Algebro-topological approach to reality problems. Real action variables in the theory of finite-gap solutions of the Sine-Gordon equations

The paper developes an algebro-topological approach to the problem of effective selection of real finite gap solutions of the sine-Gordon equation, based on the so-called $\gamma$-representation associated with a Riemann surface where action variables can be written in a closed form. The approach is a general one and applies to many other systems for which the reality problem has not yet been solved.