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Commentary on the article of Manin. (English) Zbl 0595.53071

Arbeitstag. Bonn 1984, Proc. Meet. Max-Planck-Inst. Math., Bonn 1984, Lect. Notes Math. 1111, 103-109 (1985).
[For the entire collection see Zbl 0547.00007.]
According to the introduction, this commentary on the article of Yu. I. Manin in the same volume [see ibid. 59-101 (1985; Zbl 0579.14002) and a more detailed version of that paper in Russ. Math. Surv. 39, No.6, 51- 83 (1984; Zbl 0577.14002)] ”consists of very general and speculative remarks about large areas of contemporary mathematics”. The central theme of these comments revolves not only about the resurgence of the traditional links between mathematics and physics, but is concerned also with the role played by certain branches of mathematics, such as algebraic geometry and number theory, whose relevance to physical theories had not been recognized until fairly recently. Several important examples of interactions of this kind are identified. It is suggested that the development of supermanifolds as presented by Manin should throw some light on the supersymmetry theories of physics; however, some essential ingredients still appear to be lacking, which leads the author to present a tentative proposal with regard to possible geometrical interpretations of fermionic variables. This seems to be based on the view that ”supergeometry should be some kind of limit of ordinary geometry and not an entirely different kind of entity constructed simply by formal analogy”.
Reviewer: H.Rund

MSC:

53C80 Applications of global differential geometry to the sciences
00A79 Physics
83E50 Supergravity
81T60 Supersymmetric field theories in quantum mechanics
81T08 Constructive quantum field theory