Knot invariants from topological recursion on augmentation varieties. (English) Zbl 1323.57008
This paper studies the following question: How must we extend remodeling and topological recursion methods to accommodate the calculation of colored HOMFLY polynomials? The paper observes that the usual approach, using topological recursion based directly on the Bergman kernel, cannot suffice. Instead, the paper argues, it is crucial to replace the Bergman kernel by the “calibrated annulus kernel” introduced in the paper.
The main evidence for the approach taken in the paper are detailed calculations of low-genus amplitudes for torus knots, but the authors optimistically suggest that their approach is correct to all orders for all knots. This leads them to some remarkable conjectures about knot theory, most notably that the full data of the colored HOMFLY polynomials of a knot for arbitrary representation can be extracted from the data just of the colored HOMFLY polynomials corresponding to Young diagrams with at most two rows. Connections are also made to knot contact homology augmentation varieties.
The main evidence for the approach taken in the paper are detailed calculations of low-genus amplitudes for torus knots, but the authors optimistically suggest that their approach is correct to all orders for all knots. This leads them to some remarkable conjectures about knot theory, most notably that the full data of the colored HOMFLY polynomials of a knot for arbitrary representation can be extracted from the data just of the colored HOMFLY polynomials corresponding to Young diagrams with at most two rows. Connections are also made to knot contact homology augmentation varieties.
Reviewer: Theo Johnson-Freyd (Evanston)
MSC:
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
| 53D37 | Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category |
| 14H81 | Relationships between algebraic curves and physics |
Keywords:
topological recursion; colored HOMFLY polynomials; remodeled B-model; string theory; Bergman kernel; annulus kernel; spectral curvesSoftware:
SINGULARReferences:
| [1] | Witten E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351 (1989) · Zbl 0667.57005 · doi:10.1007/BF01217730 |
| [2] | Labastida, J.: Chern-Simons gauge theory: Ten years after, AIP Conf. Proc. 484, 1(1999). arXiv:hep-th/9905057 [hep-th] · Zbl 1162.81414 |
| [3] | Mariño, M.: Chern-Simons theory, matrix models and topological strings. In: International Series of Monographs on Physics, vol. 131. The Clarendon Press Oxford University Press, Oxford (2005) · Zbl 1093.81002 |
| [4] | Gukov, S., Saberi I.: Lectures on Knot Homology and Quantum Curves, Topology and Field Theories, Contemporary Mathematics, vol. 613, pp. 41-78. American Mathematical Society, Providence (2014). arXiv:1211.6075 [hep-th] · Zbl 1320.81084 |
| [5] | Witten, E.: Chern-Simons gauge theory as a string theory. Prog. Math. 133, 637-678 (1995). arXiv:hep-th/9207094 [hep-th] · Zbl 0844.58018 |
| [6] | Ooguri, H., Vafa, C.: Knot invariants and topological strings, Nucl. Phys. B577, 419-438 (2000). arXiv:hep-th/9912123 [hep-th] · Zbl 1036.81515 |
| [7] | Gopakumar, R., Vafa, C.: On the gauge theory / geometry correspondence. Adv. Theor. Math. Phys. 3, 1415-1443 (1999). arXiv:hep-th/9811131 [hep-th] · Zbl 0972.81135 |
| [8] | Dijkgraaf, R., Vafa, C.: Matrix models, topological strings, and supersymmetric gauge theories. Nucl. Phys. B 644, 3-20 (2002). arXiv:hep-th/0206255 [hep-th] · Zbl 0999.81068 |
| [9] | Mariño, M.: Open string amplitudes and large order behavior in topological string theory, JHEP 0803, 060 (2008). arXiv:hep-th/0612127 [hep-th] |
| [10] | Bouchard, V., Klemm, A., Mariño, M., Pasquetti, S.: Remodeling the B-model. Commun. Math. Phys. 287 117-178, (2009). arXiv:0709.1453 [hep-th] · Zbl 1178.81214 |
| [11] | Eynard, B., Orantin, N.: Invariants of algebraic curves and topological expansion. Commun. Number Theory Phys. 1, 347-452 (2007). arXiv:math-ph/0702045 [math-ph] · Zbl 1161.14026 |
| [12] | Harvey R., Lawson H.B. Jr: Calibrated geometries. Acta Math. 148, 47-157 (1982) · Zbl 0584.53021 · doi:10.1007/BF02392726 |
| [13] | Aganagic, M., Vafa, C.: Mirror symmetry, D-branes and counting holomorphic discs (2000). arXiv:hep-th/0012041 [hep-th] |
| [14] | Aganagic, M., Klemm, A., Vafa, C.: Disk instantons, mirror symmetry and the duality web. Z. Naturforsch. A 57, 1-28 (2002). arXiv:hep-th/0105045 [hep-th] · Zbl 1203.81153 |
| [15] | Brini, A., Eynard, B., Mariño, M.: Torus knots and mirror symmetry. Annales Henri Poincare 13, 1873-1910 (2012). arXiv:1105.2012 [hep-th] · Zbl 1256.81086 |
| [16] | Ng, L.: Knot and braid invariants from contact homology. I. Geom. Topol. 9, 247-297 (electronic) (2005) · Zbl 1111.57011 |
| [17] | Ng, L.: Knot and braid invariants from contact homology. II. Geom. Topol. 9, 1603-1637 (electronic), With an appendix by the author and Siddhartha Gadgil. (2005) · Zbl 1112.57001 |
| [18] | Ng, L.: Framed knot contact homology, Duke Math. J. 141, 365-406 (2008). arXiv:math/0407071 [math.GT] · Zbl 1145.57010 |
| [19] | Ng, L.: Combinatorial knot contact homology and transverse knots, Adv. Math. 227, 2189-2219 (2011). arXiv:1010.0451 [math.SG] · Zbl 1271.57044 |
| [20] | Ekholm, T., Etnyre, J.B., Ng, L., Sullivan, M.G.: Knot contact homology. Geom. Topol. 17, 975-1112 (2013). arXiv:1109.1542 [math.SG] · Zbl 1267.53095 |
| [21] | Ng, L.: A topological introduction to knot contact homology. In: Contact and Symplectic Topology, Bolyai Society Mathematical Studies, vol. 26, pp 485-530. (2014). arXiv:1210.4803 [math.GT] · Zbl 1351.53094 |
| [22] | Aganagic, M., Ekholm, T., Ng, L., Vafa, C.: Topological Strings, D-Model, and Knot Contact Homology (2013). arXiv:1304.5778 [hep-th] · Zbl 1315.81076 |
| [23] | Aganagic, M., Vafa, C.: Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots. (2012). arXiv:1204.4709 [hep-th] |
| [24] | Fuji, H., Gukov, S., Sułkowski, P.: Super-A-polynomial for knots and BPS states. Nucl. Phys. B 867, 506-546 (2013). arXiv:1205.1515 [hep-th] · Zbl 1262.81170 |
| [25] | Jockers, H., Klemm, A., Soroush, M.: Torus Knots and the Topological Vertex. Lett. Math. Phys. 104, 953-989 (2014). arXiv:1212.0321 [hep-th] · Zbl 1302.57021 |
| [26] | Borot, G., Eynard, B., Orantin, N.: Abstract loop equations, topological recursion, and applications (2013). arXiv:1303.5808 [math-ph] · Zbl 1329.14074 |
| [27] | Morton H.R., Cromwell P.R.: Distinguishing mutants by knot polynomials. J. Knot Theory Ramifications 5, 225-238 (1996) · Zbl 0866.57002 · doi:10.1142/S0218216596000163 |
| [28] | Diaconescu, D., Shende, V., Vafa, C.: Large N duality, lagrangian cycles, and algebraic knots. Commun. Math. Phys. 319, 813-863 (2013). arXiv:1111.6533 [hep-th] · Zbl 1309.57011 |
| [29] | Mariño, M.: Chern-Simons theory and topological strings. Rev. Mod. Phys. 77, 675-720 (2005). arXiv:hep-th/0406005 [hep-th] · Zbl 1205.81013 |
| [30] | Hori, K., Vafa, C.: Mirror symmetry (2000). arXiv:hep-th/0002222 [hep-th] · Zbl 1044.14018 |
| [31] | Leung, N.C., Vafa, C.: Branes and toric geometry, Adv. Theor. Math. Phys. 2, 91-118 (1998). arXiv:hep-th/9711013 [hep-th] · Zbl 0914.14024 |
| [32] | Aganagic, M., Vafa, C.: G2 manifolds, mirror symmetry and geometric engineering (2001). arXiv:hep-th/0110171 [hep-th] · Zbl 1094.32006 |
| [33] | Rosso M., Jones V.: On the invariants of torus knots derived from quantum groups. J. Knot Theory Ramif. 2, 97-112 (1993) · Zbl 0787.57006 · doi:10.1142/S0218216593000064 |
| [34] | Klein F.: Ueber hyperelliptische Sigmafunctionen. Math. Ann. 27, 431-464 (1886) · JFM 18.0418.02 · doi:10.1007/BF01445285 |
| [35] | Klein F.: Ueber hyperelliptische Sigmafunctionen. Math. Ann. 32, 351-380 (1888) · JFM 20.0491.01 · doi:10.1007/BF01443606 |
| [36] | Bergman S., Schiffer M.: Kernel Functions and Elliptic Differential Equations in Mathematical Physics. Academic Press Inc., New York, N.Y. (1953) · Zbl 0053.39003 |
| [37] | Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3-1-6—A computer algebra system for polynomial computations (2012). http://www.singular.uni-kl.de |
| [38] | Ekholm, T.: Notes on topological strings and knot contact homology (2013). arXiv:1312.0800 [math.SG] · Zbl 1325.57004 |
| [39] | Bouchard, V., Florea, B., Mariño, M.: Counting higher genus curves with crosscaps in Calabi-Yau orientifolds. JHEP 0412, 035 (2004). arXiv:hep-th/0405083 [hep-th] |
| [40] | Bouchard, V., Florea, B., Mariño, M.: Topological open string amplitudes on orientifolds, JHEP 0502, 002 (2005). arXiv:hep-th/0411227 [hep-th] · Zbl 0787.57006 |
| [41] | Krefl, D., Pasquetti, S., Walcher, J.: The real topological vertex at work. Nucl. Phys. B 833, 153-198 (2010). arXiv:0909.1324 [hep-th] · Zbl 1204.81145 |
| [42] | Mariño, M.: String theory and the Kauffman polynomial. Commun. Math. Phys. 298, 613-643 (2010). arXiv:0904.1088 [hep-th] · Zbl 1207.81129 |
| [43] | Eynard, B., Orantin, N.: Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture (2012). arXiv:1205.1103 [math-ph] · Zbl 1365.14072 |
| [44] | Eynard, B., Orantin, N.: Algebraic methods in random matrices and enumerative geometry (2008). arXiv:0811.3531 [math-ph] · Zbl 1134.81040 |
| [45] | Bouchard, V., Hutchinson, J., Loliencar, P., Meiers, M., Rupert, M.: A generalized topological recursion for arbitrary ramification. Annal. Henr. Poincare 15, 143-169 (2014). arXiv:1208.6035 [math-ph] · Zbl 1291.81212 |
| [46] | Bouchard, V., Eynard, B.: Think globally, compute locally. JHEP 1302, 143 (2013). arXiv:1211.2302 [math-ph] · Zbl 1342.81513 |
| [47] | Kokotov, A., Korotkin, D.: Isomonodromic tau-function of Hurwitz Frobenius manifolds and its applications. Int. Math. Res. Not. (2006). Art. ID 18746, 34. arXiv:math-ph/0310008 [math-ph] · Zbl 1098.53066 |
| [48] | Akemann, G.: Higher genus correlators for the Hermitian matrix model with multiple cuts. Nucl. Phys. B 482, 403-430 (1996). arXiv:hep-th/9606004 [hep-th] · Zbl 0925.81311 |
| [49] | Eynard, B., Orantin, N.: Geometrical interpretation of the topological recursion, and integrable string theories (2009). arXiv:0911.5096 [math-ph] · Zbl 1177.82049 |
| [50] | Ng, L.: Home page of Lenhard Ng—Programs and packages (2013). http://www.math.duke.edu/ ng/math/programs.html |
| [51] | Bouchard, V., Sułkowski, P.: Topological recursion and mirror curves. Adv. Theor. Math. Phys. 16, 1443-1483 (2012). arXiv:1105.2052 [hep-th] · Zbl 1276.14054 |
| [52] | Ambjørn, J., Chekhov, L., Kristjansen, C., Makeenko, Y.: Matrix model calculations beyond the spherical limit. Nucl. Phys. B 404, 127-172 (1993). arXiv:hep-th/9302014 [hep-th] · Zbl 1043.81636 |
| [53] | Adams, C.C.: The knot book. American Mathematical Society, Providence, RI (2004). An elementary introduction to the mathematical theory of knots, Revised reprint of the 1994 original · Zbl 0840.57001 |
| [54] | Garoufalidis, S., Mariño, M.: On Chern-Simons matrix models. (2006). arXiv:math/0601390 [math-gt] · Zbl 0757.53049 |
| [55] | Stevan, S.: Torus Knots in Lens Spaces & Topological Strings (2013). arXiv:1308.5509 [hep-th] · Zbl 1321.81053 |
| [56] | Kontsevich, M.: Enumeration of rational curves via Torus actions. In: Dijkgraaf, R., Faber, C., Van der Greer, G. (eds.) The Moduli Space of Curves, pp. 335-368. Birkhauser, Basel (1995). arXiv:hep-th/9405035 [hep-th] · Zbl 0885.14028 |
| [57] | Graber, T., Pandharipande, R.: Localization of virtual classes. Invent. Math. 135, 487-518 (1999). arXiv:alg-geom/9708001 [math.AG] · Zbl 0953.14035 |
| [58] | Katz, S.H., Liu, C.-C.M.: Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc. Adv. Theor. Math. Phys. 5, 1-49 (2002). arXiv:math/0103074 [math-ag] · Zbl 1026.32028 |
| [59] | Li, J., Song, Y.S.: Open string instantons and relative stable morphisms. Adv. Theor. Math. Phys. 5, 67-91 (2002). arXiv:hep-th/0103100 [hep-th] · Zbl 1015.14026 |
| [60] | Graber, T., Zaslow, E.: Open string Gromov-Witten invariants: Calculations and a mirror ’theorem’. In: Orbifolds in Mathematics ans Physics, Contemporary Mathematics, vol. 310, pp. 107-122. American Mathematical Society, Providence (2002). arXiv:hep-th/0109075 [hep-th] · Zbl 1085.14518 |
| [61] | Klemm, A., Zaslow, E.: Local mirror symmetry at higher genus. In: Winter School on Mirror Symmetry, Vector Bundles ans Lagrangian submanifolds, pp. 183-207. American Mathematical Society, Providence (2001). arXiv:hep-th/9906046 [hep-th] · Zbl 1079.14523 |
| [62] | Witten E.: Two-dimensional gravity and intersection theory on moduli space. Surveys Differ. Geom. 1, 243-310 (1991) · Zbl 0757.53049 · doi:10.4310/SDG.1990.v1.n1.a5 |
| [63] | Kontsevich M.: Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. 147, 1-23 (1992) · Zbl 0756.35081 · doi:10.1007/BF02099526 |
| [64] | Faber, C.: Algorithms for computing intersection numbers on moduli spaces of curves, with an application to the class of the locus of Jacobians, New trends in algebraic geometry (Warwick, 1996) (Cambridge). In: London Mathematical Society Lecture Note Series, vol. 264, pp. 93-109. Cambridge University Press, Cambridge (1999). math.AG/9706006 · Zbl 0952.14042 |
| [65] | Kioke K.: On the decomposition of tensor products of the representations of the classical groups: by means of the universal character. Adv. Math. 74, 57 (1989) · Zbl 0681.20030 · doi:10.1016/0001-8708(89)90004-2 |
| [66] | Eynard, B., Orantin, N.: About the x-y symmetry of the Fg algebraic invariants (2013). arXiv:1311.4993 [math-ph] · Zbl 1134.81040 |
| [67] | Eynard, B., Orantin, N.: Topological expansion of mixed correlations in the hermitian 2 Matrix Model and x-y symmetry of the F(g) invariants. J. Phys. A Math. Theor. 41, 015203 (2008). arXiv:0705.0958 [math-ph] · Zbl 1134.81040 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
