A-polynomial, B-model, and Quantization

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 $\hslash \to 0$, 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.

58 pages, 5 figures, minor modifications, references added

References in corpus (7)

• Open string amplitudes and large order behavior in topological string theory
• Remodeling the B-model
• The Volume Conjecture, Perturbative Knot Invariants, and Recursion Relations for Topological Strings
• Seiberg-Witten theory and matrix models
• SL(2,C) Chern-Simons theory and the asymptotic behavior of the colored Jones polynomial
• The Volume Conjecture and Topological Strings
• All orders asymptotic expansion of large partitions

Cited by in corpus (55)

• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• Walls, Lines, and Spectral Dualities in 3d Gauge Theories
• Super-A-polynomial for knots and BPS states
• JT gravity, KdV equations and macroscopic loop operators
• 3d analogs of Argyres-Douglas theories and knot homologies
• Think globally, compute locally
• Super-A-polynomials for Twist Knots
• Reconstructing WKB from topological recursion
• Sequencing BPS Spectra
• Quantum spectral curve for the Gromov-Witten theory of the complex projective line
• Genus expansion of HOMFLY polynomials
• On dualizability of braided tensor categories
• Colored Kauffman Homology and Super-A-polynomials
• Topological strings, strips and quivers
• Topological recursion for chord diagrams, RNA complexes, and cells in moduli spaces
• 3d-3d Correspondence Revisited
• A universe field theory for JT gravity
• Exact Quantization Conditions, Toric Calabi-Yau and Nonperturbative Topological String
• $\hat{Z}$ at large $N$: from curve counts to quantum modularity
• Topological recursion for irregular spectral curves
• Large $N$ twisted partition functions in 3d-3d correspondence and Holography
• Exact eigenfunctions and the open topological string
• Perturbative and nonperturbative aspects of complex Chern-Simons Theory
• Calabi-Yau geometry and electrons on 2d lattices
• Quantum Curve and the First Painlevé Equation
• BPS counting for knots and combinatorics on words
• Quantum curves and conformal field theory
• Super-quantum curves from super-eigenvalue models
• Super Quantum Airy Structures
• Quantum curves
• Models of discretized moduli spaces, cohomological field theories, and Gaussian means
• Witten-Reshetikhin-Turaev function for a knot in Seifert manifolds
• Challenges of beta-deformation
• Elliptic Quantum Curves of Class ${\mathcal{S}}_k$
• From CFT to Ramond super-quantum curves
• Branes, quivers and wave-functions
• A Unified Approach to Holomorphic Anomaly Equations and Quantum Spectral Curves
• KP integrability of triple Hodge integrals. II. Generalized Kontsevich matrix model
• Open string amplitudes of closed topological vertex
• Topological recursion and uncoupled BPS structures II: Voros symbols and the $\tau$-function
• Nahm sums, quiver A-polynomials and topological recursion
• Resurgent Analysis of SU(2) Chern-Simons Partition Function on Brieskorn Spheres $\Sigma (2,3,6n+5)$
• Generalized Ablowitz-Ladik hierarchy in topological string theory
• $x-y$ duality in Topological Recursion for exponential variables via Quantum Dilogarithm
• Combinatorics of Nahm sums, quiver resultants and the K-theoretic condition
• A Cable Knot and BPS-Series
• Quantum mirror curve of periodic chain geometry
• Airy structures for semisimple Lie algebras
• Kac-Schwarz Operators of Type $B$, Quantum Spectral Curves, and Spin Hurwitz Numbers
• Refined topological recursion revisited -- properties and conjectures
• $q$-Difference Kac-Schwarz Operators in Topological String Theory
• Diagonal Tau-Functions of 2D Toda Lattice Hierarchy, Connected $(n,m)$-Point Functions, and Double Hurwitz Numbers
• Fermionic gluing principle of the topological vertex
• On refined Chern-Simons and refined ABJ matrix models
• Painlevé Kernels and Surface Defects at Strong Coupling