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A-Polynomial, B-Model, and Quantization

  • Sergei GukovEmail author
  • Piotr Sułkowski
Chapter
Part of the Lecture Notes of the Unione Matematica Italiana book series (UMILN, volume 15)

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 \(\hslash \rightarrow 0\), 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 geometry,” 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. The material contained in this chapter was presented at the conference Mirror Symmetry and Tropical Geometry in Cetraro (July 2011) and is based on the work: Gukov and Sułkowski, “A-polynomial, B-model, and quantization”, JHEP 1202 (2012) 070.

Keywords

Partition Function Matrix Model Topological String Perturbative Expansion Bergman Kernel 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

It is pleasure to thank Vincent Bouchard, Tudor Dimofte, Nathan Dunfield, Bertrand Eynard, Maxim Kontsevich, and Don Zagier for helpful discussions and correspondence. The work of S.G. is supported in part by DOE Grant DE-FG03-92-ER40701FG-02 and in part by NSF Grant PHY-0757647. The research of P.S. is supported by the DOE grant DE-FG03-92-ER40701FG-02 and the European Commission under the Marie-Curie International Outgoing Fellowship Programme. Opinions and conclusions expressed here are those of the authors and do not necessarily reflect the views of funding agencies.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.California Institute of TechnologyPasadenaUSA
  2. 2.Max-Planck-Institut für MathematikBonnGermany
  3. 3.Faculty of PhysicsUniversity of WarsawWarsawPoland

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