Super Liouville conformal blocks from \( \mathcal{N} = 2 \) SU(2) quiver gauge theories

Article

Abstract

The conjecture about the correspondence between instanton partition functions in the N = 2 SUSY Yang-Mills theory and conformal blocks of two-dimensional conformal field theories is extended to the case of the N = 1 supersymmetric conformal blocks. We find that the necessary modification of the moduli space of instantons requires additional restriction of Z (2)-symmetry. This leads to an explicit form of the N =1 superconformal blocks in terms of Young diagrams with two sorts of cells.

Keywords

Supersymmetric gauge theory Conformal and W Symmetry Conformal Field Models in String Theory 

References

  1. [1]
    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].MathSciNetADSMATHCrossRefGoogle Scholar
  2. [2]
    A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    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].MathSciNetADSMATHCrossRefGoogle Scholar
  4. [4]
    N. Wyllard, A N−1 conformal Toda field theory correlation functions from conformal N =2 SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    A. Mironov and A. Morozov, On AGT relation in the case of U(3), Nucl. Phys. B 825 (2010) 1 [arXiv:0908.2569] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    M. Taki, On AGT conjecture for pure super Yang-Mills and W -algebra, JHEP 05 (2011) 038 [arXiv:0912.4789] [SPIRES].ADSCrossRefMathSciNetGoogle Scholar
  7. [7]
    G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991) 365.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    H. Nakajima, Instantons on ALE spaces, quiver varietie, and Kac-Moody algebras, Duke Math. 76 (1994) 365. MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    H. Nakajima and K. Yoshioka, Instanton counting on blowup. I, math/0306198.
  10. [10]
    B. Feigin, A. Tsymbaliuk, Heisenberg action in the equivariant K-theory of Hilbert schemes via scuffe algebra, arXiv:0904.1679.
  11. [11]
    K. Nagao, K-theory of quiver varieties, q-Fock space and nonsymmetric Macdonald polynomials, arXiv:0709.1767.
  12. [12]
    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].MathSciNetADSGoogle Scholar
  13. [13]
    J.-M. Bismut, Localization formulas, superconnections and the index theorem of families, Commun. Math. Phys. 103 (1986) 127. MathSciNetADSMATHCrossRefGoogle Scholar
  14. [14]
    N. Berline, E. Getzler and M. Vergne, Heat kernels and Dirac operators, Springer, Berlin Germany (1996).Google Scholar
  15. [15]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [SPIRES].MathSciNetGoogle Scholar
  16. [16]
    D. Gaiotto, Asymptotically free N =2 theories and irregular conformal blocks, arXiv:0908.0307 [SPIRES].
  17. [17]
    A. Marshakov, A. Mironov and A. Morozov, On non-conformal limit of the AGT relations, Phys. Lett. B 682 (2009) 125 [arXiv:0909.2052] [SPIRES].MathSciNetADSGoogle Scholar
  18. [18]
    B. Kostant, On Whittaker vectors and representation theory, Invent. Math. 48 (1978) 101. MathSciNetADSMATHCrossRefGoogle Scholar
  19. [19]
    A.M. Polyakov, Quantum geometry of fermionic strings, Phys. Lett. B 103 (1981) 211 [SPIRES].MathSciNetADSGoogle Scholar
  20. [20]
    A. Belavin, V. Belavin, A. Neveu and A. Zamolodchikov, Bootstrap in supersymmetric Liouville field theory. I: NS sector, Nucl. Phys. B 784 (2007) 202 [hep-th/0703084] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    L. Hadasz, Z. Jaskólski and P. Suchanek, Recursion representation of the Neveu-Schwarz superconformal block, JHEP 03 (2007) 032 [hep-th/0611266] [SPIRES].ADSCrossRefGoogle Scholar
  22. [22]
    V.A. Belavin, N =1 SUSY conformal block recursive relations, Theor. Math. Phys. 152 (2007) 1275 [hep-th/0611295] [SPIRES].MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    V.A. Belavin, On the N =1 super Liouville four-point functions, Nucl. Phys. B 798 (2008) 423 [arXiv:0705.1983] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  24. [24]
    L. Hadasz, Z. Jaskolski and P. Suchanek, Elliptic recurrence representation of the N =1 Neveu-Schwarz blocks, Nucl. Phys. B 798 (2008) 363 [arXiv:0711.1619] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  25. [25]
    M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld and Y.I. Manin, Construction of instantons, Phys. Lett. A 65 (1978) 185 [SPIRES].MathSciNetADSGoogle Scholar
  26. [26]
    A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Y.S. Tyupkin, Pseudoparticle solutions of the Yang-Mills equations, Phys. Lett. B 59 (1975) 85 [SPIRES].MathSciNetADSGoogle Scholar
  27. [27]
    A.A. Belavin and V.E. Zakharov, Yang-Mills equations as inverse scattering problem, Phys. Lett. B 73 (1978) 53 [SPIRES].MathSciNetADSGoogle Scholar
  28. [28]
    M.F. Atiyah and R.S. Ward, Instantons and algebraic geometry, Commun. Math. Phys. 55 (1977) 117 [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  29. [29]
    V. Drinfeld and Yu. Manin, A description of instantons, Comm. Math. Phys. 6 (1978) 177. MathSciNetADSCrossRefGoogle Scholar
  30. [30]
    N. Dorey, T.J. Hollowood, V.V. Khoze and M.P. Mattis, The calculus of many instantons, Phys. Rept. 371 (2002) 231 [hep-th/0206063] [SPIRES].MathSciNetADSMATHCrossRefGoogle Scholar
  31. [31]
    A. Belavin and A. Zamolodchikov, Higher equations of motion in N =1 SUSY Liouville field theory, JETP Lett. 84 (2006) 418 [hep-th/0610316] [SPIRES].ADSCrossRefGoogle Scholar
  32. [32]
    V.A. Alba, V.A. Fateev, A.V. Litvinov and G.M. Tarnopolsky, On combinatorial expansion of the conformal blocks arising from AGT conjecture, arXiv:1012.1312 [SPIRES].
  33. [33]
    A. Belavin and V. Belavin, AGT conjecture and Integrable structure of conformal field theory for c = 1, Nucl. Phys. B 850 (2011) 199 [arXiv:1102.0343] [SPIRES].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2011

Authors and Affiliations

  1. 1.Theoretical Physics DepartementLebedev Physical Institute, RASMoscowRussia
  2. 2.Quantum Field Theory DepartmentLandau Institute for Theoretical Physics, RASChernogolovkaRussia
  3. 3.Department of MathematicsHigher School of EconomicsMoscowRussia

Personalised recommendations