Fuji, Hiroyuki and Gukov, Sergei and Sułkowski, Piotr
(2013)
*Super-A-polynomial for knots and BPS states.*
Nuclear Physics B, 867
(2).
pp. 506-546.
ISSN 0550-3213.
http://resolver.caltech.edu/CaltechAUTHORS:20130103-082610853

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## Abstract

We introduce and compute a 2-parameter family deformation of the A-polynomial that encodes the color dependence of the superpolynomial and that, in suitable limits, reduces to various deformations of the A-polynomial studied in the literature. These special limits include the t-deformation which leads to the "refined A-polynomial" introduced in the previous work of the authors and the Q-deformation which leads, by the conjecture of Aganagic and Vafa, to the augmentation polynomial of knot contact homology. We also introduce and compute the quantum version of the super-A-polynomial, an operator that encodes recursion relations for S^r-colored HOMFLY homology. Much like its predecessor, the super-A-polynomial admits a simple physical interpretation as the defining equation for the space of SUSY vacua (= critical points of the twisted superpotential) in a circle compactification of the effective 3d N = 2 theory associated to a knot or, more generally, to a 3-manifold M. Equivalently, the algebraic curve defined by the zero locus of the super-A-polynomial can be thought of as the space of open string moduli in a brane system associated with M. As an inherent outcome of this work, we provide new interesting formulas for colored superpolynomials for the trefoil and the figure-eight knot.

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Additional Information: | © 2012 Elsevier B.V. Received 6 June 2012; accepted 9 October 2012. Available online 13 October 2012. We thank M. Aganagic, R. Dijkgraaf, E. Gorsky, A. Mironov, A. Morozov, L. Ng, M. Stošić, and C. Vafa for useful discussions on related topics. 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. | ||||||||||||||

Group: | Caltech Theory | ||||||||||||||

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Record Number: | CaltechAUTHORS:20130103-082610853 | ||||||||||||||

Persistent URL: | http://resolver.caltech.edu/CaltechAUTHORS:20130103-082610853 | ||||||||||||||

Official Citation: | Hiroyuki Fuji, Sergei Gukov, Piotr Sułkowski, Super-A-polynomial for knots and BPS states, Nuclear Physics B, Volume 867, Issue 2, 11 February 2013, Pages 506-546, ISSN 0550-3213, 10.1016/j.nuclphysb.2012.10.005. | ||||||||||||||

Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||||||

ID Code: | 36133 | ||||||||||||||

Collection: | CaltechAUTHORS | ||||||||||||||

Deposited By: | Tony Diaz | ||||||||||||||

Deposited On: | 10 Jan 2013 19:30 | ||||||||||||||

Last Modified: | 03 May 2016 04:15 |

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