Advertisement

Darboux coordinates and instanton corrections in projective superspace

  • P. Marcos Crichigno
  • Dharmesh Jain
Article

Abstract

By demanding consistency of the Legendre transform construction of hyperkähler metrics in projective superspace, we derive the expression for the Darboux coordinates on the hyperkähler manifold. We apply these results to study the Coulomb branch moduli space of 4D, \( \mathcal{N}=2 \) super-Yang-Mills theory (SYM) on \( {{\mathbb{R}}^3}\times {S^1} \), recovering the results by GMN. We also apply this method to study the electric corrections to the moduli space of 5D, \( \mathcal{N}=1 \) SYM on \( {{\mathbb{R}}^3}\times {T^2} \) and give the Darboux coordinates explicitly.

Keywords

Superspaces Extended Supersymmetry Supersymmetric Effective Theories 

References

  1. [1]
    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) 485–486] [hep-th/9407087] [INSPIRE].
  2. [2]
    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].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, hep-th/9607163 [INSPIRE].
  4. [4]
    M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435.
  5. [5]
    D. Gaiotto, G.W. Moore and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Commun. Math. Phys. 299 (2010) 163 [arXiv:0807.4723] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  6. [6]
    S. Alexandrov, B. Pioline, F. Saueressig and S. Vandoren, Linear perturbations of hyper-Kähler metrics, Lett. Math. Phys. 87 (2009) 225 [arXiv:0806.4620] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  7. [7]
    S. Alexandrov, B. Pioline, F. Saueressig and S. Vandoren, Linear perturbations of quaternionic metrics, Commun. Math. Phys. 296 (2010) 353 [arXiv:0810.1675] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  8. [8]
    S. Alexandrov, B. Pioline, F. Saueressig and S. Vandoren, D-instantons and twistors, JHEP 03 (2009) 044 [arXiv:0812.4219] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    S. Alexandrov, D-instantons and twistors: some exact results, J. Phys. A 42 (2009) 335402 [arXiv:0902.2761] [INSPIRE].MathSciNetGoogle Scholar
  10. [10]
    S. Alexandrov, Twistor approach to string compactifications: a review, arXiv:1111.2892 [INSPIRE].
  11. [11]
    B. Haghighat and S. Vandoren, Five-dimensional gauge theory and compactification on a torus, JHEP 09 (2011) 060 [arXiv:1107.2847] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    A. Karlhede, U. Lindström and M. Roček, Self-interacting tensor multiplets in N = 2 superspace, Phys. Lett. B 147 (1984) 297 [INSPIRE].ADSGoogle Scholar
  13. [13]
    U. Lindström and M. Roček, N = 2 super Yang-Mills theory in projective superspace, Commun. Math. Phys. 128 (1990) 191 [INSPIRE].ADSMATHCrossRefGoogle Scholar
  14. [14]
    N.J. Hitchin, A. Karlhede, U. Lindström and M. Roček, Hyper-Kähler metrics and supersymmetry, Commun. Math. Phys. 108 (1987) 535 [INSPIRE].ADSMATHCrossRefGoogle Scholar
  15. [15]
    U. Lindström and M. Roček, Properties of hyper-Kähler manifolds and their twistor spaces, Commun. Math. Phys. 293 (2010) 257 [arXiv:0807.1366] [INSPIRE].ADSMATHCrossRefGoogle Scholar
  16. [16]
    U. Lindström and M. Roček, New hyper-Kähler metrics and new supermultiplets, Commun. Math. Phys. 115 (1988) 21 [INSPIRE].ADSMATHCrossRefGoogle Scholar
  17. [17]
    I. Ivanov and M. Roček, Supersymmetric σ-models, twistors and the Atiyah-Hitchin metric, Commun. Math. Phys. 182 (1996) 291 [hep-th/9512075] [INSPIRE].ADSMATHCrossRefGoogle Scholar
  18. [18]
    H. Ooguri and C. Vafa, Summing up D instantons, Phys. Rev. Lett. 77 (1996) 3296 [hep-th/9608079] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  19. [19]
    N. Seiberg and S.H. Shenker, Hypermultiplet moduli space and string compactification to three-dimensions, Phys. Lett. B 388 (1996) 521 [hep-th/9608086] [INSPIRE].MathSciNetADSGoogle Scholar
  20. [20]
    D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing in coupled 2D-4D systems, arXiv:1103.2598 [INSPIRE].
  21. [21]
    F. Gonzalez-Rey, M. Roček, S. Wiles, U. Lindström and R. von Unge, Feynman rules in N = 2 projective superspace: 1. Massless hypermultiplets, Nucl. Phys. B 516 (1998) 426 [hep-th/9710250] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    S.M. Kuzenko, N = 2 supersymmetric σ-models and duality, JHEP 01 (2010) 115 [arXiv:0910.5771] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    S.M. Kuzenko, Comments on N = 2 supersymmetric σ-models in projective superspace, J. Phys. A 45 (2012) 095401 [arXiv:1110.4298] [INSPIRE].MathSciNetADSGoogle Scholar
  24. [24]
    M. Roček, unpublished notes.Google Scholar
  25. [25]
    S. Cecotti, S. Ferrara and L. Girardello, Geometry of type II superstrings and the moduli of superconformal field theories, Int. J. Mod. Phys. A 4 (1989) 2475 [INSPIRE].MathSciNetADSGoogle Scholar
  26. [26]
    S. Ferrara and S. Sabharwal, Quaternionic manifolds for type II superstring vacua of Calabi-Yau spaces, Nucl. Phys. B 332 (1990) 317 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  27. [27]
    S.J. Gates Jr., T. Hubsch and S.M. Kuzenko, CNM models, holomorphic functions and projective superspace C maps, Nucl. Phys. B 557 (1999) 443 [hep-th/9902211] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  28. [28]
    M. Roček, C. Vafa and S. Vandoren, Hypermultiplets and topological strings, JHEP 02 (2006) 062 [hep-th/0512206] [INSPIRE].ADSGoogle Scholar
  29. [29]
    I. Bakas, Remarks on the Atiyah-Hitchin metric, Fortsch. Phys. 48 (2000) 9 [hep-th/9903256] [INSPIRE].MathSciNetADSMATHCrossRefGoogle Scholar
  30. [30]
    R.A. Ionas, Elliptic constructions of hyper-Kähler metrics. I. The Atiyah-Hitchin manifold, arXiv:0712.3598 [INSPIRE].
  31. [31]
    N.D. Lambert and D. Tong, Dyonic instantons in five-dimensional gauge theories, Phys. Lett. B 462 (1999) 89 [hep-th/9907014] [INSPIRE].ADSGoogle Scholar
  32. [32]
    N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett. B 388 (1996) 753 [hep-th/9608111] [INSPIRE].MathSciNetADSGoogle Scholar
  33. [33]
    A.E. Lawrence and N. Nekrasov, Instanton sums and five-dimensional gauge theories, Nucl. Phys. B 513 (1998) 239 [hep-th/9706025] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  34. [34]
    D. Robles-Llana, F. Saueressig, U. Theis and S. Vandoren, Membrane instantons from mirror symmetry, Commun. Num. Theor. Phys. 1 (2007) 681 [arXiv:0707.0838] [INSPIRE].MathSciNetGoogle Scholar
  35. [35]
    H.-Y. Chen, N. Dorey and K. Petunin, Wall crossing and instantons in compactified gauge theory, JHEP 06 (2010) 024 [arXiv:1004.0703] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  36. [36]
    A. Neitzke, On a hyperholomorphic line bundle over the Coulomb branch, arXiv:1110.1619 [INSPIRE].
  37. [37]
    S. Alexandrov, D. Persson and B. Pioline, Wall-crossing, Rogers dilogarithm and the QK/HK correspondence, JHEP 12 (2011) 027 [arXiv:1110.0466] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  1. 1.C.N. Yang Institute for Theoretical PhysicsState University of New YorkStony BrookU.S.A.

Personalised recommendations