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Topological strings, D-model, and knot contact homology

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Date
2014
Authors
Aganagic, M
Ekholm, T
Ng, L
Vafa, C
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Abstract
© 2014 International Press. We study the connection between topological strings and contact homology recently proposed in the context of knot invariants. In particular, we establish the proposed relation between the Gromov- Witten disk amplitudes of a Lagrangian associated to a knot and augmentations of its contact homology algebra. This also implies the equality between the Q-deformed A-polynomial and the augmentation polynomial of knot contact homology (in the irreducible case). We also generalize this relation to the case of links and to higher rank representations for knots. The generalization involves a study of the quantum moduli space of special Lagrangian branes with higher Betti numbers probing the Calabi-Yau. This leads to an extension of SYZ, and a new notion of mirror symmetry, involving higher dimensional mirrors. The mirror theory is a topological string, related to D-modules, which we call the "D-model". In the present setting, the mirror manifold is the augmentation variety of the link. Connecting further to contact geometry, we study intersection properties of branches of the augmentation variety guided by the relation to D-modules. This study leads us to propose concrete geometric constructions of Lagrangian fillings for links. We also relate the augmentation variety with the large N limit of the colored HOMFLY, which we conjecture to be related to a Qdeformation of the extension of A-polynomials associated with the link complement.
Type
Journal article
Subject
Science & Technology
Physical Sciences
Physics, Particles & Fields
Physics, Mathematical
Physics
SYMPLECTIC FIELD-THEORY
VOLUME CONJECTURE
D-BRANES
COMPACTNESS
INVARIANTS
Permalink
https://hdl.handle.net/10161/17785
Published Version (Please cite this version)
10.4310/ATMP.2014.v18.n4.a3
Publication Info
Aganagic, M; Ekholm, T; Ng, L; & Vafa, C (2014). Topological strings, D-model, and knot contact homology. Advances in Theoretical and Mathematical Physics, 18(4). pp. 827-956. 10.4310/ATMP.2014.v18.n4.a3. Retrieved from https://hdl.handle.net/10161/17785.
This is constructed from limited available data and may be imprecise. To cite this article, please review & use the official citation provided by the journal.
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Scholars@Duke

Ng

Lenhard Lee Ng

Professor of Mathematics
My research mainly focuses on symplectic topology and low-dimensional topology. I am interested in studying structures in symplectic and contact geometry (Weinstein manifolds, contact manifolds, Legendrian and transverse knots), especially through holomorphic-curve techniques. One particular interest is extracting topological information about knots through cotangent bundles, and exploring relations to topological string theory. I have also worked in Heegaard Floer theory, quantum topology, and
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