Open Gromov-Witten invariants from the Fukaya category
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910345852878848 |
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| author | Hugtenburg, Kai |
| author_facet | Hugtenburg, Kai |
| contents | This paper proposes a framework to show that the Fukaya category of a symplectic manifold $X$ determines the open Gromov-Witten invariants of Lagrangians $L \subset X$. We associate to an object in an $A_\infty$-category an extension of the negative cyclic homology, called \emph{relative cyclic homology}. We extend the Getzler-Gauss-Manin connection to relative cyclic homology. Then, we construct (under simplifying technical assumptions) a relative cyclic open-closed map, which maps the relative cyclic homology of a Lagrangian $L$ in the Fukaya category of a symplectic manifold $X$ to the $S^1$-equivariant relative quantum homology of $(X,L)$. Relative quantum homology is the dual to the relative quantum cohomology constructed by Solomon-Tukachinsky. This is an extension of quantum cohomology, and comes equipped with a connection extending the quantum connection. We prove that the relative open-closed map respects connections. As an application of this framework, we show, assuming a construction of the relative cyclic open-closed map in a broader technical setup, that the Fukaya category of a Calabi-Yau variety determines the open Gromov-Witten invariants with one interior marked point for any null-homologous Lagrangian brane. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_08345 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Open Gromov-Witten invariants from the Fukaya category Hugtenburg, Kai Symplectic Geometry Algebraic Geometry This paper proposes a framework to show that the Fukaya category of a symplectic manifold $X$ determines the open Gromov-Witten invariants of Lagrangians $L \subset X$. We associate to an object in an $A_\infty$-category an extension of the negative cyclic homology, called \emph{relative cyclic homology}. We extend the Getzler-Gauss-Manin connection to relative cyclic homology. Then, we construct (under simplifying technical assumptions) a relative cyclic open-closed map, which maps the relative cyclic homology of a Lagrangian $L$ in the Fukaya category of a symplectic manifold $X$ to the $S^1$-equivariant relative quantum homology of $(X,L)$. Relative quantum homology is the dual to the relative quantum cohomology constructed by Solomon-Tukachinsky. This is an extension of quantum cohomology, and comes equipped with a connection extending the quantum connection. We prove that the relative open-closed map respects connections. As an application of this framework, we show, assuming a construction of the relative cyclic open-closed map in a broader technical setup, that the Fukaya category of a Calabi-Yau variety determines the open Gromov-Witten invariants with one interior marked point for any null-homologous Lagrangian brane. |
| title | Open Gromov-Witten invariants from the Fukaya category |
| topic | Symplectic Geometry Algebraic Geometry |
| url | https://arxiv.org/abs/2212.08345 |