  The correspondence between the semiclassical limit of the DOZZ quantum
Liouville theory and the Nekrasov-Shatashvili limit of the N = 2
(Omega-deformed) U(2) super-Yang-Mills theories is used to calculate the
unknown accessory parameter of the Fuchsian uniformization of the 4-punctured
sphere. The computation is based on the saddle point method. This allows to
find an analytic expression for the N_f = 4, U(2) instanton twisted
superpotential and, in turn, to sum up the 4-point classical block. It is well
known that the critical value of the Liouville action functional is the
generating function of the accessory parameters. This statement and the
factorization property of the 4-point action allow to express the unknown
accessory parameter as the derivative of the 4-point classical block with
respect to the modular parameter of the 4-punctured sphere. It has been found
that this accessory parameter is related to the sum of all rescaled column
lengths of the so-called 'critical' Young diagram extremizing the instanton
'free energy'. It is shown that the sum over the 'critical' column lengths can
be rewritten in terms of a contour integral in which the integrand is built out
of certain special functions closely related to the ordinary Gamma function.
