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Volume conjecture: refined and categorified

Fuji, Hiroyuki and Gukov, Sergei and Sułkowski, Piotr and Awata, Hidetoshi (2012) Volume conjecture: refined and categorified. Advances in Theoretical and Mathematical Physics, 16 (6). pp. 1669-1777. ISSN 1095-0761.

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The generalized volume conjecture relates asymptotic behavior of the colored Jones polynomials to objects naturally defined on an algebraic curve, the zero locus of the A-polynomial A(x,y). Another “family version” of the volume conjecture depends on a quantization parameter, usually denoted q or ħ; this quantum volume conjecture (also known as the AJ-conjecture) can be stated in a form of a q-difference equation that annihilates the colored Jones polynomials and SL(2,C) Chern– Simons partition functions. We propose refinements/categorifications of both conjectures that include an extra deformation parameter t and describe similar properties of homological knot invariants and refined BPS invariants. Much like their unrefined/decategorified predecessors, that correspond to t=−1, the new volume conjectures involve objects naturally defined on an algebraic curve A^(ref)(x,y;t) obtained by a particular deformation of the A-polynomial, and its quantization Â^(ref)(xˆ,ŷ;q,t). We compute both classical and quantum t-deformed curves in a number of examples coming from colored knot homologies and refined BPS invariants.

Item Type:Article
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URLURL TypeDescription Paper
Gukov, Sergei0000-0002-9486-1762
Sułkowski, Piotr0000-0002-6176-6240
Additional Information:© 2012 International Press. We thank H.-J. Chung and R.H. Dijkgraaf for useful discussions during the early stages of this work. We thank M. Aganagic, A. Iqbal, D. Krefl, and Sh. Shakirov for discussions and comments. The authors would also like to thank the following institutions for their hospitality: California Institute of Technology (H.F.), the Banff International Research Station (H.F., P.S.), and the Simons Center for Geometry and Physics (H.F., S.G., P.S.). The work of H.F. is supported by the Grant-in-Aid for Young Scientists (B) [#21740179] from the Japan Ministry of Education, Culture, Sports, Science and Technology, and the Grant-in-Aid for Nagoya University Global COE Program, “Quest for Fundamental Principles in the Universe: from Particles to the Solar System and the Cosmos.” 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, the European Commission under the Marie-Curie International Outgoing Fellowship Programme, and the Foundation for Polish Science. Opinions and conclusions expressed here are those of the authors and do not necessarily reflect the views of funding agencies.
Funding AgencyGrant Number
Ministry of Education, Culture, Sports, Science and Technology (MEXT)21740179
Nagoya UniversityUNSPECIFIED
Department of Energy (DOE)DE-FG03-92-ER40701FG-02
Marie Curie FellowshipUNSPECIFIED
Foundation for Polish ScienceUNSPECIFIED
Issue or Number:6
Record Number:CaltechAUTHORS:20130805-101410619
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:39761
Deposited By: Jason Perez
Deposited On:05 Aug 2013 21:00
Last Modified:11 Feb 2020 19:21

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