The tropological vertex
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866915488830849024 |
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| author | Do, Norman Parker, Brett |
| author_facet | Do, Norman Parker, Brett |
| contents | The theory of the topological vertex was originally proposed by Aganagic, Klemm, Mariño and Vafa as a means to calculate open Gromov-Witten invariants of toric Calabi-Yau threefolds. In this paper, we place the topological vertex within the context of relative Gromov-Witten invariants of log Calabi-Yau manifolds and describe how these invariants can be effectively computed via a gluing formula for the enumeration of tropical curves in a singular integral affine space. This richer context allows us to prove that the topological vertex possesses certain tropical symmetries. These symmetries are captured by the action of a quantum torus Lie algebra that is related to a quantisation of the Lie algebra of the tropical vertex group of Gross, Pandharipande and Siebert. Finally, we demonstrate how this algebra of symmetries leads to an explicit description of the topological vertex and related Gromov-Witten invariants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_02555 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The tropological vertex Do, Norman Parker, Brett Algebraic Geometry Mathematical Physics Symplectic Geometry 14N35, 53D45 The theory of the topological vertex was originally proposed by Aganagic, Klemm, Mariño and Vafa as a means to calculate open Gromov-Witten invariants of toric Calabi-Yau threefolds. In this paper, we place the topological vertex within the context of relative Gromov-Witten invariants of log Calabi-Yau manifolds and describe how these invariants can be effectively computed via a gluing formula for the enumeration of tropical curves in a singular integral affine space. This richer context allows us to prove that the topological vertex possesses certain tropical symmetries. These symmetries are captured by the action of a quantum torus Lie algebra that is related to a quantisation of the Lie algebra of the tropical vertex group of Gross, Pandharipande and Siebert. Finally, we demonstrate how this algebra of symmetries leads to an explicit description of the topological vertex and related Gromov-Witten invariants. |
| title | The tropological vertex |
| topic | Algebraic Geometry Mathematical Physics Symplectic Geometry 14N35, 53D45 |
| url | https://arxiv.org/abs/2205.02555 |