This is in reference to the section 9.4 General Theory of Random Surfaces in Gauge Fields and Strings, Polyakov. In this particular section and the next one (specialized to 2 dimensional surfaces), path integral formulation is being developed to study, from what I understand, a quantum theory of surfaces (or equivalently, I guesss, gravity). After motivating a renormalizable action which is invariant under coordinate transformations or diffeomorphisms. Now we are restricted to only those diffeomorphisms that don't move the boundary ((9.80), although in the section on 2 dimensional surfaces, you allow for diffeomorphisms which move boundary points to boundary points).
I wanted to know if there is a coordinate independent way to formulate the problem, as should be the case for a theory of gravity. The definition of the boundary, as described in the book, makes use of coordinate dependent statements ((9.80), (9.148)). Moreover, what is wrong if I consider a coordinate transformation that completely changes the boundary? For example, between cartesian and polar coordinates, the boundary of a unit disk changes from $x^2+y^2=1$ to $r=1$, which doesn't in particular seem to satisfy (9.148).
As an aside, when can we not understand a coordinate transformation as a diffeomorphism?