Conformally Invariant Off - shell Strings ∗

Recent advances in non-critical string theory allow a unique continuation of critical Polyakov string amplitudes to off-shell momenta, while preserving conformal invari-ance. These continuations possess unusual, apparently stringy, characteristics, as we illustrate with our results for three-point functions. 1. The Polyakov functional integral The Polyakov path integral 1 appears only a recipe for perturbative computations of scattering amplitudes, yet it serves as the foundation for our entire understanding of string theory. It is natural to ask whether this framework might yield any insight into the off-shell properties of strings. We report on how the recent progress in understanding non-critical strings provides a new technique to calculate off-shell Polyakov amplitudes 2. This approach preserves conformal invariance, and so these amplitudes lend themselves to analysis by essentially the standard techniques of conformal field theory. Space-time scattering amplitudes of string excitations are calculated as correlation functions of vertex operators in a functional integral over the metric on the string world-sheet and the space-time string configurations: