Who changes the string coupling?

In general bosonic closed string backgrounds the ghost-dilaton is not the only state in the semi-relative BRST cohomology that can change the dimensionless string coupling. This fact is used to establish complete dilaton theorems in closed string field theory. The ghost-dilaton, however, is the crucial state: for backgrounds where it becomes BRST trivial we prove that the string coupling becomes an unobservable parameter of the string action. For backgrounds where the matter CFT includes free uncompactified bosons we introduce a refined BRST problem by including the zero-modes ''x'' of the bosons as legal operators on the complex. We argue that string field theory can be defined on this enlarged complex and that the BRST cohomology captures accurately the notion of a string background. In this complex the ghost-dilaton appears to be the only BRST-physical state changing the string coupling.