Abstract
We construct 3D \( \mathcal{N} \) = 2 abelian gauge theories on \( \mathbbm{S} \)2 × \( \mathbbm{S} \)1 labeled by knot diagrams whose K-theoretic vortex partition functions, each of which is a building block of twisted indices, give the colored Jones polynomials of knots in \( \mathbbm{S} \)3. The colored Jones polynomials are obtained as the Wilson loop expectation values along knots in SU(2) Chern-Simons gauge theories on \( \mathbbm{S} \)3, and then our construction provides an explicit correspondence between 3D \( \mathcal{N} \) = 2 abelian gauge theories and 3D SU(2) Chern-Simons gauge theories. We verify, in particular, the applicability of our constructions to a class of tangle diagrams of 2-bridge knots with certain specific twists.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
V. Pestun et al., Localization techniques in quantum field theories, J. Phys. A 50 (2017) 440301 [arXiv:1608.02952] [INSPIRE].
F. Benini and A. Zaffaroni, A topologically twisted index for three-dimensional supersymmetric theories, JHEP 07 (2015) 127 [arXiv:1504.03698] [INSPIRE].
S. Pasquetti, Factorisation of N = 2 theories on the squashed 3-sphere, JHEP 04 (2012) 120 [arXiv:1111.6905] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, 3-manifolds and 3d indices, Adv. Theor. Math. Phys. 17 (2013) 975 [arXiv:1112.5179] [INSPIRE].
C. Beem, T. Dimofte and S. Pasquetti, Holomorphic blocks in three dimensions, JHEP 12 (2014) 177 [arXiv:1211.1986] [INSPIRE].
S. Cecotti, D. Gaiotto and C. Vafa, tt* geometry in 3 and 4 dimensions, JHEP 05 (2014) 055 [arXiv:1312.1008] [INSPIRE].
M. Fujitsuka, M. Honda and Y. Yoshida, Higgs branch localization of 3d \( \mathcal{N} \) = 2 theories, PTEP 2014 (2014) 123B02 [arXiv:1312.3627] [INSPIRE].
F. Benini and W. Peelaers, Higgs branch localization in three dimensions, JHEP 05 (2014) 030 [arXiv:1312.6078] [INSPIRE].
F. Nieri and S. Pasquetti, Factorisation and holomorphic blocks in 4d, JHEP 11 (2015) 155 [arXiv:1507.00261] [INSPIRE].
C. Hwang, P. Yi and Y. Yoshida, Fundamental vortices, Wall-Crossing, and particle-vortex duality, JHEP 05 (2017) 099 [arXiv:1703.00213] [INSPIRE].
S. Crew, N. Dorey and D. Zhang, Factorisation of 3d \( \mathcal{N} \) = 4 twisted indices and the geometry of vortex moduli space, JHEP 08 (2020) 015 [arXiv:2002.04573] [INSPIRE].
E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
Y. Terashima and M. Yamazaki, SL(2, R) Chern-Simons, Liouville, and gauge theory on duality walls, JHEP 08 (2011) 135 [arXiv:1103.5748] [INSPIRE].
Y. Terashima and M. Yamazaki, Semiclassical analysis of the 3d/3d relation, Phys. Rev. D 88 (2013) 026011 [arXiv:1106.3066] [INSPIRE].
T. Dimofte and S. Gukov, Chern-Simons theory and S-duality, JHEP 05 (2013) 109 [arXiv:1106.4550] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, Gauge theories labelled by three-manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].
S. Cecotti, C. Cordova and C. Vafa, Braids, walls, and mirrors, arXiv:1110.2115 [INSPIRE].
H. Fuji, S. Gukov, M. Stosic and P. Sulkowski, 3d analogs of Argyres-Douglas theories and knot homologies, JHEP 01 (2013) 175 [arXiv:1209.1416] [INSPIRE].
H.-J. Chung, T. Dimofte, S. Gukov and P. Sułkowski, 3d-3d correspondence revisited, JHEP 04 (2016) 140 [arXiv:1405.3663] [INSPIRE].
T. Dimofte, S. Gukov and L. Hollands, Vortex counting and Lagrangian 3-manifolds, Lett. Math. Phys. 98 (2011) 225 [arXiv:1006.0977] [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].
V.G. Turaev, The Yang-Baxter equation and invariants of links, Invent. Math. 92 (1988) 527.
H. Murakami, An introduction to the volume conjecture, arXiv:1002.0126.
L.C. Jeffrey and F.C. Kirwan, Localization for nonabelian group actions, Topology 34 (1995) 291 [alg-geom/9307001].
M. Brion and M. Vergne, Arrangements of hyperplanes I: rational functions and Jeffrey-Kirwan residue, Ann. Sci. Ecole Norm. Sup. 32 (1999) 715 [math.DG/9903178].
A. Szenes and M. Vergne, Toric reduction and a conjecture of Batyrev and Materov, Invent. Math. 158 (2004) 453 [math.AT/0306311].
F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of 2d \( \mathcal{N} \) = 2 gauge theories, Commun. Math. Phys. 333 (2015) 1241 [arXiv:1308.4896] [INSPIRE].
M. Bullimore and A. Ferrari, Twisted Hilbert spaces of 3d supersymmetric gauge theories, JHEP 08 (2018) 018 [arXiv:1802.10120] [INSPIRE].
M. Bullimore, A. Ferrari and H. Kim, Twisted indices of 3d \( \mathcal{N} \) = 4 gauge theories and enumerative geometry of quasi-maps, JHEP 07 (2019) 014 [arXiv:1812.05567] [INSPIRE].
S. Kim, The Complete superconformal index for N = 6 Chern-Simons theory, Nucl. Phys. B 821 (2009) 241 [Erratum ibid. 864 (2012) 884] [arXiv:0903.4172] [INSPIRE].
Y. Imamura and S. Yokoyama, Index for three dimensional superconformal field theories with general R-charge assignments, JHEP 04 (2011) 007 [arXiv:1101.0557] [INSPIRE].
A. Kapustin and B. Willett, Generalized superconformal index for three dimensional field theories, arXiv:1106.2484 [INSPIRE].
M. Bullimore, A.E.V. Ferrari, H. Kim and G. Xu, The twisted index and topological saddles, arXiv:2007.11603 [INSPIRE].
F. Benini and A. Zaffaroni, Supersymmetric partition functions on Riemann surfaces, Proc. Symp. Pure Math. 96 (2017) 13 [arXiv:1605.06120] [INSPIRE].
C. Closset and H. Kim, Comments on twisted indices in 3d supersymmetric gauge theories, JHEP 08 (2016) 059 [arXiv:1605.06531] [INSPIRE].
K. Ueda and Y. Yoshida, 3d \( \mathcal{N} \) = 2 Chern-Simons-matter theory, Bethe ansatz, and quantum K -theory of Grassmannians, JHEP 08 (2020) 157 [arXiv:1912.03792] [INSPIRE].
Y. Yoshida and K. Sugiyama, Localization of three-dimensional \( \mathcal{N} \) = 2 supersymmetric theories on S1 × D2 PTEP 2020 (2020) 113B02 [arXiv:1409.6713] [INSPIRE].
S. Crew, N. Dorey and D. Zhang, Blocks and vortices in the 3d ADHM quiver gauge theory, JHEP 03 (2021) 234 [arXiv:2010.09732] [INSPIRE].
M. Bullimore, S. Crew and D. Zhang, Boundaries, vermas, and factorisation, JHEP 04 (2021) 263 [arXiv:2010.09741] [INSPIRE].
R. Kirby and P. Melvin, The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl(2, C), Invent. Math. 105 (1991) 473.
J. Cho and J. Murakami, Optimistic limits of the colored Jones polynomials, Bull. Korean Math. Soc. 50 (2013) 641 [arXiv:1009.3137].
D. Thurston, Hyperbolic volume and the Jones polynomial, lecture note at Invariants des noeuds et de variétés de dimension 3 (1999), available at https://dpthurst.pages.iu.edu/speaking/Grenoble.pdf.
R.M. Kashaev, The Hyperbolic volume of knots from quantum dilogarithm, Lett. Math. Phys. 39 (1997) 269 [INSPIRE].
H. Murakami and J. Murakami, The colored Jones polynomials and the simplicial volume of a knot, Acta Math. 186 (2001) 85 [math.GT/9905075].
Y. Yokota, On the complex volume of hyperbolic knots, J. Knot Theory Ramif. 20 (2011) 955.
H. Murakami, Kashaev’s invariant and the volume of a hyperbolic knot after Y. Yokota, Phys. Combin. (2001) 244 [math.GT/0008027].
P. Kucharski, M. Reineke, M. Stosic and P. Sułkowski, BPS states, knots and quivers, Phys. Rev. D 96 (2017) 121902 [arXiv:1707.02991] [INSPIRE].
P. Kucharski, M. Reineke, M. Stosic and P. Sułkowski, Knots-quivers correspondence, Adv. Theor. Math. Phys. 23 (2019) 1849 [arXiv:1707.04017] [INSPIRE].
T. Ekholm, P. Kucharski and P. Longhi, Physics and geometry of knots-quivers correspondence, Commun. Math. Phys. 379 (2020) 361 [arXiv:1811.03110] [INSPIRE].
T. Ekholm, P. Kucharski and P. Longhi, Multi-cover skeins, quivers, and 3d \( \mathcal{N} \) = 2 dualities, JHEP 02 (2020) 018 [arXiv:1910.06193] [INSPIRE].
J. Jankowski, P. Kucharski, H. Larraguível, D. Noshchenko and P. Sułkowski, Permutohedra for knots and quivers, Phys. Rev. D 104 (2021) 086017 [arXiv:2105.11806] [INSPIRE].
T. Ekholm, P. Kucharski and P. Longhi, Knot homologies and generalized quiver partition functions, arXiv:2108.12645 [INSPIRE].
A. Gorsky, A. Milekhin and N. Sopenko, The condensate from torus knots, JHEP 09 (2015) 102 [arXiv:1506.06695] [INSPIRE].
H. Murakami, The asymptotic behavior of the colored Jones function of a knot and its volume, math.GT/0004036.
T.T.Q. Le, Quantum invariants of 3-manifolds: integrality, splitting, and perturbative expansion, Topol. Appl. 127 (2003) 125 [math.QA/0004099].
K. Habiro, On the quantum sl2 invariants of knots and integral homology spheres, Geom. Topol. Monogr. 4 (2002) 55 [math.GT/0211044].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2110.05662
Afifliated until December 2020 to: School of Mathematics and Statistics, University of Melbourne, Royal Parade, Parkville, VIC 3010, Australia. (Masahide Manabe)
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Manabe, M., Terashima, S. & Terashima, Y. The colored Jones polynomials as vortex partition functions. J. High Energ. Phys. 2021, 197 (2021). https://doi.org/10.1007/JHEP12(2021)197
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2021)197