Semiclassical Liouville Theory on the Real Projective Plane: A Complex Interpretation of the Bootstrap

Abstract

The exact one-point function of Liouville theory on the real projective plane was derived long ago from the bootstrap, yet it has never been verified against a semiclassical path-integral computation. The check is less straightforward than one might expect. On the real projective plane every constant-curvature metric is positively curved, whereas classical Liouville theory produces metrics of constant negative curvature, so no real classical solution exists. Instead, as we show, the semiclassical path integral receives contributions from infinitely many complex saddles with a negative-definite metric. Once the integration contour is chosen so that the path integral converges, these saddles reproduce the exact one-point function. Complex saddles arise in the same way in timelike Liouville theory and in two-dimensional de~Sitter gravity, and the example treated here offers a rare opportunity to test their use against a known exact answer. \end{abstract