The symplectic arc algebra is formal
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Authors
Abouzaid, Mohammed
Smith, Ivan
Publication Date
2016-01-28Journal Title
Duke Mathematical Journal
ISSN
0012-7094
Publisher
Duke University Press
Volume
165
Pages
985-1060
Language
English
Type
Article
This Version
AM
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Abouzaid, M., & Smith, I. (2016). The symplectic arc algebra is formal. Duke Mathematical Journal, 165 985-1060. https://doi.org/10.1215/00127094-3449459
Abstract
We prove a formality theorem for the Fukaya categories of the symplectic manifolds underlying symplectic Khovanov cohomology over fields of characteristic zero. The key ingredient is the construction of a degree-one Hochschild cohomology class on a Floer A$_\infty$-algebra associated to the ($k$,$k$)-nilpotent slice $y_k$ obtained by counting holomorphic discs which satisfy a suitable conormal condition at infinity in a partial compactification $\bar y$$_k$. The space $\bar y$$_k$ is obtained as the Hilbert scheme of a partial compactification of the A$_{2k-1}$-Milnor fiber. A sequel to this paper will prove formality of the symplectic cup and cap bimodules and infer that symplectic Khovanov cohomology and Khovanov cohomology have the same total rank over characteristic zero fields.
Keywords
symplectic topology, Khovanov homology, Fukaya category, nilpotent slice
Sponsorship
National Science Foundation (Grant ID: DMS-1308179)
Identifiers
External DOI: https://doi.org/10.1215/00127094-3449459
This record's URL: https://www.repository.cam.ac.uk/handle/1810/256187
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