Quivers for 3-manifolds: the correspondence, BPS states, and 3d \(\mathcal{N} = 2\) theories. (English) Zbl 1454.83139
Summary: We introduce and explore the relation between quivers and 3-manifolds with the topology of the knot complement. This idea can be viewed as an adaptation of the knots-quivers correspondence to Gukov-Manolescu invariants of knot complements (also known as \(F_K\) or \(\hat{Z} )\). Apart from assigning quivers to complements of \(T^{(2,2p+1)}\) torus knots, we study the physical interpretation in terms of the BPS spectrum and general structure of 3d \(\mathcal{N} = 2\) theories associated to both sides of the correspondence. We also make a step towards categorification by proposing a \(t\)-deformation of all objects mentioned above.
MSC:
| 83E30 | String and superstring theories in gravitational theory |
| 58J28 | Eta-invariants, Chern-Simons invariants |
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
References:
| [1] | P. Kucharski, M. Reineke, M. Stošić and P. Sułkowski, BPS states, knots and quivers, Phys. Rev. D96 (2017) 121902 [arXiv:1707.02991] [INSPIRE]. |
| [2] | P. Kucharski, M. Reineke, M. Stošić and P. Sułkowski, Knots-quivers correspondence, Adv. Theor. Math. Phys.23 (2019) 1849 [arXiv:1707.04017] [INSPIRE]. · Zbl 1521.57006 |
| [3] | S. Gukov and C. Manolescu, A two-variable series for knot complements, arXiv:1904.06057 [INSPIRE]. |
| [4] | Ooguri, H.; Vafa, C., Knot invariants and topological strings, Nucl. Phys. B, 577, 419 (2000) · Zbl 1036.81515 · doi:10.1016/S0550-3213(00)00118-8 |
| [5] | J.M.F. Labastida and M. Mariño, Polynomial invariants for torus knots and topological strings, Commun. Math. Phys.217 (2001) 423 [hep-th/0004196] [INSPIRE]. · Zbl 1018.81049 |
| [6] | J.M.F. Labastida, M. Mariño and C. Vafa, Knots, links and branes at large N, JHEP11 (2000) 007 [hep-th/0010102] [INSPIRE]. · Zbl 0990.81545 |
| [7] | M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435 [INSPIRE]. · Zbl 1248.14060 |
| [8] | M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, Commun. Num. Theor. Phys.5 (2011) 231 [arXiv:1006.2706] [INSPIRE]. · Zbl 1248.14060 |
| [9] | Efimov, AI, Cohomological Hall algebra of a symmetric quiver, Compos. Math., 148, 1133 (2012) · Zbl 1273.14113 · doi:10.1112/S0010437X12000152 |
| [10] | M. Stošić and P. Wedrich, Rational links and DT invariants of quivers, arXiv:1711.03333 [INSPIRE]. |
| [11] | M. Stošić and P. Wedrich, Tangle addition and the knots-quivers correspondence, arXiv:2004.10837 [INSPIRE]. |
| [12] | T. Ekholm, P. Kucharski and P. Longhi, Physics and geometry of knots-quivers correspondence, arXiv:1811.03110 [INSPIRE]. · Zbl 1435.81171 |
| [13] | Ekholm, T.; Kucharski, P.; Longhi, P., Multi-cover skeins, quivers, and 3d \(\mathcal{N} = 2\) dualities, JHEP, 02, 018 (2020) · Zbl 1435.81171 · doi:10.1007/JHEP02(2020)018 |
| [14] | H. Awata, S. Gukov, P. Sułkowski and H. Fuji, Volume Conjecture: Refined and Categorified, Adv. Theor. Math. Phys.16 (2012) 1669 [arXiv:1203.2182] [INSPIRE]. · Zbl 1282.57016 |
| [15] | M. Aganagic and C. Vafa, Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots, arXiv:1204.4709 [INSPIRE]. |
| [16] | H. Fuji, S. Gukov and P. Sułkowski, Super-A-polynomial for knots and BPS states, Nucl. Phys. B867 (2013) 506 [arXiv:1205.1515] [INSPIRE]. · Zbl 1262.81170 |
| [17] | S. Garoufalidis, A.D. Lauda and T.T.Q. Lê, The colored HOMFLYPT function is q-holonomic, Duke Math. J.167 (2018) 397 [arXiv:1604.08502] [INSPIRE]. · Zbl 1414.57012 |
| [18] | H. Fuji, S. Gukov, M. Stošić and P. Sułkowski, 3d analogs of Argyres-Douglas theories and knot homologies, JHEP01 (2013) 175 [arXiv:1209.1416] [INSPIRE]. |
| [19] | M. Panfil, M. Stošić and P. Sułkowski, Donaldson-Thomas invariants, torus knots, and lattice paths, Phys. Rev. D98 (2018) 026022 [arXiv:1802.04573] [INSPIRE]. |
| [20] | Panfil, M.; Sułkowski, P., Topological strings, strips and quivers, JHEP, 01, 124 (2019) · Zbl 1409.83193 · doi:10.1007/JHEP01(2019)124 |
| [21] | Larraguivel, H.; Noshchenko, D.; Panfil, M.; Sułkowski, P., Nahm sums, quiver A-polynomials and topological recursion, JHEP, 07, 151 (2020) · Zbl 1451.83099 · doi:10.1007/JHEP07(2020)151 |
| [22] | Gukov, S.; Putrov, P.; Vafa, C., Fivebranes and 3-manifold homology, JHEP, 07, 071 (2017) · Zbl 1380.58018 · doi:10.1007/JHEP07(2017)071 |
| [23] | S. Gukov, D. Pei, P. Putrov and C. Vafa, BPS spectra and 3-manifold invariants, J. Knot Theor. Ramifications29 (2020) 2040003 [arXiv:1701.06567] [INSPIRE]. · Zbl 1448.57020 |
| [24] | S. Gukov, M. Mariño and P. Putrov, Resurgence in complex Chern-Simons theory, arXiv:1605.07615 [INSPIRE]. |
| [25] | Cheng, MCN; Chun, S.; Ferrari, F.; Gukov, S.; Harrison, SM, 3d Modularity, JHEP, 10, 010 (2019) · Zbl 1427.81118 · doi:10.1007/JHEP10(2019)010 |
| [26] | Kucharski, P., \( \hat{Z}\) invariants at rational τ, JHEP, 09, 092 (2019) · Zbl 1423.81171 · doi:10.1007/JHEP09(2019)092 |
| [27] | H.-J. Chung, BPS Invariants for 3-Manifolds at Rational Level K , arXiv:1906.12344 [INSPIRE]. |
| [28] | T. Dimofte, D. Gaiotto and S. Gukov, Gauge Theories Labelled by Three-Manifolds, Commun. Math. Phys.325 (2014) 367 [arXiv:1108.4389] [INSPIRE]. · Zbl 1292.57012 |
| [29] | S. Cecotti, C. Cordova and C. Vafa, Braids, Walls, and Mirrors, arXiv:1110.2115 [INSPIRE]. |
| [30] | T. Dimofte, D. Gaiotto and S. Gukov, 3-Manifolds and 3d Indices, Adv. Theor. Math. Phys.17 (2013) 975 [arXiv:1112.5179] [INSPIRE]. · Zbl 1297.81149 |
| [31] | Gadde, A.; Gukov, S.; Putrov, P., Walls, Lines, and Spectral Dualities in 3d Gauge Theories, JHEP, 05, 047 (2014) · doi:10.1007/JHEP05(2014)047 |
| [32] | S. Chun, S. Gukov, S. Park and N. Sopenko, 3d-3d correspondence for mapping tori, arXiv:1911.08456 [INSPIRE]. |
| [33] | H.-J. Chung, Index for a Model of 3d-3d Correspondence for Plumbed 3-Manifolds, arXiv:1912.13486 [INSPIRE]. |
| [34] | K. Bringmann, K. Mahlburg and A. Milas, Quantum modular forms and plumbing graphs of 3-manifolds, arXiv:1810.05612 [INSPIRE]. · Zbl 1444.11112 |
| [35] | K. Bringmann, K. Mahlburg and A. Milas, Higher depth quantum modular forms and plumbed 3-manifolds, arXiv:1906.10722. · Zbl 1444.11112 |
| [36] | M.C.N. Cheng, F. Ferrari and G. Sgroi, Three-Manifold Quantum Invariants and Mock Theta Functions, Phil. Trans. Roy. Soc. Lond.378 (2019) 20180439 [arXiv:1912.07997] [INSPIRE]. · Zbl 1462.11037 |
| [37] | Melvin, P.; Morton, H., The coloured Jones function, Commun. Math. Phys., 169, 501 (1995) · Zbl 0845.57007 · doi:10.1007/BF02099310 |
| [38] | Bar-Natan, D.; Garoufalidis, S., On the Melvin-Morton-Rozansky conjecture, Invent. Math., 125, 103 (1996) · Zbl 0855.57004 · doi:10.1007/s002220050070 |
| [39] | L. Rozansky, A Contribution to the trivial connection to Jones polynomial and Witten’s invariant of 3-D manifolds. 1., Commun. Math. Phys.175 (1996) 275 [hep-th/9401061] [INSPIRE]. · Zbl 0872.57010 |
| [40] | L. Rozansky, The universal R-matrix, Burau representation, and the Melvin-Morton expansion of the colored Jones polynomial, Adv. Math.134 (1998) 1. · Zbl 0949.57006 |
| [41] | S. Garoufalidis, On the characteristic and deformation varieties of a knot, Geom. Topol. Monographs7 (2004) 291 [math/0306230]. · Zbl 1080.57014 |
| [42] | Gukov, S., Three-dimensional quantum gravity, Chern-Simons theory, and the A polynomial, Commun. Math. Phys., 255, 577 (2005) · Zbl 1115.57009 · doi:10.1007/s00220-005-1312-y |
| [43] | T. Ekholm, A. Gruen, S. Gukov, P. Kucharski, S. Park and P. Sułkowski, \( \hat{Z}\) at large N: from curve counts to quantum modularity, arXiv:2005.13349 [INSPIRE]. |
| [44] | L. Diogo and T. Ekholm, Augmentations, annuli, and Alexander polynomials, arXiv:2005.09733 [INSPIRE]. |
| [45] | Park, S., Higher rank \(\hat{Z}\) and F_K, SIGMA, 16, 044 (2020) · Zbl 1460.57013 |
| [46] | S. Park, Large color R-matrix for knot complements and strange identities, arXiv:2004.02087 [INSPIRE]. |
| [47] | S. Gukov, P.-S. Hsin, H. Nakajima, S. Park, D. Pei and N. Sopenko, Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants, arXiv:2005.05347 [INSPIRE]. |
| [48] | N.M. Dunfield, S. Gukov and J. Rasmussen, The Superpolynomial for knot homologies, math/0505662 [INSPIRE]. · Zbl 1118.57012 |
| [49] | Freyd, P.; Yetter, D.; Hoste, J.; Lickorish, WBR; Millett, K.; Ocneanu, A., A new polynomial invariant of knots and links, Bull. Am. Math. Soc., 12, 239 (1985) · Zbl 0572.57002 · doi:10.1090/S0273-0979-1985-15361-3 |
| [50] | Przytycki, J.; Traczyk, P., Invariants of links of Conway type, Kobe J. Math., 4, 115 (1987) · Zbl 0655.57002 |
| [51] | Witten, E., Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys., 121, 351 (1989) · Zbl 0667.57005 · doi:10.1007/BF01217730 |
| [52] | S. Garoufalidis, P. Kucharski and P. Sułkowski, Knots, BPS states, and algebraic curves, Commun. Math. Phys.346 (2016) 75 [arXiv:1504.06327] [INSPIRE]. · Zbl 1365.57015 |
| [53] | P. Kucharski and P. Sułkowski, BPS counting for knots and combinatorics on words, JHEP11 (2016) 120 [arXiv:1608.06600] [INSPIRE]. · Zbl 1390.81448 |
| [54] | S. Meinhardt and M. Reineke, Donaldson-Thomas invariants versus intersection cohomology of quiver moduli, arXiv:1411.4062. · Zbl 1439.14160 |
| [55] | Franzen, H.; Reineke, M., Semi-stable Chow-Hall algebras of quivers and quantized Donaldson-Thomas invariants, Alg. Numb. Theor., 12, 1001 (2018) · Zbl 1423.14314 · doi:10.2140/ant.2018.12.1001 |
| [56] | T. Dimofte, S. Gukov and L. Hollands, Vortex Counting and Lagrangian 3-manifolds, Lett.Math. Phys.98 (2011) 225 [arXiv:1006.0977] [INSPIRE]. · Zbl 1239.81057 |
| [57] | Y. Terashima and M. Yamazaki, Semiclassical Analysis of the 3d/3d Relation, Phys. Rev. D88 (2013) 026011 [arXiv:1106.3066] [INSPIRE]. |
| [58] | Gopakumar, R.; Vafa, C., On the gauge theory/geometry correspondence, AMS/IP Stud. Adv. Math., 23, 45 (2001) · Zbl 1026.81029 · doi:10.1090/amsip/023/03 |
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