Fukaya Categories as Categorical Morse Homology
The Fukaya category of a Weinstein manifold is an intricate symplectic invariant of high interest in mirror symmetry and geometric representation theory. This paper informally sketches how, in analogy with Morse homology, the Fukaya category might result from gluing together Fukaya categories of Wei...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2014 |
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| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2014
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/146836 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Fukaya Categories as Categorical Morse Homology / D. Nadler // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 75 назв. — англ. |
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Nadler, D. 2019-02-11T16:52:22Z 2019-02-11T16:52:22Z 2014 Fukaya Categories as Categorical Morse Homology / D. Nadler // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 75 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53D37 DOI:10.3842/SIGMA.2014.018 https://nasplib.isofts.kiev.ua/handle/123456789/146836 The Fukaya category of a Weinstein manifold is an intricate symplectic invariant of high interest in mirror symmetry and geometric representation theory. This paper informally sketches how, in analogy with Morse homology, the Fukaya category might result from gluing together Fukaya categories of Weinstein cells. This can be formalized by a recollement pattern for Lagrangian branes parallel to that for constructible sheaves. Assuming this structure, we exhibit the Fukaya category as the global sections of a sheaf on the conic topology of the Weinstein manifold. This can be viewed as a symplectic analogue of the well-known algebraic and topological theories of (micro)localization. This paper is a contribution to the Special Issue “Mirror Symmetry and Related Topics”. The full collection is available at http://www.emis.de/journals/SIGMA/mirror symmetry.html. I am indebted to D. Ben-Zvi, P. Seidel and E. Zaslow for the impact they have had on my thinking about symplectic and homotopical geometry. I am grateful to T. Perutz and D. Treumann for many stimulating discussions, both of a technical and philosophical nature. I am grateful to M. Abouzaid and D. Auroux for their patient explanations of foundational issues and related questions in mirror symmetry. I am also grateful to the anonymous referees for their thoughtful reading and generous investment in improving the paper. I would like to thank A. Preygel for sharing his perspective on ind-coherent sheaves. I am also pleased to acknowledge the motivating influence of a question asked by C. Teleman at ESI in Vienna in January 2007. Finally, I am grateful to the participants of the June 2011 MIT RTG Geometry retreat for their inspiring interest in this topic. This work was supported by NSF grant DMS-0600909. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Fukaya Categories as Categorical Morse Homology Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Fukaya Categories as Categorical Morse Homology |
| spellingShingle |
Fukaya Categories as Categorical Morse Homology Nadler, D. |
| title_short |
Fukaya Categories as Categorical Morse Homology |
| title_full |
Fukaya Categories as Categorical Morse Homology |
| title_fullStr |
Fukaya Categories as Categorical Morse Homology |
| title_full_unstemmed |
Fukaya Categories as Categorical Morse Homology |
| title_sort |
fukaya categories as categorical morse homology |
| author |
Nadler, D. |
| author_facet |
Nadler, D. |
| publishDate |
2014 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
The Fukaya category of a Weinstein manifold is an intricate symplectic invariant of high interest in mirror symmetry and geometric representation theory. This paper informally sketches how, in analogy with Morse homology, the Fukaya category might result from gluing together Fukaya categories of Weinstein cells. This can be formalized by a recollement pattern for Lagrangian branes parallel to that for constructible sheaves. Assuming this structure, we exhibit the Fukaya category as the global sections of a sheaf on the conic topology of the Weinstein manifold. This can be viewed as a symplectic analogue of the well-known algebraic and topological theories of (micro)localization.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/146836 |
| citation_txt |
Fukaya Categories as Categorical Morse Homology / D. Nadler // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 75 назв. — англ. |
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2025-11-29T12:08:04Z |
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2025-11-29T12:08:04Z |
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