Advertisement

Journal of High Energy Physics

, 2019:67 | Cite as

Deconstructing defects

  • Joseph Hayling
  • Vasilis Niarchos
  • Constantinos PapageorgakisEmail author
Open Access
Regular Article - Theoretical Physics

Abstract

We use the exact-deconstruction prescription to lift various squashed-S3 partition functions with supersymmetric-defect insertions to four-dimensional superconformal indices. Starting from three-dimensional circular-quiver theories with vortex-loop-operator insertions, we recover the index of four-dimensional theories in the presence of codimension-two surface defects with (2,2) supersymmetry. The case of deconstruction with Wilson-loop insertions is discussed separately. We provide evidence that a certain prescription leads to the index of four-dimensional theories in the presence of surface defects with (4,0) supersymmetry. In addition, we deconstruct the index of four-dimensional gauge theories with codimension-one \( \frac{1}{2} \)-BPS defects, starting from three-dimensional circular-quiver theories containing localised matter/gauge-field insertions at specific nodes. We also clarify certain calculational and conceptual points related to exact deconstruction.

Keywords

Extended Supersymmetry Supersymmetric Gauge Theory Wilson, ’t Hooft and Polyakov loops Brane Dynamics in Gauge Theories 

Notes

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.

References

  1. [1]
    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].ADSMathSciNetzbMATHGoogle Scholar
  2. [2]
    C. Romelsberger, Counting chiral primaries in N = 1, d = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  3. [3]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  4. [4]
    V. Pestun et al., Localization techniques in quantum field theories, J. Phys. A 50 (2017) 440301 [arXiv:1608.02952] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  5. [5]
    D. Gang, E. Koh and K. Lee, Superconformal index with duality domain wall, JHEP 10 (2012) 187 [arXiv:1205.0069] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  6. [6]
    D. Gaiotto, L. Rastelli and S.S. Razamat, Bootstrapping the superconformal index with surface defects, JHEP 01 (2013) 022 [arXiv:1207.3577] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  7. [7]
    A. Gadde and S. Gukov, 2d index and surface operators, JHEP 03 (2014) 080 [arXiv:1305.0266] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  8. [8]
    N. Drukker, T. Okuda and F. Passerini, Exact results for vortex loop operators in 3d supersymmetric theories, JHEP 07 (2014) 137 [arXiv:1211.3409] [INSPIRE].ADSGoogle Scholar
  9. [9]
    B. Assel and J. Gomis, Mirror symmetry and loop operators, JHEP 11 (2015) 055 [arXiv:1506.01718] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  10. [10]
    J. Gomis, B. Le Floch, Y. Pan and W. Peelaers, Intersecting surface defects and two-dimensional CFT, Phys. Rev. D 96 (2017) 045003 [arXiv:1610.03501] [INSPIRE].ADSMathSciNetGoogle Scholar
  11. [11]
    C. Cordova, D. Gaiotto and S.-H. Shao, Surface defect indices and 2d-4d BPS states, JHEP 12 (2017) 078 [arXiv:1703.02525] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  12. [12]
    F.A.H. Dolan, V.P. Spiridonov and G.S. Vartanov, From 4d superconformal indices to 3d partition functions, Phys. Lett. B 704 (2011) 234 [arXiv:1104.1787] [INSPIRE].ADSGoogle Scholar
  13. [13]
    A. Gadde and W. Yan, Reducing the 4d index to the S 3 partition function, JHEP 12 (2012) 003 [arXiv:1104.2592] [INSPIRE].ADSzbMATHGoogle Scholar
  14. [14]
    V. Niarchos, Seiberg dualities and the 3d/4d connection, JHEP 07 (2012) 075 [arXiv:1205.2086] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  15. [15]
    O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3d dualities from 4d dualities, JHEP 07 (2013) 149 [arXiv:1305.3924] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  16. [16]
    A. Amariti, D. Orlando and S. Reffert, String theory and the 4D/3D reduction of Seiberg duality. A review, Phys. Rept. 705-706 (2017) 1 [arXiv:1611.04883] [INSPIRE].
  17. [17]
    N. Arkani-Hamed, A.G. Cohen and H. Georgi, (De)constructing dimensions, Phys. Rev. Lett. 86 (2001) 4757 [hep-th/0104005] [INSPIRE].
  18. [18]
    C.T. Hill, S. Pokorski and J. Wang, Gauge invariant effective lagrangian for Kaluza-Klein modes, Phys. Rev. D 64 (2001) 105005 [hep-th/0104035] [INSPIRE].ADSGoogle Scholar
  19. [19]
    D.B. Kaplan, E. Katz and M. Ünsal, Supersymmetry on a spatial lattice, JHEP 05 (2003) 037 [hep-lat/0206019] [INSPIRE].
  20. [20]
    A. Bourget, A. Pini and D. Rodriguez-Gomez, Towards deconstruction of the type D (2, 0) theory, JHEP 12 (2017) 146 [arXiv:1710.10247] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  21. [21]
    K. Aitken, A. Karch and B. Robinson, Deconstructing S-duality, SciPost Phys. 4 (2018) 032 [arXiv:1802.01592] [INSPIRE].ADSGoogle Scholar
  22. [22]
    N. Arkani-Hamed et al., Deconstructing (2, 0) and little string theories, JHEP 01 (2003) 083 [hep-th/0110146] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  23. [23]
    N. Lambert, C. Papageorgakis and M. Schmidt-Sommerfeld, Deconstructing (2, 0) Proposals, Phys. Rev. D 88 (2013) 026007 [arXiv:1212.3337] [INSPIRE].ADSGoogle Scholar
  24. [24]
    J. Hayling, C. Papageorgakis, E. Pomoni and D. Rodríguez-Gómez, Exact deconstruction of the 6D (2, 0) theory, JHEP 06 (2017) 072 [arXiv:1704.02986] [INSPIRE].
  25. [25]
    J. Hayling, R. Panerai and C. Papageorgakis, Deconstructing little strings with \( \mathcal{N} \) = 1 gauge theories on ellipsoids, SciPost Phys. 4 (2018) 042 [arXiv:1803.06177] [INSPIRE].ADSGoogle Scholar
  26. [26]
    J. Gomis and F. Passerini, Holographic Wilson loops, JHEP 08 (2006) 074 [hep-th/0604007] [INSPIRE].ADSMathSciNetGoogle Scholar
  27. [27]
    E.I. Buchbinder, J. Gomis and F. Passerini, Holographic gauge theories in background fields and surface operators, JHEP 12 (2007) 101 [arXiv:0710.5170] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  28. [28]
    C. Csáki, J. Erlich, C. Grojean and G.D. Kribs, 4D constructions of supersymmetric extra dimensions and gaugino mediation, Phys. Rev. D 65 (2002) 015003 [hep-ph/0106044] [INSPIRE].
  29. [29]
    L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  30. [30]
    A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [INSPIRE].ADSMathSciNetGoogle Scholar
  31. [31]
    J. de Boer, K. Hori and Y. Oz, Dynamics of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 500 (1997) 163 [hep-th/9703100] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  32. [32]
    O. Aharony et al., Aspects of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  33. [33]
    D. Gaiotto and E. Witten, S-duality of boundary conditions in N = 4 super Yang-Mills theory, Adv. Theor. Math. Phys. 13 (2009) 721 [arXiv:0807.3720] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  34. [34]
    M.R. Douglas and G.W. Moore, D-branes, quivers and ALE instantons, hep-th/9603167 [INSPIRE].
  35. [35]
    A. Giveon and D. Kutasov, Brane dynamics and gauge theory, Rev. Mod. Phys. 71 (1999) 983 [hep-th/9802067] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  36. [36]
    A. Kapustin, B. Willett and I. Yaakov, Exact results for Wilson loops in superconformal Chern-Simons theories with matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  37. [37]
    N. Hama, K. Hosomichi and S. Lee, Notes on SUSY gauge theories on three-sphere, JHEP 03 (2011) 127 [arXiv:1012.3512] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  38. [38]
    N. Hama, K. Hosomichi and S. Lee, SUSY gauge theories on squashed three-spheres, JHEP 05 (2011) 014 [arXiv:1102.4716] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  39. [39]
    F.A. 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].ADSMathSciNetzbMATHGoogle Scholar
  40. [40]
    G. Festuccia and N. Seiberg, Rigid supersymmetric theories in curved superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  41. [41]
    M. Bullimore, M. Fluder, L. Hollands and P. Richmond, The superconformal index and an elliptic algebra of surface defects, JHEP 10 (2014) 062 [arXiv:1401.3379] [INSPIRE].ADSGoogle Scholar
  42. [42]
    C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, Supersymmetric field theories on three-manifolds, JHEP 05 (2013) 017 [arXiv:1212.3388] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  43. [43]
    F. Benini and W. Peelaers, Higgs branch localization in three dimensions, JHEP 05 (2014) 030 [arXiv:1312.6078] [INSPIRE].ADSGoogle Scholar
  44. [44]
    F. Aprile and V. Niarchos, \( \mathcal{N} \) = 2 supersymmetric field theories on 3-manifolds with A-type boundaries, JHEP 07 (2016) 126 [arXiv:1604.01561] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  45. [45]
    K. Hori, H. Kim and P. Yi, Witten index and wall crossing, JHEP 01 (2015) 124 [arXiv:1407.2567] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  46. [46]
    Y. Ito and Y. Yoshida, Superconformal index with surface defects for class \( \mathcal{S} \) k, arXiv:1606.01653 [INSPIRE].
  47. [47]
    F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of 2d \( \mathcal{N} \) = 2 gauge theories, Commun. Math. Phys. 333 (2015) 1241 [arXiv:1308.4896] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  48. [48]
    F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of two-dimensional N = 2 gauge theories with rank-one gauge groups, Lett. Math. Phys. 104 (2014) 465 [arXiv:1305.0533] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  49. [49]
    R. Lodin, F. Nieri and M. Zabzine, Elliptic modular double and 4d partition functions, J. Phys. A 51 (2018) 045402 [arXiv:1703.04614] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  50. [50]
    F. Nieri, An elliptic Virasoro symmetry in 6d, Lett. Math. Phys. 107 (2017) 2147 [arXiv:1511.00574] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  51. [51]
    J.A. Harvey and A.B. Royston, Localized modes at a D-brane-O-plane intersection and heterotic Alice atrings, JHEP 04 (2008) 018 [arXiv:0709.1482] [INSPIRE].ADSzbMATHGoogle Scholar
  52. [52]
    J. Gomis and F. Passerini, Wilson loops as D3-branes, JHEP 01 (2007) 097 [hep-th/0612022] [INSPIRE].ADSMathSciNetGoogle Scholar
  53. [53]
    P. Agarwal, J. Kim, S. Kim and A. Sciarappa, Wilson surfaces in M5-branes, JHEP 08 (2018) 119 [arXiv:1804.09932] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  54. [54]
    Y. Tachikawa, On W-algebras and the symmetries of defects of 6d N = (2, 0) theory, JHEP 03 (2011) 043 [arXiv:1102.0076] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  55. [55]
    C. Lawrie, S. Schäfer-Nameki and T. Weigand, Chiral 2d theories from N = 4 SYM with varying coupling, JHEP 04 (2017) 111 [arXiv:1612.05640] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  56. [56]
    S.S. Razamat, Flavored surface defects in 4d \( \mathcal{N} \) = 1 SCFTs, arXiv:1808.09509 [INSPIRE].
  57. [57]
    M. Bullimore and H.-C. Kim, The superconformal index of the (2, 0) theory with defects, JHEP 05 (2015) 048 [arXiv:1412.3872] [INSPIRE].ADSMathSciNetGoogle Scholar
  58. [58]
    T. Bourton, A. Pini and E. Pomoni, 4d \( \mathcal{N} \) = 3 indices via discrete gauging, JHEP 10 (2018) 131 [arXiv:1804.05396] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  59. [59]
    I. García-Etxebarria and D. Regalado, \( \mathcal{N} \) = 3 four dimensional field theories, JHEP 03 (2016) 083 [arXiv:1512.06434] [INSPIRE].
  60. [60]
    O. Aharony and Y. Tachikawa, S-folds and 4d N = 3 superconformal field theories, JHEP 06 (2016) 044 [arXiv:1602.08638] [INSPIRE].ADSzbMATHGoogle Scholar
  61. [61]
    S. Ruijsenaars, First order analytic difference equations and integrable quantum systems, J. Math. Phys. 38 (1997) 1069.ADSMathSciNetzbMATHGoogle Scholar
  62. [62]
    F. van de Bult, Hyperbolic hypergeometric functions, Ph.D. thesis, University of Amsterdam, Amsterdam Netherlands (2007).Google Scholar
  63. [63]
    S.N.M. Ruijsenaars, Complete integrability of relativistic Calogero-Moser systems and elliptic function identities, Comm. Math. Phys. 110 (1987) 191.ADSMathSciNetzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.CRST and School of Physics and AstronomyQueen Mary University of LondonLondonU.K.
  2. 2.Department of Mathematical Sciences and Centre for Particle TheoryDurham UniversityDurhamU.K.

Personalised recommendations