Skip to main content
SpringerLink
Log in

Vortex counting from field theory

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

The vortex partition function in 2d \( \mathcal{N} \)= (2, 2) U(N ) gauge theory is derived from the field theoretical point of view by using the moduli matrix approach. The character for the tangent space at each moduli space fixed point is written in terms of the moduli matrix, and then the vortex partition function is obtained by applying the localization formula. We find that dealing with the fermionic zero modes is crucial to obtain the vortex partition function with the anti-fundamental and adjoint matters in addition to the fundamental chiral multiplets. The orbifold vortex partition function is also investigated from the field theoretical point of view.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  2. N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238 [INSPIRE].

  3. G.W. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs branches, Commun. Math. Phys. 209 (2000) 97 [hep-th/9712241] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. 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].

    Article  MathSciNet  ADS  Google Scholar 

  5. 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].

    Article  MathSciNet  ADS  Google Scholar 

  6. 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].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  8. D. Gaiotto, \( \mathcal{N} \)= 2 dualities, arXiv:0904.2715 [INSPIRE].

  9. S. Shadchin, On F-term contribution to effective action, JHEP 08 (2007) 052 [hep-th/0611278] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  10. T. Dimofte, S. Gukov and L. Hollands, Vortex counting and lagrangian 3-manifolds, Lett. Math. Phys. 98 (2011) 225 [arXiv:1006.0977] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Y. Yoshida, Localization of vortex partition functions in \( \mathcal{N} \)= (2, 2) super Yang-Mills theory, arXiv:1101.0872 [INSPIRE].

  12. G. Bonelli, A. Tanzini and J. Zhao, Vertices, vortices and interacting surface operators, arXiv:1102.0184 [INSPIRE].

  13. G. Bonelli, A. Tanzini and J. Zhao, The Liouville side of the Vortex, JHEP 09 (2011) 096 [arXiv:1107.2787] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  14. A. Hanany and D. Tong, Vortices, instantons and branes, JHEP 07 (2003) 037 [hep-th/0306150] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  15. R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi and A. Yung, NonAbelian superconductors: vortices and confinement in N = 2 SQCD, Nucl. Phys. (2003) 187 [hep-th/0307287] [INSPIRE].

  16. D. Tong, TASI lectures on solitons: instantons, monopoles, vortices and kinks, hep-th/0509216 [INSPIRE].

  17. M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, Solitons in the Higgs phase: the moduli matrix approach, J. Phys. A 39 (2006) R315 [hep-th/0602170] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  18. M. Shifman and A. Yung, Supersymmetric solitons and how they help us understand non-abelian gauge theories, Rev. Mod. Phys. 79 (2007) 1139 [hep-th/0703267] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. D. Tong, Quantum vortex strings: a review, Annals Phys. 324 (2009) 30 [arXiv:0809.5060] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  20. A. Miyake, K. Ohta and N. Sakai, Volume of moduli space of vortex equations and localization, Prog. Theor. Phys. 126 (2012) 637 [arXiv:1105.2087] [INSPIRE].

    Article  ADS  Google Scholar 

  21. T. Fujimori, G. Marmorini, M. Nitta, K. Ohashi and N. Sakai, The moduli space metric for well-separated non-abelian vortices, Phys. Rev. D 82 (2010) 065005 [arXiv:1002.4580] [INSPIRE].

    ADS  Google Scholar 

  22. F. Fucito, J.F. Morales and R. Poghossian, Multi instanton calculus on ALE spaces, Nucl. Phys. B 703 (2004) 518 [hep-th/0406243] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  23. V. Belavin and B. Feigin, Super Liouville conformal blocks from N = 2 SU(2) quiver gauge theories, JHEP 07 (2011) 079 [arXiv:1105.5800] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  24. T. Nishioka and Y. Tachikawa, Central charges of para-Liouville and Toda theories from M 5-branes, Phys. Rev. D 84 (2011) 046009 [arXiv:1106.1172] [INSPIRE].

    ADS  Google Scholar 

  25. G. Bonelli, K. Maruyoshi and A. Tanzini, Instantons on ALE spaces and super Liouville conformal field theories, JHEP 08 (2011) 056 [arXiv:1106.2505] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  26. A. Belavin, V. Belavin and M. Bershtein, Instantons and 2D superconformal field theory, JHEP 09 (2011) 117 [arXiv:1106.4001] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  27. G. Bonelli, K. Maruyoshi and A. Tanzini, Gauge theories on ALE space and super Liouville correlation functions, arXiv:1107.4609 [INSPIRE].

  28. N. Wyllard, Coset conformal blocks and \( \mathcal{N} \)= 2 gauge theories, arXiv:1109.4264 [INSPIRE].

  29. T. Kimura, Matrix model from \( \mathcal{N} \)= 2 orbifold partition function, JHEP 09 (2011) 015 [arXiv:1105.6091] [INSPIRE].

    Article  ADS  Google Scholar 

  30. T. Kimura, β-ensembles for toric orbifold partition function, Prog. Theor. Phys. 127 (2012) 271 [arXiv:1109.0004] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  31. T. Kimura and M. Nitta, Vortices on orbifolds, JHEP 09 (2011) 118 [arXiv:1108.3563] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  32. J. Zhao, Orbifold vortex and super Liouville theory, arXiv:1111.7095 [INSPIRE].

  33. M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, Moduli space of non-abelian vortices, Phys. Rev. Lett. 96 (2006) 161601 [hep-th/0511088] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  34. M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, Manifestly supersymmetric effective Lagrangians on BPS solitons, Phys. Rev. D 73 (2006) 125008 [hep-th/0602289] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  35. M. Eto et al., Non-abelian vortices of higher winding numbers, Phys. Rev. D 74 (2006) 065021 [hep-th/0607070] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  36. M. Eto et al., Universal reconnection of non-abelian cosmic strings, Phys. Rev. Lett. 98 (2007) 091602 [hep-th/0609214] [INSPIRE].

    Article  ADS  Google Scholar 

  37. M. Eto, T. Fujimori, M. Nitta, K. Ohashi and N. Sakai, Dynamics of non-abelian vortices, Phys. Rev. D 84 (2011) 125030 [arXiv:1105.1547] [INSPIRE].

    ADS  Google Scholar 

  38. M. Eto et al., Group theory of non-abelian vortices, JHEP 11 (2010) 042 [arXiv:1009.4794] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  39. U. Bruzzo, F. Fucito, J.F. Morales and A. Tanzini, Multiinstanton calculus and equivariant cohomology, JHEP 05 (2003) 054 [hep-th/0211108] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  40. Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, All exact solutions of a 1/4 Bogomol’nyi-Prasad-Sommerfield equation, Phys. Rev. D 71 (2005) 065018 [hep-th/0405129] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  41. C.-S. Lin and Y. Yang, Non-abelian multiple vortices in supersymmetric field theory, Commun. Math. Phys. 304 (2011) 433 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  42. S.C. Davis, A.-C. Davis and M. Trodden, N = 1 supersymmetric cosmic strings, Phys. Lett. B 405 (1997) 257 [hep-ph/9702360] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  43. A. Penin, V. Rubakov, P. Tinyakov and S.V. Troitsky, What becomes of vortices in theories with flat directions, Phys. Lett. B 389 (1996) 13 [hep-ph/9609257] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  44. A. Achucarro, A. Davis, M. Pickles and J. Urrestilla, Vortices in theories with flat directions, Phys. Rev. D 66 (2 002) 105013 [hep-th/0109097] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  45. M. Shifman and A. Yung, Non-Abelian flux tubes in N = 1 SQCD: supersizing world-sheet supersymmetry, Phys. Rev. D 72 (2005) 085017 [hep-th/0501211] [INSPIRE].

    ADS  Google Scholar 

  46. M. Edalati and D. Tong, Heterotic vortex strings, JHEP 05 (2007) 005 [hep-th/0703045] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  47. A. Adams, J. Polchinski and E. Silverstein, Don’t panic! Closed string tachyons in ALE space-times, JHEP 10 (2001) 029 [hep-th/0108075] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  48. T. Vachaspati and A. Achucarro, Semilocal cosmic strings, Phys. Rev. D 44 (1991) 3067 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  49. A. Achucarro and T. Vachaspati, Semilocal and electroweak strings, Phys. Rept. 327 (2000) 347 [hep-ph/9904229] [INSPIRE].

    Article  ADS  Google Scholar 

  50. M. Shifman and A. Yung, Non-abelian semilocal strings in N = 2 supersymmetric QCD, Phys. Rev. D 73 (2006) 125012 [hep-th/0603134] [INSPIRE].

    ADS  Google Scholar 

  51. M. Eto et al., On the moduli space of semilocal strings and lumps, Phys. Rev. D 76 (2007) 105002 [arXiv:0704.2218] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  52. N. Dorey, The BPS spectra of two-dimensional supersymmetric gauge theories with twisted mass terms, JHEP 11 (1998) 005 [hep-th/9806056] [INSPIRE].

    ADS  Google Scholar 

  53. N. Dorey, T.J. Hollowood and D. Tong, The BPS spectra of gauge theories in two-dimensions and four-dimensions, JHEP 05 (1999) 006 [hep-th/9902134] [INSPIRE].

    Article  ADS  Google Scholar 

  54. M. Shifman and A. Yung, Non-abelian string junctions as confined monopoles, Phys. Rev. D 70 (2004) 045004 [hep-th/0403149] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  55. A. Hanany and D. Tong, Vortex strings and four-dimensional gauge dynamics, JHEP 04 (2004) 066 [hep-th/0403158] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  56. M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, Instantons in the Higgs phase, Phys. Rev. D 72 (2005) 025011 [hep-th/0412048] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  57. T. Fujimori, M. Nitta, K. Ohta, N. Sakai and M. Yamazaki, Intersecting solitons, amoeba and tropical geometry, Phys. Rev. D 78 (2008) 105004 [arXiv:0805.1194] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  58. 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].

    Article  MathSciNet  ADS  Google Scholar 

  59. H.-Y. Chen, N. Dorey, T.J. Hollowood and S. Lee, A new 2D/4D duality via integrability, JHEP 09 (2011) 040 [arXiv:1104.3021] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  60. M. Eto et al., Non-abelian vortices in SO(N ) and USp(N ) gauge theories, JHEP 06 (2009) 004 [arXiv:0903.4471] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  61. M. Eto, T. Fujimori, S.B. Gudnason, M. Nitta and K. Ohashi, SO and USp Kähler and hyper-Kähler quotients and lumps, Nucl. Phys. B 815 (2009) 495 [arXiv:0809.2014] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  62. M. Eto et al., Vortices and monopoles in mass-deformed SO and USp gauge theories, JHEP 12 (2011) 017 [arXiv:1108.6124] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  63. M. Eto et al., Constructing non-abelian vortices with arbitrary gauge groups, Phys. Lett. B 669 (2008) 98 [arXiv:0802.1020] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  64. N. Nekrasov and S. Shadchin, ABCD of instantons, Commun. Math. Phys. 252 (2004) 359 [hep-th/0404225] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  65. 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 

  66. M. Eto et al., Non-abelian vortices on cylinder: duality between vortices and walls, Phys. Rev. D 73 (2006) 085008 [hep-th/0601181] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  67. M. Eto et al., Statistical mechanics of vortices from D-branes and T-duality, Nucl. Phys. B 788 (2008) 120 [hep-th/0703197] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  68. G. Lozano, D. Marques and F. Schaposnik, Non-abelian vortices on the torus, JHEP 09 (2007) 095 [arXiv:0708.2386] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  69. N.S. Manton and N. Sakai, Maximally non-abelian vortices from self-dual Yang-Mills fields, Phys. Lett. B 687 (2010) 395 [arXiv:1001.5236] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  70. A.D. Popov, Non-abelian vortices on riemann surfaces: an integrable case, Lett. Math. Phys. 84 (2008) 139 [arXiv:0801.0808] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  71. J. Baptista, Non-abelian vortices on compact Riemann surfaces, Commun. Math. Phys. 291 (2009) 799 [arXiv:0810.3220] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  72. J. Baptista, On the L 2 -metric of vortex moduli spaces, Nucl. Phys. B 844 (2011) 308 [arXiv:1003.1296] [INSPIRE]

    Article  MathSciNet  ADS  Google Scholar 

  73. N.S. Manton and N.A. Rink, Geometry and energy of non-abelian vortices, J. Math. Phys. 52 (2011)043511 [arXiv:1012.3014] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  74. V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, arXiv:0712.2824 [INSPIRE].

  75. K. Ohta, Counting BPS solitons and applications, arXiv:0710.4011 [INSPIRE].

Download references

Author information

Authors and Affiliations

  1. INFN — Sezione di Pisa, Largo B. Pontecorvo, 3, Ed. C, 56127, Pisa, Italy

    Toshiaki Fujimori

  2. Department of Physics, “E. Fermi”, University of Pisa, Largo B. Pontecorvo, 3, Ed. C, 56127, Pisa, Italy

    Toshiaki Fujimori

  3. Department of Basic Science, University of Tokyo, Tokyo, 153-8902, Japan

    Taro Kimura

  4. Mathematical Physics Laboratory, RIKEN Nishina Center, Saitama, 351-0198, Japan

    Taro Kimura

  5. Department of Physics, and Research and Education Center for Natural Sciences, Keio University, Hiyoshi 4-4-1, Kanagawa, 223-8521, Japan

    Muneto Nitta

  6. Department of Physics, Kyoto University, Kyoto, 606-8502, Japan

    Keisuke Ohashi

Authors
  1. Toshiaki Fujimori
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Taro Kimura
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Muneto Nitta
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Keisuke Ohashi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Taro Kimura.

Additional information

ArXiv ePrint: 1204.1968

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fujimori, T., Kimura, T., Nitta, M. et al. Vortex counting from field theory. J. High Energ. Phys. 2012, 28 (2012). https://doi.org/10.1007/JHEP06(2012)028

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP06(2012)028

Keywords

Search

Navigation