Winding vacuum energies in a deformed O(4)O(4) sigma model

We consider the problem of calculating the Casimir energies in the winding sectors of Fateev's SS  -model, which is an integrable two-parameter deformation of the O(4)O(4) non-linear sigma model in two dimensions. This problem lies beyond the scope of all traditional methods of integrable quantum field theory including the thermodynamic Bethe ansatz and non-linear integral equations. Here we propose a solution based on a remarkable correspondence between classical and quantum integrable systems and express the winding energies in terms of certain solutions of the classical sinh-Gordon equation.