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We consider various trace formulas for the cubic Schrodinger equa- tion 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 ex- plicitly these symplectic structures in QP coordinates. We compute the ratio of two symplectic volume elements in the infinite genus limit.

Symplectic structures and volume elements in the function space for the cubic Schrödinger equation