  Combining the semiclassical/Nekrasov-Shatashvili limit of the AGT conjecture
and the Bethe/gauge correspondence results in a triple correspondence which
identifies classical conformal blocks with twisted superpotentials and then
with Yang-Yang functions. In this paper the triple correspondence is studied in
the simplest, yet not completely understood case of pure SU(2) super-Yang-Mills
gauge theory. A missing element of that correspondence is identified with the
classical irregular block. Explicit tests provide a convincing evidence that
such a function exists. In particular, it has been shown that the classical
irregular block can be recovered from classical blocks on the torus and sphere
in suitably defined decoupling limits of classical external conformal weights.
These limits are "classical analogues" of known decoupling limits for
corresponding quantum blocks. An exact correspondence between the classical
irregular block and the SU(2) gauge theory twisted superpotential has been
obtained as a result of another consistency check. The latter determines the
spectrum of the 2-particle periodic Toda (sin-Gordon) Hamiltonian in accord
with the Bethe/gauge correspondence. An analogue of this statement is found
entirely within 2d CFT. Namely, considering the classical limit of the null
vector decoupling equation for the degenerate irregular block a celebrated
Mathieu's equation is obtained with an eigenvalue determined by the classical
irregular block. As it has been checked this result reproduces a well known
weak coupling expansion of Mathieu's eigenvalue. Finally, yet another new
formulae for Mathieu's eigenvalue relating the latter to a solution of certain
Bethe-like equation are found.
