Bases in coset conformal field theory from AGT correspondence and Macdonald polynomials at the roots of unity

  • A. A. Belavin
  • M. A. Bershtein
  • G. M. Tarnopolsky


We continue our study of the AGT correspondence between instanton counting on \( {{{{{\mathbb{C}}^2}}} \left/ {{{{\mathbb{Z}}_p}}} \right.} \) and Conformal field theories with the symmetry algebra \( \mathcal{A}\left( {r,p} \right) \). In the cases r = 1, p = 2 and r = 2, p = 2 this algebra specialized to: \( \mathcal{A}\left( {1,2} \right)=\mathcal{H}\oplus \widehat{\mathfrak{sl}}{(2)_1} \) and \( \mathcal{A}\left( {2,2} \right)=\mathcal{H}\oplus \widehat{\mathfrak{sl}}{(2)_2}\oplus \mathrm{NSR} \).

As the main tool we use a new construction of the algebra A(r, 2) as the limit of the toroidal \( \mathfrak{g}\mathfrak{l}(1) \) algebra for q, t tend to −1. We claim that the basis of the representation of the algebra \( \mathcal{A}\left( {r,2} \right) \) (or equivalently, of the space of the local fields of the corresponding CFT) can be expressed through Macdonald polynomials with the parameters q, t go to −1. The vertex operator which naturally arises in this construction has factorized matrix elements in this basis. We also argue that the singular vectors of the \( \mathcal{N}=1 \) Super Virasoro algebra can be realized in terms of Macdonald polynomials for a rectangular Young diagram and parameters q, t tend to −1.


Conformal and W Symmetry Quantum Groups Supersymmetric gauge theory 


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Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  • A. A. Belavin
    • 1
    • 2
    • 3
  • M. A. Bershtein
    • 1
    • 2
    • 3
    • 4
  • G. M. Tarnopolsky
    • 1
    • 3
    • 5
  1. 1.Landau Institute for Theoretical Physics, RASChernogolovkaRussia
  2. 2.Institute for Information Transmission Problems, RASMoscowRussia
  3. 3.Moscow Institute of Physics and TechnologyDolgoprudnyRussia
  4. 4.Independent University of MoscowMoscowRussia
  5. 5.Department of PhysicsPrinceton UniversityPrincetonU.S.A.

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