On the Polyakov theory of open string off-shell green functions

A Polyakov theory of oriented open-bosonic-string off-shell Green functions is illustrated. It is shown that the relevant world sheets are manifolds with corners. The structure of the gauge group in relation to the corners is investigated. In particular, it is shown that the mapping-class group factorizes into the semidirect product of the subgroup of all mapping-classes which leave the corners fixed with a finite group whose definition and properties are explicitly given. The gauge volume of the latter is divided out, leading to a simplified starting expression. Further, it is shown that the final expression is an integral over an extended moduli space, defined as the quotient of the space of all admissible metrics by the semidirect product of the Weyl group with the subgroup of all diffeomorphisms which leave the corners fixed.