HMS symmetries of toric boundary divisors
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917620976975872 |
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| author | Špenko, Špela Bergh, Michel Van den |
| author_facet | Špenko, Špela Bergh, Michel Van den |
| contents | Let $X$ be a projective crepant resolution of a Gorenstein affine toric variety and let $((\mathbb{C}^*)^k,f)$ be the LG-model which is the Hori-Vafa mirror dual of $X$. Let ${D}$ be a generic fiber of $f$ equipped with the restriction of the standard Liouville form on $(\mathbb{C}^*)^k$. Let $\mathcal{K}_A$ be the so-called "stringy Kähler moduli space" of $X$. We show that $π_1(\mathcal{K}_A)$ acts on the wrapped Fukaya category of $D$. Using results by Gammage - Shende and Zhou, this result implies that $π_1(\mathcal{K}_A)$ acts on $D^b(\operatorname{coh}(\partial X))$ where $\partial X$ is the toric boundary divisor of $X$. We show that the induced action of $π_1(\mathcal{K}_A)$ on $K_0(\operatorname{coh}(\partial X))$ may be extended in a natural way to an action on $K_0(X)$ which corresponds to a GKZ system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15660 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | HMS symmetries of toric boundary divisors Špenko, Špela Bergh, Michel Van den Algebraic Geometry Symplectic Geometry Let $X$ be a projective crepant resolution of a Gorenstein affine toric variety and let $((\mathbb{C}^*)^k,f)$ be the LG-model which is the Hori-Vafa mirror dual of $X$. Let ${D}$ be a generic fiber of $f$ equipped with the restriction of the standard Liouville form on $(\mathbb{C}^*)^k$. Let $\mathcal{K}_A$ be the so-called "stringy Kähler moduli space" of $X$. We show that $π_1(\mathcal{K}_A)$ acts on the wrapped Fukaya category of $D$. Using results by Gammage - Shende and Zhou, this result implies that $π_1(\mathcal{K}_A)$ acts on $D^b(\operatorname{coh}(\partial X))$ where $\partial X$ is the toric boundary divisor of $X$. We show that the induced action of $π_1(\mathcal{K}_A)$ on $K_0(\operatorname{coh}(\partial X))$ may be extended in a natural way to an action on $K_0(X)$ which corresponds to a GKZ system. |
| title | HMS symmetries of toric boundary divisors |
| topic | Algebraic Geometry Symplectic Geometry |
| url | https://arxiv.org/abs/2403.15660 |