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2d index and surface operators

Abstract

In this paper we compute the superconformal index of 2d (2, 2) supersymmetric gauge theories. The 2d superconformal index, a.k.a. flavored elliptic genus, is computed by a unitary matrix integral much like the matrix integral that computes the 4d superconformal index. We compute the 2d index explicitly for a number of examples. In the case of abelian gauge theories we see that the index is invariant under flop transition and under CY-LG correspondence. The index also provides a powerful check of the Seiberg-type duality for non-abelian gauge theories discovered by Hori and Tong.

In the later half of the paper, we study half-BPS surface operators in \( \mathcal{N} \) = 2 super-conformal gauge theories. They are engineered by coupling the 2d (2, 2) supersymmetric gauge theory living on the support of the surface operator to the 4d \( \mathcal{N} \) = 2 theory, so that different realizations of the same surface operator with a given Levi type are related by a 2d analogue of the Seiberg duality. The index of this coupled system is computed by using the tools developed in the first half of the paper. The superconformal index in the presence of surface defect is expected to be invariant under generalized S-duality. We demonstrate that it is indeed the case. In doing so the Seiberg-type duality of the 2d theory plays an important role.

References

  1. [1]

    C. Romelsberger, Counting chiral primaries in N = 1, D = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  2. [2]

    C. Romelsberger, Calculating the superconformal index and Seiberg duality, arXiv:0707.3702 [INSPIRE].

  3. [3]

    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].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  4. [4]

    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].

    ADS  Article  MathSciNet  Google Scholar 

  5. [5]

    V. Spiridonov and G. Vartanov, Superconformal indices for N = 1 theories with multiple duals, Nucl. Phys. B 824 (2010) 192 [arXiv:0811.1909] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  6. [6]

    V. Spiridonov and G. Vartanov, Elliptic hypergeometry of supersymmetric dualities, Commun. Math. Phys. 304 (2011) 797 [arXiv:0910.5944] [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  7. [7]

    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].

    ADS  Article  MathSciNet  Google Scholar 

  8. [8]

    A. Gadde, E. Pomoni, L. Rastelli and S.S. Razamat, S-duality and 2d topological QFT, JHEP 03 (2010) 032 [arXiv:0910.2225] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  9. [9]

    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].

    ADS  Article  MathSciNet  Google Scholar 

  10. [10]

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

    ADS  Article  Google Scholar 

  11. [11]

    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].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  12. [12]

    D. Gaiotto, L. Rastelli and S.S. Razamat, Bootstrapping the superconformal index with surface defects, arXiv:1207.3577 [INSPIRE].

  13. [13]

    C. Beem and A. Gadde, The superconformal index of N = 1 class S fixed points, arXiv:1212.1467 [INSPIRE].

  14. [14]

    T. Dimofte, D. Gaiotto and S. Gukov, 3-manifolds and 3d indices, arXiv:1112.5179 [INSPIRE].

  15. [15]

    A. Schellekens and N. Warner, Anomalies and modular invariance in string theory, Phys. Lett. B 177 (1986) 317 [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  16. [16]

    A. Schellekens and N. Warner, Anomalies, characters and strings, Nucl. Phys. B 287 (1987) 317 [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  17. [17]

    E. Witten, Elliptic genera and quantum field theory, Commun. Math. Phys. 109 (1987) 525.

    ADS  Article  MATH  MathSciNet  Google Scholar 

  18. [18]

    E. Witten, On the Landau-Ginzburg description of N = 2 minimal models, Int. J. Mod. Phys. A 9 (1994) 4783 [hep-th/9304026] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  19. [19]

    P. Berglund and M. Henningson, Landau-Ginzburg orbifolds, mirror symmetry and the elliptic genus, Nucl. Phys. B 433 (1995) 311 [hep-th/9401029] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  20. [20]

    P. Berglund and M. Henningson, On the elliptic genus and mirror symmetry, hep-th/9406045 [INSPIRE].

  21. [21]

    M. Henningson, N = 2 gauged WZW models and the elliptic genus, Nucl. Phys. B 413 (1994) 73 [hep-th/9307040] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  22. [22]

    J. Troost, The non-compact elliptic genus: mock or modular, JHEP 06 (2010) 104 [arXiv:1004.3649] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  23. [23]

    S.K. Ashok and J. Troost, A twisted non-compact elliptic genus, JHEP 03 (2011) 067 [arXiv:1101.1059] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  24. [24]

    S.K. Ashok and J. Troost, Elliptic genera of non-compact Gepner models and mirror symmetry, JHEP 07 (2012) 005 [arXiv:1204.3802] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  25. [25]

    T. Eguchi and Y. Sugawara, SL(2, \( \mathbb{R} \))/U(1) supercoset and elliptic genera of noncompact Calabi-Yau manifolds, JHEP 05 (2004) 014 [hep-th/0403193] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  26. [26]

    T. Eguchi, Y. Sugawara and A. Taormina, Liouville field, modular forms and elliptic genera, JHEP 03 (2007) 119 [hep-th/0611338] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  27. [27]

    T. Eguchi, Y. Sugawara and A. Taormina, Modular forms and elliptic genera for ALE spaces, arXiv:0803.0377 [INSPIRE].

  28. [28]

    M.R. Gaberdiel, S. Gukov, C.A. Keller, G.W. Moore and H. Ooguri, Extremal N = (2, 2) 2D conformal field theories and constraints of modularity, Commun. Num. Theor. Phys. 2 (2008) 743 [arXiv:0805.4216] [INSPIRE].

    Article  MATH  MathSciNet  Google Scholar 

  29. [29]

    S. Gukov and E. Witten, Gauge theory, ramification, and the geometric Langlands program, hep-th/0612073 [INSPIRE].

  30. [30]

    S. Gukov and E. Witten, Rigid surface operators, Adv. Theor. Math. Phys. 14 (2010) [arXiv:0804.1561] [INSPIRE].

  31. [31]

    L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in N = 2 gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [arXiv:0909.0945] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  32. [32]

    M. Shifman and A. Yung, Non-Abelian semilocal strings in N = 2 supersymmetric QCD, Phys. Rev. D 73 (2006) 125012 [hep-th/0603134] [INSPIRE].

    ADS  Google Scholar 

  33. [33]

    A. Hanany and D. Tong, Vortices, instantons and branes, JHEP 07 (2003) 037 [hep-th/0306150] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  34. [34]

    P. Di Francesco and S. Yankielowicz, Ramond sector characters and N = 2 Landau-Ginzburg models, Nucl. Phys. B 409 (1993) 186 [hep-th/9305037] [INSPIRE].

    ADS  Article  Google Scholar 

  35. [35]

    E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  36. [36]

    F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of two-dimensional N = 2 gauge theories with rank-one gauge groups, arXiv:1305.0533 [INSPIRE].

  37. [37]

    K. Hori and D. Tong, Aspects of non-abelian gauge dynamics in two-dimensional N = (2, 2) theories, JHEP 05 (2007) 079 [hep-th/0609032] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  38. [38]

    D. Kutasov and A. Schwimmer, On duality in supersymmetric Yang-Mills theory, Phys. Lett. B 354 (1995) 315 [hep-th/9505004] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  39. [39]

    T. Dimofte and S. Gukov, Chern-Simons theory and S-duality, JHEP 05 (2013) 109 [arXiv:1106.4550] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  40. [40]

    T. Dimofte, S. Gukov and L. Hollands, Vortex counting and lagrangian 3-manifolds, Lett. Math. Phys. 98 (2011) 225 [arXiv:1006.0977] [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  41. [41]

    S. Gukov, Three-dimensional quantum gravity, Chern-Simons theory and the A polynomial, Commun. Math. Phys. 255 (2005) 577 [hep-th/0306165] [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  42. [42]

    N. Marcus, The other topological twisting of N = 4 Yang-Mills, Nucl. Phys. B 452 (1995) 331 [hep-th/9506002] [INSPIRE].

    ADS  Article  Google Scholar 

  43. [43]

    A. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands program, Commun. Num. Theor. Phys. 1 (2007) 1 [hep-th/0604151] [INSPIRE].

    Article  MATH  MathSciNet  Google Scholar 

  44. [44]

    N.J. Hitchin, The selfduality equations on a Riemann surface, Proc. Lond. Math. Soc. 55 (1987) 59.

    Article  MATH  MathSciNet  Google Scholar 

  45. [45]

    S. Gukov, Gauge theory and knot homologies, Fortsch. Phys. 55 (2007) 473 [arXiv:0706.2369] [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  46. [46]

    N.R. Constable, J. Erdmenger, Z. Guralnik and I. Kirsch, Intersecting D3 branes and holography, Phys. Rev. D 68 (2003) 106007 [hep-th/0211222] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  47. [47]

    D. Collingwood and W. McGovern, Nilpotent orbits in semisimple Lie algebras, CRC Press, U.S.A. (1993).

    MATH  Google Scholar 

  48. [48]

    P.Z. Kobzk and A.F. Swann, Classical nilpotent orbits as hyper-Kähler quotients, Int. J. Math. 7 (1996) 193.

    Article  Google Scholar 

  49. [49]

    D. Gaiotto and E. Witten, Supersymmetric boundary conditions in N = 4 super Yang-Mills theory, J. Statist. Phys. 135 (2009) 789 [arXiv:0804.2902] [INSPIRE].

    ADS  Article  MATH  MathSciNet  Google Scholar 

  50. [50]

    S. Elitzur, A. Giveon, D. Kutasov, E. Rabinovici and A. Schwimmer, Brane dynamics and N =1 supersymmetric gauge theory, Nucl. Phys. B 505 (1997) 202 [hep-th/9704104] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  51. [51]

    D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].

    ADS  Article  MathSciNet  Google Scholar 

  52. [52]

    V.P. Spiridonov, Elliptic hypergeometric functions, arXiv:0704.3099.

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Correspondence to Abhijit Gadde.

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ArXiv ePrint: 1305.0266

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Gadde, A., Gukov, S. 2d index and surface operators. J. High Energ. Phys. 2014, 80 (2014). https://doi.org/10.1007/JHEP03(2014)080

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Keywords

  • Supersymmetric gauge theory
  • Brane Dynamics in Gauge Theories