Deformation of Dirac Structures via L∞ Algebras
| Autor: | Mykola Matviichuk, Marco Gualtieri, Geoffrey Scott |
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| Rok vydání: | 2018 |
| Předmět: | |
| Zdroj: | International Mathematics Research Notices. 2020:4295-4323 |
| ISSN: | 1687-0247 1073-7928 |
| DOI: | 10.1093/imrn/rny134 |
| Popis: | The deformation theory of a Dirac structure is controlled by a differential graded Lie algebra that depends on the choice of an auxiliary transversal Dirac structure; if the transversal is not involutive, one obtains an $L_\infty $ algebra instead. We develop a simplified method for describing this $L_\infty $ algebra and use it to prove that the $L_\infty $ algebras corresponding to different transversals are canonically $L_\infty $–isomorphic. In some cases, this isomorphism provides a formality map, as we show in several examples including (quasi)-Poisson geometry, Dirac structures on Lie groups, and Lie bialgebras. Finally, we apply our result to a classical problem in the deformation theory of complex manifolds; we provide explicit formulas for the Kodaira–Spencer deformation complex of a fixed small deformation of a complex manifold, in terms of the deformation complex of the original manifold. |
| Databáze: | OpenAIRE |
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