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The correspondence between the semiclassical limit of the DOZZ quantumLiouville theory and the Nekrasov-Shatashvili limit of the N = 2(Omega-deformed) U(2) super-Yang-Mills theories is used to calculate theunknown accessory parameter of the Fuchsian uniformization of the 4-puncturedsphere. The computation is based on the saddle point method. This allows tofind an analytic expression for the N_f = 4, U(2) instanton twistedsuperpotential and, in turn, to sum up the 4-point classical block. It is wellknown that the critical value of the Liouville action functional is thegenerating function of the accessory parameters. This statement and thefactorization property of the 4-point action allow to express the unknownaccessory parameter as the derivative of the 4-point classical block withrespect to the modular parameter of the 4-punctured sphere. It has been foundthat this accessory parameter is related to the sum of all rescaled columnlengths of the so-called 'critical' Young diagram extremizing the instanton'free energy'. It is shown that the sum over the 'critical' column lengths canbe rewritten in terms of a contour integral in which the integrand is built outof certain special functions closely related to the ordinary Gamma function.

Liouville theory, N=2 gauge theories and accessory parameters