\( \mathcal{N} \) = 2 supersymmetric gauge theories and quantum integrable systems

  • Yuan Luo
  • Meng-Chwan Tan
  • Junya Yagi
Open Access


We study \( \mathcal{N} \) = 2 supersymmetric gauge theories on the product of a twosphere and a cylinder. We show that the low-energy dynamics of a BPS sector of such a theory is described by a quantum integrable system, with the Planck constant set by the inverse of the radius of the sphere. If the sphere is replaced with a hemisphere, then our system reduces to an integrable system of the type studied by Nekrasov and Shatashvili. In this case we establish a correspondence between the effective prepotential of the gauge theory and the Yang-Yang function of the integrable system.


Supersymmetric gauge theory Bethe Ansatz 


Open Access

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  1. [1]
    N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485] [hep-th/9407087] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  2. [2]
    N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  3. [3]
    A. Gorsky, I. Krichever, A. Marshakov, A. Mironov and A. Morozov, Integrability and Seiberg-Witten exact solution, Phys. Lett. B 355 (1995) 466 [hep-th/9505035] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  4. [4]
    E.J. Martinec and N.P. Warner, Integrable systems and supersymmetric gauge theory, Nucl. Phys. B 459 (1996) 97 [hep-th/9509161] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  5. [5]
    T. Nakatsu and K. Takasaki, Whitham-Toda hierarchy and N = 2 supersymmetric Yang-Mills theory, Mod. Phys. Lett. A 11 (1996) 157 [hep-th/9509162] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  6. [6]
    R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996) 299 [hep-th/9510101] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  7. [7]
    E.J. Martinec, Integrable structures in supersymmetric gauge and string theory, Phys. Lett. B 367 (1996) 91 [hep-th/9510204] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  8. [8]
    A. Gorsky and A. Marshakov, Towards effective topological gauge theories on spectral curves, Phys. Lett. B 375 (1996) 127 [hep-th/9510224] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  9. [9]
    H. Itoyama and A. Morozov, Integrability and Seiberg-Witten theory: Curves and periods, Nucl. Phys. B 477 (1996) 855 [hep-th/9511126] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  10. [10]
    H. Itoyama and A. Morozov, Prepotential and the Seiberg-Witten theory, Nucl. Phys. B 491 (1997) 529 [hep-th/9512161] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  11. [11]
    N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, arXiv:0908.4052 [INSPIRE].
  12. [12]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].CrossRefMathSciNetGoogle Scholar
  13. [13]
    N. Nekrasov and E. Witten, The Omega Deformation, Branes, Integrability and Liouville Theory, JHEP 09 (2010) 092 [arXiv:1002.0888] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  14. [14]
    N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, hep-th/9607163 [INSPIRE].
  15. [15]
    D. Gaiotto, G.W. Moore and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Commun. Math. Phys. 299 (2010) 163 [arXiv:0807.4723] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  16. [16]
    D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
  17. [17]
    R.Y. Donagi, Seiberg-Witten integrable systems, alg-geom/9705010 [INSPIRE].
  18. [18]
    F. Benini and S. Cremonesi, Partition functions of N = (2,2) gauge theories on S 2 and vortices, arXiv:1206.2356 [INSPIRE].
  19. [19]
    N. Doroud, J. Gomis, B. Le Floch and S. Lee, Exact Results in D = 2 Supersymmetric Gauge Theories, JHEP 05 (2013) 093 [arXiv:1206.2606] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    K. Hori and M. Romo, Exact Results In Two-Dimensional (2,2) Supersymmetric Gauge Theories With Boundary, arXiv:1308.2438 [INSPIRE].
  21. [21]
    K. Hori et al., Mirror symmetry, Clay Mathematics Monographs, vol 1., American Mathematical Society, Providence, RI, 2003.Google Scholar
  22. [22]
    J. Gomis and S. Lee, Exact Kähler Potential from Gauge Theory and Mirror Symmetry, JHEP 04 (2013) 019 [arXiv:1210.6022] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  23. [23]
    B. Jia and E. Sharpe, Curvature Couplings in \( \mathcal{N} \) = (2, 2) Nonlinear σ-models on S 2, JHEP 09 (2013) 031 [arXiv:1306.2398] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    S. Sugishita and S. Terashima, Exact Results in Supersymmetric Field Theories on Manifolds with Boundaries, JHEP 11 (2013) 021 [arXiv:1308.1973] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  25. [25]
    D. Honda and T. Okuda, Exact results for boundaries and domain walls in 2d supersymmetric theories, arXiv:1308.2217 [INSPIRE].

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Department of PhysicsNational University of SingaporeSingaporeSingapore
  2. 2.International School for Advanced Studies (SISSA)TriesteItaly
  3. 3.INFN, Sezione di TriesteTriesteItaly

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