  In this work the correspondence between the semiclassical limit of the DOZZ
quantum Liouville theory on the torus and the Nekrasov-Shatashvili limit of the
N=2* (Omega-deformed) U(2) super-Yang-Mills theory is used to propose new
formulae for the accessory parameter of the Lame equation. This quantity is in
particular crucial for solving the problem of uniformization of the
one-punctured torus. The computation of the accessory parameters for torus and
sphere is an open longstanding problem which can however be solved if one
succeeds to derive an expression for the so-called classical Liouville action.
The method of calculation of the latter has been proposed some time ago by
Zamolodchikov brothers. Studying the semiclassical limit of the four-point
function of the quantum Liouville theory on the sphere they have derived the
classical action for the Riemann sphere with four punctures. In the present
work Zamolodchikovs idea is exploited in the case of the Liouville field theory
on the torus. It is found that the Lame accessory parameter is determined by
the classical Liouville action on the one-punctured torus or more concretely by
the torus classical block evaluated on the saddle point intermediate classical
weight. Secondly, as an implication of the aforementioned correspondence it is
obtained that the torus accessory parameter is related to the sum of all
rescaled column lengths of the so-called "critical" Young diagrams extremizing
the instanton "free energy" for the N=2* gauge theory. Finally, it is pointed
out that thanks to the known relation the sum over the "critical" column
lengths can be expressed in terms of a contour integral in which the integrand
is built out of certain special functions.
