The structure of supermanifolds

Batchelor, Marjorie

doi:10.1090/S0002-9947-1979-0536951-0

The structure of supermanifolds

Author: Marjorie Batchelor
Journal: Trans. Amer. Math. Soc. 253 (1979), 329-338
MSC: Primary 58A05; Secondary 83E99
DOI: https://doi.org/10.1090/S0002-9947-1979-0536951-0
MathSciNet review: 536951
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The increasing recognition of Lie superalgebras and their importance in physics inspired a search to find an object, a ``supermanifold", which would realize the geometry implicit in Lie superalgebras. This paper analyzes the structure of supermanifolds as defined by B. Kostant. The result is the following structure theorem.

The Main Theorem. If E is a real vector bundle over the smooth manifold X, let $\Lambda E$ be the associated exterior bundle and let $\Gamma (\Lambda E)$ be the sheaf of sections of $\Lambda E$ . Then every supermanifold over X is isomorphic to $\Gamma (\Lambda E)$ for some vector bundle E over X.

Although the vector bundle E is not unique but is determined only up to isomorphism, and the isomorphism guaranteed is not canonical, the existence of the isomorphism provides a base for a better understanding of geometry in the graded setting.

[1] F. A. Berezin, The method of second quantization, Translated from the Russian by Nobumichi Mugibayashi and Alan Jeffrey. Pure and Applied Physics, Vol. 24, Academic Press, New York-London, 1966. MR 0208930
[2] F. A. Berezin and G. I. Kac, Lie groups with commuting and anticommuting parameters, Mat. Sb. (N.S.) 82 (124) (1970), 343–359 (Russian). MR 0265520
[3] F. Berezin and D. Leites, Supervarieties, Soviet Math. Dokl. 16 (1975), 1218-1222.
[4] L. Corwin, Y. Ne’eman, and S. Sternberg, Graded Lie algebras in mathematics and physics (Bose-Fermi symmetry), Rev. Modern Phys. 47 (1975), 573–603. MR 0438925, https://doi.org/10.1103/RevModPhys.47.573
[5] J. Dell and L. Smolin, A differential-geometric formulation of supersymmetry, Harvard Preprint (in preparation).
[6] Friedrich Hirzebruch, Topological methods in algebraic geometry, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Translated from the German and Appendix One by R. L. E. Schwarzenberger; With a preface to the third English edition by the author and Schwarzenberger; Appendix Two by A. Borel; Reprint of the 1978 edition. MR 1335917
[7] V. G. Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8–96. MR 486011, https://doi.org/10.1016/0001-8708(77)90017-2
[8] Bertram Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential geometrical methods in mathematical physics (Proc. Sympos., Univ. Bonn, Bonn, 1975) Springer, Berlin, 1977, pp. 177–306. Lecture Notes in Math., Vol. 570. MR 0580292