Abstract
We study the solution of the Schlesinger system for the 4-point \( \mathfrak{s}{\mathfrak{l}}_N \) isomonodromy problem and conjecture an expression for the isomonodromic τ-function in terms of 2d conformal field theory beyond the known N = 2 Painlevé VI case. We show that this relation can be used as an alternative definition of conformal blocks for the W N algebra and argue that the infinite number of arbitrary constants arising in the algebraic construction of W N conformal block can be expressed in terms of only a finite set of parameters of the monodromy data of rank N Fuchsian system with three regular singular points. We check this definition explicitly for the known conformal blocks of the W 3 algebra and demonstrate its consistency with the conjectured form of the structure constants.
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Gavrylenko, P. Isomonodromic τ-functions and W N conformal blocks. J. High Energ. Phys. 2015, 167 (2015). https://doi.org/10.1007/JHEP09(2015)167
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DOI: https://doi.org/10.1007/JHEP09(2015)167