On effective theories of topological strings

We study the construction of effective target-space theories of topological string theories. The example of the CP1 topological sigma model is analysed in detail. An effective target-space theory whose correlation functions are defined by the sum over connected Riemann surfaces of all genera is found to be itself topological. The values of the couplings of this effective theory are expressed in terms of those of the world-sheet theory for general CP1-like world-sheet model. Any model of this type can be obtained as an effective theory. The definition of the effective theory's expectation values as a sum over disconnected surfaces as well, is shown not to be compatible with those of a topological thoery, at least as long as the connectivity of the target space is kept fixed. Dilaton-type couplings emerge in the full lagrangian realization of the moduli space of topological theories with n observables.En route, we encounter a nonperturbative duality, an equivalence of theories with different world-sheets and discuss the relation between the cosmological constant in these finite theories and the zero-point function.