Link Homology from Homological Mirror Symmetry

We explain how to calculate link homology for a Lie algebra \(\mathfrak{g}\) using the Fukaya category associated to a 2d A-model. Links are represented as configurations of particular A-branes and link homology is given by Homs between these A-branes. In the case of \(\mathfrak{g}=\mathfrak{su}_2\)...

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Bibliographic Details
Published in:arXiv.org 2023-07
Main Author: LePage, Elise
Format: Article
Language:English
Subjects:
Algebra
Algorithms
Branes
Homology
Lie groups
Two dimensional models
Online Access:Get full text
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Summary:We explain how to calculate link homology for a Lie algebra \(\mathfrak{g}\) using the Fukaya category associated to a 2d A-model. Links are represented as configurations of particular A-branes and link homology is given by Homs between these A-branes. In the case of \(\mathfrak{g}=\mathfrak{su}_2\), we explain how to explicitly construct projective resolutions of the relevant A-branes in terms of thimbles, whose algebra is known. This gives an explicit algorithm for computing Khovanov homology. This algorithm can be extended to all Lie algebras.
ISSN:2331-8422