Abstract
We revisit Dimofte-Gaiotto-Gukov’s construction of 3d gauge theories associated to 3-manifolds with a torus boundary. After clarifying their construction from a viewpoint of compactification of a 6d \( \mathcal{N}=\left(2,0\right) \) theory of A1-type on a 3-manifold, we propose a topological criterion for SU(2)/SO(3) flavor symmetry enhancement for the u(1) symmetry in the theory associated to a torus boundary, which is expected from the 6d viewpoint. Base on the understanding of symmetry enhancement, we generalize the construction to closed 3-manifolds by identifying the gauge theory counterpart of Dehn filling operation. The generalized construction predicts infinitely many 3d dualities from surgery calculus in knot theory. Moreover, by using the symmetry enhancement criterion, we show that theories associated to all hyperboilc twist knots have surprising SU(3) symmetry enhancement which is unexpected from the 6d viewpoint.
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K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].
J. de Boer, K. Hori, H. Ooguri and Y. Oz, Mirror symmetry in three-dimensional gauge theories, quivers and D-branes, Nucl. Phys. B 493 (1997) 101 [hep-th/9611063] [INSPIRE].
O. Aharony et al., Aspects of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110] [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin systems and the WKB approximation, arXiv:0907.3987 [INSPIRE].
J. Yagi, 3d TQFT from 6d SCFT, JHEP 08 (2013) 017 [arXiv:1305.0291] [INSPIRE].
C. Cordova and D.L. Jafferis, Complex Chern-Simons from M5-branes on the squashed three-sphere, JHEP 11 (2017) 119 [arXiv:1305.2891] [INSPIRE].
S. Lee and M. Yamazaki, 3d Chern-Simons theory from M5-branes, JHEP 12 (2013) 035 [arXiv:1305.2429] [INSPIRE].
T. Dimofte, Complex Chern-Simons theory at level k via the 3d-3d correspondence, Commun. Math. Phys. 339 (2015) 619 [arXiv:1409.0857] [INSPIRE].
D. Gang et al., Aspects of defects in 3d-3d correspondence, JHEP 10 (2016) 062 [arXiv:1510.05011] [INSPIRE].
K. Hikami, Generalized volume conjecture and the A-polynomials: the Neumann-Zagier potential function as a classical limit of quantum invariant, J. Geom. Phys. 57 (2007) 1895 [math/0604094] [INSPIRE].
T. Dimofte, Quantum Riemann surfaces in Chern-Simons theory, Adv. Theor. Math. Phys. 17 (2013) 479 [arXiv:1102.4847] [INSPIRE].
J. Ellegaard Andersen and R. Kashaev, A TQFT from quantum teichmüller theory, Commun. Math. Phys. 330 (2014) 887 [arXiv:1109.6295] [INSPIRE].
T.D. Dimofte and S. Garoufalidis, The quantum content of the gluing equations, Geom. Topol. 17 (2013) 1253 [arXiv:1202.6268] [INSPIRE].
S. Kim, The complete superconformal index for N = 6 Chern-Simons theory, Nucl. Phys. B 821 (2009) 241 [Erratum ibid. B 864 (2012) 884] [arXiv:0903.4172] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Exact results for Wilson loops in superconformal Chern-Simons theories with matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, Notes on SUSY gauge theories on three-sphere, JHEP 03 (2011) 127 [arXiv:1012.3512] [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].
T. Dimofte, D. Gaiotto and S. Gukov, Gauge theories labelled by three-manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].
H.-J. Chung, T. Dimofte, S. Gukov and P. Sulkowski, 3d-3d correspondence revisited, JHEP 04 (2016) 140 [arXiv:1405.3663] [INSPIRE].
D. Gaiotto and E. Witten, S-duality of boundary conditions in N = 4 super Yang-Mills theory, Adv. Theor. Math. Phys. 13 (2009) 721 [arXiv:0807.3720] [INSPIRE].
Y. Tachikawa, N = 2 supersymmetric dynamics for pedestrians, Lecture Notes in Physics volume 890, Springer, Germany (2014), arXiv:1312.2684.
D. Gang, Y. Tachikawa and K. Yonekura, Smallest 3d hyperbolic manifolds via simple 3d theories, Phys. Rev. D 96 (2017) 061701 [arXiv:1706.06292] [INSPIRE].
F. Benini, S. Benvenuti and Y. Tachikawa, Webs of five-branes and N = 2 superconformal field theories, JHEP 09 (2009) 052 [arXiv:0906.0359] [INSPIRE].
F. Benini, Y. Tachikawa and D. Xie, Mirrors of 3d Sicilian theories, JHEP 09 (2010) 063 [arXiv:1007.0992] [INSPIRE].
D. Gaiotto, G.W. Moore and Y. Tachikawa, On 6d N = (2, 0) theory compactified on a Riemann surface with finite area, PTEP 2013 (2013) 013B03 [arXiv:1110.2657] [INSPIRE].
O. Chacaltana, J. Distler and Y. Tachikawa, Nilpotent orbits and codimension-two defects of 6d N = (2, 0) theories, Int. J. Mod. Phys. A 28 (2013) 1340006 [arXiv:1203.2930] [INSPIRE].
K. Yonekura, Supersymmetric gauge theory, (2, 0) theory and twisted 5d Super-Yang-Mills, JHEP 01 (2014) 142 [arXiv:1310.7943] [INSPIRE].
D. Xie and K. Yonekura, The moduli space of vacua of \( \mathcal{N}=2 \) class \( \mathcal{S} \) theories, JHEP 10 (2014) 134 [arXiv:1404.7521] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, SUSY gauge theories on squashed three-spheres, JHEP 05 (2011) 014 [arXiv:1102.4716] [INSPIRE].
D. Gang, N. Kim and S. Lee, Holography of 3d-3d correspondence at large N, JHEP 04 (2015) 091 [arXiv:1409.6206] [INSPIRE].
J.-B. Bae, D. Gang and J. Lee, 3d \( \mathcal{N}=2 \) minimal SCFTs from wrapped M5-branes, JHEP 08 (2017) 118 [arXiv:1610.09259] [INSPIRE].
V. Mikhaylov, Teichmüller TQFT vs. Chern-Simons theory, JHEP 04 (2018) 085 [arXiv:1710.04354] [INSPIRE].
D. Gang and Y. Hatsuda, S-duality resurgence in SL(2) Chern-Simons theory, JHEP 07 (2018) 053 [arXiv:1710.09994] [INSPIRE].
S. Tillmann, Degenerations of ideal hyperbolic triangulations, Math. Z. 272 (2012) 793.
W.D. Neumann, Combinatorics of triangulations and the Chern-Simons invariant for hyperbolic 3-manifolds, Topology 90 (1992) 243.
W.D. Neumann and D. Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985) 307.
E. Witten, SL(2, ℤ) action on three-dimensional conformal field theories with Abelian symmetry, hep-th/0307041 [INSPIRE].
S. Garoufalidis, C.D. Hodgson, J.H. Rubinstein and H. Segerman, 1-efficient triangulations and the index of a cusped hyperbolic 3-manifold, arXiv:1303.5278 [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, 3-manifolds and 3d indices, Adv. Theor. Math. Phys. 17 (2013) 975 [arXiv:1112.5179] [INSPIRE].
D. Rolfsen, Rational surgery calculus: extension of Kirby’s theorem, Pacific J. Math. 110 (1984) 377.
B. Martelli and C. Petronio, Dehn filling of the “magic” 3-manifold, math/0204228.
D. Pei and K. Ye, A 3d-3d appetizer, JHEP 11 (2016) 008 [arXiv:1503.04809] [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, Fivebranes and 4-manifolds, arXiv:1306.4320 [INSPIRE].
S. Gukov, D. Pei, P. Putrov and C. Vafa, BPS spectra and 3-manifold invariants, arXiv:1701.06567 [INSPIRE].
L.F. Alday, P. Benetti Genolini, M. Bullimore and M. van Loon, Refined 3d-3d correspondence, JHEP 04 (2017) 170 [arXiv:1702.05045] [INSPIRE].
O. Aharony and D. Fleischer, IR dualities in general 3d supersymmetric SU(N) QCD theories, JHEP 02 (2015) 162 [arXiv:1411.5475] [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Graduate Texts in Contemporary Physics, Springer, U.S.A. (1997).
E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
N. Seiberg and E. Witten, Gapped boundary phases of topological insulators via weak coupling, PTEP 2016 (2016) 12C101 [arXiv:1602.04251] [INSPIRE].
Y. Tachikawa and K. Yonekura, On time-reversal anomaly of 2 + 1d topological phases, PTEP 2017 (2017) 033B04 [arXiv:1610.07010] [INSPIRE].
Y. Tachikawa and K. Yonekura, More on time-reversal anomaly of 2 + 1d topological phases, Phys. Rev. Lett. 119 (2017) 111603 [arXiv:1611.01601] [INSPIRE].
W.B.R. Lickorish, A representation of orientable combinatorial 3-manifolds, Ann. Math. 76 (1962) 531.
A.H. Wallace, Modifications and cobounding manifolds, Canad. J. Math. 12 (1960) 503.
D. Gang, M. Romo and M. Yamazaki, All-order volume conjecture for closed 3-manifolds from complex Chern-Simons theory, Commun. Math. Phys. 359 (2018) 915 [arXiv:1704.00918] [INSPIRE].
F. Benini, P.-S. Hsin and N. Seiberg, Comments on global symmetries, anomalies and duality in (2 + 1)d, JHEP 04 (2017) 135 [arXiv:1702.07035] [INSPIRE].
Z. Komargodski, A. Sharon, R. Thorngren and X. Zhou, Comments on abelian Higgs models and persistent order, arXiv:1705.04786 [INSPIRE].
Z. Komargodski, T. Sulejmanpasic and M. Unsal, Walls, anomalies and deconfinement in quantum antiferromagnets, Phys. Rev. B 97 (2018) 054418 [arXiv:1706.05731] [INSPIRE].
H. Shimizu and K. Yonekura, Anomaly constraints on deconfinement and chiral phase transition, Phys. Rev. D 97 (2018) 105011 [arXiv:1706.06104] [INSPIRE].
D. Gaiotto, Z. Komargodski and N. Seiberg, Time-reversal breaking in QCD 4 , walls and dualities in 2 + 1 dimensions, JHEP 01 (2018) 110 [arXiv:1708.06806] [INSPIRE].
Y. Tanizaki, T. Misumi and N. Sakai, Circle compactification and ’t Hooft anomaly, JHEP 12 (2017) 056 [arXiv:1710.08923] [INSPIRE].
Y. Tanizaki, Y. Kikuchi, T. Misumi and N. Sakai, Anomaly matching for the phase diagram of massless ℤN -QCD, Phys. Rev. D 97 (2018) 054012 [arXiv:1711.10487] [INSPIRE].
T. Sulejmanpasic, H. Shao, A. Sandvik and M. Ünsal, Confinement in the bulk, deconfinement on the wall: infrared equivalence between compactified QCD and quantum magnets, Phys. Rev. Lett. 119 (2017) 091601 [arXiv:1608.09011] [INSPIRE].
T. Sulejmanpasic and Y. Tanizaki, C-P-T anomaly matching in bosonic quantum field theory and spin chains, Phys. Rev. B 97 (2018) 144201 [arXiv:1802.02153] [INSPIRE].
E. Witten, AdS/CFT correspondence and topological field theory, JHEP 12 (1998) 012 [hep-th/9812012] [INSPIRE].
Y. Tachikawa, On the 6d origin of discrete additional data of 4d gauge theories, JHEP 05 (2014) 020 [arXiv:1309.0697] [INSPIRE].
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Gang, D., Yonekura, K. Symmetry enhancement and closing of knots in 3d/3d correspondence. J. High Energ. Phys. 2018, 145 (2018). https://doi.org/10.1007/JHEP07(2018)145
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DOI: https://doi.org/10.1007/JHEP07(2018)145