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Why is Landau-Ginzburg link cohomology equivalent to Khovanov homology?

  • Dmitry GalakhovEmail author
Open Access
Regular Article - Theoretical Physics
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Abstract

In this note we make an attempt to compare a cohomological theory of Hilbert spaces of ground states in the \( \mathcal{N}=\left(2,2\right) \) 2d Landau-Ginzburg theory in models describing link embeddings in ℝ3 to Khovanov and Khovanov-Rozansky homologies. To confirm the equivalence we exploit the invariance of Hilbert spaces of ground states for interfaces with respect to homotopy. In this attempt to study solitons and instantons in the LandauGiznburg theory we apply asymptotic analysis also known in the literature as exact WKB method, spectral networks method, or resurgence. In particular, we associate instantons in LG model to specific WKB line configurations we call null-webs.

Keywords

Chern-Simons Theories Differential and Algebraic Geometry Sigma Models Topological Field Theories 

Notes

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.

References

  1. [1]
    M. Aganagic, E. Frenkel and A. Okounkov, Quantum q-Langlands Correspondence, Trans. Moscow Math. Soc. 79 (2018) 1 [arXiv:1701.03146] [INSPIRE].
  2. [2]
    M. Aganagic and A. Okounkov, Elliptic stable envelopes, arXiv:1604.00423 [INSPIRE].
  3. [3]
    M. Aganagic and S. Shakirov, Knot Homology and Refined Chern-Simons Index, Commun. Math. Phys. 333 (2015) 187 [arXiv:1105.5117] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    A. Anokhina and A. Morozov, Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary T -deformation of HOMFLY, JHEP 07 (2014) 063 [arXiv:1403.8087] [INSPIRE].
  5. [5]
    V.I. Arnold, On a characteristic class entering into conditions of quantisation, (english translation), Funct. Anal. Appl. 1 (1967) 1.Google Scholar
  6. [6]
    S. Arthamonov and S. Shakirov, Refined Chern-Simons Theory in Genus Two, arXiv:1504.02620 [INSPIRE].
  7. [7]
    M.F. Atiyah and N.J. Hitchin, The Geometry And Dynamics Of Magnetic Monopoles. M.B. Porter Lectures, Princeton University Press, (1988).Google Scholar
  8. [8]
    D. Bar-Natan, On Khovanovs categorification of the Jones polynomial, Algebr. Geom. Topol. 2 (2002)337 [math/0201043].
  9. [9]
    S. Bigelow, A homological definition of the Jones polynomial, Geom. Topol. Monographs 4 (2002)29 [math/0201221].
  10. [10]
    A. Braverman, G. Dobrovolska and M. Finkelberg, Gaiotto-Witten superpotential and Whittaker D-modules on monopoles, Adv. Math. 300 (2016) 451 [arXiv:1406.6671] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    S. Cecotti, P. Fendley, K.A. Intriligator and C. Vafa, A new supersymmetric index, Nucl. Phys. B 386 (1992) 405 [hep-th/9204102] [INSPIRE].
  12. [12]
    S. Cecotti and C. Vafa, Topological-anti-topological fusion, Nucl. Phys. B 367 (1991) 359 [INSPIRE].
  13. [13]
    S. Cecotti and C. Vafa, On classification of N = 2 supersymmetric theories, Commun. Math. Phys. 158 (1993) 569 [hep-th/9211097] [INSPIRE].
  14. [14]
    S. Chun, S. Gukov and D. Roggenkamp, Junctions of surface operators and categorification of quantum groups, arXiv:1507.06318 [INSPIRE].
  15. [15]
    V. Dolotin and A. Morozov, Introduction to Khovanov Homologies. I. Unreduced Jones superpolynomial, JHEP 01 (2013) 065 [arXiv:1208.4994] [INSPIRE].
  16. [16]
    V. Dolotin and A. Morozov, Introduction to Khovanov Homologies. II. Reduced Jones superpolynomials, J. Phys. Conf. Ser. 411 (2013) 012013 [arXiv:1209.5109] [INSPIRE].
  17. [17]
    V. Dolotin and A. Morozov, Introduction to Khovanov Homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants, Nucl. Phys. B 878 (2014) 12 [arXiv:1308.5759] [INSPIRE].
  18. [18]
    S.K. Donaldson, NahmS equations and the classification of monopoles, Commun. Math. Phys. 96 (1984) 387 [INSPIRE].
  19. [19]
    V.G. Drinfeld, Quasi-Hopf algebras and Knizhnik-Zamolodchikov equations, in Problems of Modern Quantum Field Theory, Springer, (1989), pp. 1-13.Google Scholar
  20. [20]
    O. Dumitrescu, L. Fredrickson, G. Kydonakis, R. Mazzeo, M. Mulase and A. Neitzke, Opers versus nonabelian Hodge, arXiv:1607.02172.
  21. [21]
    N.M. Dunfield, S. Gukov and J. Rasmussen, The Superpolynomial for Knot Homologies, Exper. Math. 15 (2006) 129 [math/0505662].
  22. [22]
    G.V. Dunne and M. Ünsal, Deconstructing zero: resurgence, supersymmetry and complex saddles, JHEP 12 (2016) 002 [arXiv:1609.05770] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    R. Eager, S.A. Selmani and J. Walcher, Exponential Networks and Representations of Quivers, JHEP 08 (2017) 063 [arXiv:1611.06177] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    S. Elitzur, G.W. Moore, A. Schwimmer and N. Seiberg, Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory, Nucl. Phys. B 326 (1989) 108 [INSPIRE].
  25. [25]
    P. Elyutin, V. Krivchenkov and N. Bogolyubov, Quantum mechanics with tasks (in Russian), Fizmatlit, (2001).Google Scholar
  26. [26]
    P. Etingof, I. Frenkel and A. Kirillov, Lectures on representation theory and Knizhnik-Zamolodchikov equations, No. 58., AMS, U.S.A., (1998).Google Scholar
  27. [27]
    P.J. Fréyd and D.N. Yetter, Braided compact closed categories with applications to low dimensional topology, Adv. Math. 77 (1989) 156.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    P. Fréyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K. Millett and A. Ocneanu, A new polynomial invariant of knots and links, Bull. Am. Math. Soc. 12 (1985) 239 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    D. Gaiotto and E. Witten, Knot Invariants from Four-Dimensional Gauge Theory, Adv. Theor. Math. Phys. 16 (2012) 935 [arXiv:1106.4789] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    D. Gaiotto, G.W. Moore and E. Witten, An Introduction To The Web-Based Formalism, arXiv:1506.04086 [INSPIRE].
  31. [31]
    D. Gaiotto, G.W. Moore and E. Witten, Algebra of the Infrared: String Field Theoretic Structures in Massive \( \mathcal{N}=\left(2,2\right) \) Field Theory In Two Dimensions, arXiv:1506.04087 [INSPIRE].
  32. [32]
    D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
  33. [33]
    D. Gaiotto, G.W. Moore and A. Neitzke, Framed BPS States, Adv. Theor. Math. Phys. 17 (2013) 241 [arXiv:1006.0146] [INSPIRE].
  34. [34]
    D. Gaiotto, G.W. Moore and A. Neitzke, Wall-Crossing in Coupled 2d-4d Systems, JHEP 12 (2012) 082 [arXiv:1103.2598] [INSPIRE].
  35. [35]
    D. Gaiotto, G.W. Moore and A. Neitzke, Spectral networks, Annales Henri Poincaré 14 (2013)1643 [arXiv:1204.4824] [INSPIRE].
  36. [36]
    D. Gaiotto, G.W. Moore and A. Neitzke, Spectral Networks and Snakes, Annales Henri Poincaré 15 (2014) 61 [arXiv:1209.0866] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    D. Galakhov, P. Longhi and G.W. Moore, Spectral Networks with Spin, Commun. Math. Phys. 340 (2015) 171 [arXiv:1408.0207] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    D. Galakhov, A. Mironov and A. Morozov, Wall Crossing Invariants: from quantum mechanics to knots, J. Exp. Theor. Phys. 120 (2015) 549 [Zh. Eksp. Teor. Fiz. 147 (2015) 623] [arXiv:1410.8482] [INSPIRE].
  39. [39]
    D. Galakhov and G.W. Moore, Comments On The Two-Dimensional Landau-Ginzburg Approach To Link Homology, arXiv:1607.04222 [INSPIRE].
  40. [40]
    A. Gerasimov, A. Morozov, M. Olshanetsky, A. Marshakov and S.L. Shatashvili, Wess-Zumino-Witten model as a theory of free fields, Int. J. Mod. Phys. A 5 (1990) 2495 [INSPIRE].
  41. [41]
    S. Gukov, A. Iqbal, C. Kozcaz and C. Vafa, Link Homologies and the Refined Topological Vertex, Commun. Math. Phys. 298 (2010) 757 [arXiv:0705.1368] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    S. Gukov, M. Mariño and P. Putrov, Resurgence in complex Chern-Simons theory, arXiv:1605.07615 [INSPIRE].
  43. [43]
    S. Gukov, P. Putrov and C. Vafa, Fivebranes and 3-manifold homology, JHEP 07 (2017) 071 [arXiv:1602.05302] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    S. Gukov, A.S. Schwarz and C. Vafa, Khovanov-Rozansky homology and topological strings, Lett. Math. Phys. 74 (2005) 53 [hep-th/0412243] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    A. Haydys, Fukaya-Seidel category and gauge theory, J. Sympl. Geom. 13 (2015) 151 [arXiv:1010.2353] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    L. Hollands and A. Neitzke, BPS states in the Minahan-Nemeschansky E 6 theory, Commun. Math. Phys. 353 (2017) 317 [arXiv:1607.01743] [INSPIRE].
  47. [47]
    K. Hori, A. Iqbal and C. Vafa, D-branes and mirror symmetry, hep-th/0005247 [INSPIRE].
  48. [48]
    K. Hori et al., Mirror symmetry, in Clay mathematics monographs, vol. 1, AMS, Providence, U.S.A., (2003).Google Scholar
  49. [49]
    J. Hurtubise, Monopoles and Rational Maps: A Note on a Theorem of Donaldson, Commun. Math. Phys. 100 (1985) 191 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    J. Hurtubise, The Classification of Monopoles for the Classical Groups, Commun. Math. Phys. 120 (1989) 613 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    J. Hurtubise and M.K. Murray, On the Construction of Monopoles for the Classical Groups, Commun. Math. Phys. 122 (1989) 35 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    A. Kapustin and E. Witten, Electric-Magnetic Duality And The Geometric Langlands Program, Commun. Num. Theor. Phys. 1 (2007) 1 [hep-th/0604151] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101 (2000) 359 [math/9908171] [INSPIRE].
  54. [54]
    M. Khovanov, sl(3) link homology, Algebr. Geom. Topol. 4 (2004) 1045 [math/0304375].
  55. [55]
    M. Khovanov and L. Rozansky, Matrix factorizations and link homology, Fund. Math. 199 (2008)1 [math/0401268].
  56. [56]
    P.B. Kronheimer and T.S. Mrowka, Khovanov homology is an unknot-detector, arXiv:1005.4346.
  57. [57]
    A.N. Kirilov and N. Yu. Reshetikhin, Representations of the algebra Uq (sl(2)), q-orthogonal polynomials and invariants of links, in Infinite Dimensional Lie Algebras and Groups, World Scientific, (1988), pp. 285-339.Google Scholar
  58. [58]
    L.D. Landau and E.M. Lifshitz, Quantum mechanics, non-relativistic theory (in Russian), 6th edition, Fizmatlit, (2004).Google Scholar
  59. [59]
    L.D. Landau, E.M. Lifshitz, J.B. Sykes, J.S. Bell and M.E. Rose, Quantum mechanics, non-relativistic theory, Phys. Today 11 (1958) 56.ADSCrossRefGoogle Scholar
  60. [60]
    A.D. Lauda, H. Queffelec and D.E.V. Rose, Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m), arXiv:1212.6076.
  61. [61]
    R.J. Lawrence, Homological representations of the Hecke algebra, Commun. Math. Phys. 135 (1990) 141.Google Scholar
  62. [62]
    P. Longhi and C.Y. Park, ADE Spectral Networks, JHEP 08 (2016) 087 [arXiv:1601.02633] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  63. [63]
  64. [64]
    M. Mackaay and M. Stošić and P. Vaz, Sl(N) link homology using foams and the Kapustin-Li formula, Geom. Topol. 13 (2009) 1075 [arXiv:0708.2228].
  65. [65]
    A. Matsuo, Jackson integrals of Jordan-Pochhammer type and quantum Knizhnik-Zamolodchikov equations, Commun. Math. Phys. 151 (1993) 263.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  66. [66]
    D. Maulik and A. Okounkov, Quantum Groups and Quantum Cohomology, arXiv:1211.1287 [INSPIRE].
  67. [67]
    R. Mazzeo and E. Witten, The Nahm Pole Boundary Condition, arXiv:1311.3167 [INSPIRE].
  68. [68]
    A. Mironov, A. Morozov and Y. Zenkevich, Ding-Iohara-Miki symmetry of network matrix models, Phys. Lett. B 762 (2016) 196 [arXiv:1603.05467] [INSPIRE].
  69. [69]
    G.W. Moore and N. Seiberg, Lectures On RCFT, RU-89-32, YCTP-P13-89, C89-08-14.Google Scholar
  70. [70]
    A. Yu. Morozov, Unitary Integrals and Related Matrix Models, Theor. Math. Phys. 162 (2010) 1 [arXiv:0906.3518] [INSPIRE].
  71. [71]
    A. Okounkov, Lectures on K-theoretic computations in enumerative geometry, arXiv:1512.07363 [INSPIRE].
  72. [72]
    J.H. Przytycki and P. Traczyk, Conway algebras and skein equivalence of links, Proc. Am. Math. Soc. 100 (1987) 744.MathSciNetCrossRefzbMATHGoogle Scholar
  73. [73]
    J. Rasmussen, Khovanov-Rozansky homology of two-bridge knots and links, math/0508510.
  74. [74]
    N. Yu. Reshetikhin and V.G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Commun. Math. Phys. 127 (1990) 1 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  75. [75]
    P. Turner, A hitchhikers guide to Khovanov homology, arXiv:1409.6442 [INSPIRE].
  76. [76]
    E. Witten, Supersymmetry and Morse theory, J. Diff. Geom. 17 (1982) 661 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  77. [77]
    E. Witten, Instantons, the Quark Model and the 1/n Expansion, Nucl. Phys. B 149 (1979) 285 [INSPIRE].
  78. [78]
    E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
  79. [79]
    E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].
  80. [80]
    E. Witten, Analytic Continuation Of Chern-Simons Theory, AMS/IP Stud. Adv. Math. 50 (2011) 347 [arXiv:1001.2933] [INSPIRE].
  81. [81]
    E. Witten, A New Look At The Path Integral Of Quantum Mechanics, arXiv:1009.6032 [INSPIRE].
  82. [82]
    E. Witten, Fivebranes and Knots, arXiv:1101.3216 [INSPIRE].
  83. [83]
    E. Witten, Khovanov Homology And Gauge Theory, arXiv:1108.3103 [INSPIRE].
  84. [84]
    E. Witten, Two Lectures On The Jones Polynomial And Khovanov Homology, arXiv:1401.6996 [INSPIRE].
  85. [85]
    A.B. Zamolodchikov, Conformal symmetry in two-dimensional space: recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088.CrossRefGoogle Scholar
  86. [86]
    Y. Zenkevich, Quantum spectral curve for (q, t)-matrix model, Lett. Math. Phys. 108 (2018) 413 [arXiv:1507.00519] [INSPIRE].

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© The Author(s) 2019

Authors and Affiliations

  1. 1.Center for Theoretical PhysicsUniversity of CaliforniaBerkeleyU.S.A.
  2. 2.Institute for Information Transmission ProblemsMoscowRussia

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