On odd parameters in geometry
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916404186316800 |
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| author | Leites, Dimitry |
| author_facet | Leites, Dimitry |
| contents | 1) In 1976, looking at simple finite-dimensional complex Lie superalgebras, J.~Bernstein and I, and independently M.~Duflo, observed that certain divergence-free vectorial Lie superalgebras have deformations with odd parameters and conjectured that other simple Lie superalgebras have no such deformations (unpublished). Here, I prove this conjecture and overview the known classification of simple finite-dimensional complex Lie superalgebras, their presentations, realizations, and (very sketchily) relations with simple Lie (super)algebras over fields of positive characteristic.
2) Any supermanifold which is a ringed space of the form (a manifold $M$, the sheaf of sections of the exterior algebra of a vector bundle over $M$) is called split. Gawȩdzki (1977) and Batchelor (1979) proved that every smooth supermanifolds is split. In 1982, P. Green and Palamodov showed that a~complex-analytic supermanifold can be non-split, i.e., not diffeomorphic to a split supermanifold. So far, researchers considered, mostly, even obstructions to splitness. This lead them to the conclusion that any supermanifolds of superdimension $m|1$ is split. I'll show that there are non-split supermanifolds of superdimension $m|1$; for example, certain $1|1$-dimensional superstrings, the obstructions to their splitness correspond to odd parameters. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_17096 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On odd parameters in geometry Leites, Dimitry Representation Theory Mathematical Physics Primary 17A70, Secondary 14M30, 17B20 1) In 1976, looking at simple finite-dimensional complex Lie superalgebras, J.~Bernstein and I, and independently M.~Duflo, observed that certain divergence-free vectorial Lie superalgebras have deformations with odd parameters and conjectured that other simple Lie superalgebras have no such deformations (unpublished). Here, I prove this conjecture and overview the known classification of simple finite-dimensional complex Lie superalgebras, their presentations, realizations, and (very sketchily) relations with simple Lie (super)algebras over fields of positive characteristic. 2) Any supermanifold which is a ringed space of the form (a manifold $M$, the sheaf of sections of the exterior algebra of a vector bundle over $M$) is called split. Gawȩdzki (1977) and Batchelor (1979) proved that every smooth supermanifolds is split. In 1982, P. Green and Palamodov showed that a~complex-analytic supermanifold can be non-split, i.e., not diffeomorphic to a split supermanifold. So far, researchers considered, mostly, even obstructions to splitness. This lead them to the conclusion that any supermanifolds of superdimension $m|1$ is split. I'll show that there are non-split supermanifolds of superdimension $m|1$; for example, certain $1|1$-dimensional superstrings, the obstructions to their splitness correspond to odd parameters. |
| title | On odd parameters in geometry |
| topic | Representation Theory Mathematical Physics Primary 17A70, Secondary 14M30, 17B20 |
| url | https://arxiv.org/abs/2210.17096 |