Advertisement

Journal of High Energy Physics

, 2019:159 | Cite as

Chiral rings for surface operators in 4d and 5d SQCD

  • Jong-Hyun BaekEmail author
Open Access
Regular Article - Theoretical Physics
  • 15 Downloads

Abstract

Chiral rings of two-dimensional (2,2) theories coupled to 4d \( \mathcal{N} \) = 2 theories with matter hypermultiplets are studied. Specifically, the vacua of the twisted superpotential of the 2d theories with vanishing sum of matter charges are computed by considering the resolvent of the bulk theory. The solutions to the chiral ring equations are also obtained from the instanton partition function via Higgsing for simple surface operators and via the orbifold description for full surface operators. These 2d/4d coupled theories are lifted to 3d/5d theories and vacua are found similarly in two different methods: by solving the 3d chiral ring equations taking into account the effect of 5d resolvent and by computing the 5d instanton partition function in the presence of a surface operator. We also check the Seiberg-like duality for both 2d/4d and 3d/5d coupled systems with a specific Chern-Simons coefficient for the latter.

Keywords

Sigma Models Supersymmetric Gauge Theory Supersymmetry and Duality Solitons Monopoles and Instantons 

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]
    S. Gukov and E. Witten, Gauge Theory, Ramification, And The Geometric Langlands Program, hep-th/0612073 [INSPIRE].
  2. [2]
    S. Gukov and E. Witten, Rigid Surface Operators, Adv. Theor. Math. Phys. 14 (2010) 87 [arXiv:0804.1561] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    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].MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    M. Taki, On AGT Conjecture for Pure Super Yang-Mills and W-algebra, JHEP 05 (2011) 038 [arXiv:0912.4789] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    C. Kozcaz, S. Pasquetti and N. Wyllard, A & B model approaches to surface operators and Toda theories, JHEP 08 (2010) 042 [arXiv:1004.2025] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    L.F. Alday and Y. Tachikawa, Affine SL(2) conformal blocks from 4d gauge theories, Lett. Math. Phys. 94 (2010) 87 [arXiv:1005.4469] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    T. Dimofte, S. Gukov and L. Hollands, Vortex Counting and Lagrangian 3-manifolds, Lett. Math. Phys. 98 (2011) 225 [arXiv:1006.0977] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    K. Maruyoshi and M. Taki, Deformed Prepotential, Quantum Integrable System and Liouville Field Theory, Nucl. Phys. B 841 (2010) 388 [arXiv:1006.4505] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    M. Taki, Surface Operator, Bubbling Calabi-Yau and AGT Relation, JHEP 07 (2011) 047 [arXiv:1007.2524] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    H. Awata, H. Fuji, H. Kanno, M. Manabe and Y. Yamada, Localization with a Surface Operator, Irregular Conformal Blocks and Open Topological String, Adv. Theor. Math. Phys. 16 (2012)725 [arXiv:1008.0574] [INSPIRE].
  11. [11]
    C. Kozcaz, S. Pasquetti, F. Passerini and N. Wyllard, Affine sl(N) conformal blocks from N = 2 SU(N) gauge theories, JHEP 01 (2011) 045 [arXiv:1008.1412] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    N. Wyllard, W-algebras and surface operators in N = 2 gauge theories, J. Phys. A 44 (2011) 155401 [arXiv:1011.0289] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  13. [13]
    A. Marshakov, A. Mironov and A. Morozov, On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles, J. Geom. Phys. 61 (2011) 1203 [arXiv:1011.4491] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    N. Wyllard, Instanton partition functions in N = 2 SU(N) gauge theories with a general surface operator and their W-algebra duals, JHEP 02 (2011) 114 [arXiv:1012.1355] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    H. Kanno and Y. Tachikawa, Instanton counting with a surface operator and the chain-saw quiver, JHEP 06 (2011) 119 [arXiv:1105.0357] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    D. Gaiotto, S. Gukov and N. Seiberg, Surface Defects and Resolvents, JHEP 09 (2013) 070 [arXiv:1307.2578] [INSPIRE].CrossRefGoogle Scholar
  17. [17]
    J. Gomis and B. Le Floch, M2-brane surface operators and gauge theory dualities in Toda, JHEP 04 (2016) 183 [arXiv:1407.1852] [INSPIRE].zbMATHGoogle Scholar
  18. [18]
    S. Nawata, Givental J-functions, Quantum integrable systems, AGT relation with surface operator, Adv. Theor. Math. Phys. 19 (2015) 1277 [arXiv:1408.4132] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    D. Gaiotto and H.-C. Kim, Surface defects and instanton partition functions, JHEP 10 (2016) 012 [arXiv:1412.2781] [INSPIRE].
  20. [20]
    M. Bullimore, H.-C. Kim and P. Koroteev, Defects and Quantum Seiberg-Witten Geometry, JHEP 05 (2015) 095 [arXiv:1412.6081] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    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].MathSciNetGoogle Scholar
  22. [22]
    Y. Pan and W. Peelaers, Intersecting Surface Defects and Instanton Partition Functions, JHEP 07 (2017) 073 [arXiv:1612.04839] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    S.K. Ashok, M. Billó, E. Dell’Aquila, M. Frau, R.R. John and A. Lerda, Modular and duality properties of surface operators in N = 2* gauge theories, JHEP 07 (2017) 068 [arXiv:1702.02833] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    D. Gaiotto, Surface Operators in N = 2 4d Gauge Theories, JHEP 11 (2012) 090 [arXiv:0911.1316] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    S. Gukov, Surface Operators, in New Dualities of Supersymmetric Gauge Theories, J. Teschner ed., pp. 223–259 (2016) [DOI: https://doi.org/10.1007/978-3-319-18769-3_8] [arXiv:1412.7127] [INSPIRE].
  26. [26]
    E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    A. Hanany and K. Hori, Branes and N = 2 theories in two-dimensions, Nucl. Phys. B 513 (1998) 119 [hep-th/9707192] [INSPIRE].
  28. [28]
    F. Cachazo, M.R. Douglas, N. Seiberg and E. Witten, Chiral rings and anomalies in supersymmetric gauge theory, JHEP 12 (2002) 071 [hep-th/0211170] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  29. [29]
    S.K. Ashok et al., Surface operators, chiral rings and localization in \( \mathcal{N} \) = 2 gauge theories, JHEP 11 (2017) 137 [arXiv:1707.08922] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    A. Braverman, Instanton counting via affine Lie algebras. 1. Equivariant J functions of (affine) flag manifolds and Whittaker vectors, CRM Proc. Lect. Notes 38 (2004) [math/0401409].
  31. [31]
    A. Braverman and P. Etingof, Instanton counting via affine Lie algebras II: From Whittaker vectors to the Seiberg-Witten prepotential, in Studies in Lie theory, Birkhauser Boston (2006) [math/0409441].
  32. [32]
    B. Feigin, M. Finkelberg, A. Negut and R. Leonid, Yangians and cohomology rings of Laumon spaces, Selecta Math. 17 (2011) 573 [arXiv:0812.4656].MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    E. Frenkel, S. Gukov and J. Teschner, Surface Operators and Separation of Variables, JHEP 01 (2016) 179 [arXiv:1506.07508] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    A. Gorsky, B. Le Floch, A. Milekhin and N. Sopenko, Surface defects and instanton-vortex interaction, Nucl. Phys. B 920 (2017) 122 [arXiv:1702.03330] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    N. Seiberg, Electric-magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].CrossRefzbMATHGoogle Scholar
  36. [36]
    F. Benini, D.S. Park and P. Zhao, Cluster Algebras from Dualities of 2d \( \mathcal{N} \) = (2, 2) Quiver Gauge Theories, Commun. Math. Phys. 340 (2015) 47 [arXiv:1406.2699] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    C. Closset, S. Cremonesi and D.S. Park, The equivariant A-twist and gauged linear σ-models on the two-sphere, JHEP 06 (2015) 076 [arXiv:1504.06308] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    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].MathSciNetCrossRefGoogle Scholar
  39. [39]
    N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. Proc. Suppl. 192-193 (2009) 91 [arXiv:0901.4744] [INSPIRE].
  40. [40]
    N.A. Nekrasov and S.L. Shatashvili, Quantum integrability and supersymmetric vacua, Prog. Theor. Phys. Suppl. 177 (2009) 105 [arXiv:0901.4748] [INSPIRE].CrossRefzbMATHGoogle Scholar
  41. [41]
    N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, in Proceedings, 16th International Congress on Mathematical Physics (ICMP09), Prague, Czech Republic, August 3–8, 2009, pp. 265–289 (2009) [DOI: https://doi.org/10.1142/9789814304634_0015] [arXiv:0908.4052] [INSPIRE].
  42. [42]
    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].
  43. [43]
    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].MathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    S.K. Ashok et al., Surface operators in 5d gauge theories and duality relations, JHEP 05 (2018) 046 [arXiv:1712.06946] [INSPIRE].
  45. [45]
    H.-Y. Chen, T.J. Hollowood and P. Zhao, A 5d/3d duality from relativistic integrable system, JHEP 07 (2012) 139 [arXiv:1205.4230] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  46. [46]
    N. Nekrasov, Five dimensional gauge theories and relativistic integrable systems, Nucl. Phys. B 531 (1998) 323 [hep-th/9609219] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
  49. [49]
    M. Wijnholt, Five-dimensional gauge theories and unitary matrix models, hep-th/0401025 [INSPIRE].
  50. [50]
    N. Dorey, S. Lee and T.J. Hollowood, Quantization of Integrable Systems and a 2d/4d Duality, JHEP 10 (2011) 077 [arXiv:1103.5726] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    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].MathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    T.J. Hollowood, A. Iqbal and C. Vafa, Matrix models, geometric engineering and elliptic genera, JHEP 03 (2008) 069 [hep-th/0310272] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  53. [53]
    Y. Tachikawa, Five-dimensional Chern-Simons terms and Nekrasov’s instanton counting, JHEP 02 (2004) 050 [hep-th/0401184] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  54. [54]
    A.N. Redlich, Gauge Noninvariance and Parity Violation of Three-Dimensional Fermions, Phys. Rev. Lett. 52 (1984) 18 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  55. [55]
    A.N. Redlich, Parity Violation and Gauge Noninvariance of the Effective Gauge Field Action in Three-Dimensions, Phys. Rev. D 29 (1984) 2366 [INSPIRE].MathSciNetGoogle Scholar
  56. [56]
    S.K. Ashok et al., Surface operators, dual quivers and contours, arXiv:1807.06316 [INSPIRE].
  57. [57]
    G. Bonelli, A. Sciarappa, A. Tanzini and P. Vasko, The Stringy Instanton Partition Function, JHEP 01 (2014) 038 [arXiv:1306.0432] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    N. Nekrasov, V. Pestun and S. Shatashvili, Quantum geometry and quiver gauge theories, Commun. Math. Phys. 357 (2018) 519 [arXiv:1312.6689] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    A. Sciarappa, Exact relativistic Toda chain eigenfunctions from Separation of Variables and gauge theory, JHEP 10 (2017) 116 [arXiv:1706.05142] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    N. Nekrasov, BPS/CFT correspondence IV: σ-models and defects in gauge theory, arXiv:1711.11011 [INSPIRE].
  61. [61]
    N. Nekrasov, BPS/CFT correspondence V: BPZ and KZ equations from qq-characters, arXiv:1711.11582 [INSPIRE].
  62. [62]
    S. Jeong and N. Nekrasov, Opers, surface defects and Yang-Yang functional, arXiv:1806.08270 [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.SeoulKorea

Personalised recommendations