The \( \mathcal{N} \) = 1 superconformal index for class \( \mathcal{S} \) fixed points

  • Christopher BeemEmail author
  • Abhijit Gadde
Open Access


We investigate the superconformal index of four-dimensional superconformal field theories that arise on coincident M5 branes wrapping a holomorphic curve in a local Calabi-Yau three-fold. The structure of the index is very similar to that which appears in the special case preserving \( \mathcal{N} \) = 2 supersymmetry. We first compute the index for the fixed points that admit a known four-dimensional ultraviolet description and prove infrared equivalence at the level of the index for all such constructions. These results suggest a formulation of the index as a two-dimensional topological quantum field theory that generalizes the one that computes the \( \mathcal{N} \) = 2 index. The TQFT structure leads to an expression for the index of a much larger family of \( \mathcal{N} \) = 1 class S fixed points in terms of the index of the \( \mathcal{N} \) = 2 theories. Calculations of simple quantities with the index suggests a connection between these families of fixed points and the mathematics of SU(2) Yang-Mills theory on the wrapped curve.


Supersymmetric gauge theory Supersymmetry and Duality Duality in Gauge Field Theories 


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.


  1. [1]
    D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  2. [2]
    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].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    T. Dimofte, D. Gaiotto and S. Gukov, Gauge theories labelled by three-manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    S. Cecotti, C. Cordova and C. Vafa, Braids, walls and mirrors, arXiv:1110.2115 [INSPIRE].
  5. [5]
    I. Bah, C. Beem, N. Bobev and B. Wecht, Four-dimensional SCFTs from M 5-branes, JHEP 06 (2012) 005 [arXiv:1203.0303] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  6. [6]
    I. Bah, C. Beem, N. Bobev and B. Wecht, AdS/CFT dual pairs from M 5-branes on Riemann surfaces, Phys. Rev. D 85 (2012) 121901 [arXiv:1112.5487] [INSPIRE].ADSGoogle Scholar
  7. [7]
    F. Benini, Y. Tachikawa and B. Wecht, Sicilian gauge theories and N = 1 dualities, JHEP 01 (2010) 088 [arXiv:0909.1327] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  8. [8]
    B.S. Acharya, J.P. Gauntlett and N. Kim, Five-branes wrapped on associative three cycles, Phys. Rev. D 63 (2001) 106003 [hep-th/0011190] [INSPIRE].ADSMathSciNetGoogle Scholar
  9. [9]
    F. Benini and N. Bobev, Two-dimensional c-extremization for wrapped branes, to appear.Google Scholar
  10. [10]
    F. Benini and N. Bobev, Exact two-dimensional superconformal R-symmetry and c-extremization, Phys. Rev. Lett. 110 (2013) 061601 [arXiv:1211.4030] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    A. Gadde, K. Maruyoshi, Y. Tachikawa and W. Yan, New N = 1 dualities, JHEP 06 (2013) 056 [arXiv:1303.0836] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  12. [12]
    K.A. Intriligator and B. Wecht, The exact superconformal R symmetry maximizes a, Nucl. Phys. B 667 (2003) 183 [hep-th/0304128] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  13. [13]
    J.M. Maldacena and C. Núñez, Supergravity description of field theories on curved manifolds and a no go theorem, Int. J. Mod. Phys. A 16 (2001) 822 [hep-th/0007018] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    D.M. Hofman and J. Maldacena, Conformal collider physics: energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    M. Kulaxizi and A. Parnachev, Energy flux positivity andl unitarity in CFTs, Phys. Rev. Lett. 106 (2011) 011601 [arXiv:1007.0553] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  16. [16]
    F. Dolan and H. Osborn, On short and semi-short representations for four-dimensional superconformal symmetry, Annals Phys. 307 (2003) 41 [hep-th/0209056] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  17. [17]
    J. Kinney, J.M. Maldacena, S. Minwalla and S. Raju, An index for 4 dimensional super conformal theories, Commun. Math. Phys. 275 (2007) 209 [hep-th/0510251] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  18. [18]
    A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, On the superconformal index of N = 1 IR fixed points: a holographic check, JHEP 03 (2011) 041 [arXiv:1011.5278] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  19. [19]
    D. Green, Z. Komargodski, N. Seiberg, Y. Tachikawa and B. Wecht, Exactly marginal deformations and global symmetries, JHEP 06 (2010) 106 [arXiv:1005.3546] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  20. [20]
    C. Romelsberger, Counting chiral primaries in N = 1, D = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  21. [21]
    C. Romelsberger, Calculating the superconformal index and Seiberg duality, arXiv:0707.3702 [INSPIRE].
  22. [22]
    F. Dolan and H. Osborn, Applications of the superconformal index for protected operators and q-hypergeometric identities to N = 1 dual theories, Nucl. Phys. B 818 (2009) 137 [arXiv:0801.4947] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  23. [23]
    A. Gadde, E. Pomoni, L. Rastelli and S.S. Razamat, S-duality and 2d topological QFT, JHEP 03 (2010) 032 [arXiv:0910.2225] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  24. [24]
    F.J. van de Bult, An elliptic hypergeometric beta integral transformation, arXiv:0912.3812.
  25. [25]
    V.P. Spiridonov, Theta hypergeometric integrals, math/0303205.
  26. [26]
    V. Spiridonov and G. Vartanov, Superconformal indices for N = 1 theories with multiple duals, Nucl. Phys. B 824 (2010) 192 [arXiv:0811.1909] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  27. [27]
    D. Gaiotto, L. Rastelli and S.S. Razamat, Bootstrapping the superconformal index with surface defects, arXiv:1207.3577 [INSPIRE].
  28. [28]
    A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge theories and Macdonald polynomials, Commun. Math. Phys. 319 (2013) 147 [arXiv:1110.3740] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar
  29. [29]
    A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, The superconformal index of the E 6 SCFT, JHEP 08 (2010) 107 [arXiv:1003.4244] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  30. [30]
    J. Bryan and R. Pandharipande, The local Gromov-Witten theory of curves, math/0411037 [INSPIRE].
  31. [31]
    M. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. Lond. A 308 (1982) 523.ADSMathSciNetGoogle Scholar
  32. [32]
    M.T. Anderson, C. Beem, N. Bobev and L. Rastelli, Holographic uniformization, Commun. Math. Phys. 318 (2013) 429 [arXiv:1109.3724] [INSPIRE].ADSCrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Simons Center for Geometry and PhysicsState University of New YorkStony BrookU.S.A
  2. 2.California Institute of TechnologyPasadenaU.S.A

Personalised recommendations