Topological terms and the global symplectic geometry of the phase space in string theory

Using an imbedding supported background tensor approach for the differential geometry of an imbedded surface in an arbitrary background, we show that the topological terms associated with the inner and outer curvature scalars of the string worldsheet, have a dramatic effect on the global symplectic geometry of the phase space of the theory. By identifying the global symplectic potential of each Lagrangian term in the string action as the argument of the corresponding pure divergence term in a variational principle, we show that those topological terms contribute explicitly to the symplectic potential of any action describing strings, without modifying the string dynamics and the phase space itself. The variation (the exterior derivative on the phase space) of the symplectic potential generates the integral kernel of a covariant and gauge invariant symplectic structure for the theory, changing thus the global symplectic geometry of the phase space. Similar results for non-Abelian gauge theories and General Relativity are briefly discussed