Correlation functions with fusion-channel multiplicity in \( {\mathcal{W}}_3 \) Toda field theory

  • Vladimir Belavin
  • Benoit Estienne
  • Omar Foda
  • Raoul Santachiara
Open Access
Regular Article - Theoretical Physics


Current studies of \( {\mathcal{W}}_N \) Toda field theory focus on correlation functions such that the \( {\mathcal{W}}_N \) highest-weight representations in the fusion channels are multiplicity-free. In this work, we study \( {\mathcal{W}}_3 \) Toda 4-point functions with multiplicity in the fusion channel. The conformal blocks of these 4-point functions involve matrix elements of a fully-degenerate primary field with a highest-weight in the adjoint representation of \( \mathfrak{s}{\mathfrak{l}}_3 \), and a fully-degenerate primary field with a highest-weight in the fundamental representation of \( \mathfrak{s}{\mathfrak{l}}_3 \). We show that, when the fusion rules do not involve multiplicities, the matrix elements of the fully-degenerate adjoint field, between two arbitrary descendant states, can be computed explicitly, on equal footing with the matrix elements of the semi-degenerate fundamental field. Using null-state conditions, we obtain a fourth-order Fuchsian differential equation for the conformal blocks. Using Okubo theory, we show that, due to the presence of multiplicities, this differential equation belongs to a class of Fuchsian equations that is different from those that have appeared so far in \( {\mathcal{W}}_N \) theories. We solve this equation, compute its monodromy group, and construct the monodromy-invariant correlation functions. This computation shows in detail how the ambiguities that are caused by the presence of multiplicities are fixed by requiring monodromy-invariance.


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


Open Access

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Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Vladimir Belavin
    • 1
    • 2
  • Benoit Estienne
    • 3
  • Omar Foda
    • 4
  • Raoul Santachiara
    • 5
  1. 1.I.E. Tamm Department of Theoretical PhysicsP.N. Lebedev Physical InstituteMoscowRussia
  2. 2.Department of Quantum PhysicsInstitute for Information Transmission ProblemsMoscowRussia
  3. 3.LPTHE, CNRS and Université Pierre et Marie CurieSorbonne UniversitésParis Cedex 05France
  4. 4.School of Mathematics and StatisticsUniversity of MelbourneParkvilleAustralia
  5. 5.LPTMS, CNRS (UMR 8626), Université Paris-SaclayOrsayFrance

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