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Rigid Fuchsian Systems in 2-Dimensional Conformal Field Theories

  • Vladimir Belavin
  • Yoshishige Haraoka
  • Raoul Santachiara
Article
  • 19 Downloads

Abstract

We investigate Fuchsian equations arising in the context of 2-dimensional conformal field theory (CFT) and we apply the Katz theory of Fucshian rigid systems to solve some of these equations. We show that the Katz theory provides a precise mathematical framework to answer the question whether the fusion rules of degenerate primary fields are enough for determining the differential equations satisfied by their correlation functions. We focus on the case of \({\mathcal{W}_{3}}\) Toda CFT: we argue that the differential equations arising for four-point conformal blocks with one nth level semi-degenerate field and a fully-degenerate one in the fundamental sl3 representation are associated to Fuchsian rigid systems. We show how to apply Katz theory to determine the explicit form of the differential equations, the integral expression of solutions and the monodromy group representation. The theory of twisted homology is also used in the analysis of the integral expression. The computation of the connection coefficients is done for the first time in the case of a Katz system with multiplicities, thus extending the work done by Oshima in the multiplicity free case. This approach allows us to construct the corresponding fusion matrices and to perform the whole bootstrap program: new explicit factorization of \({\mathcal{W}_{3}}\) correlation functions as well as shift relations between structure constants for general Toda theories are also provided.

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Notes

Acknowledgement

We thank T. Dupic, P. Gavrylenko, N. Iorgov, O. Lisovyy, Y. Matsuo, and S. Ribault for discussions and X. Cao, B. Estienne, and O. Foda for preliminary contributions and previous results on which the study of this manuscript is based. The second author is supported by the JSPS grants-in-aid for scientific research B, No. 15H03628.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Particle Physics and AstrophysicsWeizmann Institute of ScienceRehovotIsrael
  2. 2.Department of MathematicsKumamoto UniversityKumamotoJapan
  3. 3.LPTMS, CNRS (UMR 8626), Université Paris-SaclayOrsayFrance

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