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Scheme dependence of instanton counting in ALE spaces

Abstract

There have been two distinct schemes studied in the literature for instanton counting in A p−1 asymptotically locally Euclidean (ALE) spaces. We point out that the two schemes — namely the counting of orbifolded instantons and instanton counting in the resolved space — lead in general to different results for partition functions. We illustrate this observation in the case of \( \mathcal{N}=2 \) U(N) gauge theory with 2N flavors on the A p−1 ALE space. We propose simple relations between the instanton partition functions given by the two schemes and test them by explicit calculations.

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References

  1. [1]

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

    MathSciNet  Google Scholar 

  2. [2]

    P.B. Kronheimer and H. Nakajima, Yang-Mills instantons on ALE gravitational instantons, Math. Ann. 288 (1990) 263.

    MathSciNet  MATH  Article  Google Scholar 

  3. [3]

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

    MathSciNet  ADS  Article  Google Scholar 

  4. [4]

    N. Nekrasov, Localizing gauge theories, XIVth International Congress On Mathematical Physics (2003) 645.

  5. [5]

    T. Sasaki, O(−2) blow-up formula via instanton calculus on affine C 2 /Z 2 and weil conjecture, hep-th/0603162 [INSPIRE].

  6. [6]

    E. Gasparim and C.-C.M. Liu, The Nekrasov conjecture for toric surfaces, Commun. Math. Phys. 293 (2010) 661 [arXiv:0808.0884] [INSPIRE].

    MathSciNet  MATH  Article  Google Scholar 

  7. [7]

    U. Bruzzo, R. Poghossian and A. Tanzini, Poincaré polynomial of moduli spaces of framed sheaves on (stacky) Hirzebruch surfaces, Commun. Math. Phys. 304 (2011) 395 [arXiv:0909.1458] [INSPIRE].

    MathSciNet  ADS  MATH  Article  Google Scholar 

  8. [8]

    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].

    MathSciNet  ADS  Article  Google Scholar 

  9. [9]

    G. Bonelli, K. Maruyoshi and A. Tanzini, Gauge theories on ALE space and super Liouville correlation functions, Lett. Math. Phys. 101 (2012) 103 [arXiv:1107.4609] [INSPIRE].

    MathSciNet  ADS  MATH  Article  Google Scholar 

  10. [10]

    M. Cirafici and R.J. Szabo, Curve counting, instantons and McKay correspondences, arXiv:1209.1486 [INSPIRE].

  11. [11]

    G. Bonelli, K. Maruyoshi, A. Tanzini and F. Yagi, N = 2 gauge theories on toric singularities, blow-up formulae and W-algebrae, JHEP 01 (2013) 014 [arXiv:1208.0790] [INSPIRE].

    ADS  Article  Google Scholar 

  12. [12]

    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].

    MathSciNet  ADS  MATH  Article  Google Scholar 

  13. [13]

    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].

    MathSciNet  ADS  Article  Google Scholar 

  14. [14]

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

    MathSciNet  ADS  Article  Google Scholar 

  15. [15]

    N. Wyllard, Coset conformal blocks and N = 2 gauge theories, arXiv:1109.4264 [INSPIRE].

  16. [16]

    Y. Ito, Ramond sector of super Liouville theory from instantons on an ALE space, Nucl. Phys. B 861 (2012) 387 [arXiv:1110.2176] [INSPIRE].

    ADS  Article  Google Scholar 

  17. [17]

    M. Alfimov and G. Tarnopolsky, Parafermionic Liouville field theory and instantons on ALE spaces, JHEP 02 (2012) 036 [arXiv:1110.5628] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  18. [18]

    A. Belavin, M. Bershtein, B. Feigin, A. Litvinov and G. Tarnopolsky, Instanton moduli spaces and bases in coset conformal field theory, Comm. Math. Phys. 319 1, pp 269301 (2013) 269 [arXiv:1111.2803] [INSPIRE].

  19. [19]

    D. Krefl and S.-Y.D. Shih, Holomorphic anomaly in gauge theory on ALE space, arXiv:1112.2718 [INSPIRE].

  20. [20]

    V. Belavin and N. Wyllard, N = 2 superconformal blocks and instanton partition functions, JHEP 06 (2012) 173 [arXiv:1205.3091] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  21. [21]

    A. Belavin and B. Mukhametzhanov, N = 1 superconformal blocks with Ramond fields from AGT correspondence, JHEP 01 (2013) 178 [arXiv:1210.7454] [INSPIRE].

    ADS  Article  Google Scholar 

  22. [22]

    A. Belavin, M. Bershtein and G. Tarnopolsky, Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity, JHEP 03 (2013) 019 [arXiv:1211.2788] [INSPIRE].

    ADS  Article  Google Scholar 

  23. [23]

    F. Fucito, J.F. Morales and R. Poghossian, Instanton on toric singularities and black hole countings, JHEP 12 (2006) 073 [hep-th/0610154] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  24. [24]

    H. Nakajima and K. Yoshioka, Lectures on instanton counting, math/0311058 [INSPIRE].

  25. [25]

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

  26. [26]

    L. Gottsche, H. Nakajima and K. Yoshioka, Instanton counting and Donaldson invariants, math/0606180 [INSPIRE].

  27. [27]

    P. Kronheimer, Monopoles and Taub-NUT metrics, M.Sc. Thesis, Oxford University, Oxford U.K. (1986) [http://www.math.harvard.edu/~kronheim/papers.html].

  28. [28]

    J. Gomis, T. Okuda and V. Pestun, Exact results fort Hooft loops in gauge theories on S 4, JHEP 05 (2012) 141 [arXiv:1105.2568] [INSPIRE].

    ADS  Article  Google Scholar 

  29. [29]

    Y. Ito, T. Okuda and M. Taki, Line operators on S 1× R 3 and quantization of the Hitchin moduli space, JHEP 04 (2012) 010 [arXiv:1111.4221] [INSPIRE].

    MathSciNet  ADS  Article  Google Scholar 

  30. [30]

    D. Gang, E. Koh and K. Lee, Line operator index on S 1× S 3, JHEP 05 (2012) 007 [arXiv:1201.5539] [INSPIRE].

    ADS  Article  Google Scholar 

  31. [31]

    Y. Ito and T. Okuda, work in progress.

  32. [32]

    R. Flume and R. Poghossian, An algorithm for the microscopic evaluation of the coefficients of the Seiberg-Witten prepotential, Int. J. Mod. Phys. A 18 (2003) 2541 [hep-th/0208176] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  33. [33]

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

    MathSciNet  ADS  Article  Google Scholar 

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Correspondence to Takuya Okuda.

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ArXiv ePrint: 1303.5765

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Ito, Y., Maruyoshi, K. & Okuda, T. Scheme dependence of instanton counting in ALE spaces. J. High Energ. Phys. 2013, 45 (2013). https://doi.org/10.1007/JHEP05(2013)045

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Keywords

  • Supersymmetric gauge theory
  • Solitons Monopoles and Instantons
  • Topological Field Theories