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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The axioms of supermanifolds and a new structure arising from them

Author: Mitchell J. Rothstein
Journal: Trans. Amer. Math. Soc. 297 (1986), 159-180
MSC: Primary 58A50; Secondary 58C50
MathSciNet review: 849473
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Abstract: An analysis of supermanifolds over an arbitrary graded-commmutative algebra is given, proceeding from a set of axioms the first of which is that the derivations of the structure sheaf of a supermanifold are locally free. These axioms are satisfied not by the sheaf of $ {G^\infty }$ functions, as has been asserted elsewhere, but by an extension of this sheaf. A given $ {G^\infty }$ manifold may admit many supermanifold extensions, and it is unknown at present whether there are $ {G^\infty }$ manifolds that admit no such extension. When the underlying graded-commutative algebra is commutative, the axioms reduce to the Berezin-Kostant supermanifold theory.

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  • [1] M. Batchelor, The structure of supermanifolds, Trans. Amer. Math. Soc. 253 (1979), 329-338. MR 536951 (80h:58002)
  • [2] -, Two approaches to supermanifolds, Trans. Amer. Math. Soc. 258 (1980), 257-270. MR 554332 (81f:58001)
  • [3] F. A. Berezin and D. A. Leites, Supermanifolds, Soviet Math. Dokl. 16 (1975), 1218-1221. MR 0402795 (53:6609)
  • [4] C. P. Boyer and S. Gitler, The theory of $ {\mathcal{G}^\infty }$ supermanifolds, Tráns. Amer. Math. Soc. 285 (1984), 241-267. MR 748840 (87i:58010)
  • [5] B. S. DeWitt, Supermanifolds, Cambridge Univ. Press, Cambridge, 1984. MR 778559 (87b:58007)
  • [6] K. Gawedzki, Supersymmetries--mathematics of supergeometry, Ann. Inst. H. Poincaré, Sect. A 27 (1977), 335-366. MR 0489701 (58:9092)
  • [7] A. Jadczyk and K. Pilch, Superspaces and supersymmetries, Comm. Math. Phys. 78 (1980), 373-390. MR 603500 (82e:58002)
  • [8] B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential Geometric Methods in Mathematical Physics (Proc. Sympos. Univ. Bonn, Bonn, 1975), Lecture Notes in Math., vol. 570, Springer-Verlag, Berlin, 1977, pp. 177-306. MR 0580292 (58:28326)
  • [9] D. A. Leites, Introduction to supermanifold theory (preprint).
  • [10] J. M. Rabin and L. Crane, Global properties of supermanifolds, Comm. Math. Phys. 100 (1985), 141-160. MR 796165 (86m:58019)
  • [11] A. Rogers, A global theory of supermanifolds, J. Math. Phys. 21 (1980), 1352-1365. MR 574696 (82d:58001)
  • [12] M. Rothstein, Deformations of complex supermanifolds, Proc. Amer. Math. Soc. 95 (1985), 255-260. MR 801334 (87d:32041)

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Article copyright: © Copyright 1986 American Mathematical Society

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