Franco Ferrari, Marcin Piatek
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. According
to the so-called Polyakov conjecture, the unknown accessory parameter is given
by the derivative of the 4-point classical block with respect to the moduli 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.
View original:
http://arxiv.org/abs/1202.2149
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