Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 047, 141 pages      arXiv:1211.2240
Contribution to the Special Issue on Differential Geometry Inspired by Mathematical Physics in honor of Jean-Pierre Bourguignon for his 75th birthday

Seiberg-Witten Geometry of Four-Dimensional $\mathcal N=2$ Quiver Gauge Theories

Nikita Nekrasov a and Vasily Pestun b
a) Simons Center for Geometry and Physics, Stony Brook University, Stony Brook, NY 11794-3636, USA
b) Institut des Hautes Etudes Scientifiques, 91440 Bures-sur-Yvette, France

Received December 19, 2022, in final form June 20, 2023; Published online July 16, 2023

Seiberg-Witten geometry of mass deformed $\mathcal N=2$ superconformal ADE quiver gauge theories in four dimensions is determined. We solve the limit shape equations derived from the gauge theory and identify the space $\mathfrak M$ of vacua of the theory with the moduli space of the genus zero holomorphic (quasi)maps to the moduli space ${\rm Bun}_{\mathbf G} (\mathcal E)$ of holomorphic $G^{\mathbb C}$-bundles on a (possibly degenerate) elliptic curve $\mathcal E$ defined in terms of the microscopic gauge couplings, for the corresponding simple ADE Lie group $G$. The integrable systems $\mathfrak P$ underlying the special geometry of $\mathfrak M$ are identified. The moduli spaces of framed $G$-instantons on ${\mathbb R}^{2} \times {\mathbb T}^{2}$, of $G$-monopoles with singularities on ${\mathbb R}^{2} \times {\mathbb S}^{1}$, the Hitchin systems on curves with punctures, as well as various spin chains play an important rôle in our story. We also comment on the higher-dimensional theories.

Key words: low-energy theory; instantons; monopoles; integrability.

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  1. Alday L.F., Gaiotto D., Tachikawa Y., Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010), 167-197, arXiv:0906.3219.
  2. Alim M., Cecotti S., Cordova C., Espahbodi S., Rastogi A., Vafa C., BPS quivers and spectra of complete ${\mathcal{N}}=2$ quantum field theories, Commun. Math. Phys. 323 (2013), 1185-1227, arXiv:1109.4941.
  3. Arnold V.I., Gusein-Zade S.M., Varchenko A.N., Singularities of differentiable maps. Vol. I. The classification of critical points, caustics and wave fronts, Monogr. Math., Vol. 82, Birkhäuser, Boston, MA, 1985.
  4. Astashkevich A., Nekrasov N., Schwarz A., On noncommutative Nahm transform, Comm. Math. Phys. 211 (2000), 167-182, arXiv:hep-th/9810147.
  5. Atiyah M.F., Instantons in two and four dimensions, Comm. Math. Phys. 93 (1984), 437-451.
  6. Atiyah M.F., Magnetic monopoles and the Yang-Baxter equations, Internat. J. Modern Phys. A 6 (1991), 2761-2774.
  7. Atiyah M.F., Murray M., Monopoles and Yang-Baxter equations, in Further Advances in Twistor Theory, Vol. II. Integrable systems, conformal geometry and gravitation, Pitman Res. Notes Math. Ser., Vol. 232,Longman Scientific & Technical, Harlow, 1995, 13-14.
  8. Baranovsky V., Ginzburg V., Conjugacy classes in loop groups and $G$-bundles on elliptic curves, Int. Math. Res. Not. 1996 (1996), 733-751, arXiv:alg-geom/9607008.
  9. Bardakci K., Halpern M.B., New dual quark models, Phys. Rev. D 3 (1971), 2493-2506.
  10. Baxter R.J., Exactly solved models in statistical mechanics, in Integrable Systems in Statistical Mechanics, Ser. Adv. Statist. Mech., Vol. 1, World Sci. Publishing, Singapore, 1985, 5-63.
  11. Belavin A.A., Drinfeld V.G., Solutions of the classical Yang-Baxter equation for simple Lie algebras, Funct. Anal. Appl. 16 (1982), 159-180.
  12. Bernshtein I.N., Shvartsman O.V., Chevalley's theorem for complex crystallographic Coxeter groups, Funct. Anal. Appl. 12 (1978), 308-310.
  13. Blum J.D., Intriligator K., New phases of string theory and 6d RG fixed points via branes at orbifold singularities, Nuclear Phys. B 506 (1997), 199-222, arXiv:hep-th/9705044.
  14. Braam P.J., van Baal P., Nahm's transformation for instantons, Comm. Math. Phys. 122 (1989), 267-280.
  15. Cartier P., Jacobiennes généralisées, monodromie unipotente et intégrales itérées, Astérisque 161-162 (1988), Exp. No. 687, 31-52.
  16. Chalmers G., Hanany A., Three-dimensional gauge theories and monopoles, Nuclear Phys. B 489 (1997), 223-244, arXiv:hep-th/9608105.
  17. Chen H.-Y., Dorey N., Hollowood T.J., Lee S., A new 2d/4d duality via integrability, J. High Energy Phys. 2011 (2011), no. 9, 040, 16 pages, arXiv:1104.3021.
  18. Cherkis S., Kapustin A., Nahm transform for periodic monopoles and $\mathcal{N}=2$ super Yang-Mills theory, Comm. Math. Phys. 218 (2001), 333-371, arXiv:hep-th/0006050.
  19. Cherkis S.A., Kapustin A., Singular monopoles and supersymmetric gauge theories in three dimensions, Nuclear Phys. B 525 (1998), 215-234, arXiv:hep-th/9711145.
  20. Cherkis S., Kapustin A., Hyper-Kähler metrics from periodic monopoles, Phys. Rev. D 65 (2002), 084015, 10 pages, arXiv:hep-th/0109141.
  21. Connes A., Douglas M.R., Schwarz A., Noncommutative geometry and matrix theory: compactification on tori, J. High Energy Phys. 1998 (1998), no. 2, 003, 35 pages, arXiv:hep-th/9711162.
  22. Corrigan E., Goddard P., Construction of instanton and monopole solutions and reciprocity, Ann. Physics 154 (1984), 253-279.
  23. Costello K., Yagi J., Unification of integrability in supersymmetric gauge theories, Adv. Theor. Math. Phys. 24 (2020), 1931-2041, arXiv:1810.01970.
  24. Cremmer E., Kounnas C., Van Proeyen A., Derendinger J.-P., Ferrara S., de Wit B., Girardello L., Vector multiplets coupled to $N=2$ supergravity: super-Higgs effect, flat potentials and geometric structure, Nuclear Phys. B 250 (1985), 385-426.
  25. Curio G., Donagi R., Moduli in $\mathcal{N}=1$ heterotic/F-theory duality, Nuclear Phys. B 518 (1998), 603-631, arXiv:hep-th/9801057.
  26. de Boer J., Hori K., Ooguri H., Oz Y., Yin Z., Mirror symmetry in three-dimensional gauge theories, ${\rm SL}(2, Z)$ and D-brane moduli spaces, Nuclear Phys. B 493 (1997), 148-176, arXiv:hep-th/9612131.
  27. Demidov S.V., Dubovsky S.L., Rubakov V.A., Sibiryakov S.M., Gauge theory solitons on the noncommutative cylinder, Theoret. and Math. Phys. 138 (2004), 269-283, arXiv:hep-th/0301047.
  28. Dolan L., A new symmetry group of real self-dual Yang-Mills theory, Phys. Lett. B 113 (1982), 387-390.
  29. Dolan L., Goddard P., Current algebra on the torus, Comm. Math. Phys. 285 (2009), 219-264, arXiv:0710.3743.
  30. Donagi R., Spectral covers, in Current Topics in Complex Algebraic Geometry (Berkeley, CA, 1992/93), Math. Sci. Res. Inst. Publ., Vol. 28, Cambridge University Press, Cambridge, 1995, 65-86, arXiv:alg-geom/9505009.
  31. Donagi R., Principal bundles on elliptic fibrations, Asian J. Math. 1 (1997), 214-223, arXiv:alg-geom/9702002.
  32. Donagi R., Seiberg-Witten integrable systems, in Algebraic Geometry - Santa Cruz 1995, Proc. Sympos. Pure Math., Vol. 62, Amer. Math. Soc., Providence, RI, 1997, 3-43, arXiv:alg-geom/9705010.
  33. Donagi R., Taniguchi lectures on principal bundles on elliptic fibrations, in Integrable Systems and Algebraic Geometry (Kobe/Kyoto, 1997), World Sci. Publ., River Edge, NJ, 1998, 33-46, arXiv:hep-th/9802094.
  34. Donagi R., Gaitsgory D., The gerbe of Higgs bundles, Transform. Groups 7 (2002), 109-153, arXiv:math.AG/0005132.
  35. Donagi R., Markman E., Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles, in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 1-119, arXiv:alg-geom/9507017.
  36. Donagi R., Wijnholt M., Breaking GUT groups in $F$-theory, Adv. Theor. Math. Phys. 15 (2011), 1523-1603, arXiv:0808.2223.
  37. Donagi R., Wijnholt M., Model building with $F$-theory, Adv. Theor. Math. Phys. 15 (2011), 1237-1317, arXiv:0802.2969.
  38. Donagi R., Witten E., Supersymmetric Yang-Mills theory and integrable systems, Nuclear Phys. B 460 (1996), 299-334, arXiv:hep-th/9510101.
  39. Donaldson S.K., Nahm's equations and the classification of monopoles, Comm. Math. Phys. 96 (1984), 387-407.
  40. Donaldson S.K., Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. Lond. Math. Soc. 50 (1985), 1-26.
  41. Dorey N., Lee S., Hollowood T.J., Quantization of integrable systems and a 2d/4d duality, J. High Energy Phys. 2011 (2011), no. 10, 077, 42 pages, arXiv:1103.5726.
  42. Douglas M.R., Moore G.W., D-branes, quivers, and ALE instantons, arXiv:hep-th/9603167.
  43. Douglas M.R., Nekrasov N., Noncommutative field theory, Rev. Modern Phys. 73 (2001), 977-1029, arXiv:hep-th/0106048.
  44. Drinfeld V.G., Quantum groups, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, 798-820.
  45. Duistermaat J.J., Heckman G.J., On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982), 259-268.
  46. Enriquez B., Feigin B., Rubtsov V., Separation of variables for Gaudin-Calogero systems, Compositio Math. 110 (1998), 1-16, arXiv:q-alg/9605030.
  47. Faddeev L., Integrable models in $(1+1)$-dimensional quantum field theory, in Recent Advances in Field Theory and Statistical Mechanics (Les Houches, 1982), North-Holland, Amsterdam, 1984, 561-608.
  48. Feigin B., Frenkel E., Reshetikhin N., Gaudin model, Bethe ansatz and critical level, Comm. Math. Phys. 166 (1994), 27-62, arXiv:hep-th/9402022.
  49. Felder G., Elliptic quantum groups, in XIth International Congress of Mathematical Physics (Paris, 1994), Int. Press, Cambridge, MA, 1995, 211-218, arXiv:hep-th/9412207.
  50. Felder G., Varchenko A., Integral representation of solutions of the elliptic Knizhnik-Zamolodchikov-Bernard equations, Int. Math. Res. Not. 1995 (1995), 221-233, arXiv:hep-th/9502165.
  51. Felder G., Varchenko A., On representations of the elliptic quantum group $E_{\tau,\eta}({\rm sl}_2)$, Comm. Math. Phys. 181 (1996), 741-761, arXiv:q-alg/9601003.
  52. Frenkel E., Affine algebras, Langlands duality and Bethe ansatz, in XIth International Congress of Mathematical Physics (Paris, 1994), Int. Press, Cambridge, MA, 1995, 606-642, arXiv:q-alg/9506003.
  53. Frenkel E., Opers on the projective line, flag manifolds and Bethe ansatz, Mosc. Math. J. 4 (2004), 655-705, arXiv:math.QA/0308269.
  54. Frenkel E., Reshetikhin N., The $q$-characters of representations of quantum affine algebras and deformations of $\mathcal{W}$-algebras, in Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), Contemp. Math., Vol. 248, Amer. Math. Soc., Providence, RI, 1999, 163-205, arXiv:math.QA/9810055.
  55. Friedman R., Morgan J., Witten E., Vector bundles and ${\rm F}$ theory, Comm. Math. Phys. 187 (1997), 679-743, arXiv:hep-th/9701162.
  56. Friedman R., Morgan J.W., Holomorphic principal bundles over elliptic curves. II. The parabolic construction, J. Differential Geom. 56 (2000), 301-379, arXiv:math.AG/0006174.
  57. Friedman R., Morgan J.W., Exceptional groups and del Pezzo surfaces, in Symposium in Honor of C.H. Clemens (Salt Lake City, UT, 2000),Contemp. Math., Vol. 312, Amer. Math. Soc., Providence, RI, 2002, 101-116, arXiv:math.AG/0009155.
  58. Friedman R., Morgan J.W., Automorphism sheaves, spectral covers, and the Kostant and Steinberg sections, in Vector Bundles and Representation Theory (Columbia, MO, 2002), Contemp. Math., Vol. 322, Amer. Math. Soc., Providence, RI, 2003, 217-244, arXiv:math.AG/0209053.
  59. Friedman R., Morgan J.W., Witten E., Principal $G$-bundles over elliptic curves, Math. Res. Lett. 5 (1998), 97-118, arXiv:alg-geom/9707004.
  60. Friedman R., Morgan J.W., Witten E., Vector bundles over elliptic fibrations, J. Algebraic Geom. 8 (1999), 279-401, arXiv:alg-geom/9709029.
  61. Gaiotto D., $\mathcal{N}=2$ dualities, J. High Energ. Phys. 2012 (2012), no. 8, 034, 58 pages, arXiv:0904.2715.
  62. Gaiotto D., Moore G.W., Neitzke A., Framed BPS States, Adv. Theor. Math. Phys 17 (2013), 241-397, arXiv:1006.0146.
  63. Gaiotto D., Moore G.W., Neitzke A., Wall-crossing, Hitchin systems, and the WKB approximation, Adv. Math. 234 (2013), 239-403, arXiv:0907.3987.
  64. Gaiotto D., Witten E., $S$-duality of boundary conditions in $\mathcal{N}=4$ super Yang-Mills theory, Adv. Theor. Math. Phys. 13 (2009), 721-896, arXiv:0807.3720.
  65. Garland H., Murray M.K., Kac-Moody monopoles and periodic instantons, Comm. Math. Phys. 120 (1988), 335-351.
  66. Gerasimov A., Degenerate surfaces and solitons: Handle gluing operator, Sov. J. Nucl. Phys. 49 (1989), 553-557.
  67. Gerasimov A., Kharchev S., Lebedev D., Oblezin S., On a class of representations of the Yangian and moduli space of monopoles, Comm. Math. Phys. 260 (2005), 511-525, arXiv:math.AG/0409031.
  68. Gerasimov A., Shatashvili S., Higgs bundles, gauge theories and quantum groups, Comm. Math. Phys. 277 (2008), 323-367, arXiv:hep-th/0609024.
  69. Gerasimov A., Shatashvili S., Two-dimensional gauge theories and quantum integrable systems, in From Hodge Theory to Integrability and TQFT tt*-geometry, Proc. Sympos. Pure Math., Vol. 78, Amer. Math. Soc., Providence, RI, 2008, 239-262, arXiv:0711.1472.
  70. Ginzburg V., Kapranov M., Vasserot E., Elliptic algebras and equivariant elliptic cohomology, arXiv:q-alg/9505012.
  71. Givental A., A mirror theorem for toric complete intersections, in Topological Field Theory, Primitive Forms and Related Topics (Kyoto, 1996), Progr. Math., Vol. 160, Birkhäuser, Boston, MA, 1998, 141-175, arXiv:alg-geom/9701016.
  72. Gorsky A., Krichever I.M., Marshakov A., Mironov A., Morozov A., Integrability and Seiberg-Witten exact solution, Phys. Lett. B 355 (1995), 466-474, arXiv:hep-th/9505035.
  73. Gorsky A., Marshakov A., Mironov A., Morozov A., A note on spectral curve for the periodic homogeneous $XYZ$-spin chain, arXiv:hep-th/9604078.
  74. Gorsky A., Marshakov A., Mironov A., Morozov A., $\mathcal{N}=2$ supersymmetric QCD and integrable spin chains: rational case $N_f < 2N_c$, Phys. Lett. B 380 (1996), 75-80, arXiv:hep-th/9603140.
  75. Gorsky A., Nekrasov N., Elliptic Calogero-Moser system from two-dimensional current algebra, arXiv:hep-th/9401021.
  76. Gorsky A., Nekrasov N., Rubtsov V., Hilbert schemes, separated variables, and D-branes, Comm. Math. Phys. 222 (2001), 299-318, arXiv:hep-th/9901089.
  77. Grojnowski I., Delocalised equivariant elliptic cohomology, in Elliptic Cohomology, London Math. Soc. Lecture Note Ser., Vol. 342, Cambridge University Press, Cambridge, 2007, 114-121.
  78. Gross D.J., Nekrasov N., Monopoles and strings in non-commutative gauge theory, J. High Energy Phys. 2000 (2000), no. 7, 034, 34 pages, arXiv:hep-th/0005204.
  79. Gukov S., Witten E., Gauge theory, ramification, and the geometric Langlands program, in Current Developments in Mathematics, 2006, Int. Press, Somerville, MA, 2008, 35-180, arXiv:hep-th/0612073.
  80. Gukov S., Witten E., Rigid surface operators, Adv. Theor. Math. Phys. 14 (2010), 87-177, arXiv:0804.1561.
  81. Hanany A., Witten E., Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics, Nuclear Phys. B 492 (1997), 152-190, arXiv:hep-th/9611230.
  82. Hitchin N.J., On the construction of monopoles, Comm. Math. Phys. 89 (1983), 145-190.
  83. Hitchin N.J., Stable bundles and integrable systems, Duke Math. J. 54 (1987), 91-114.
  84. Hitchin N.J., Karlhede A., Lindström U., Roček M., Hyper-Kähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), 535-589.
  85. Howe P.S., Stelle K.S., West P.C., A class of finite four-dimensional supersymmetric field theories, Phys. Lett. B 124 (1983), 55-58.
  86. Intriligator K., Seiberg N., Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996), 513-519, arXiv:hep-th/9607207.
  87. Jeong S., Lee N., Nekrasov N., Parallel surface defects, Hecke operators, and quantum Hitchin system, arXiv:2304.04656.
  88. Jeong S., Lee N., Nekrasov N., Intersecting defects in gauge theory, quantum spin chains, and Knizhnik-Zamolodchikov equations, J. High Energy Phys. 2021 (2021), no. 10, 120, 82 pages, arXiv:2103.17186.
  89. Jeong S., Nekrasov N., Opers, surface defects, and Yang-Yang functional, Adv. Theor. Math. Phys. 24 (2020), 1789-1916, arXiv:1806.08270.
  90. Jeong S., Nekrasov N., Riemann-Hilbert correspondence and blown up surface defects, J. High Energy Phys. 2020 (2020), no. 12, 006, 81 pages, arXiv:2007.03660.
  91. Jeong S., Zhang X., BPZ equations for higher degenerate fields and non-perturbative Dyson-Schwinger equations, arXiv:1710.06970.
  92. Johnson C.V., Myers R.C., Aspects of type IIB theory on asymptotically locally Euclidean spaces, Phys. Rev. D 55 (1997), 6382-6393, arXiv:hep-th/9610140.
  93. Kac V.G., Peterson D.H., Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. Math. 53 (1984), 125-264.
  94. Kanev V., Spectral curves, simple Lie algebras, and Prym-Tjurin varieties, in Theta Functions - Bowdoin 1987, Part 1 (Brunswick, ME, 1987), Proc. Sympos. Pure Math., Vol. 49, Amer. Math. Soc., Providence, RI, 1989, 627-645.
  95. Kapustin A., $D_n$ quivers from branes, J. High Energy Phys. 1998 (1998), no. 12, 015, 17 pages, arXiv:hep-th/9806238.
  96. Kapustin A., Solution of $\mathcal{N}=2$ gauge theories via compactification to three dimensions, Nuclear Phys. B 534 (1998), 531-545, arXiv:hep-th/9804069.
  97. Kapustin A., Orlov D., Remarks on A-branes, mirror symmetry, and the Fukaya category, J. Geom. Phys. 48 (2003), 84-99, arXiv:hep-th/0109098.
  98. Kapustin A., Sethi S., The Higgs branch of impurity theories, Adv. Theor. Math. Phys. 2 (1998), 571-591, arXiv:hep-th/9804027.
  99. Kapustin A., Witten E., Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007), 1-236, arXiv:hep-th/0604151.
  100. Katz S., Klemm A., Vafa C., Geometric engineering of quantum field theories, Nuclear Phys. B 497 (1997), 173-195, arXiv:hep-th/9609239.
  101. Katz S., Mayr P., Vafa C., Mirror symmetry and exact solution of $4$D $N=2$ gauge theories. I, Adv. Theor. Math. Phys. 1 (1997), 53-114, arXiv:hep-th/9706110.
  102. Kimura T., Pestun V., Fractional quiver W-algebras, Lett. Math. Phys. 108 (2018), 2425-2451, arXiv:1705.04410.
  103. Kimura T., Pestun V., Quiver elliptic W-algebras, Lett. Math. Phys. 108 (2018), 1383-1405, arXiv:1608.04651.
  104. Kimura T., Pestun V., Quiver W-algebras, Lett. Math. Phys. 108 (2018), 1351-1381, arXiv:1512.08533.
  105. Klein F., Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade, Birkhäuser, Basel, 1993.
  106. Knizhnik V.G., Analytic fields on Riemann surfaces. II, Comm. Math. Phys. 112 (1987), 567-590.
  107. Kontsevich M., Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), 157-216, arXiv:q-alg/9709040.
  108. Kronheimer P.B., The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. 29 (1989), 665-683.
  109. Kronheimer P.B., A hyper-Kählerian structure on coadjoint orbits of a semisimple complex group, J. Lond. Math. Soc. 42 (1990), 193-208.
  110. Kronheimer P.B., Instantons and the geometry of the nilpotent variety, J. Differential Geom. 32 (1990), 473-490.
  111. Lawrence A., Nekrasov N., Vafa C., On conformal field theories in four dimensions, Nuclear Phys. B 533 (1998), 199-209, arXiv:hep-th/9803015.
  112. Lee N., Nekrasov N., Quantum spin systems and supersymmetric gauge theories. Part I, J. High Energy Phys. 2021 (2021), no. 3, 093, 86 pages, arXiv:2009.11199.
  113. Lerche W., Introduction to Seiberg-Witten theory and its stringy origin, Nuclear Phys. B Proc. Suppl. 55B (1997), 83-117, arXiv:hep-th/9611190.
  114. Levin A., Olshanetsky M., Zotov A., Hitchin systems - symplectic Hecke correspondence and two-dimensional version, arXiv:nlin.SI/0110045.
  115. Levin A., Olshanetsky M., Smirnov A., Zotov A., Characteristic classes and integrable systems. General construction, arXiv:1006.0702.
  116. Levin A., Olshanetsky M., Smirnov A., Zotov A., Characteristic classes and integrable systems for simple Lie groups, arXiv:1007.4127.
  117. Licata A.M., Moduli spaces of sheaves on surfaces in geometric representation theory, Ph.D. Thesis, Yale University, 2007.
  118. Looijenga E., Root systems and elliptic curves, Invent. Math. 38 (1976), 17-32.
  119. Losev A., Marshakov A., Nekrasov N., Small instantons, little strings and free fermions, in From Fields to Strings: Circumnavigating Theoretical Physics. Vol. 1, World Sci. Publ., Singapore, 2005, 581-621, arXiv:hep-th/0302191.
  120. Losev A., Nekrasov N., Shatashvili S., Issues in topological gauge theory, Nuclear Phys. B 534 (1998), 549-611, arXiv:hep-th/9711108.
  121. Losev A., Nekrasov N., Shatashvili S., Freckled instantons in two and four dimensions, Classical Quantum Gravity 17 (2000), 1181-1187, arXiv:hep-th/9911099.
  122. Lurie J., A survey of elliptic cohomology, in Algebraic Topology, Abel Symp., Vol. 4, Springer, Berlin, 2009, 219-277.
  123. Manschot J., Moore G.W., Zhang X., Effective gravitational couplings of four-dimensional $\mathcal{N} = 2$ supersymmetric gauge theories, J. High Energy Phys. 2020 (2020), no. 6, 150, 40 pages, arXiv:1912.04091.
  124. Martinec E.J., Warner N.P., Integrable systems and supersymmetric gauge theory, Nuclear Phys. B 459 (1996), 97-112, arXiv:hep-th/9509161.
  125. McKay J., Graphs, singularities, and finite groups, in The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), Proc. Sympos. Pure Math., Vol. 37, Amer. Math. Soc., Providence, RI, 1980, 183-186.
  126. Mironov A., Morozov A., Runov B., Zenkevich Y., Zotov A., Spectral duality between Heisenberg chain and Gaudin model, Lett. Math. Phys. 103 (2013), 299-329, arXiv:1206.6349.
  127. Moore G., Nekrasov N., Shatashvili S., Integrating over Higgs branches, Comm. Math. Phys. 209 (2000), 97-121, arXiv:hep-th/9712241.
  128. Muneyuki K., Tai T.-S., Yonezawa N., Yoshioka R., Baxter's T-Q equation, ${\rm SU}(N)/{\rm SU}(2)^{N-3}$ correspondence and $\Omega$-deformed Seiberg-Witten prepotential, J. High Energy Phys. 2011 (2011), no. 9, 125, 17 pages, arXiv:1107.3756.
  129. Nahm W., All self-dual multimonopoles for arbitrary gauge groups, in Structural Elements in Particle Physics and Statistical Mechanics (Freiburg, 1981), NATO Adv. Study Inst. Ser. B: Physics, Vol. 82, Plenum, New York, 1983, 301-310.
  130. Nakajima H., Quiver varieties and finite-dimensional representations of quantum affine algebras, J. Amer. Math. Soc. 14 (2001), 145-238, arXiv:math.QA/9912158.
  131. Nanopoulos D., Xie D., $\mathcal{N}=2$ SU quiver with USP ends or SU ends with antisymmetric matter, J. High Energy Phys. 2009 (2009), no. 8, 108, 21 pages, arXiv:0907.1651.
  132. Nanopoulos D., Xie D., Hitchin equation, singularity, and $\mathcal{N}=2$ superconformal field theories, J. High Energy Phys. 2010 (2010), no. 3, 043, 43 pages, arXiv:0911.1990.
  133. Nanopoulos D., Xie D., Hitchin equation, irregular singularity, and $\mathcal{N}=2$ asymptotical free theories, arXiv:1005.1350.
  134. Nanopoulos D., Xie D., $N=2$ generalized superconformal quiver gauge theory, J. High Energy Phys. 2012 (2012), no. 9, 127, 19 pages, arXiv:1006.3486.
  135. Narasimhan M.S., Seshadri C.S., Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. 82 (1965), 540-567.
  136. Nekrasov N., Four-dimensional holomorphic theories, Ph.D. Thesis, Princeton University, 1996, available at href
  137. Nekrasov N., Holomorphic bundles and many-body systems, Comm. Math. Phys. 180 (1996), 587-603, arXiv:hep-th/9503157.
  138. Nekrasov N., Five-dimensional gauge theories and relativistic integrable systems, Nuclear Phys. B 531 (1998), 323-344, arXiv:hep-th/9609219.
  139. Nekrasov N., Noncommutative instantons revisited, Comm. Math. Phys. 241 (2003), 143-160, arXiv:hep-th/0010017.
  140. Nekrasov N., Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003), 831-864, arXiv:hep-th/0206161.
  141. Nekrasov N., Seiberg-Witten prepotential, higher Casimirs, and free fermions, in Progress in String, Field and Particle Theory, NATO Science Series, Vol. 104, Springer, Dordrecht, 2003, 263-274.
  142. Nekrasov N., Solution of $\mathcal{N}=2$ theories via instanton counting, Ann. Henri Poincaré 4 (2003), S129-S145.
  143. Nekrasov N., BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and $qq$-characters, J. High Energy Phys. 2016 (2016), no. 3, 181, 70 pages, arXiv:1512.05388.
  144. Nekrasov N., BPS/CFT correspondence II: Instantons at crossroads, moduli and compactness theorem, Adv. Theor. Math. Phys. 21 (2017), 503-583, arXiv:1608.07272.
  145. Nekrasov N., BPS/CFT correspondence III: Gauge origami partition function and $qq$-characters, Comm. Math. Phys. 358 (2018), 863-894, arXiv:1701.00189.
  146. Nekrasov N., BPS/CFT correspondence IV: sigma models and defects in gauge theory, Lett. Math. Phys. 109 (2019), 579-622, arXiv:1711.11011.
  147. Nekrasov N., BPS/CFT correspondence V: BPZ and KZ equations from $qq$-characters, arXiv:1711.11582.
  148. Nekrasov N., Blowups in BPS/CFT correspondence, and Painlevé VI, Ann. Henri Poincaré, to appear, arXiv:2007.03646.
  149. Nekrasov N., Okounkov A., Seiberg-Witten theory and random partitions, in The Unity of Mathematics: In Honor of the Ninetieth Birthday of I.M. Gelfand, Birkhäuser, Boston, 2006, 525-596, arXiv:hep-th/0306238.
  150. Nekrasov N., Pestun V., Seiberg-Witten geometry of four dimensional $\mathcal{N}=2$ quiver gauge theories, arXiv:1211.2240.
  151. Nekrasov N., Pestun V., Shatashvili S., Quantum geometry and quiver gauge theories, Comm. Math. Phys. 357 (2018), 519-567, arXiv:1312.6689.
  152. Nekrasov N., Rosly A., Shatashvili S., Darboux coordinates, Yang-Yang functional, and gauge theory, Nuclear Phys. B Proc. Suppl. 216 (2011), 69-93, arXiv:1103.3919.
  153. Nekrasov N., Schwarz A., Instantons on noncommutative $\mathbb{R}^4$, and $(2,0)$ superconformal six-dimensional theory, Comm. Math. Phys. 198 (1998), 689-703, arXiv:hep-th/9802068.
  154. Nekrasov N., Shadchin S., ABCD of instantons, Comm. Math. Phys. 252 (2004), 359-391, arXiv:hep-th/0404225.
  155. Nekrasov N., Shatashvili S., Bethe ansatz and supersymmetric vacua, AIP Conf. Proc. 1134 (2009), 154-169.
  156. Nekrasov N., Shatashvili S., Quantum integrability and supersymmetric vacua, Prog. Theor. Phys. Suppl. 177 (2009), 105-119, arXiv:0901.4748.
  157. Nekrasov N., Shatashvili S., Quantization of integrable systems and four dimensional gauge theories, in XVIth International Congress on Mathematical Physics, World Sci. Publ., Hackensack, NJ, 2010, 265-289, arXiv:0908.4052.
  158. Nekrasov N., Tsymbaliuk A., Surface defects in gauge theory and KZ equation, Lett. Math. Phys. 112 (2022), 28, 53 pages, arXiv:2103.12611.
  159. Nekrasov N., Witten E., The omega deformation, branes, integrability and Liouville theory, J. High Energy Phys. 2010 (2010), no. 9, 092, 82 pages, arXiv:1002.0888.
  160. Nieri F., Zenkevich Y., Quiver ${\rm W}_{\epsilon_1,\epsilon_2}$ algebras of 4D $\mathcal{N}=2$ gauge theories, J. Phys. A 53 (2020), 275401, 43 pages, arXiv:1912.09969.
  161. Novikov V.A., Shifman M.A., Vainshtein A.I., Zakharov V.I., Exact Gell-Mann-Low function of supersymmetric Yang-Mills theories from instanton calculus, Nuclear Phys. B 229 (1983), 381-393.
  162. Olshanetsky M., Classical integrable systems and gauge field theories, Phys. Part. Nuclei 40 (2009), 93-114.
  163. Pestun V., Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Comm. Math. Phys. 313 (2012), 71-129, arXiv:0712.2824.
  164. Ramanathan A., Stable principal bundles on a compact Riemann surface, Math. Ann. 213 (1975), 129-152.
  165. Schwarz A., Noncommutative instantons: a new approach, Comm. Math. Phys. 221 (2001), 433-450, arXiv:hep-th/0102182.
  166. Seiberg N., Witten E., Electric-magnetic duality, monopole condensation, and confinement in $\mathcal{N}=2$ supersymmetric Yang-Mills theory, Nuclear Phys. B 426 (1994), 19-52, arXiv:hep-th/9407087.
  167. Seiberg N., Witten E., Monopoles, duality and chiral symmetry breaking in $\mathcal{N}=2$ supersymmetric QCD, Nuclear Phys. B 431 (1994), 484-550, arXiv:hep-th/9408099.
  168. Shadchin S., On certain aspects of string theory/gauge theory correspondence, Ph.D. Thesis, Université Paris-Sud, 2005, arXiv:hep-th/0502180.
  169. Shadchin S., Cubic curves from instanton counting, J. High Energy Phys. 2006 (2006), no. 3, 046, 27 pages, arXiv:hep-th/0511132.
  170. Shadchin S., Status report on the instanton counting, SIGMA 2 (2006), 008, 11 pages, arXiv:hep-th/0601167.
  171. Sklyanin E.K., Separation of variables in the Gaudin model, J. Sov. Math. 47 (1989), 2473-2488.
  172. Sklyanin E.K., Separation of variables& - new trends, Progr. Theoret. Phys. Suppl. 118 (1995), 35-60, arXiv:solv-int/9504001.
  173. Sklyanin E.K., Takhtadzhyan L.A., Faddeev L.D., Quantum inverse problem method. I, Theoret. and Math. Phys. 40 (1979), 688-706.
  174. Steinberg R., Regular elements of semisimple algebraic groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 49-80.
  175. Witten E., Solutions of four-dimensional field theories via $M$-theory, Nuclear Phys. B 500 (1997), 3-42, arXiv:hep-th/9703166.
  176. Witten E., New ''gauge'' theories in six dimensions, J. High Energy Phys. 1998 (1998), no. 1, 001, 26 pages, arXiv:hep-th/9710065.
  177. Witten E., Toroidal compactification without vector structure, J. High Energy Phys. 1998 (1998), no. 2, 006, 43 pages, arXiv:hep-th/9712028.
  178. Witten E., Gauge theory and wild ramification, Anal. Appl. (Singap.) 6 (2008), 429-501, arXiv:0710.0631.

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