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Fukaya Categories as Categorical Morse Homology
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2014, том 10
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Fukaya Categories as Categorical Morse Homology
Інший ID:
2010 Mathematics Subject Classification: 53D37
DOI:10.3842/SIGMA.2014.018
DOI:10.3842/SIGMA.2014.018
Посилання:
Fukaya Categories as Categorical Morse Homology / D. Nadler // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 75 назв. — англ.
Підтримка:
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.
Дата:
2014
Переглядів:
651
Завантажень:
257
Анотація:
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.
