Y-algebroids and $E_{7(7)} \times \mathbb{R}^+$-generalised geometry
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| Format: | Preprint |
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2023
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| _version_ | 1866910719124963328 |
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| author | Hulik, Ondrej Malek, Emanuel Valach, Fridrich Waldram, Daniel |
| author_facet | Hulik, Ondrej Malek, Emanuel Valach, Fridrich Waldram, Daniel |
| contents | We define the notion of Y-algebroids, generalising the Lie, Courant, and exceptional algebroids that have been used to capture the local symmetry structure of type II string theory and M-theory compactifications to $D \geq 5$ dimensions. Instead of an invariant inner product, or its generalisation arising in exceptional algebroids, Y-algebroids are built around a specific type of tensor, denoted $Y$, that provides exactly the necessary properties to also describe compactifications to $D=4$ dimensions. We classify "M-exact" $E_7$-algebroids and show that this precisely matches the form of the generalised tangent space of $E_{7(7)} \times \mathbb{R}^+$-generalised geometry, with possible twists due to 1-, 4- and 7-form fluxes, corresponding physically to the derivative of the warp factor and the M-theory fluxes. We translate the notion of generalised Leibniz parallelisable spaces, relevant to consistent truncations, into this language, where they are mapped to so-called exceptional Manin pairs. We also show how to understand Poisson-Lie U-duality and exceptional complex structures using Y-algebroids. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_01130 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Y-algebroids and $E_{7(7)} \times \mathbb{R}^+$-generalised geometry Hulik, Ondrej Malek, Emanuel Valach, Fridrich Waldram, Daniel High Energy Physics - Theory Mathematical Physics Differential Geometry We define the notion of Y-algebroids, generalising the Lie, Courant, and exceptional algebroids that have been used to capture the local symmetry structure of type II string theory and M-theory compactifications to $D \geq 5$ dimensions. Instead of an invariant inner product, or its generalisation arising in exceptional algebroids, Y-algebroids are built around a specific type of tensor, denoted $Y$, that provides exactly the necessary properties to also describe compactifications to $D=4$ dimensions. We classify "M-exact" $E_7$-algebroids and show that this precisely matches the form of the generalised tangent space of $E_{7(7)} \times \mathbb{R}^+$-generalised geometry, with possible twists due to 1-, 4- and 7-form fluxes, corresponding physically to the derivative of the warp factor and the M-theory fluxes. We translate the notion of generalised Leibniz parallelisable spaces, relevant to consistent truncations, into this language, where they are mapped to so-called exceptional Manin pairs. We also show how to understand Poisson-Lie U-duality and exceptional complex structures using Y-algebroids. |
| title | Y-algebroids and $E_{7(7)} \times \mathbb{R}^+$-generalised geometry |
| topic | High Energy Physics - Theory Mathematical Physics Differential Geometry |
| url | https://arxiv.org/abs/2308.01130 |