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Supersymmetric vortex defects in two dimensions

Open Access
Regular Article - Theoretical Physics

Abstract

We study codimension-two BPS defects in 2d \( \mathcal{N} = \left(2,\ 2\right) \) supersymmetric gauge theories, focusing especially on those characterized by vortex-like singularities in the dynamical or non-dynamical gauge field. We classify possible SUSY-preserving boundary conditions on charged matter fields around the vortex defects, and derive a formula for defect correlators on the squashed sphere. We also prove an equivalence relation between vortex defects and 0d-2d coupled systems. Our defect correlators are shown to be consistent with the mirror symmetry duality between Abelian gauged linear sigma models and Landau-Ginzburg models, as well as that between the minimal model and its orbifold. We also study the vortex defects inserted at conical singularities.

Keywords

Conformal Field Theory Field Theories in Lower Dimensions Supersymmetric Gauge Theory Supersymmetry and Duality 

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, Surface Operators, in New Dualities of Supersymmetric Gauge Theories, J. Teschner eds., Springer, Heidelberg Germany (2016), pg. 223 [arXiv:1412.7127] [INSPIRE].
  2. [2]
    K.G. Wilson, Confinement of Quarks, Phys. Rev. D 10 (1974) 2445 [INSPIRE].ADSGoogle Scholar
  3. [3]
    G. ’t Hooft, On the Phase Transition Towards Permanent Quark Confinement, Nucl. Phys. B 138 (1978) 1 [INSPIRE].
  4. [4]
    S. Gukov and E. Witten, Gauge Theory, Ramification, And The Geometric Langlands Program, hep-th/0612073 [INSPIRE].
  5. [5]
    S. Gukov and A. Kapustin, Topological Quantum Field Theory, Nonlocal Operators and Gapped Phases of Gauge Theories, arXiv:1307.4793 [INSPIRE].
  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].ADSMathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    E. Frenkel, S. Gukov and J. Teschner, Surface Operators and Separation of Variables, JHEP 01 (2016) 179 [arXiv:1506.07508] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    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].ADSMathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    B. Assel and J. Gomis, Mirror Symmetry And Loop Operators, JHEP 11 (2015) 055 [arXiv:1506.01718] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    J. Gomis and B. Le Floch, M2-brane surface operators and gauge theory dualities in Toda, JHEP 04 (2016) 183 [arXiv:1407.1852] [INSPIRE].ADSMATHGoogle Scholar
  11. [11]
    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].ADSGoogle Scholar
  12. [12]
    Y. Pan and W. Peelaers, Intersecting Surface Defects and Instanton Partition Functions, JHEP 07 (2017) 073 [arXiv:1612.04839] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    H.-Y. Chen and T.-H. Tsai, On Higgs branch localization of Seiberg-Witten theories on an ellipsoid, PTEP 2016 (2016) 013B09 [arXiv:1506.04390] [INSPIRE].
  14. [14]
    Y. Pan and W. Peelaers, Ellipsoid partition function from Seiberg-Witten monopoles, JHEP 10 (2015) 183 [arXiv:1508.07329] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    A. Kapustin, B. Willett and I. Yaakov, Exact results for supersymmetric abelian vortex loops in 2+1 dimensions, JHEP 06 (2013) 099 [arXiv:1211.2861] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    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].ADSCrossRefGoogle Scholar
  17. [17]
    S. Nawata, Givental J-functions, Quantum integrable systems, AGT relation with surface operator, Adv. Theor. Math. Phys. 19 (2015) 1277 [arXiv:1408.4132] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    F. Benini and S. Cremonesi, Partition Functions of \( \mathcal{N}=\left(2,\ 2\right) \) Gauge Theories on S 2 and Vortices, Commun. Math. Phys. 334 (2015) 1483 [arXiv:1206.2356] [INSPIRE].
  19. [19]
    N. Doroud, J. Gomis, B. Le Floch and S. Lee, Exact Results in D = 2 Supersymmetric Gauge Theories, JHEP 05 (2013) 093 [arXiv:1206.2606] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    K. Hori and C. Vafa, Mirror symmetry, hep-th/0002222 [INSPIRE].
  21. [21]
    J. Gomis and S. Lee, Exact Kähler Potential from Gauge Theory and Mirror Symmetry, JHEP 04 (2013) 019 [arXiv:1210.6022] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  22. [22]
    T. Nishioka and I. Yaakov, Supersymmetric Renyi Entropy, JHEP 10 (2013) 155 [arXiv:1306.2958] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  23. [23]
    X. Huang, S.-J. Rey and Y. Zhou, Three-dimensional SCFT on conic space as hologram of charged topological black hole, JHEP 03 (2014) 127 [arXiv:1401.5421] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    X. Huang and Y. Zhou, \( \mathcal{N} = 4 \) super-Yang-Mills on conic space as hologram of STU topological black hole, JHEP 02 (2015) 068 [arXiv:1408.3393] [INSPIRE].
  25. [25]
    M. Crossley, E. Dyer and J. Sonner, Super-Rényi entropy & Wilson loops for \( \mathcal{N} = 4 \) SYM and their gravity duals, JHEP 12 (2014) 001 [arXiv:1409.0542] [INSPIRE].
  26. [26]
    L.F. Alday, P. Richmond and J. Sparks, The holographic supersymmetric Renyi entropy in five dimensions, JHEP 02 (2015) 102 [arXiv:1410.0899] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  27. [27]
    N. Hama, T. Nishioka and T. Ugajin, Supersymmetric Rényi entropy in five dimensions, JHEP 12 (2014) 048 [arXiv:1410.2206] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    T. Nishioka and I. Yaakov, Supersymmetric Rényi entropy and defect operators, JHEP 11 (2017) 071 [arXiv:1612.02894] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  29. [29]
    K. Hosomichi, Orbifolds, Defects and Sphere Partition Function, JHEP 02 (2016) 155 [arXiv:1507.07650] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  30. [30]
    C. Closset and S. Cremonesi, Comments on \( \mathcal{N}=\left(2,\ 2\right) \) supersymmetry on two-manifolds, JHEP 07 (2014) 075 [arXiv:1404.2636] [INSPIRE].
  31. [31]
    A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and Duality, JHEP 04 (2014) 001 [arXiv:1401.0740] [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    T. Okuda, Mirror symmetry and the flavor vortex operator in two dimensions, JHEP 10 (2015) 174 [arXiv:1508.07179] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    V. Pestun et al., Localization techniques in quantum field theories, J. Phys. A 50 (2017) 440301 [arXiv:1608.02952] [INSPIRE].MathSciNetMATHGoogle Scholar
  34. [34]
    T. Okuda, to appear.Google Scholar
  35. [35]
    E. Witten, Holomorphic Morse inequalities, in Teubner-Texte zur Mathematik. Vol. 70: Algebraic and differential topology — global differential geometry, Teubner, Leipzig Germany (1984), pg. 318.Google Scholar
  36. [36]
    A. Bilal, Lectures on Anomalies, arXiv:0802.0634 [INSPIRE].
  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].ADSMathSciNetCrossRefMATHGoogle Scholar
  38. [38]
    A.B. Givental, Homological geometry. I. Projective hypersurfaces, Selecta Math. (N.S.) 1 (1995) 325.Google Scholar
  39. [39]
    E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].
  40. [40]
    D.R. Morrison and M.R. Plesser, Summing the instantons: Quantum cohomology and mirror symmetry in toric varieties, Nucl. Phys. B 440 (1995) 279 [hep-th/9412236] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  41. [41]
    C. Vafa, String Vacua and Orbifoldized L-G Models, Mod. Phys. Lett. A 4 (1989) 1169 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    K. Hori, A. Iqbal and C. Vafa, D-branes and mirror symmetry, hep-th/0005247 [INSPIRE].
  43. [43]
    H. Mori, Supersymmetric Rényi entropy in two dimensions, JHEP 03 (2016) 058 [arXiv:1512.02829] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  44. [44]
    W. Chen and Y. Ruan, Orbifold Gromov-Witten Theory, math/0103156.
  45. [45]
    D. Cheong, I. Ciocan-Fontanine and B. Kim, Orbifold quasimap theory, Math. Ann. 363 (2015) 777.MathSciNetCrossRefMATHGoogle Scholar
  46. [46]
    Y. Ruan, The cohomology ring of crepant resolutions of orbifolds, in Contemporary Mathematics. Vol. 403: Gromov-Witten theory of spin curves and orbifolds, AMS Press, Providence U.S.A. (2000), pg. 117.Google Scholar
  47. [47]
    D.A. Cox and S. Katz, Mirror symmetry and algebraic geometry, AMS Press, Providence U.S.A. (2000).Google Scholar
  48. [48]
    H. Jockers, V. Kumar, J.M. Lapan, D.R. Morrison and M. Romo, Two-Sphere Partition Functions and Gromov-Witten Invariants, Commun. Math. Phys. 325 (2014) 1139 [arXiv:1208.6244] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Department of Applied PhysicsNational Defense AcademyKanagawaJapan
  2. 2.Korea Institute for Advanced StudySeoulSouth Korea
  3. 3.University of Tokyo, KomabaTokyoJapan

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