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Global properties of supermanifolds and their bodies

Published online by Cambridge University Press:  24 October 2008

Peter Bryant
Peter Bryant
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge CB2 1SB
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Abstract

In supersymmetric field theories, the use of Berezin-like integrals for curved superspace presupposes the existence of a smooth manifold structure on the underlying curved space (the body). The aim of this paper is to address the following question: when does a Rogers supermanifold have a body manifold?

Elementary techniques from the theory of foliations are used to derive some necessary and sufficient conditions. The main result is the necessity of regularity of the soul foliation in the sense of Catenacci, Reina and Teofilatto. Regularity is not sufficient (for example, punctured superspace is regular with non-Hausdorff body). A sufficient condition (albeit highly sensitive to the choice of charts) is the transitivity of the body relation ˜, a property enjoyed by De Witt supermanifolds.

We show that the body of a quotient supermanifold S/Γ is the quotient of the body (under the induced action). Several examples in the text illustrate that this is extremely useful.

New characterizations ρ-supermanifolds and their saturations are given and the saturation construction is extended to all bodied supermanifolds.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 107 , Issue 3 , May 1990 , pp. 501 - 523
Copyright
Copyright © Cambridge Philosophical Society 1990

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References

REFERENCES

[1]Abraham, R., Marsden, J. and Ratiu, T.. Manifolds, Tensor Analysis and Applications (Addison-Wesley, 1983).Google Scholar
[2]Batchelor, M.. Graded manifolds and supermanifolds. In Mathematical Aspects of Superspace (editors Clarke, C. J. S., Rosenblum, A. and Seifert, H.-J.) (Reidel, 1981), pp. 91–133.Google Scholar
[3]Bruzzo, U. and Cianci, R.. Structure of supermanifolds and supersymmetry transformations. Comm. Math. Phys. 95 (1984), 393–400.Google Scholar
[4]Bruzzo, U. and Cianci, R.. On the structure of superfields in a field theory on a supermanifold. Lett. Math. Phys. 11 (1986), 21–26.CrossRefGoogle Scholar
[5]Bryant, P.. De Witt supermanifolds and infinite-dimensional ground rings. J. London Math. Soc. 39 (1989), 347–368.Google Scholar
[6]Bryant, P.. The structure of De Witt supermanifolds. Riv. Mat. Pura Appl. 3 (1988), 57–79.Google Scholar
[7]Catenacci, R., Reina, C. and Teofilatto, P.. On the body of supermanifolds. J. Math. Phys. 26 (1985), 671–674.CrossRefGoogle Scholar
[8]Camacho, C. and Lins Neto, A.. Geometric Theory of Foliations (Birkhäuser, 1985).CrossRefGoogle Scholar
[9]De Witt, B.. Supermanifolds (Cambridge University Press, 1984).Google Scholar
[10]Freund, P.. Introduction to Supersymmetry (Cambridge University Press, 1983).Google Scholar
[11]Hoyos, J., Quirós, M., Mittelbrunn, J. and De Urríes, F.. Generalised supermanifolds II: Analysis on superspace and III: ρ-supermanifolds. J. Math. Phys. 25 (1984), 841–854.Google Scholar
[12]Molino, P.. Riemannian Foliations (Birkhäuser, 1988).Google Scholar
[13]Rabin, J. and Crane, L.. Global properties of supermanifolds. Comm. Math. Phys. 100 (1985), 141–160.Google Scholar
[14]Rabin, J. and Crane, L.. How different are the supermanifolds-of Rogers and De Witt? Comm. Math. Phys. 102 (1985), 123–137.Google Scholar
[15]Rogers, A.. A global theory of supermanifolds. J. Math. Phys. 21 (1980), 1352–1365.Google Scholar
[16]Rogers, A.. Some examples of compact supermanifolds with non-abelian fundamental group. J. Math. Phys. 22 (1981), 443–444.Google Scholar
[17]Rogers, A.. The existence of global integral forms on supermanifolds. J. Math. Phys. 26 (1985), 2749–2753.Google Scholar
[18]Rogers, A.. Integration on supermanifolds. In Mathematical Aspects of Superspace (editors Clarke, C. J. S., Rosenblum, A. and Seifert, H.-J.) (Reidel, 1981), pp. 149–160.Google Scholar
[19]Rothstein, M.. Integration on non-compact supermanifolds. Trans. Amer. Math. Soc. 299 (1987), 387.Google Scholar