Homological vector fields over differentiable stacks
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916172025298944 |
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| author | Álvarez, Daniel Cueca, Miquel |
| author_facet | Álvarez, Daniel Cueca, Miquel |
| contents | In this work we solve the problem of providing a Morita invariant definition of Lie and Courant algebroids over Lie groupoids. By relying on supergeometry, we view these structures as instances of vector fields on graded groupoids which are homological up to homotopy. We describe such vector fields in general from two complementary viewpoints: firstly, as Maurer-Cartan elements in a differential graded Lie algebra of multivector fields and, secondly, we also view them from a categorical approach, in terms of functors and natural transformations. Thereby, we obtain a unifying conceptual framework for studying LA-groupoids, $L_2$-algebroids (including semistrict Lie 2-algebras and 2-term representations up to homotopy), infinitesimal gerbe prequantizations, higher gauge theory (specifically, 2-connections on 2-bundles), quasi-Poisson groupoids and (twisted) multiplicative Courant algebroids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_14871 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Homological vector fields over differentiable stacks Álvarez, Daniel Cueca, Miquel Differential Geometry Mathematical Physics Symplectic Geometry 18F20, 58A50, 53D17, 58H05 In this work we solve the problem of providing a Morita invariant definition of Lie and Courant algebroids over Lie groupoids. By relying on supergeometry, we view these structures as instances of vector fields on graded groupoids which are homological up to homotopy. We describe such vector fields in general from two complementary viewpoints: firstly, as Maurer-Cartan elements in a differential graded Lie algebra of multivector fields and, secondly, we also view them from a categorical approach, in terms of functors and natural transformations. Thereby, we obtain a unifying conceptual framework for studying LA-groupoids, $L_2$-algebroids (including semistrict Lie 2-algebras and 2-term representations up to homotopy), infinitesimal gerbe prequantizations, higher gauge theory (specifically, 2-connections on 2-bundles), quasi-Poisson groupoids and (twisted) multiplicative Courant algebroids. |
| title | Homological vector fields over differentiable stacks |
| topic | Differential Geometry Mathematical Physics Symplectic Geometry 18F20, 58A50, 53D17, 58H05 |
| url | https://arxiv.org/abs/2403.14871 |