  The critical values of the Yang-Yang functional corresponding to the vacuum
states of the sine-Gordon QFT in the finite-volume are studied. Two major
applications are discussed: (i) generalization of Fendley-Saleur-Zamolodchikov
relations to arbitrary values of the sine-Gordon coupling constant, and (ii)
connection problem for a certain two-parameter family of solutions of the
Painleve III equation.
