Off-shell strings I: S-matrix and action

We explain why Tseytlin’s off-shell formulation of string theory is well-defined. Although quantizing strings on an off-shell background requires an arbitrary choice of Weyl frame, this choice is not physically significant since it can be absorbed into a field redefinition of the target space fields. The off-shell formalism is particularly subtle at tree-level, due to the treatment of the noncompact conformal Killing group SL(2, C ℂ ) of the sphere. We prove that Tseytlin’s sphere prescriptions recover the standard tree-level Lorentzian S-matrix, and show how to extract the stringy iε iε prescription from the UV cutoff on the worldsheet. We also demonstrate that the correct tree-level equations of motion are obtained to all orders in perturbation theory in g_s gs and α^{} α′ , and illuminate the close connection between the string action and the c-theorem.