Symplectic Structures and Volume Elements in the Function Space for the Cubic Schrodinger Equation

We consider various trace formulas for the cubic Schrodinger equation in the space of infinitely smooth functions subject to periodic boundary conditions. The formulas relate conventional integrals of motion to the periods of some Abelian differentials (holomorphic one-forms) on the spectral curve. We show that the periods of Abelian differentials are global coordinates on the moduli space of spectral curves. The exterior derivatives of the holomorphic one-forms are the basic and higher symplectic structures on the phase space. We write explicitly these symplectic structures in $QP$ coordinates. We compute the ratio of two symplectic volume elements in the infinite genus limit.