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Topological strings, D-model, and knot contact homology
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 articleSubject
Science & TechnologyPhysical Sciences
Physics, Particles & Fields
Physics, Mathematical
Physics
SYMPLECTIC FIELD-THEORY
VOLUME CONJECTURE
D-BRANES
COMPACTNESS
INVARIANTS
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https://hdl.handle.net/10161/17785Published Version (Please cite this version)
10.4310/ATMP.2014.v18.n4.a3Publication 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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Show full item recordScholars@Duke
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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