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A-polynomial, B-model, and quantization

Gukov, Sergei and Sułkowski, Piotr (2012) A-polynomial, B-model, and quantization. Journal of High Energy Physics, 2012 (2). Art. No. 070. ISSN 1126-6708. http://resolver.caltech.edu/CaltechAUTHORS:20120511-113608838

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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 ħ → 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 Â(^x, ^y) using the technique of the topological recursion. This leads to a powerful and systematic algorithm for computing  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 Ktheory 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.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1007/JHEP02(2012)070DOIArticle
http://www.springerlink.com/content/yp417pm104288l53/?MUD=MPPublisherArticle
http://arxiv.org/abs/1108.0002arXivDiscussion Paper
Additional Information:© 2012 Springer, Part of Springer Science+Business Media. Published for SISSA by Springer. This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Received: November 2, 2011; accepted: January 31, 2012; published: February 20, 2012. 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.
Group:Caltech Theory
Funders:
Funding AgencyGrant Number
Department of Energy (DOE)DE-FG03-92-ER40701FG-02
NSFPHY-0757647
Marie Curie FellowshipUNSPECIFIED
Subject Keywords:Matrix Models; Non-Commutative Geometry; Chern-Simons Theories; Topological Strings
Record Number:CaltechAUTHORS:20120511-113608838
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20120511-113608838
Usage Policy:This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
ID Code:31436
Collection:CaltechAUTHORS
Deposited By: Jason Perez
Deposited On:11 May 2012 20:54
Last Modified:03 May 2016 04:27

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