A-polynomial, B-model, and Quantization (1108.0002v2)

Abstract: Exact solution to many problems in mathematical physics and quantum field theory often can be expressed in terms of an algebraic curve equipped with a meromorphic differential. Typically, the geometry of the curve can be seen most clearly in a suitable semi-classical limit, as $\hbar \to 0$, and becomes non-commutative or "quantum" away from this limit. For a classical curve defined by the zero locus of a polynomial $A(x,y)$, we provide a construction of its non-commutative counterpart $\hat{A} (\hat x, \hat y)$ using the technique of the topological recursion. This leads to a powerful and systematic algorithm for computing $\hat{A}$ that, surprisingly, turns out to be much simpler than any of the existent methods. In particular, as a bonus feature of our approach comes a curious observation that, for all curves that come from knots or topological strings, their non-commutative counterparts can be determined just from the first few steps of the topological recursion. We also propose a K-theory criterion for a curve to be "quantizable," and then apply our construction to many examples that come from applications to knots, strings, instantons, and random matrices.

Overview of "A-polynomial, B-model, and Quantization"

This paper by Sergei Gukov and Piotr Sulkowski explores several intricate relationships between mathematical physics, knot theory, quantum field theory, and string theory by focusing on the quantization of algebraic curves. The authors offer a systematic approach to derive quantum operators from classical algebraic curves, specifically those related to knots, topological strings, and matrix models, through the use of topological recursion and a novel criterion for "quantizability."

The cornerstone of the paper revolves around classical curves defined by algebraic equations $A(x,y) = 0$, where $A(x,y)$ is a polynomial. In the quantum field, variables transform into non-commutative operators leading to a quantum curve described by an operator $\hat{A}$. The authors introduce a procedure that simplifies this quantization process compared to previous methods, emphasizing the extraction of the quantum $\hat{A}$-polynomial just by examining early terms in a topological recursion process.

Quantum Curves and Topological Recursion

Gukov and Sulkowski systematically analyze the process of quantizing classical algebraic curves by elaborating on the role of topological recursion. Fundamentally, the recursion begins with the definition of a spectral curve and moves forward by calculating perturbative coefficients of partition functions, primarily through the integration of Bergman kernels and vertices defined on these curves.

One of the notable theoretical achievements presented in the paper is the formulation of quantum operators as polynomials in non-commutative variables $\hat{x}$, $\hat{y}$, and $q$, where $q = e^{\hbar}$. The paper provides exact formulas like equation (3.3) and formula (1.16) for calculating coefficients that dictate the quantum corrections necessary for developing the $\hat{A}$-polynomial. The investigation extends to diverse examples such as the figure-8 knot, conifold, torus knots, and classical matrix models like the Airy curve.

Quantizability and the K-Theory Criterion

A crucial conceptual development in this research is the introduction of a K-theory-based criterion for assessing the quantizability of these curves. Not all classical algebraic curves have a clear quantum counterpart. The authors argue that a curve is quantizable if the corresponding K-theory symbol $\{x, y\}$ is a torsion class, guiding the quantization feasibility based on deeper algebraic properties.

Numerical and Theoretical Implications

The paper provides a plethora of examples demonstrating how these theoretical constructs translate into real computations: from Airy models and c=1 theories to torus knots and generalized conifold geometries. Each example's classical curve is meticulously quantized, showcasing consistency and the elegance of the authors' technique.

By introducing a streamlined methodology for quantization, this paper holds potential implications for better understanding partition functions and quantum invariants in string theory and knot theory. Gukov and Sulkowski's work also suggests intriguing possibilities for future exploration in areas such as integrating algebraic K-theory with quantum field theories and deepening our understanding of quantum dilogarithms.

Future Prospects

Given the robust framework presented, future work may involve the extension of these quantization methods to higher-genus curves and exploring additional connections with modular forms or exotic symplectic manifolds. Researchers might explore practical applications in geometry and topology, especially considering systems with complex moduli spaces where non-trivial quantum corrections exhibit broader implications.

In summary, the paper provides a rich, fundamental contribution to the field of quantization within mathematical physics, offering both novel theoretical insights and a variety of practical computational techniques that expand our understanding of classical versus quantum correspondences.