Off-Shell Strings I: S-matrix and Action

Abstract

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,$\mathbb{C}$) 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 $i\epsilon$ 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 $g_s$ and $\hat{I}{\pm }^{\prime }$, and illuminate the close connection between the string action and the c-theorem.

39 pages, 9 figures. v3: edited for clarity, corrected typos, made major changes to sections III.B and III.C, and added more details to sections VII.A and VII.D