A Symplectic Structure for String Theory on Integrable Backgrounds

We define regularised Poisson brackets for the monodromy matrix of classical string theory on R x S^3. The ambiguities associated with Non-Ultra Locality are resolved using the symmetrisation prescription of Maillet. The resulting brackets lead to an infinite tower of Poisson-commuting conserved charges as expected in an integrable system. The brackets are also used to obtain the correct symplectic structure on the moduli space of finite-gap solutions and to define the corresponding action-angle variables. The canonically-normalised action variables are the filling fractions associated with each cut in the finite-gap construction. Our results are relevant for the leading-order semiclassical quantisation of string theory on AdS_5 x S^5 and lead to integer-valued filling fractions in this context.