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Instantons on ALE spaces and super Liouville conformal field theories

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Abstract

We provide evidence that the conformal blocks of \( \mathcal{N} = 1 \) super Liouville conformal field theory are described in terms of the SU(2) Nekrasov partition function on the ALE space \( {\mathcal{O}_{{\mathbb{P}^1}}}\left( { - 2} \right) \).

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References

  1. L.F. Alday, F. Benini and Y. Tachikawa, Liouville/Toda central charges from M5-branes, Phys. Rev. Lett. 105 (2010) 141601 [arXiv:0909.4776] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  2. 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] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  3. 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] [SPIRES].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. L.F. Alday and Y. Tachikawa, Affine SL(2) conformal blocks from 4 d gauge theories, Lett. Math. Phys. 94 (2010) 87 [arXiv:1005.4469] [SPIRES].

    Article  MATH  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  Google Scholar 

  6. G. Bonelli, K. Maruyoshi and A. Tanzini, Quantum Hitchin Systems via beta-deformed Matrix Models, arXiv:1104.4016 [SPIRES].

  7. G. Bonelli, K. Maruyoshi and A. Tanzini, Gauge Theories on ALE Space and Super Liouville Correlation Functions, work in progress.

  8. G. Bonelli and A. Tanzini, Hitchin systems, N = 2 gauge theories and W-gravity, Phys. Lett. B 691 (2010) 111 [arXiv:0909.4031] [SPIRES].

    ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  10. 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] [SPIRES].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. R. Dijkgraaf, L. Hollands, P. Sulkowski and C. Vafa, Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions, JHEP 02 (2008) 106 [arXiv:0709.4446] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  12. R. Dijkgraaf and P. Sulkowski, Instantons on ALE spaces and orbifold partitions, JHEP 03 (2008) 013 [arXiv:0712.1427] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  13. R. Dijkgraaf and C. Vafa, Toda Theories, Matrix Models, Topological Strings and N = 2 Gauge Systems, arXiv:0909.2453 [SPIRES].

  14. 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] [SPIRES].

    ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  16. D. Gaiotto, Asymptotically free N = 2 theories and irregular conformal blocks, arXiv:0908.0307 [SPIRES].

  17. T. Kimura, Matrix model from N = 2 orbifold partition function, arXiv:1105.6091 [SPIRES].

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

    Article  MATH  MathSciNet  Google Scholar 

  19. 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] [SPIRES].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  20. K. Maruyoshi and M. Taki, Deformed Prepotential, Quantum Integrable System and Liouville Field Theory, Nucl. Phys. B 841 (2010) 388 [arXiv:1006.4505] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  21. H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994) 365.

    Article  MATH  MathSciNet  Google Scholar 

  22. H. Nakajima, Sheaves on ALE spaces and quiver varieties, Moscow Math. J. 7 (2007) 699.

    MATH  MathSciNet  Google Scholar 

  23. H. Nakajima and K. Yoshioka, Lectures on instanton counting, in CRM Proc. Lecture Notes. Vol. 38: Algebraic structures and moduli spaces, Amer. Math. Soc., Providence U.S.A. (2004), pg. 31.

    Google Scholar 

  24. H. Nakajima and K. Yoshioka, Instanton counting on blowup. I: 4-dimensional pure gauge theory, Invent. Math. 162 (2005) 313 [math/0306198].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  25. N.A. Nekrasov, Seiberg-Witten Prepotential From Instanton Counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [SPIRES].

    MathSciNet  Google Scholar 

  26. T. Nishioka and Y. Tachikawa, Para-Liouville/Toda central charges from M5-branes, arXiv:1106.1172 [SPIRES].

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

  28. R.H. Poghosian, Structure constants in the N = 1 super-Liouville field theory, Nucl. Phys. B 496 (1997) 451 [hep-th/9607120] [SPIRES].

    Article  ADS  Google Scholar 

  29. R.C. Rashkov and M. Stanishkov, Three-point correlation functions in N = 1 Super Lioville Theory, Phys. Lett. B 380 (1996) 49 [hep-th/9602148] [SPIRES].

    ADS  MathSciNet  Google Scholar 

  30. T. Sasaki, \( \mathcal{O}\left( { - 2} \right) \) blow-up formula via instanton calculus on C**2/Z(2)-hat and Weil conjecture, hep-th/0603162 [SPIRES].

  31. J. Teschner, Quantization of the Hitchin moduli spaces, Liouville theory and the geometric Langlands correspondence I, arXiv:1005.2846 [SPIRES].

  32. C. Vafa and E. Witten, A Strong coupling test of S duality, Nucl. Phys. B 431 (1994) 3 [hep-th/9408074] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Scuola Internazionale Superiore di Studi Avanzati and Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, Via Bonomea 256, I-34136, Trieste, Italia

    Giulio Bonelli, Kazunobu Maruyoshi & Alessandro Tanzini

Authors
  1. Giulio Bonelli
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  2. Kazunobu Maruyoshi
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  3. Alessandro Tanzini
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Corresponding author

Correspondence to Kazunobu Maruyoshi.

Additional information

This research was partly supported by the INFN Research Project PI14 “Nonperturbative dynamics of gauge theory”, by PRIN “Geometria delle varietà algebriche” and by the INFN Research Project TV12.

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Bonelli, G., Maruyoshi, K. & Tanzini, A. Instantons on ALE spaces and super Liouville conformal field theories. J. High Energ. Phys. 2011, 56 (2011). https://doi.org/10.1007/JHEP08(2011)056

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