Physical spectrum of timelike Liouville theory

Determine the physical momentum spectrum of timelike Liouville theory and establish whether it is shifted from the imaginary axis by a prescription of the form \(\hat p\in i\mathbb{R}-\epsilon\).

Background

Timelike Liouville theory has poles in its structure constants on the same imaginary momentum axis that would ordinarily contain the physical spectrum. This creates an ambiguity in defining a consistent physical contour or spectrum.

The paper reports a conjecture that the spectrum is displaced slightly off the imaginary axis, but does not prove this proposal or derive it from a complete nonperturbative formulation.