Abstract
It is clear from both the non-perturbative and perturbative approaches to two-dimensional quantum gravity that a new strong coupling regime is setting in at d = 1, independent of the genus of the worldsheet being considered. It has been suggested that a Kosterlitz–Thouless (KT) phase transition in the Liouville theory is the cause of this behavior. However, it has recently been pointed out that the XY-model, which displays a KT transition on the plane and the sphere, is always in the strong coupling, disordered phase on a surface of constant negative curvature. A higher genus worldsheet can be represented as a fundamental region on just such a surface, which might seem to suggest that the KT picture predicts a strong coupling region for arbitrary d, contradicting the known results. We resolve the apparent paradox.
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