Letters in Mathematical Physics

, Volume 109, Issue 3, pp 579–622 | Cite as

BPS/CFT correspondence IV: sigma models and defects in gauge theory

  • Nikita NekrasovEmail author


Quantum field theory \(L_1\) on spacetime \(X_{1}\) can be coupled to another quantum field theory \(L_2\) on a spacetime \(X_{2}\) via the third quantum field theory \(L_{12}\) living on \(X_{12} = X_{1} \cap X_{2}\). We explore several such constructions with two- and four-dimensional \(X_{1}, X_{2}\)’s and zero- and two-dimensional \(X_{12}\)’s, in the context of \({\mathcal {N}}=2\) supersymmetry, non-perturbative Dyson–Schwinger equations, and BPS/CFT correspondence. The companion paper (Nekrasov, “BPS/CFT correspondence V: BPZ and KZ equations from qq-characters”, 2017. arXiv:1711.11582 [hep-th]) will show that the BPZ and KZ equations of two-dimensional conformal field theory are obeyed by the half-BPS surface defects in quiver \({\mathcal {N}}=2\) gauge theories.


Supersymmetry BPS/CFT correspondence Hidden symmetry Defects in quantum field theory 

Mathematics Subject Classification

81T60 81T13 81T45 81T30 



Research was partly supported by the National Science Foundation under Grant No. NSF-PHY/1404446. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The author thanks A. Losev for discussions about two-dimensional topological theories in 1992–1994, E. Frenkel for the invitation to the DARPA program ‘Langlands Program and Physics’ conference at the IAS in March 8–10, 2004, and for the opportunity to present there some of the ideas developed in this paper, as well as for numerous patient explanations on [35] and other topics. The author is grateful to A. Okounkov, V. Pestun and A. Rosly for numerous useful discussions concerning the topics of this paper over the recent years, as well as to S. Jeong and O. Tsymbaliuk for their collaboration of the related projects. The paper was finished while the author visited the IHES (Bures-sur-Yvette). We thank this remarkable institution for its hospitality.


  1. 1.
    Aganagic, M., Okounkov, A.: Quasimap counts and Bethe eigenfunctions. arXiv:1704.08746 [math-ph]
  2. 2.
    Alday, L.F., Gaiotto, D., Tachikawa, Y.: Liouville correlation functions from four-dimensional gauge theories. Lett. Math. Phys. 91, 167 (2010). [arXiv:0906.3219 [hep-th]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alday, L.F., Gaiotto, D., Gukov, S., Tachikawa, Y., Verlinde, H.: Loop and surface operators in \(\text{ N }=2\) gauge theory and Liouville modular geometry. JHEP 1001, 113 (2010). [arXiv:0909.0945 [hep-th]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Alday, L.F., Tachikawa, Y.: Affine SL(2) conformal blocks from 4d gauge theories. Lett. Math. Phys. 94, 87 (2010). [arXiv:1005.4469 [hep-th]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Arutyunov, G., Frolov, S., Medvedev, P.: Elliptic Ruijsenaars–Schneider model from the cotangent bundle over the two-dimensional current group. J. Math. Phys. 38, 5682 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ashok, S.K., Billo, M., Dell’Aquila, E., Frau, M., Gupta, V., John, R.R., Lerda, A.: Surface operators, chiral rings, and localization in \({\cal{N}}=2\) gauge theories. JHEP 1711, 137 (2017). [arXiv:1707.08922 [hep-th]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Awata, H., Tsuchiya, A., Yamada, Y.: Integral formulas for the WZNW correlation functions. Nucl. Phys. B 365, 680–698 (1991)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Awata, H., Fuji, H., Kanno, H., Manabe, M., Yamada, Y.: Localization with a surface operator, irregular conformal blocks and open topological string. Adv. Theor. Math. Phys. 16(3), 725 (2012). [arXiv:1008.0574 [hep-th]]MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Babujian, H.M.: Off-shell Bethe Ansatz equation and N point correlators in SU(2) WZNW theory. J. Phys. A 26, 6981 (1993). [arXiv:hep-th/9307062]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Babujian, H.M., Flume, R.: Off-shell Bethe Ansatz equation for Gaudin magnets and solutions of Knizhnik–Zamolodchikov equations. Mod. Phys. Lett. A 9, 2029 (1994). [arXiv:hep-th/9310110]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Barns-Graham, A., Dorey, N., Lohitsiri, N., Tong, D., Turner, C.: ADHM and the 4d quantum hall effect. arXiv:1710.09833 [hep-th]
  12. 12.
    Baulieu, L., Losev, A., Nekrasov, N.: Chern–Simons and twisted supersymmetry in various dimensions. Nucl. Phys. B 522, 82 (1998). [arXiv:hep-th/9707174]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bershadsky, M., Johansen, A., Sadov, V., Vafa, C.: Topological reduction of 4-d SYM to 2-d sigma models. Nucl. Phys. B 448, 166 (1995). [arXiv:hep-th/9501096]ADSCrossRefzbMATHGoogle Scholar
  14. 14.
    Bertram, A., Ciocan-Fontanine, I., Kim, B.S.: Two proofs of a conjecture of Hori and Vafa. arXiv:math/0304403 [math-ag]
  15. 15.
    Biquard, O.: Sur les Fibrés Paraboliques sur une Surface Complexe. J. Lond. Math. Soc. 53, 302 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Biswas, I.: Parabolic bundles as orbifold bundles. Duke Math J. 88(2), 305–325 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Blau, M., Thompson, G.: Derivation of the Verlinde formula from Chern–Simons theory and the G/G model. Nucl. Phys. B 408, 345 (1993). [arXiv:hep-th/9305010]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Blau, M., Thompson, G.: Lectures on 2-d gauge theories: topological aspects and path integral techniques. arXiv:hep-th/9310144
  19. 19.
    Braverman, A.: Instanton counting via affine Lie algebras. 1. Equivariant \(J\)-functions of (affine) flag manifolds and Whittaker vectors. arXiv:math/0401409 [math-ag]
  20. 20.
    Braverman, A., Maulik, D., Okounkov, A.: Quantum cohomology of the Springer resolution. arXiv:1001.0056 [math.AG]
  21. 21.
    Bruzzo, U., Chuang, W-y, Diaconescu, D.-E., Jardim, M., Pan, G., Zhang, Y.: D-branes, surface operators, and ADHM quiver representations. Adv. Theor. Math. Phys. 15(3), 849 (2011). [arXiv:1012.1826 [hep-th]]MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Bullimore, M., Kim, H.C., Koroteev, P.: Defects and quantum Seiberg–Witten geometry. JHEP 1505, 095 (2015). [arXiv:1412.6081 [hep-th]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Bullimore, M., Kim, H.C., Lukowski, T.: Expanding the Bethe/gauge dictionary. JHEP 1711, 055 (2017). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Chang, C. K., Chen, H. Y., Jain, D., Lee, N.: Connecting localization and wall-crossing via D-branes. arXiv:1512.02645 [hep-th]
  25. 25.
    Chen, H.Y., Dorey, N., Hollowood, T.J., Lee, S.: A New 2d/4d duality via integrability. JHEP 1109, 040 (2011). [arXiv:1104.3021 [hep-th]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Costello, K.: Supersymmetric gauge theory and the Yangian. arXiv:1303.2632 [hep-th]
  27. 27.
    Costello, K.: Integrable lattice models from four-dimensional field theories. Proc. Symp. Pure Math. 88, 3 (2014). [arXiv:1308.0370 [hep-th]]MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Dotsenko, V.S., Fateev, V.A.: Conformal algebra and multipoint correlation functions in two-dimensional statistical models. Nucl. Phys. B 240, 312 (1984). ADSCrossRefGoogle Scholar
  29. 29.
    Douglas, M.R., Moore, G.W.: D-branes, quivers, and ALE instantons. arXiv:hep-th/9603167
  30. 30.
    Dorey, N., Lee, S., Hollowood, T.J.: Quantization of integrable systems and a 2d/4d duality. JHEP 1110, 077 (2011). [arXiv:1103.5726 [hep-th]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Dorey, N., Zhao, P.: Solution of quantum integrable systems from quiver gauge theories. JHEP 1702, 118 (2017). [arXiv:1512.09367 [hep-th]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Duistermaat, J.J., Heckman, G.J.: On the variation in the cohomology in the symplectic form of the reduced phase space. Invent. Math. 69, 259 (1982)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Frenkel, E.: Free field realizations in representation theory and conformal field theory. arXiv:hep-th/9408109
  34. 34.
    Frenkel, E., Gukov, S., Teschner, J.: Surface operators and separation of variables. JHEP 1601, 179 (2016). [arXiv:1506.07508 [hep-th]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Frenkel, E., Reshetikhin, N.: The \(q\)-characters of representations of quantun affine algebras and deformations of \(W\)-algebras. arXiv:math/9810055v5 [math.QA]
  36. 36.
    Frenkel, I.B., Kac, V.G.: Basic representations of affine Lie algebras and dual resonance models. Invent. Math. 62, 23 (1980)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Gaiotto, D.: Surface operators in \(\text{ N } = 2\) 4d gauge theories. JHEP 1211, 090 (2012). [arXiv:0911.1316 [hep-th]]ADSMathSciNetCrossRefGoogle Scholar
  38. 38.
    Gaiotto, D., Gukov, S., Seiberg, N.: Surface defects and resolvents. JHEP 1309, 070 (2013). [arXiv:1307.2578 [hep-th]]ADSCrossRefGoogle Scholar
  39. 39.
    Gaiotto, D., Moore, G.W., Neitzke, A.: Framed BPS states. Adv. Theor. Math. Phys. 17(2), 241 (2013). [arXiv:1006.0146 [hep-th]]MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Gamayun, O., Iorgov, N., Lisovyy, O.: Conformal field theory of Painlevé VI, JHEP 1210, 038 (2012) Erratum: [JHEP 1210, 183 (2012)] ,, [arXiv:1207.0787 [hep-th]]
  41. 41.
    Gerasimov, A., Morozov, A., Olshanetsky, M., Marshakov, A., Shatashvili, S.L.: Wess-Zumino–Witten model as a theory of free fields. Int. J. Mod. Phys. A 5, 2495 (1990). (originally published as 4 preprints in April 1989)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    Gerasimov, A.: Localization in GWZW and Verlinde formula. arXiv:hep-th/9305090
  43. 43.
    Givental, A.: Equivariant Gromov–Witten invariants. arXiv:alg-geom/9603021
  44. 44.
    Givental, A.: A mirror theorem for toric complete intersections. arXiv:alg-geom/9701016v2
  45. 45.
    Givental, A.: The mirror formula for quintic threefolds. arXiv:math/9807070
  46. 46.
    Gomis, J., Le Floch, B.: M2-brane surface operators and gauge theory dualities in Toda. JHEP 1604, 183 (2016). [arXiv:1407.1852 [hep-th]]ADSzbMATHGoogle Scholar
  47. 47.
    Gomis, J., Okuda, T., Pestun, V.: Exact results for ’t Hooft loops in gauge theories on \(\text{ S }^{\wedge }4\). JHEP 1205, 141 (2012). [arXiv:1105.2568 [hep-th]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Gorsky, A., Le Floch, B., Milekhin, A., Sopenko, N.: Surface defects and instanton–vortex interaction. Nucl. Phys. B 920, 122 (2017). [arXiv:1702.03330 [hep-th]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Gorsky, A., Nekrasov, N.: Elliptic Calogero–Moser system from two dimensional current algebra. arXiv:hep-th/9401021
  50. 50.
    Gukov, S.: Gauge theory and knot homologies. Fortsch. Phys. 55, 473 (2007). [arXiv:0706.2369 [hep-th]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Gukov, S., Witten, E.: Rigid surface operators. Adv. Theor. Math. Phys. 14(1), 87 (2010). [arXiv:0804.1561 [hep-th]]MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Ito, Y., Okuda, T., Taki, M.: Line operators on \(\text{ S }^{\wedge }1\text{ xR }^{\wedge }3\) and quantization of the Hitchin moduli space. JHEP 1204, 010 (2012) Erratum: [JHEP 1603, 085 (2016)] ,, [arXiv:1111.4221 [hep-th]]
  53. 53.
    Hosomichi, K., Lee, S., Okuda, T.: Supersymmetric vortex defects in two dimensions. arXiv:1705.10623 [hep-th]
  54. 54.
    Hori, K., Vafa, C.: Mirror symmetry. arXiv:hep-th/0002222
  55. 55.
    Jeong, S., Nekrasov, N.: Opers, surface defects, and Yang–Yang functional. arXiv:1806.08270 [hep-th]
  56. 56.
    Kanno, H., Tachikawa, Y.: Instanton counting with a surface operator and the chain-saw quiver. JHEP 1106, 119 (2011). [arXiv:1105.0357 [hep-th]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    King, A.: Instantons and holomorphic bundles on the blown up plane, Ph.D. thesis, Oxford (1989)Google Scholar
  58. 58.
    Koroteev, P., Pushkar, P.P., Smirnov, A., Zeitlin, A.M.: Quantum K-theory of quiver varieties and many-body systems. arXiv:1705.10419 [math.AG]
  59. 59.
    Koroteev, P., Zeitlin, A.M.: Difference equations for K-theoretic vertex functions of type-A Nakajima varieties. arXiv:1802.04463 [math.AG]
  60. 60.
    Kronheimer, P., Mrowka, T.: Gauge theory for embedded surfaces, I. Topology 32, 773 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Kronheimer, P., Mrowka, T.: Gauge theory for embedded surfaces, II. Topology 34, 37 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Kozcaz, C., Pasquetti, S., Passerini, F., Wyllard, N.: Affine sl(N) conformal blocks from \(\text{ N }=2\) SU(N) gauge theories. JHEP 1101, 045 (2011). [arXiv:1008.1412 [hep-th]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Kronheimer, P., Nakajima, H.: Yang–Mills instantons on ALE gravitational instantons. Math. Ann. 288, 263–307 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Losev, A., Marshakov, A., Nekrasov, N.A.: Small instantons, little strings and free fermions, In: Shifman M et al. (eds.) From Fields to Strings, vol. 1, pp. 581-621. [arXiv:hep-th/0302191]
  65. 65.
    Losev, A., Moore, G.W., Nekrasov, N., Shatashvili, S.: Four-dimensional avatars of two-dimensional RCFT. Nucl. Phys. Proc. Suppl. 46, 130 (1996). [arXiv:hep-th/9509151]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Losev, A., Nekrasov, N., Shatashvili, S.L.: The Freckled instantons, In: Shifman MA (ed.) The Many Faces of the Superworld, pp. 453–475. [arXiv:hep-th/9908204]
  67. 67.
    Losev, A., Nekrasov, N., Shatashvili, S.L.: Freckled instantons in two-dimensions and four-dimensions. Class. Quantum Grav. 17, 1181 (2000). [arXiv:hep-th/9911099]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Maulik, D., Okounkov, A.: Quantum groups and quantum cohomology. arXiv:1211.1287 [math.AG]
  69. 69.
    Moore, G.W., Nekrasov, N., Shatashvili, S.: Integrating over Higgs branches. Commun. Math. Phys. 209, 97 (2000). [arXiv:hep-th/9712241]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Nakajima, H.: Instantons on ALE spaces, quiver varieties, and Kac–Moody algebras. Duke Math. J. 76(2), 365–416 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    Nakajima, H.: Instantons and affine Lie algebras. Nucl. Phys. B Proc. Suppl. 46(1–3), 154–161 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    Nakajima, H.: Quiver varieties and Kac–Moody algebras. Duke Math. J. 91(3), 515–560 (1998). MathSciNetCrossRefzbMATHGoogle Scholar
  73. 73.
    Nakajima, H., Yoshioka, K.: Instanton counting on blowup. 1. Invent. Math. 162, 313 (2005). [arXiv:math/0306198 [math.AG]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  74. 74.
    Nakajima, H., Yoshioka, K.: Perverse coherent sheaves on blow-up. I. A Quiver description. arXiv:0802.3120 [math.AG]
  75. 75.
    Nakajima, H., Yoshioka, K.: Perverse coherent sheaves on blowup, III: blow-up formula from wall-crossing. Kyoto J. Math. 51(2), 263 (2011). [arXiv:0911.1773 [math.AG]]MathSciNetCrossRefzbMATHGoogle Scholar
  76. 76.
    Nawata, S.: Givental J-functions, quantum integrable systems, AGT relation with surface operator. Adv. Theor. Math. Phys. 19, 1277 (2015). [arXiv:1408.4132 [hep-th]]MathSciNetCrossRefzbMATHGoogle Scholar
  77. 77.
    Nekrasov, N.: Four dimensional holomorphic theories, PhD thesis, Princeton University, 1996.
  78. 78.
    Nekrasov, N.: On the BPS/CFT correspondence, Lecture at the University of Amsterdam string theory group seminar (2004)Google Scholar
  79. 79.
    Nekrasov, N.A.: Seiberg–Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7(5), 831 (2003). [arXiv:hep-th/0206161]MathSciNetCrossRefzbMATHGoogle Scholar
  80. 80.
    Nekrasov, N., Rosly, A., Shatashvili, S.: Darboux coordinates, Yang–Yang functional, and gauge theory. Nucl. Phys. Proc. Suppl. 216, 69 (2011). [arXiv:1103.3919 [hep-th]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  81. 81.
    Nekrasov, N.: BPS/CFT correspondence: non-perturbative Dyson–Schwinger equations and qq-characters. JHEP 1603, 181 (2016). arxiv:1512.05388 [hep-th]ADSCrossRefzbMATHGoogle Scholar
  82. 82.
    Nekrasov, N.: BPS/CFT correspondence II: instantons at crossroads, moduli and compactness theorem. arXiv:1608.07272 [hep-th]
  83. 83.
    Nekrasov, N.: BPS/CFT correspondence III: gauge origami partition function and \(qq\)-characters. arXiv:1701.00189 [hep-th]
  84. 84.
    Nekrasov, N.: BPS/CFT correspondence V: BPZ and KZ equations from \(qq\)-characters. arXiv:1711.11582 [hep-th] (2017)
  85. 85.
    Nekrasov, N.: Bethe states as defects in gauge theories, Bethe wavefunctions from gauged linear sigma models via Bethe/gauge correspondence, talks at the SCGP, delivered on 2013-10-02 and 2014-11-03.
  86. 86.
    Nekrasov, N.: Supersymmetric gauge theories and quantization of integrable systems, lecture at Strings’2009, Rome.
  87. 87.
    Nekrasov, N., Okounkov, A.: Seiberg–Witten theory and random partitions. Prog. Math. 244, 525 (2006). [arXiv:hep-th/0306238]MathSciNetCrossRefzbMATHGoogle Scholar
  88. 88.
    Nekrasov, N., Pestun, V.: Seiberg–Witten geometry of four dimensional \(\text{ N }=2\) quiver gauge theories. arXiv:1211.2240 [hep-th]
  89. 89.
    Nekrasov, N., Pestun, V., Shatashvili, S.: Quantum geometry and quiver gauge theories. arXiv:1312.6689 [hep-th]
  90. 90.
    Nekrasov, N., Prabhakar, N.S.: Spiked instantons from intersecting D-branes. Nucl. Phys. B 914, 257 (2017). arXiv:1611.03478 [hep-th]ADSCrossRefzbMATHGoogle Scholar
  91. 91.
    Nekrasov, N.A., Shatashvili, S.L.: Supersymmetric vacua and Bethe ansatz. Nucl. Phys. Proc. Suppl. 192–193, 91 (2009). arXiv:0901.4744 [hep-th]MathSciNetCrossRefzbMATHGoogle Scholar
  92. 92.
    Nekrasov, N.A., Shatashvili, S.L.: Quantum integrability and supersymmetric vacua. Prog. Theor. Phys. Suppl. 177, 105 (2009). arXiv:0901.4748 [hep-th]ADSCrossRefzbMATHGoogle Scholar
  93. 93.
    Nekrasov, N.A., Shatashvili, S.L.: Quantization of integrable systems and four dimensional gauge theories. arXiv:0908.4052 [hep-th]
  94. 94.
    Okounkov, A., Smirnov, A.: Quantum difference equation for Nakajima varieties. arXiv:1602.09007 [math-ph]
  95. 95.
    Pan, Y., Peelaers, W.: Intersecting surface defects and instanton partition functions. JHEP 1707, 073 (2017). [arXiv:1612.04839 [hep-th]]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  96. 96.
    Poghossian, R.: Deformed SW curve and the null vector decoupling equation in Toda field theory. JHEP 1604, 070 (2016). [arXiv:1601.05096 [hep-th]]ADSMathSciNetzbMATHGoogle Scholar
  97. 97.
    Poghosyan, G., Poghossian, R.: VEV of Baxter’s Q-operator in \(\text{ N }=2\) gauge theory and the BPZ differential equation. JHEP 1611, 058 (2016). [arXiv:1602.02772 [hep-th]]MathSciNetCrossRefzbMATHGoogle Scholar
  98. 98.
    Seiberg, N., Witten, E.: String theory and noncommutative geometry. JHEP 9909, 032 (1999). [arXiv:hep-th/9908142]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  99. 99.
    Shifman, M., Yung, A.: Quantum deformation of the effective theory on non-abelian string and 2D–4D correspondence. Phys. Rev. D 89(6), 065035 (2014). [arXiv:1401.1455 [hep-th]]ADSCrossRefGoogle Scholar
  100. 100.
    Tarasov, V., Varchenko, A.: Jackson integral representations for solutions of the quantized Knizhnik–Zamolodchikov equation. [arXiv:hep-th/9311040]
  101. 101.
    Verlinde, E.P.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B 300, 360 (1988)ADSCrossRefzbMATHGoogle Scholar
  102. 102.
    Witten, E.: Two-dimensional gauge theory revisited. J. Geom. Phys. 9, 303–368 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  103. 103.
    Witten, E.: Supersymmetric Yang–Mills theory on a four manifold. J. Math. Phys. 35, 5101 (1994). [arXiv:hep-th/9403195]ADSMathSciNetCrossRefzbMATHGoogle Scholar
  104. 104.
    Witten, E.: Integrable lattice models from gauge theory. arXiv:1611.00592 [hep-th]
  105. 105.
    Witten, E.: The Verlinde algebra and the cohomology of the Grassmannian, In: Geometry, Topology, and Physics, pp. 357–422. Cambridge (1993). [arXiv:hep-th/9312104]

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Simons Center for Geometry and PhysicsStony Brook UniversityStony BrookUSA

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