Journal of High Energy Physics

, 2012:70 | Cite as

A-polynomial, B-model, and quantization

Open Access
Article

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 \( \widehat{A}\left( {\widehat{x},\widehat{y}} \right) \) using the technique of the topological recursion. This leads to a powerful and systematic algorithm for computing \( \widehat{A} \) 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.

Keywords

Matrix Models Non-Commutative Geometry Chern-Simons Theories Topological Strings 

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Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  1. 1.Division of Physics, Mathematics and AstronomyCalifornia Institute of TechnologyPasadenaU.S.A.
  2. 2.Max-Planck-Institut für MathematikBonnGermany
  3. 3.Faculty of PhysicsUniversity of WarsawWarsawPoland

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