Splitting the diagonal for broken maps
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910916356866048 |
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| author | Venugopalan, Sushmita Woodward, Chris |
| author_facet | Venugopalan, Sushmita Woodward, Chris |
| contents | In previous work, we introduced a version of the Fukaya algebra associated to a degeneration of a symplectic manifold, whose structure maps count collections of maps in the components of the degeneration satisfying matching conditions. In this paper, we introduce a further degeneration of the matching conditions (similar in spirit to Bourgeois' version of symplectic field theory) which results in a "split Fukaya algebra" whose structure maps are, in good cases, sums of products over vertices of tropical graphs. In the case of toric Lagrangians contained in a toric component of the degeneration, an invariance argument implies the existence of projective Maurer-Cartan solutions, which gives an alternate proof of the unobstructedness result of Fukaya-Oh-Ohta-Ono for toric manifolds. Our result also proves unobstructedness in more general cases, such as for toric Lagrangians in almost toric four-manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_15583 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Splitting the diagonal for broken maps Venugopalan, Sushmita Woodward, Chris Symplectic Geometry 53D37, 53D40, 53D42 In previous work, we introduced a version of the Fukaya algebra associated to a degeneration of a symplectic manifold, whose structure maps count collections of maps in the components of the degeneration satisfying matching conditions. In this paper, we introduce a further degeneration of the matching conditions (similar in spirit to Bourgeois' version of symplectic field theory) which results in a "split Fukaya algebra" whose structure maps are, in good cases, sums of products over vertices of tropical graphs. In the case of toric Lagrangians contained in a toric component of the degeneration, an invariance argument implies the existence of projective Maurer-Cartan solutions, which gives an alternate proof of the unobstructedness result of Fukaya-Oh-Ohta-Ono for toric manifolds. Our result also proves unobstructedness in more general cases, such as for toric Lagrangians in almost toric four-manifolds. |
| title | Splitting the diagonal for broken maps |
| topic | Symplectic Geometry 53D37, 53D40, 53D42 |
| url | https://arxiv.org/abs/2504.15583 |