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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2410.15664 |
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| _version_ | 1866912440695914496 |
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| author | Shemyakova, Ekaterina Yilmaz, Yagmur |
| author_facet | Shemyakova, Ekaterina Yilmaz, Yagmur |
| contents | It is well known that the chain map between the de Rham and Poisson complexes on a Poisson manifold also maps the Koszul bracket of differential forms into the Schouten bracket of multivector fields.
In the generalized case of a $P_\infty$-structure, where a Poisson bivector $P$ is replaced by an arbitrary even multivector obeying $[[P,P]]=0$, an analog of the chain map and an $L_\infty$-morphism from the higher Koszul brackets into the Schouten bracket are also known; however, they differ significantly in nature.
In the present paper, we address the problem of quantizing this picture. In particular, we show that the $L_\infty$-morphism is quantized into a single linear operator, which is a formal Fourier integral operator.
This paper employs Voronov's thick morphism technique and quantum Mackenzie-Xu transformations in the framework of $L_\infty$-algebroids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_15664 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A quantum anchor for higher Koszul brackets Shemyakova, Ekaterina Yilmaz, Yagmur Mathematical Physics Differential Geometry It is well known that the chain map between the de Rham and Poisson complexes on a Poisson manifold also maps the Koszul bracket of differential forms into the Schouten bracket of multivector fields. In the generalized case of a $P_\infty$-structure, where a Poisson bivector $P$ is replaced by an arbitrary even multivector obeying $[[P,P]]=0$, an analog of the chain map and an $L_\infty$-morphism from the higher Koszul brackets into the Schouten bracket are also known; however, they differ significantly in nature. In the present paper, we address the problem of quantizing this picture. In particular, we show that the $L_\infty$-morphism is quantized into a single linear operator, which is a formal Fourier integral operator. This paper employs Voronov's thick morphism technique and quantum Mackenzie-Xu transformations in the framework of $L_\infty$-algebroids. |
| title | A quantum anchor for higher Koszul brackets |
| topic | Mathematical Physics Differential Geometry |
| url | https://arxiv.org/abs/2410.15664 |