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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2511.05206 |
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| _version_ | 1866912693457256448 |
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| author | Kim, Taesu |
| author_facet | Kim, Taesu |
| contents | We introduce $L_{\infty}$-Kuranishi spaces by associating, to each chart, $L_{\infty}[1]$-algebras defined on open neighborhoods of the zero points of the Kuranishi section. We show that these objects collectively form a category, which naturally embeds the category of smooth manifolds. Certain notions in \cite{FOOO1} are modified to achieve desired categorical structures; for instance, the tangent bundle condition is interpreted as a quasi-isomorphism condition for the $L_{\infty}$-structures. In this process, the originally strict and rigid cocycle condition for coordinate changes is replaced by more flexible homotopy-theoretic compatibilities. To this end, a model of higher homotopy theory for $L_{\infty}[1]$-morphisms is proposed. Moreover, the moduli space of pseudoholomorphic disks with Lagrangian boundary condition is shown to serve as an example of $L_{\infty}$-Kuranishi spaces, provided that a Whitney stratification with a compatible system of tubular neighborhoods exists on each chart. Finally, the forgetful and evaluation maps for the moduli space are lifted to morphisms between $L_{\infty}$-Kuranishi spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_05206 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $L_\infty$-Kuranishi spaces and the moduli space of pseudoholomorphic maps Kim, Taesu Symplectic Geometry We introduce $L_{\infty}$-Kuranishi spaces by associating, to each chart, $L_{\infty}[1]$-algebras defined on open neighborhoods of the zero points of the Kuranishi section. We show that these objects collectively form a category, which naturally embeds the category of smooth manifolds. Certain notions in \cite{FOOO1} are modified to achieve desired categorical structures; for instance, the tangent bundle condition is interpreted as a quasi-isomorphism condition for the $L_{\infty}$-structures. In this process, the originally strict and rigid cocycle condition for coordinate changes is replaced by more flexible homotopy-theoretic compatibilities. To this end, a model of higher homotopy theory for $L_{\infty}[1]$-morphisms is proposed. Moreover, the moduli space of pseudoholomorphic disks with Lagrangian boundary condition is shown to serve as an example of $L_{\infty}$-Kuranishi spaces, provided that a Whitney stratification with a compatible system of tubular neighborhoods exists on each chart. Finally, the forgetful and evaluation maps for the moduli space are lifted to morphisms between $L_{\infty}$-Kuranishi spaces. |
| title | $L_\infty$-Kuranishi spaces and the moduli space of pseudoholomorphic maps |
| topic | Symplectic Geometry |
| url | https://arxiv.org/abs/2511.05206 |