On Combinatorial Expansion of the Conformal Blocks Arising from AGT Conjecture

  • Vasyl A. Alba
  • Vladimir A. Fateev
  • Alexey V. Litvinov
  • Grigory M. Tarnopolskiy


In their recent paper, Alday et al. (Lett Math Phys 91:167–197, 2010) proposed a relation between \({\mathcal{N}=2}\) four-dimensional supersymmetric gauge theories and two-dimensional conformal field theories. As part of their conjecture they gave an explicit combinatorial formula for the expansion of the conformal blocks inspired by the exact form of the instanton part of the Nekrasov partition function. In this paper we study the origin of such an expansion from a CFT point of view. We consider the algebra \({\mathcal{A}={\sf Vir} \otimes\mathcal{H}}\) which is the tensor product of mutually commuting Virasoro and Heisenberg algebras and discover the special orthogonal basis of states in the highest weight representations of \({\mathcal{A}}\). The matrix elements of primary fields in this basis have a very simple factorized form and coincide with the function called \({Z_{{\sf bif}}}\) appearing in the instanton counting literature. Having such a simple basis, the problem of computation of the conformal blocks simplifies drastically and can be shown to lead to the expansion proposed in Alday et al. (2010). We found that this basis diagonalizes an infinite system of commuting Integrals of Motion related to Benjamin–Ono integrable hierarchy.

Mathematics Subject Classification (2010)

81T40 81T60 


conformal field theory gauge theory 


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Copyright information

© Springer 2011

Authors and Affiliations

  • Vasyl A. Alba
    • 1
    • 2
    • 3
    • 4
  • Vladimir A. Fateev
    • 1
    • 5
  • Alexey V. Litvinov
    • 1
  • Grigory M. Tarnopolskiy
    • 1
    • 3
  1. 1.Landau Institute for Theoretical PhysicsChernogolovkaRussia
  2. 2.Bogolyubov Institute for Theoretical Physics NASUKievUkraine
  3. 3.Department of General and Applied PhysicsMoscow Institute of Physics and TechnologyDolgoprudnyRussia
  4. 4.Institute for Theoretical and Experimental PhysicsMoscowRussia
  5. 5.Laboratoire de Physique Théorique et AstroparticulesUniversité Montpellier II, UMR5207 CNRS-UM2MontpellierFrance

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