Coset conformal field theory and instanton counting on ℂ2/ℤ p

  • M. N. Alfimov
  • A. A. Belavin
  • G. M. Tarnopolsky
Open Access


We study conformal field theory with the symmetry algebra \( \mathcal{A}\left( {2,\ p} \right)={{{\widehat{\mathfrak{gl}}{(n)_2}}} \left/ {{\widehat{\mathfrak{gl}}{{{\left( {n-p} \right)}}_2}}} \right.} \). In order to support the conjecture that this algebra acts on the moduli space of instantons on ℂ2/ℤ p , we calculate the characters of its representations and check their coincidence with the generating functions of the fixed points of the moduli space of instantons.

We show that the algebra \( \mathcal{A}\left( {2,\ p} \right) \) can be realized in two ways. The first realization is connected with the cross-product of p Virasoro and p Heisenberg algebras: \( {{\mathcal{H}}^p} \) × Vir p . The second realization is connected with: \( {{\mathcal{H}}^p} \times \widehat{\mathfrak{sl}}{(p)_2}\times \left( {\widehat{\mathfrak{sl}}{(2)_p}\times {{{\widehat{\mathfrak{sl}}{(2)_{n-p }}}} \left/ {{\widehat{\mathfrak{sl}}{(2)_n}}} \right.}} \right) \). The equivalence of these two realizations provides the non-trivial identity for the characters of \( \mathcal{A}\left( {2,\ p} \right) \).

The moduli space of instantons on ℂ2/ℤ p admits two different compactifications. This leads to two different bases for the representations of \( \mathcal{A}\left( {2,\ p} \right) \). We use this fact to explain the existence of two forms of the instanton pure partition functions.


Supersymmetric gauge theory Conformal and W Symmetry 


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Copyright information

© SISSA 2013

Authors and Affiliations

  • M. N. Alfimov
    • 1
    • 2
    • 3
    • 4
  • A. A. Belavin
    • 4
    • 5
    • 6
  • G. M. Tarnopolsky
    • 6
    • 7
  1. 1.LPT, Ecole Normale SuperieureParisFrance
  2. 2.Institut de Physique ThéoriqueOrme des Merisiers, CEA SaclayGif-sur-Yvette CedexFrance
  3. 3.P.N. Lebedev Physical InstituteMoscowRussia
  4. 4.Moscow Institute of Physics and TechnologyDolgoprudnyRussia
  5. 5.Institute for Information Transmission ProblemsMoscowRussia
  6. 6.Landau Institute for Theoretical PhysicsChernogolovkaRussia
  7. 7.Department of PhysicsPrinceton UniversityPrincetonU.S.A.

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