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2d TQFT structure of the superconformal indices with outer-automorphism twists

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Abstract

We study the superconformal indices of 4d theories coming from 6d \( \mathcal{N} \) = (2, 0) theory of type Γ on a Riemann surface, with the action of the outer-automorphism σ in the trace. We find that the indices are given by the partition function of a deformed 2d Yang-Mills on the Riemann surface with gauge group G which is S-dual to the subgroup of Γ fixed by σ. In the 2-parameter deformed version, we find that it is governed not by Macdonald polynomials of type G, but by Macdonald polynomials associated to twisted affine root systems.

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References

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

    Article  MathSciNet  ADS  Google Scholar 

  2. 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].

    Article  ADS  Google Scholar 

  3. 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].

    Article  MathSciNet  Google Scholar 

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

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

    Article  ADS  Google Scholar 

  6. D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin systems and the WKB approximation, arXiv:0907.3987 [INSPIRE].

  7. 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].

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  9. T. Kawano and N. Matsumiya, 5D SYM on 3D sphere and 2D YM, Phys. Lett. B 716 (2012) 450 [arXiv:1206.5966] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  10. Y. Fukuda, T. Kawano and N. Matsumiya, 5D SYM and 2D q-deformed YM, Nucl. Phys. B 869 (2013) 493 [arXiv:1210.2855] [INSPIRE].

    Article  ADS  Google Scholar 

  11. C. Vafa, Geometric origin of Montonen-Olive duality, Adv. Theor. Math. Phys. 1 (1998) 158 [hep-th/9707131] [INSPIRE].

    MathSciNet  Google Scholar 

  12. Y. Tachikawa, On S-duality of 5D super Yang-Mills on S 1, JHEP 11 (2011) 123 [arXiv:1110.0531] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  13. B.I. Zwiebel, Charging the superconformal index, JHEP 01 (2012) 116 [arXiv:1111.1773] [INSPIRE].

    Article  ADS  Google Scholar 

  14. Y. Tachikawa, Six-dimensional D(N ) theory and four-dimensional SO-USp quivers, JHEP 07 (2009) 067 [arXiv:0905.4074] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  15. Y. Tachikawa, N = 2 S-duality via outer-automorphism twists, J. Phys. A 44 (2011) 182001 [arXiv:1009.0339] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  16. M. Lemos, W. Peelaers and L. Rastelli, The superconformal index of class S theories of type D, arXiv:1212.1271 [INSPIRE].

  17. B. Feng, A. Hanany and Y.-H. He, Counting gauge invariants: the plethystic program, JHEP 03 (2007) 090 [hep-th/0701063] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  18. C.A. Keller, N. Mekareeya, J. Song and Y. Tachikawa, The ABCDEFG of instantons and W-algebras, JHEP 03 (2012) 045 [arXiv:1111.5624] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  19. I.G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics 157, Cambridge University Press, Cambridge U.K. (2003).

  20. V.G. Kac, Infinite-dimensional Lie algebras, Cambridge University Press, Cambridge U.K. (1990).

    Book  MATH  Google Scholar 

  21. D. Gaiotto and E. Witten, S-duality of boundary conditions in N = 4 super Yang-Mills theory, Adv. Theor. Math. Phys. 13 (2009) [arXiv:0807.3720] [INSPIRE].

  22. O. Chacaltana, J. Distler and Y. Tachikawa, Nilpotent orbits and codimension-two defects of 6D N = (2, 0) theories, arXiv:1203.2930 [INSPIRE].

  23. 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].

    Article  MathSciNet  ADS  Google Scholar 

  24. V.P. Spiridonov and S.O. Warnaar, Inversions of integral operators and elliptic beta integrals on root systems, arXiv:math/0411044.

  25. P.C. Argyres and N. Seiberg, S-duality in N = 2 supersymmetric gauge theories, JHEP 12 (2007) 088 [arXiv:0711.0054] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  26. Y. Tachikawa, 4D partition function on S 1 × S 3 and 2D Yang-Mills with nonzero area, PTEP 2013 (2013) 013B01 [arXiv:1207.3497] [INSPIRE].

    Google Scholar 

  27. J. Fuchs, B. Schellekens and C. Schweigert, From Dynkin diagram symmetries to fixed point structures, Commun. Math. Phys. 180 (1996) 39 [hep-th/9506135] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. J.F. van Diejen, L. Lapointe and J. Morse, Determinantal construction of orthogonal polynomials associated with root systems, math/0303263.

  29. J.F. van Diejen and E. Emsiz, A Pieri formula for MacDonald’s spherical functions and polynomials, arXiv:1009.4482.

  30. T.H. Koornwinder, Askey-Wilson polynomials, http://www.scholarpedia.org/article/Askey-Wilson_polynomial.

  31. D. Nanopoulos and D. Xie, N = 2 SU quiver with USp ends or SU ends with antisymmetric matter, JHEP 08 (2009) 108 [arXiv:0907.1651] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  32. O. Chacaltana and J. Distler, Tinkertoys for Gaiotto duality, JHEP 11 (2010) 099 [arXiv:1008.5203] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  33. Y. Tachikawa and S. Terashima, Seiberg-Witten geometries revisited, JHEP 09 (2011) 010 [arXiv:1108.2315] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, 80805, München, Germany

    Noppadol Mekareeya

  2. Department of Physics, University of California, San Diego, La Jolla, CA, 92093, U.S.A.

    Jaewon Song

  3. Department of Physics, Faculty of Science, University of Tokyo, Bunkyo-ku, Tokyo, 133-0022, Japan

    Yuji Tachikawa

  4. Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba, 277-8583, Japan

    Yuji Tachikawa

Authors
  1. Noppadol Mekareeya
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  2. Jaewon Song
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  3. Yuji Tachikawa
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Correspondence to Noppadol Mekareeya.

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ArXiv ePrint: 1212.0545

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Mekareeya, N., Song, J. & Tachikawa, Y. 2d TQFT structure of the superconformal indices with outer-automorphism twists. J. High Energ. Phys. 2013, 171 (2013). https://doi.org/10.1007/JHEP03(2013)171

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  • DOI: https://doi.org/10.1007/JHEP03(2013)171

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