Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Topological Strings, D-Model, and Knot Contact Homology

Mina Aganagic, Tobias Ekholm, Lenhard Ng, Cumrun Vafa

TL;DR

This work builds a unified bridge between topological string theory and knot/contact geometry by proving that disk amplitudes of Lagrangian fillings associated to knots encode augmentations of knot contact homology, thereby equating the $Q$-deformed $A$-polynomial with the augmentation polynomial in irreducible cases. The authors generalize to links and higher-rank representations by analyzing a quantum moduli space of branes with higher first Betti number, which leads to a higher-dimensional mirror geometry they formulate as the D-model. Central constructs include the augmentation variety $V_K$, its decomposition into components $V_K(P)$ labeled by primitive partitions, and the D-mirror viewpoint that ties these to a union over fillings; the Hopf link, Whitehead, and Borromean links serve as key test cases. The paper also develops a rigorous mathematical program connecting knot contact homology to Lagrangian fillings via exact and non-exact GW-potentials, obstruction chains, and Legendre duality of superpotentials, and it introduces a D-module quantization framework for the higher-dimensional mirrors of links. Overall, the work provides a deep synthesis of Chern–Simons theory, open/topological strings, and knot contact homology, with substantial implications for computing link invariants, understanding mirror symmetry in noncompact Calabi–Yau settings, and formulating higher-rank and multi-component generalizations via D-modules.

Abstract

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 Q-deformation of the extension of A-polynomials associated with the link complement.

Topological Strings, D-Model, and Knot Contact Homology

TL;DR

This work builds a unified bridge between topological string theory and knot/contact geometry by proving that disk amplitudes of Lagrangian fillings associated to knots encode augmentations of knot contact homology, thereby equating the -deformed -polynomial with the augmentation polynomial in irreducible cases. The authors generalize to links and higher-rank representations by analyzing a quantum moduli space of branes with higher first Betti number, which leads to a higher-dimensional mirror geometry they formulate as the D-model. Central constructs include the augmentation variety , its decomposition into components labeled by primitive partitions, and the D-mirror viewpoint that ties these to a union over fillings; the Hopf link, Whitehead, and Borromean links serve as key test cases. The paper also develops a rigorous mathematical program connecting knot contact homology to Lagrangian fillings via exact and non-exact GW-potentials, obstruction chains, and Legendre duality of superpotentials, and it introduces a D-module quantization framework for the higher-dimensional mirrors of links. Overall, the work provides a deep synthesis of Chern–Simons theory, open/topological strings, and knot contact homology, with substantial implications for computing link invariants, understanding mirror symmetry in noncompact Calabi–Yau settings, and formulating higher-rank and multi-component generalizations via D-modules.

Abstract

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 Q-deformation of the extension of A-polynomials associated with the link complement.

Paper Structure

This paper contains 49 sections, 15 theorems, 343 equations, 25 figures.

Table of Contents

  1. Introduction
  2. Review
  3. Chern-Simons theory and topological string
  4. Higher representations and multiple branes
  5. Large $N$ duality
  6. SYZ mirror symmetry for knots
  7. Chern-Simons theory and Lagrangian fillings
  8. Geometric transition between $L_K$ and $M_K$
  9. Framing of the Lagrangians
  10. Large $N$ duality, mirror symmetry, and HOMFLY invariant for links
  11. Conormal bundle $L_K$
  12. Link-complement-like fillings $M_K$
  13. General fillings
  14. D-mirror variety
  15. An example: Hopf link
  16. ...and 34 more sections

Key Result

Proposition 5.2

Suppose that $K = K_1 \cup K_2$ is a split link; that is, there is an embedded $S^2$ in $S^3$ that separates $K_1$ from $K_2$. Then any augmentation of the DGA of $K$ splits into augmentations of the DGAs of $K_1$ and $K_2$, and thus $V_K = V_{K_1} \times_Q V_{K_2}$.

Figures (25)

  • Figure 1: Holomorphic disks contributing terms $1$, $a_j$, $a_{j_1j_2}$, respectively, to $\partial(a_i)$. The cylinders represent $\mathbb{R}\times\Lambda \subset \mathbb{R}\times V$, with top and bottom corresponding to $+\infty$ and $-\infty$ in $\mathbb{R}$, respectively.
  • Figure 2: An exact Lagrangian cobordism.
  • Figure 3: A two level holomorphic building in the boundary of a 1-dimensional moduli space of disks with one positive puncture.
  • Figure 4: The front of the standard immersed Lagrangian $n$-sphere: two smooth sheets come together over a cusp edge that locally is a product of a semi-cubical cusp $x^{3}=z^{2}$ and $\mathbb{R}^{n-1}$. The intersection of the Lagrangian immersion with a symplectic coordinate plane is shown on the right.
  • Figure 5: A Lagrangian cone.
  • ...and 20 more figures

Theorems & Definitions (32)

  • Conjecture 5.1
  • Proposition 5.2
  • proof
  • Conjecture 5.3
  • Conjecture 5.4
  • Theorem 5.5
  • proof
  • Theorem 6.1
  • proof
  • Theorem 6.2
  • ...and 22 more