A-coordinate on cluster integrable systems are special values of a tau-function

he statement that a-coordinates on cluster integrable system has something to do with tau-functions since they are given by infinite dimensional determinants was around for several years. We will try to give to it a precise meaning. Namely we will give a “bosonic” definition of a tau-function as a limit of generating function of flat Abelian connections on a Riemann surface and defined on generalized divisors. Then we show that the value of this function on ordinary divisors supported at infinity are just cluster variables in the corresponding integrable system.