Torus knots and mirror symmetry (1105.2012v1)

Abstract: We propose a spectral curve describing torus knots and links in the B-model. In particular, the application of the topological recursion to this curve generates all their colored HOMFLY invariants. The curve is obtained by exploiting the full Sl(2, Z) symmetry of the spectral curve of the resolved conifold, and should be regarded as the mirror of the topological D-brane associated to torus knots in the large N Gopakumar-Vafa duality. Moreover, we derive the curve as the large N limit of the matrix model computing torus knot invariants.

Overview of the Paper: "Torus Knots and Mirror Symmetry"

The paper, "Torus knots and mirror symmetry," presents an intriguing examination of torus knots and links from the perspective of the B-model in topological string theory. The authors propose a framework to describe torus knots and links using a spectral curve derived from the B-model, which generates the colored HOMFLY polynomials. This exploration leverages mirror symmetry and builds on the large $N$ duality, known as Gopakumar-Vafa duality, to establish a connection between knot theory and topological string theory.

Foundational Concepts

1. Gopakumar-Vafa Duality: This duality relates Chern-Simons theory on the three-sphere to topological string theory on the resolved conifold. It translates knot invariants into enumerative geometric invariants, offering a path to understand knot invariants through the geometry of string theory.
2. Mirror Symmetry: This concept suggests that the geometry of knots can be represented by holomorphic curves in the B-model, providing a dual description to the traditional A-model approaches.
3. Spectral Curves and Topological Recursion: The paper posits a spectral curve that aligns with torus knots in the context of the B-model. By applying the topological recursion framework, the authors derive the knot invariants reflected in the HOMFLY polynomials.

Key Results

• Spectral Curve Description: The authors derive a spectral curve for torus knots, portraying them as modular transformations in the context of resolved conifold geometry. This curve capably reproduces the entire set of colored HOMFLY invariants via the topological recursion method.
• Matrix Model Representation: By examining the large $N$ limit of matrix models, the authors validate their spectral curve proposal. They derive that the spectral curve obtained is a symplectic transformation of the canonical curve for the resolved conifold, having crucial implications for the partition functions and knot invariants.
• Connection with Chern-Simons Theory: Utilizing a matrix model that computes torus knot invariants, the authors map these results to the Chern-Simons realization of the Verlinde algebra, aligning knot invariants with a broader-framework using symplectic geometry.

Implications and Future Directions

This paper extends the string-theory toolkit for evaluating knot invariants beyond the simpler knots like the unknot and Hopf link. It creates avenues for future work to explore more complex knots, potentially incorporating new structures beyond known methods. The framework also gives insight into how large $N$ asymptotic expansions could be universally applied in topological strings, suggesting a broader potential to encapsulate string dualities and symmetries inherently tied to geometric representations.

Additionally, the intersection of matrix models and geometric topology presented in this paper poses future investigations into the role of quantum algebras in string theory, and possible connections to refined topological string theories hint at deeper symmetries that are not yet fully understood.

In conclusion, the work captures foundational insights bridging knot theory with topological string theory, offering a stronger theoretical structure to investigate knots through the lens of B-model geometry and uncovering critical relations between physical theories and mathematical invariants.