2011

Torus knots and mirror symmetry

Brini, Andrea, Eynard, Bertrand, Marino, Marcos

Understand

We propose a spectral curve describing torus knots and links in the B-model.

In particular, the application of the topological recursion to this curve generates all their colored HOMFLY invariants.
The curve is obtained by exploiting the full Sl(2, Z) symmetry of the spectral curve of the resolved conifold, and should be regarded as the mirror of the topological D-brane associated to torus knots in the large N Gopakumar-Vafa duality.
Moreover, we derive the curve as the large N limit of the matrix model computing torus knot invariants.

Built on

V.F.R. Jones, “Hecke algebra representations of braid groups and link polynomials,” Ann. Math. 121
1987
Earlier work this paper cites.
M. Bos, V. P. Nair, “U(1) Chern-Simons Theory and c=1 Conformal Blocks,” Phys. Lett. B223
1989
Earlier work this paper cites.
S. Elitzur, G. W. Moore, A. Schwimmer, N. Seiberg, “Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory,” Nucl. Phys. B326
1989
Earlier work this paper cites.
E. Witten, “Quantum field theory and the Jones polynomial,” Commun. Math. Phys. 121
1989
Earlier work this paper cites.
E. Witten, “Gauge Theories And Integrable Lattice Models,” Nucl. Phys. B322
1989
Earlier work this paper cites.
S. Axelrod, S. Della Pietra, E. Witten, “Geometric Quantization Of Chern-simons Gauge Theory,” J. Diff. Geom. 33
1991
Earlier work this paper cites.
J. M. F. Labastida, P. M. Llatas, A. V. Ramallo, “Knot operators in Chern-Simons gauge theory,” Nucl. Phys. B348
1991
Earlier work this paper cites.
P. Traczyk, “Periodic knots and the skein polynomial,” Invent. Math. 106
1991
Earlier work this paper cites.
P. Rama Devi, T. R. Govindarajan, R. K. Kaul, “Three-dimensional Chern-Simons theory as a theory of knots and links. 3. Compact semisimple group,” Nucl. Phys. B402
1993
Earlier work this paper cites.
J. M. Isidro, J. M. F. Labastida, A. V. Ramallo, “Polynomials for torus links from Chern-Simons gauge theories,” Nucl. Phys. B398
1993
Earlier work this paper cites.
M. Rosso and V. F. R. Jones, “On the invariants of torus knots derived from quantum groups,” J. Knot Theory Ramifications 2
1993
Earlier work this paper cites.
R. Correale, E. Guadagnini, “Large N Chern-Simons field theory,” Phys. Lett. B337
1994
Earlier work this paper cites.
J. M. F. Labastida, M. Mariño, “The HOMFLY polynomial for torus links from Chern-Simons gauge theory,” Int. J. Mod. Phys. A10
1995
Earlier work this paper cites.

Similar

W. B. R. Lickorish, An introduction to knot theory
1997
Cited alongside, same era.
R. Gopakumar, C. Vafa, “On the gauge theory / geometry correspondence,” Adv. Theor. Math. Phys. 3
1999
Cited alongside, same era.
R. Lawrence and L. Rozansky, “Witten-Reshetikhin-Turaev invariants of Seifert manifolds,” Comm. Math. Phys. 205
1999
Cited alongside, same era.
J. M. F. Labastida, M. Mariño, C. Vafa, “Knots, links and branes at large N,” JHEP 0011
2000
Cited alongside, same era.
H. Ooguri, C. Vafa, “Knot invariants and topological strings,” Nucl. Phys. B577
2000
Cited alongside, same era.
C. H. Taubes, “Lagrangians for the Gopakumar-Vafa conjecture,” Adv. Theor. Math. Phys. 5
2001
Cited alongside, same era.
M. Aganagic, A. Klemm, C. Vafa, “Disk instantons, mirror symmetry and the duality web,” Z. Naturforsch. A57
2002
Cited alongside, same era.
S. H. Katz, C. -C. M. Liu, “Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc,” Adv. Theor. Math. Phys. 5
2002
Cited alongside, same era.
M. Mariño, “Knot invariants, matrix models, and open strings,” unpublished (2002)
2002
Cited alongside, same era.
M. Mariño, “Chern–Simons theory, matrix integrals, and perturbative three-manifold invariants,” Commun. Math. Phys. 253
2004
Cited alongside, same era.
M. Tierz, “Soft matrix models and Chern-Simons partition functions,” Mod. Phys. Lett. A19
2004
Cited alongside, same era.
M. Aganagic, A. Klemm, M. Mariño, C. Vafa, “The Topological vertex,” Commun. Math. Phys. 254
2005
Cited alongside, same era.

Then

S. Gukov, A. S. Schwarz, C. Vafa, “Khovanov-Rozansky homology and topological strings,” Lett. Math. Phys. 74
2005
Later among the works it cites.
A. Iqbal, A. -K. Kashani-Poor, “The Vertex on a strip,” Adv. Theor. Math. Phys. 10
2006
Later among the works it cites.
Y. Dolivet, M. Tierz, “Chern-Simons matrix models and Stieltjes-Wigert polynomials,” J. Math. Phys. 48
2007
Later among the works it cites.
S. Koshkin, “Conormal bundles to knots and the Gopakumar-Vafa conjecture,” Adv. Theor. Math. Phys. 11
2007
Later among the works it cites.
M. Mariño, “Open string amplitudes and large order behavior in topological string theory,” JHEP 0803
2008
Later among the works it cites.
H. R. Morton and P. M. G. Manchón, “Geometrical relations and plethysms in the Homfly skein of the annulus,” J. Lond. Math. Soc. 78
2008
Later among the works it cites.
V. Bouchard, A. Klemm, M. Mariño, S. Pasquetti, “Remodeling the B-model,” Commun. Math. Phys. 287
Original
2009
Later among the works it cites.
A. Iqbal, C. Kozcaz, C. Vafa, “The Refined topological vertex,” JHEP 0910
2009
Later among the works it cites.
X.-S. Lin and H. Zheng, “On the Hecke algebras and the colored HOMFLY polynomial,” Trans. Amer. Math. Soc. 362
2010
Later among the works it cites.
S. Stevan, “Chern-Simons Invariants of Torus Links,” Annales Henri Poincaré 11
Original
2010
Later among the works it cites.
M. Tierz, “Schur polynomials and biorthogonal random matrix ensembles,” J. Math. Phys. 51
2010
Later among the works it cites.
J. Zhou, “A proof of the full Mariño–Vafa conjecture,” Math. Res. Lett. 17
Original
2092
Closest in time.

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

Reading the bibliography…