The Euler characteristic and the first Chern number in the covariant   phase space formulation of string theory
  Using a covariant description of the geometry of deformations for extendons, it is shown that the topological corrections for the string action associated with the Euler characteristic and the first Chern number of the normal bundle of the worldsheet, although do not give dynamics to the string, modify the symplectic properties of the covariant phase space of the theory. Future extensions of the present results are outlined. 
R. Cartas-Fuentevilla