22]. Berezinian integration has been also used to define other volume sheaves: e.g., Berezinian integration is the origin of the pioneering work of [1] wherer | s-forms first appeared defined as objects whose restriction on compact (r, 5)-dimensional subsupermanifolds are sections of the berezinian sheaf (see also [29] [30] for a developement of these ideas); the complex of integral forms is defined as a tensor power of the berezinian sheaf with the dual of forms on supermanifolds. Thanks to this complex, a Stokes formula can be proved for the berezinian integration 6[]; the integral of the pseu­ dodifferential forms introduced in [7] coincides whit the berezinian integration on the (0, l)-tangent superbundles (see [7] and [15]). On the other hand, the integral of the superforms introduced in [9] is interpreted as the berezinian integral of the corresponding r | s-form by the Radon supertransformation (see [9] for details). The berezinian integration, however, does not satisfy the de Rham theorem, which constitutes one of its main drawbacks. Besides, its corresponding determinant - the berezinian - is not suitable for important algebraic operations such as Lagrange expansion. (See [26] for an alternative superdeterminant that does). There is also available an integration of 1-superforms along (1,1)-dimensional Berezin-Leites-Kostant supermanifolds introduced in [24] and used in [17] as an alternative to berezinian integration to perform calculus of variations. It does satisfy the de Rham theorem, but no generalization of it is known to higher dimen­ sions. This integration emerges from the theorem of existence and uniqueness for ordinary differential equations on Berezin-Leites-Kostant supermanifolds proved in [16]. This theorem generalizes and completes some results of [25] on integration of vector fields on supermanifolds (see problem (10) in [14] and also [15, Chp. 4, Section 4]). In [28] a Z2-graded de Rham cohomology functor - not determined by the usual de Rham cohomology of the underlying manifold - was introduced in the category of Berezin-Leites-Kostant super manifolds. It is an answer to problem (8) in the list of [14]. It is also to be said that a Z-graded, but not a Z2-graded, de Rham cohomology for G-supermanifolds can be found in [2] (see also [3]). In fact, for general G-supermanifolds this de Rham cohomology is not determined by the usual de Rham cohomology of the body either. However, this de Rham INTEGRATION AND BORDISM ON SUPERMANIFOLDS 127 cohomology of DeWitt supermanifolds is proved in [3] to be determined by the usual de Rham cohomology of its body. Thus, aside from the impossibility of a functorial comparison of the cohomologies of [28] and of [2] [3] (first because the categories of (isomorphism classes of) Berezin-Leites-Kostant supermanifolds and (isomorphism classes of) DeWitt supermanifolds are not equivalent, and moreover, because the cohomology in [28] takes values in a category of Z2-graded groups whereas that in [2] [3] does in Z-graded groups) the cohomology groups of a DeWitt supermanifold and of its corresponding Berezin-Leites-Kostant supermanifold are in general different. Here is now a brief description of the specific contents of this paper. In Section 1 we fix the notation, and recall from [28] the basic results on Z2-forms. In Section 2 we recall the more traditional description of integration of compactly supported berezinian sections (see [4] [5] [13] and also [15]) and the alternative description given in [11]. We also recall the integration of the non necessarily compactly case constructed in [22]. In Section 3 we define the integration of the Z2-graded forms on Berezin-Leites-Kostant supermanifolds (see Definition 3.3). This integration scheme does not use the berezinian integration at all, thus marking a major con­ trast with the integration theory for other types of forms on Berezin-Leites-Kostant supermanifolds introduced earlier in the literature (e.g., pseudodifferential forms, r | s-forms, superforms, etc.). Nevertheless, it emerges as a pleasant surprise the fact that the integration theory introduced here can in fact be related with the usual berezinian integration in the exact sense stated in Corollary 3.5. Stokes formula is proved in Section 4 (Theorem 4.2). This allows to define a pairing between the cohomology groups of Z2-graded forms constructed in [28] and the bordism groups introduced in Section 5. It must be mentioned that this pairing is different from that given for r | s-forms in [31] because of several reasons: Firstly, because the coho­ mology and the bordism groups used in [31] are defined only for the subcategory of Berezin-Leites-Kostant super manifolds with “proper” morphisms (in the sense of [29] [30]). By way of contrast, the cohomology introduced in [28] and the bordism introduced here, are well defined for the full category of Berezin-Leites-Kostant supermanifolds. Secondly, becasuse the restriction of the cohomology of Z2-forms to the subcategory of Berezin-Leites-Kostant supermanifolds with “proper” mor­ phisms is different from the cohomology of [29] [30], as it was proved in [28]. At any rate, one similarity is observed indeed: the bordism groups we introduce here are proved to depend only on the split class of the given supermanifold (see Theorem 5.2); a similar result holds true for the bordism groups of [31]. From now on all supermanifolds are assumed to be real differentiable (smooth) Berezin-Leites-Kostant supermanifolds. The contents of this work are part of the author’s Ph.D. dissertation [27]. The author deeply acknowledges the encouragement and support received by his advisor S. Xambo-Descamps. Special thanks are given to O.A. Sanchez-Valenzuela for all those helpful comments and criticisms. Last, but not least, the author would also like to thank the valuable suggestions made by the referee; they greatly contributed to the enlightening of the points made by this paper, by putting them into a wider perspective.
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