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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2403.15515 |
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| _version_ | 1866929428266745856 |
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| author | Kobayashi, Kazushi |
| author_facet | Kobayashi, Kazushi |
| contents | Let $(X^n,\check{X}^n)$ be a mirror pair of an $n$-dimensional complex torus $X^n$ and its mirror partner $\check{X}^n$. Then, by SYZ transform, we can construct a holomorphic line bundle with an integrable connection from each pair of a Lagrangian section of $\check{X}^n\to \mathbb{R}^n/\mathbb{Z}^n$ and a unitary local system along it, and those holomorphic line bundles with integrable connections forms a dg-category $DG_{X^n}$. In this paper, we focus on a certain B-field transform of the generalized complex structure induced from the complex structure on $X^n$, and interpret it as the deformation $X_{\mathcal{G}}^n$ of $X^n$ by a flat gerbe $\mathcal{G}$. Moreover, we construct the deformation of $DG_{X^n}$ associated to the deformation from $X^n$ to $X_{\mathcal{G}}^n$, and also discuss the homological mirror symmetry between $X_{\mathcal{G}}^n$ and its mirror partner on the object level. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15515 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a B-field transform of generalized complex structures over complex tori Kobayashi, Kazushi Differential Geometry High Energy Physics - Theory 14J33, 53D37, 53D18, 14F08, 53C08 Let $(X^n,\check{X}^n)$ be a mirror pair of an $n$-dimensional complex torus $X^n$ and its mirror partner $\check{X}^n$. Then, by SYZ transform, we can construct a holomorphic line bundle with an integrable connection from each pair of a Lagrangian section of $\check{X}^n\to \mathbb{R}^n/\mathbb{Z}^n$ and a unitary local system along it, and those holomorphic line bundles with integrable connections forms a dg-category $DG_{X^n}$. In this paper, we focus on a certain B-field transform of the generalized complex structure induced from the complex structure on $X^n$, and interpret it as the deformation $X_{\mathcal{G}}^n$ of $X^n$ by a flat gerbe $\mathcal{G}$. Moreover, we construct the deformation of $DG_{X^n}$ associated to the deformation from $X^n$ to $X_{\mathcal{G}}^n$, and also discuss the homological mirror symmetry between $X_{\mathcal{G}}^n$ and its mirror partner on the object level. |
| title | On a B-field transform of generalized complex structures over complex tori |
| topic | Differential Geometry High Energy Physics - Theory 14J33, 53D37, 53D18, 14F08, 53C08 |
| url | https://arxiv.org/abs/2403.15515 |