Skip to main content
SpringerLink
Log in

Exact Results on \({\mathcal N}=2\) Supersymmetric Gauge Theories

  • Chapter
  • First Online:
New Dualities of Supersymmetric Gauge Theories

Part of the book series: Mathematical Physics Studies ((MPST))

Abstract

The following is meant to give an overview over our special volume. The first three Sects. 1–3 are intended to give a general overview over the physical motivations behind this direction of research, and some of the developments that initiated this project.

A citation of the form [V:x] refers to article number x in this volume.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The lacing number \(n_{\scriptscriptstyle G}\) is equal to 1 is the Lie-algebra of G is simply-laced, 2 if it is of type \(B_n\), \(C_n\) and \(F_4\), and 3 if it is of type \(G_2\).

  2. 2.

    The result of [Sen] furnishes a nontrivial check of a prediction following from the Montonen-Olive conjecture.

  3. 3.

    A fairly extensive list of references to the early literature can be found e.g. in [Le].

  4. 4.

    In other cases \({\mathcal M}\) may a union of infinitely many finite-dimensional components of increasing dimensions, as happens in the cases discussed in Sect. 1.5.

  5. 5.

    The regularisation introduced in [N] provides a physical interpretation of a regularisation for integrals over instanton moduli spaces previously used in [LNS, MNS1].

  6. 6.

    The results presented in [N, NO03] were based in particular on the previous work [LNS, MNS1, MNS2]. Similar results were presented in [FPS, Ho1, Ho2, FP, BFMT]; for a review see [V:4].

  7. 7.

    The infinite series (1.12) are probably convergent. This was verified explicitly for the example of pure SU(2) Super-Yang-Mills theory in [ILTy], and it is expected to follow for UV finite gauge theories from the relations with conformal field theory to be discussed in the next section.

  8. 8.

    A concise description of the definition of the conformal blocks can be found in ([V:12], Sect. 2.5).

  9. 9.

    The IR duality conjectures can be used to describe the moduli space of vacua as manifold covered by charts with local coordinates \(a_r\), \(a_r^D\). The transition functions between different charts define a Riemann-Hilbert problem. The solution to this problem defines the function \({\mathcal F}(a)\). It was shown in [N, NO03, NY, BE] that the series expansion of \({\mathcal F}(a)\) around one of the singular points on the moduli space of vacua satisfies (1.13). Taken together, one obtains a highly nontrivial check of the IR-duality conjectures underlying Seiberg-Witten theory.

  10. 10.

    This limit is easier to define in the A-model, but the definition can be translated to the B-model using mirror symmetry.

  11. 11.

    This paper is part of a program initiated in [NSa, NSb] investigating even more general connections between field theories with \({\mathcal N}=2\)-supersymmetry and integrable models.

  12. 12.

    The relations between the topological vertex and free fermion theories discussed in [ADKMV] imply general relations between topological string partition functions of local Calabi-Yau manifolds, integrable models and theories of free fermions on certain Riemann surfaces; possible implications for four-dimensional gauge theories were discussed in [DHSV, DHS].

References

  1. Andersen, M., Beem, C., Bobev, N., Rastelli, L.: Holographic uniformization. Commun. Math. Phys. 318, 429–471 (2013). arXiv:1109.3724

    Article  ADS  Google Scholar 

  2. Argyres, P.C., Douglas, M.R.: New phenomena in SU(3) supersymmetric gauge theory. Nucl. Phys. B448, 93–126 (1995). arXiv:hep-th/9505062

    Article  MathSciNet  ADS  Google Scholar 

  3. Aganagic, M., Dijkgraaf, R., Klemm, A., Marino, M., Vafa, C.: Topological strings and integrable hierarchies. Commun. Math. Phys. 261, 451–516 (2006). arXiv:hep-th/0312085

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Antoniadis, I., Florakis, I., Hohenegger, S., Narain, K.S., Zein Assi, A.: Worldsheet realization of the refined topological string. Nucl. Phys. B875, 101–133 (2013). arXiv:1302.6993

    Article  MathSciNet  ADS  Google Scholar 

  5. Antoniadis, I., Florakis, I., Hohenegger, S., Narain, K.S., Zein Assi, A.: Non-perturbative Nekrasov partition function from string theory. Nucl. Phys. B880, 87–108 (2014). arXiv:1309.6688

    Article  MathSciNet  ADS  Google Scholar 

  6. Alba, V.A., Fateev, V.A., Litvinov, A.V., Tarnopolsky, G.M.: On combinatorial expansion of the conformal blocks arising from AGT conjecture. Lett. Math. Phys. 98, 33–64 (2011). arXiv:1012.1312

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Aharony, O., Gubser, S.S., Maldacena, J., Ooguri, H., Oz, Y.: Large N field theories, string theory and gravity. Phys. Rep. 323, 183–386 (2000). arXiv:hep-th/9905111

    Article  MathSciNet  ADS  Google Scholar 

  8. Alday, L.F., Gaiotto, D., Tachikawa, Y.: Liouville correlation functions from four-dimensional gauge theories. Lett. Math. Phys. 91, 167–197 (2010). arXiv:0906.3219

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Aganagic, M., Klemm, A., Marino, M., Vafa, C.: The topological vertex. Commun. Math. Phys. 254, 425–478 (2005). arXiv:hep-th/0305132

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. Aganagic, M., Ooguri, H., Saulina, N., Vafa, C.: Black holes, q-deformed 2d Yang-Mills, and non-perturbative topological strings. Nucl. Phys. B 715, 304 (2005). arXiv:hep-th/0411280

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Argyres, P.C., Plesser, M.R., Seiberg, N., Witten, E.: New \(N=2\) superconformal field theories in four-dimensions. Nucl. Phys. B461, 71–84 (1996). arXiv:hep-th/9511154

    Article  MathSciNet  ADS  Google Scholar 

  12. Argyres, P.C., Seiberg, N.: S-duality in \(N=2\) supersymmetric gauge theories. JHEP 0712, 088 (2007). arXiv:0711.0054

  13. Aganagic, M., Schaeffer, K.: Refined black hole ensembles and topological strings. JHEP 1301, 060 (2013). arXiv:1210.1865

  14. Aganagic, M., Shakirov, S.: Refined Chern-Simons theory and topological string. Preprint arXiv:1210.2733 [hep-th]

  15. Belavin, A.A., Bershtein, M.A., Feigin, B.L., Litvinov, A.V., Tarnopolsky, G.M.: Instanton moduli spaces and bases in coset conformal field theory. Commun. Math. Phys. 319, 269–301 (2013). arXiv:1111.2803

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Braverman, A., Etingof, A.: Instanton Counting Via Affine Lie Algebras. II. From Whittaker Vectors to the Seiberg-Witten Prepotential. Studies in Lie Theory. Progress in Mathematics, vol. 243, pp. 61–78. Birkhäuser, Boston (2006). arXiv:math/0409441

  17. Bobev, N., Elvang, H., Freedman, D.Z., Pufu, S.S.: Holography for \(N=2\) on \(S^4\). JHEP 1407, 1 (2014). arXiv:1311.1508

  18. Beisert, N., et al.: Review of AdS/CFT integrability: an overview. Lett. Math. Phys. 99, 3 (2012). arXiv:1012.3982

    Article  MathSciNet  ADS  Google Scholar 

  19. Bruzzo, U., Fucito, F., Morales, J.F., Tanzini, A.: Multiinstanton calculus and equivariant cohomology. JHEP 0305, 054 (2003). arXiv:hep-th/0211108

    Google Scholar 

  20. Braverman, A., Finkelberg, M., Nakajima, H.: Instanton moduli spaces and W-algebras. Preprint arXiv:1406.2381

  21. Bilal, A.: Duality in \(N=2\) SUSY SU(2) Yang-Mills theory: a pedagogical introduction to the work of Seiberg and Witten. In: G. ’t Hooft et al. (eds.) Proceedings, NATO Advanced Study Institute, Quantum Fields and Quantum Space Time, Cargese, France, 22 July–3 August 1996. Plenum, New York (1997). arXiv:hep-th/9601007

  22. Bao, L., Mitev, V., Pomoni, E., Taki, M., Yagi, F.: Non-Lagrangian theories from Brane junctions. JHEP 1401, 175 (2014). arXiv:1310.3841

  23. Bonelli, G., Maruyoshi, K., Tanzini, A.: Wild quiver gauge theories. JHEP 1202, 031 (2012). arXiv:1112.1691

  24. Baggio, M., Niarchos, V., Papadodimas, K.: Exact correlation functions in SU(2) \(N=2\) superconformal QCD. Preprint arXiv:1409.4217

  25. Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B241, 333–380 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  26. Buchel, A., Russo, J.G., Zarembo, K.: Rigorous test of non-conformal holography: Wilson loops in \(N=2^*\) theory. JHEP 1303, 062 (2013). arXiv:1301.1597

  27. Chacaltana, O., Distler, J.: Tinkertoys for Gaiotto duality. JHEP 1011, 099 (2010). arXiv:1008.5203

  28. Cordova, C., Jafferis, D.L.: Complex Chern-Simons from M5-branes on the squashed three-sphere. Preprint arXiv:1305.2891

  29. Cordova, C., Jafferis, D.: Talk at Strings. Princeton, June 25 (2014)

    Google Scholar 

  30. Choi, J., Katz, S., Klemm, A.: The refined BPS index from stable pair invariants. Commun. Math. Phys. 328, 903–954 (2014). arXiv:1210.4403

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. Carlsson, E., Nekrasov, N., Okounkov, A.: Five dimensional gauge theories and vertex operators. Moscow Math. J. 14, 39–61 (2014). arXiv:1308.2465

    MathSciNet  MATH  Google Scholar 

  32. Curtright, T.L., Thorn, C.B.: Conformally invariant quantization of the Liouville theory. Phys. Rev. Lett. 48, 1309 (1982) (Erratum-ibid. 48, 1768 (1982))

    Google Scholar 

  33. Dorey, Nick, Hollowood, Timothy J., Khoze, Valentin V., Mattis, Michael P.: The calculus of many instantons. Phys. Rept. 371, 231–459 (2002). arXiv:hep-th/0206063

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. Dijkgraaf, R., Hollands, L., Sułkowski, P.: Quantum curves and \({\cal D}\)-modules. JHEP 0911, 047 (2009). arXiv:0810.4157

  35. Dijkgraaf, R., Hollands, L., Sułkowski, P., Vafa, C.: Supersymmetric gauge theories, intersecting branes and free fermions. JHEP 0802, 106 (2008). arXiv:.0709.4446

    Google Scholar 

  36. Dorn, H., Otto, H.-J.: Two and three-point functions in Liouville theory. Nucl. Phys. B 429, 375–388 (1994). arXiv:hep-th/9403141

    Article  MathSciNet  ADS  MATH  Google Scholar 

  37. Dolan, F.A., Osborn, H.: Applications of the superconformal index for protected operators and q-hypergeometric identities to \(N=1\) dual theories. Nucl. Phys. B818, 137–178 (2009). arXiv:0801.4947

    Article  MathSciNet  ADS  Google Scholar 

  38. D’Hoker, E., Phong, D.H.: Lectures on supersymmetric Yang-Mills theory and integrable systems. In: Saint-Aubin, Y., Vinet, L. (ed.) Proceedings. 9th CRM Summer School, Theoretical Physics at the End of the Twentieth Century, Banff, Canada, 1999. Springer, New York (2002). arXiv:hep-th/9912271

    Google Scholar 

  39. Donagi, R., Witten, E.: Supersymmetric Yang-Mills theory and integrable systems. Nucl. Phys. B460, 299–334 (1996). arXiv:hep-th/9510101

    Article  MathSciNet  ADS  Google Scholar 

  40. Eguchi, T., Kanno, H.: Topological strings and Nekrasov’s formulas. JHEP 0312, 006 (2003). arXiv:hep-th/0310235

    Article  MathSciNet  ADS  Google Scholar 

  41. Fateev, A.V., Litvinov, A.V.: Integrable structure, W-symmetry and AGT relation. JHEP 1201, 051 (2012). arXiv:1109.4042

  42. Flume, R., Poghossian, R.: An algorithm for the microscopic evaluation of the coefficients of the Seiberg-Witten prepotential. Int. J. Mod. Phys. A18, 2541 (2003). arXiv:hep-th/0208176

    Article  MathSciNet  ADS  Google Scholar 

  43. Flume, R., Poghossian, R., Storch, H.: The Seiberg-Witten prepotential and the Euler class of the reduced moduli space of instantons. Mod. Phys. Lett. A17, 327–340 (2002). arXiv:hep-th/0112211

    Google Scholar 

  44. Freed, D.: Special Kähler manifolds. Commun. Math. Phys. 203, 31–52 (1999). arXiv:hep-th/9712042

    Article  MathSciNet  ADS  MATH  Google Scholar 

  45. Gaiotto, D.: \(N=2\) dualities. JHEP 1208, 034 (2012). arXiv:0904.2715

  46. Gorsky, A., Krichever, I., Marshakov, A., Mironov, A., Morozov, A.: Integrability and Seiberg-Witten exact solution. Phys. Lett. B355, 466–474 (1995). arXiv:hep-th/9505035

    Article  MathSciNet  ADS  Google Scholar 

  47. Gaiotto, D., Moore, G., Neitzke, A.: Four-dimensional wall-crossing via three-dimensional field theory. Commun. Math. Phys. 299, 163–224 (2010). arXiv:0807.4723

    Article  MathSciNet  ADS  MATH  Google Scholar 

  48. Gaiotto, D., Moore, G., Neitzke, A.: Wall-crossing, Hitchin systems, and the WKB approximation. Adv. Math. 234, 239–403 (2013). arXiv:0907.3987

    Article  MathSciNet  MATH  Google Scholar 

  49. Gaiotto, D., Moore, G., Neitzke, A.: Framed BPS states. Adv. Theor. Math. Phys. 17, 241–397 (2013). arXiv:1006.0146

    Article  MathSciNet  MATH  Google Scholar 

  50. Goddard, P., Nuyts, J., Olive, D.I.: Gauge theories and magnetic charge. Nucl. Phys. B125, 1 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  51. Gomis, J., Okuda, T., Pestun, V.: Exact results for ’t Hooft loops in gauge theories on \(S^4\), JHEP 1205, 141 (2012). arXiv:1105.2568

  52. Gadde, A., Pomoni, E., Rastelli, L., Razamat, S.S.: S-duality and 2d topological QFT. JHEP 1003, 032 (2010). arXiv:0910.2225

  53. Gaiotto, D., Rastelli, L., Razamat, S.S.: Bootstrapping the superconformal index with surface defects. JHEP 1301, 022 (2013). arXiv:1207.3577

  54. Gadde, A., Rastelli, L., Razamat, S.S., Yan, W.: The 4d superconformal index from q-deformed 2d Yang-Mills. Phys. Rev. Lett. 106, 241602 (2011). arXiv:1104.3850

    Article  ADS  Google Scholar 

  55. Gadde, A., Rastelli, L., Razamat, S.S., Yan, W.: Gauge theories and Macdonald polynomials. Commun. Math. Phys. 319, 147 (2013). arXiv:1110.3740

    Article  MathSciNet  ADS  MATH  Google Scholar 

  56. Gaiotto, D., Teschner, J.: Irregular singularities in Liouville theory and Argyres-Douglas type gauge theories. JHEP 1212, 050 (2012). arXiv:1203.1052

  57. Hama, N., Hosomichi, K.: Seiberg-Witten theories on ellipsoids. JHEP 1209, 033 (2012). Addendum-ibid. 1210, 051 (2012). arXiv:1206.6359

  58. Hitchin, N.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. 55(3), 59–126 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  59. Hollowood, T.J., Iqbal, A., Vafa, C.: Matrix models, geometric engineering and elliptic genera. JHEP 0803, 069 (2008). arXiv:hep-th/0310272

    Article  MathSciNet  ADS  Google Scholar 

  60. Huang, M.-X., Klemm, A.: Direct integration for general \(\Omega \) backgrounds. Adv. Theor. Math. Phys. 16, 805–849 (2012). arXiv:1009.1126

    Article  MathSciNet  MATH  Google Scholar 

  61. Huang, M.-X., Kashani-Poor, A.-K., Klemm, A.: The \(\Omega \) deformed B-model for rigid \({\cal {N}}=2\) theories. Annales Henri Poincare 14, 425–497 (2013). arXiv:1109.5728

    Article  MathSciNet  ADS  MATH  Google Scholar 

  62. Hayashi, H., Kim, H.-C., Nishinaka, T.: Topological strings and 5d TN partition functions. JHEP 1406, 014 (2014). arXiv:1310.3854

  63. Hollowood, T.J.: Calculating the prepotential by localization on the moduli space of instantons. JHEP 03, 038 (2002). arXiv:hep-th/0201075

    Article  MathSciNet  ADS  Google Scholar 

  64. Hollowood, T.J.: Testing Seiberg-Witten theory to all orders in the instanton expansion. Nucl. Phys. B639, 66–94 (2002). arXiv:hep-th/0202197

    Article  MathSciNet  ADS  Google Scholar 

  65. Iqbal, A., Kashani-Poor, A.-K.: Instanton counting and Chern-Simons theory. Adv. Theor. Math. Phys. 7, 457–497 (2004). arXiv:hep-th/0212279

    Article  MathSciNet  Google Scholar 

  66. Iqbal, A., Kashani-Poor, A.-K.: SU(N) geometries and topological string amplitudes. Adv. Theor. Math. Phys. bf 10, 1–32 (2006). arXiv:hep-th/0306032

    Article  MathSciNet  MATH  Google Scholar 

  67. Iqbal, A., Kozcaz, C., Vafa, C.: The refined topological. JHEP 0910, 069 (2009). arXiv:hep-th/0701156

    Article  MathSciNet  ADS  Google Scholar 

  68. Iorgov, N., Lisovyy, O., Teschner, J.: Isomonodromic tau-functions from Liouville conformal blocks. Preprint arXiv:1401.6104

  69. Its, A., Lisovyy, O., Tykhyy, Yu.: Connection problem for the sine-Gordon/Painlevé III tau function and irregular conformal blocks. Preprint arXiv:1403.1235

  70. Katz, S.H., Klemm, A., Vafa, C.: Geometric engineering of quantum field theories. Nucl. Phys. B497, 173–195 (1997). arXiv:hep-th/9609239

    Article  MathSciNet  ADS  Google Scholar 

  71. Klemm, A., Lerche, W., Mayr, P., Vafa, C., Warner, N.P.: Selfdual strings and \(N=2\) supersymmetric field theory. Nucl. Phys. B477, 746–766 (1996). arXiv:hep-th/9604034

    Article  MathSciNet  ADS  Google Scholar 

  72. Kinney, J., Maldacena, J., Minwalla, S., Raju, S.: An Index for 4 dimensional super conformal theories. Commun. Math. Phys. 275, 209–254 (2007). arXiv:hep-th/0510251

    Article  MathSciNet  ADS  MATH  Google Scholar 

  73. Kanno, H., Maruyoshi, K., Shiba, S., Taki, M.: \(W_3\) irregular states and isolated \(N=2\) superconformal field theories. JHEP 1303, 147 (2013). arXiv:1301.0721

  74. Katz, S., Mayr, P., Vafa, C.: Mirror symmetry and exact solution of 4D \(N=2\) gauge theories. I. Adv. Theor. Math. Phys. 1, 53–114 (1997). arXiv:hep-th/9706110

    MathSciNet  MATH  Google Scholar 

  75. Kanno, S., Matsuo, Y., Zhang, H.: Extended conformal symmetry and recursion formulae for nekrasov partition function. JHEP 1308, 028 (2013). arXiv:1306.1523

  76. Kozcaz, C., Pasquetti, S., Wyllard, N.: A & B model approaches to surface operators and Toda theories. JHEP 1008, 042 (2010). arXiv:1004.2025

  77. Krefl, D., Walcher, J.: Extended holomorphic anomaly in gauge theory. Lett. Math. Phys. 95, 67–88 (2011). arXiv:1007.0263

    Article  MathSciNet  ADS  MATH  Google Scholar 

  78. Lerche, W.: Introduction to Seiberg-Witten theory and its stringy origin. Nucl. Phys. Proc. Suppl. 55B, 83–117 (1997). Fortsch. Phys. 45, 293–340 (1997). arXiv:hep-th/9611190

  79. Losev, A.S., Marshakov, A.V., Nekrasov, N.A.: Small instantons, little strings and free fermions. From Fields to Strings: Circumnavigating Theoretical Physics, vol. 1, pp. 581–621. World Scientific Publishing, Singapore (2005). arXiv:hep-th/0302191

    Google Scholar 

  80. Losev, A.S., Nekrasov, N.A., Shatashvili, S.: Testing Seiberg-Witten solution. Strings, Branes and Dualities (Cargèse, 1997). NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 520, pp. 359–372. Kluwer Acadamic Publishing, Dordrecht (1999). arXiv:hep-th/9801061

    Google Scholar 

  81. Lee, S., Yamazaki, M.: 3d Chern-Simons theory from M5-branes. JHEP 1312, 035 (2013). arXiv:1305.2429

  82. Maldacena, J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231–252 (1998). arXiv:hep-th/9711200

    MathSciNet  ADS  MATH  Google Scholar 

  83. Maulik, D., Okounkov, A.: Quantum groups and quantum cohomology. Preprint arXiv:1211.1287

  84. Mironov, A., Morozov, A., Shakirov, Sh.: A direct proof of AGT conjecture at beta = 1. JHEP 1102, 067 (2011). arXiv:1012.3137

  85. Moore, G., Nekrasov, N.A., Shatashvili, S.: Integrating over Higgs branches. Commun. Math. Phys. 209, 97–121 (2000). arXiv:hep-th/9712241

    Article  MathSciNet  ADS  MATH  Google Scholar 

  86. Moore, G., Nekrasov, N.A., Shatashvili, S.: D-particle bound states and generalized instantons. Commun. Math. Phys. 209, 77–95 (2000). arXiv:hep-th/9803265

    Article  MathSciNet  ADS  MATH  Google Scholar 

  87. Montonen, C., Olive, D.: Magnetic monopoles as gauge particles? Phys. Lett. B72, 117 (1977)

    Article  ADS  Google Scholar 

  88. Mitev, V., Pomoni, E.: The exact effective couplings of 4D \(N=2\) Gauge theories. Preprint arXiv:1406.3629

  89. Mitev, V., Pomoni, E.: Toda 3-point functions from topological strings. Preprint arXiv:1409.6313

  90. Morozov, A., Smirnov, A.: Towards the proof of AGT relations with the help of the generalized Jack polynomials. Lett. Math. Phys. 104, 585–612 (2014). arXiv:1307.2576

    Google Scholar 

  91. Martinec, E.J., Warner, N.P.: Integrable systems and supersymmetric gauge theory. Nucl. Phys. B459, 97–112 (1996). arXiv:hep-th/9509161

    Article  MathSciNet  ADS  Google Scholar 

  92. Nekrasov, N.A.: Seiberg-Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831–864 (2003). arXiv:hep-th/0206161

    Article  MathSciNet  MATH  Google Scholar 

  93. Nekrasov, N., Okounkov, A.: Seiberg-Witten Theory and Random Partitions. The Unity of Mathematics. Progress in Mathematics, vol. 244, pp. 525–596. Birkhäauser, Boston (2006) arXiv:hep-th/0306238

  94. Nekrasov, N., Okounkov, A.: Membranes and sheaves. Preprint arXiv:1404.2323

  95. Nekrasov, N., Pestun, V., Shatashvili, S.: Quantum geometry and quiver gauge theories. Preprint arXiv:1312.6689

  96. Nekrasov, N., Pestun, V.: Seiberg-Witten geometry of four dimensional \(N=2\) quiver gauge theories. Preprint arXiv:1211.2240

  97. Nekrasov, N., Rosly, A., Shatashvili, S.: Darboux coordinates, Yang-Yang functional, and gauge theory. Nucl. Phys. Proc. Suppl. 216, 69–93 (2011). arXiv:1103.3919

    Article  MathSciNet  ADS  Google Scholar 

  98. Nekrasov, N., Shatashvili, S.: Supersymmetric vacua and Bethe Ansatz. Nucl. Phys. Proc. Suppl. 192–193, 91–112 (2009). arXiv:0901.4744

    Article  MathSciNet  Google Scholar 

  99. Nekrasov, N., Shatashvili, S.: Quantum integrability and supersymmetric vacua. Prog. Theor. Phys. Suppl. 177, 105–119 (2009). arXiv:0901.4748

    Article  ADS  MATH  Google Scholar 

  100. Nekrasov, N., Shatashvili, S.: Quantization of integrable systems and four dimensional gauge theories. In: Exner, P. (ed.) Proceedings of the 16th International Congress on Mathematical Physics, Prague, August 2009, pp. 265–289. World Scientific, Singapore (2010). arXiv:0908.4052

  101. Nekrasov, N., Witten, E.: The omega deformation, branes, integrability, and Liouville theory. JHEP 1009, 092 (2010). arXiv:1002.0888

  102. Nakajima, H., Yoshioka, K.: Instanton counting on blowup. I. 4-dimensional pure gauge theory. Invent. Math. 162, 313–355 (2005). arXiv:math/0306198

    Article  MathSciNet  ADS  MATH  Google Scholar 

  103. Osborn, H.: Topological charges for \(N=4\) supersymmetric gauge theories and monopoles of spin 1. Phys. Lett. B83, 321 (1979)

    Article  ADS  Google Scholar 

  104. Peskin, M.E.: Duality in supersymmetric Yang-Mills theory. In: Efthimiou, C., Greene, B. (eds.) Proceedings, Summer School TASI’96, Fields, Strings and Duality. World Scientific, Singapore (1997). arXiv:hep-th/9702094

  105. Pestun, V.: Localization of gauge theory on a four-sphere and supersymmetric Wilson loops. Commun. Math. Phys. 313, 71–129 (2012). arXiv:0712.2824

    Article  MathSciNet  ADS  MATH  Google Scholar 

  106. Romelsberger, C.: Counting chiral primaries in \(N =1\), \(d=4\) superconformal field theories. Nucl. Phys. B747, 329–353 (2006). arXiv:hep-th/0510060

    Article  MathSciNet  ADS  Google Scholar 

  107. Russo, J.G., Zarembo, K.: Evidence for large-N phase transitions in \(N=2^*\) theory. JHEP 1304, 065 (2013). arXiv:1302.6968

  108. Russo, J.G., Zarembo, K.: Massive \(N=2\) gauge theories at large N. JHEP 1311, 130 (2013). arXiv:1309.1004

  109. Schwarz, A., Tang, X.: Quantization and holomorphic anomaly. JHEP 0703, 062 (2007). arXiv:hep-th/0611281

    Article  MathSciNet  ADS  Google Scholar 

  110. Schiffmann, O., Vasserot, E.: Cherednik algebras, W-algebras and the equivariant cohomology of the moduli space of instantons on \({\mathbf{A}}^2\), Publ. IHÉS 118, 213–342 (2013). arXiv:1202.2756

  111. Seiberg, N.: Notes on theories with 16 supercharges. Nucl. Phys. Proc. Suppl. 67, 158–171 (1998). arXiv:hep-th/9705117

    Article  MathSciNet  ADS  MATH  Google Scholar 

  112. Sen, A.: Dyon—monopole bound states, self-dual harmonic forms on the multi-monopole moduli space, and SL(2, Z) invariance in string theory. Phys. Lett. B329, 217–221 (1994). arXiv:hep-th/9402032

    Article  ADS  Google Scholar 

  113. Spiridonov, V.P., Vartanov, G.S.: Superconformal indices for \(N=1\) theories with multiple duals. Nucl. Phys. B824, 192–216 (2010). arXiv:0811.1909

    Article  MathSciNet  ADS  Google Scholar 

  114. Strominger, A.: Open p-branes. Phys. Lett. B383, 44–47 (1996). arXiv:hep-th/9512059

    Article  MathSciNet  ADS  Google Scholar 

  115. Seiberg, N., Witten, E.: Monopole condensation, and confinement in \(N=2\) supersymmetric Yang-Mills theory. Nucl. Phys. B426, 19–52 (1994). arXiv:hep-th/9407087

    Article  MathSciNet  ADS  Google Scholar 

  116. Seiberg, N., Witten, E.: Monopoles, duality and chiral symmetry breaking in \(N=2\) supersymmetric QCD. Nucl. Phys. B431, 484–550 (1994). arXiv:hep-th/9408099

    Article  MathSciNet  ADS  Google Scholar 

  117. Tachikawa, Y.: \(N=2\) supersymmetric dynamics for pedestrians. Lect. Notes Phys. 890, (2014). arXiv:1312.2684

  118. Tan, M.-C.: M-theoretic derivations of 4d-2d dualities: from a geometric langlands duality for surfaces, to the AGT correspondence, to integrable systems. JHEP 1307, 171 (2013). arXiv:1301.1977

  119. Teschner, J.: Liouville theory revisited. Class. Quantum Gravity 18, R153–R222 (2001). arXiv:hep-th/0104158

    Article  MathSciNet  ADS  MATH  Google Scholar 

  120. Teschner, J.: Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I. Adv. Theor. Math. Phys. 15, 471–564 (2011). arXiv:1005.2846

    Article  MathSciNet  MATH  Google Scholar 

  121. Teschner, J., Vartanov, G.: Supersymmetric gauge theories, quantisation of moduli spaces of flat connections, and conformal field theory. Adv. Theor. Math. Phys. 19, 1–135 (2015). arXiv:1302.3778

  122. Vafa, C., Witten, E.: A strong coupling test of S-duality. Nucl. Phys. B431, 3–77 (1994). arXiv:hep-th/9408074

    Article  MathSciNet  ADS  Google Scholar 

  123. Witten, E.: Topological quantum field theory. Commun. Math. Phys. 117, 353 (1988)

    Article  ADS  MATH  Google Scholar 

  124. Witten, E.: Quantum background independence in string theory. Published in Salamfest, 0257–275 (1993). arXiv:hep-th/9306122

  125. Witten, E.: Some comments on string dynamics. In: Bars, I., Bouwknegt, P., Minahan, J., Nemeschansky, D., Pilch, K., Saleur, H., Warner N. (eds.) Proceedings. Future Perspectives in String Theory (STRINGS’95). World Scientific, River Edge (1996). arXiv:hep-th/9507121

  126. Witten, E.: Five-branes and M-theory on an orbifold. Nucl. Phys. B463, 383–397 (1996). arXiv:hep-th/9512219

    Article  ADS  Google Scholar 

  127. Witten, E.: Geometric langlands from six dimensions, a celebration of the mathematical legacy of Raoul Bott. In: CRM Proceedings of the Lecture Notes, vol. 50, pp. 281–310. American Mathematical Society, Providence (2010). arXiv:0905.2720

  128. Witten, E., Olive, D.: Supersymmetry algebras that include topological charges. Phys. Lett. B78, 97 (1978)

    Article  ADS  Google Scholar 

  129. Yagi, Y.: Compactification on the \(\Omega \)-background and the AGT correspondence. JHEP 1209, 101 (2012). arXiv:1205.6820

  130. Yagi, Y.: 3d TQFT from 6d SCFT. JHEP 1308, 017 (2013). arXiv:1305.0291

  131. Zamolodchikov, A.B., Zamolodchikov, Al.B.: Structure constants and conformal bootstrap in Liouville field theory. Nucl. Phys. B477, 577–605 (1996). arXiv:hep-th/9506136

    Google Scholar 

Download references

Acknowledgments

The author is grateful to D. Krefl, K. Maruyoshi, E. Pomoni, L. Rastelli, S. Razamat, Y. Tachikawa and J. Walcher for very useful comments and suggestions on a previous draft of this article.

Author information

Authors and Affiliations

  1. DESY Theory, Notkestr. 85, 22603, Hamburg, Germany

    Jörg Teschner

Authors
  1. Jörg Teschner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jörg Teschner .

Editor information

Editors and Affiliations

  1. String Theory Group, DESY String Theory Group, Hamburg, Hamburg, Germany

    Jörg Teschner

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Teschner, J. (2016). Exact Results on \({\mathcal N}=2\) Supersymmetric Gauge Theories. In: Teschner, J. (eds) New Dualities of Supersymmetric Gauge Theories. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-18769-3_1

Download citation

Publish with us

Policies and ethics

Search

Navigation