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.

References

It has hence been conjectured that the physical spectrum gets shifted \hat p\in i \mathbb{R} - \epsilon.

— Notes on de Sitter space  (2609.19454 - Mühlmann, 16 Sep 2026) in Section 10, subsection 'Liouville theory'

Understanding the timelike analog (cf. for preliminary semiclassical analysis), and constructing a bona-fide string by coupling to 2d worldsheet supergravity remains open.

— Super Liouville theory on the torus: bootstrap and the super Virasoro minimal string  (2610.01909 - Collier et al., 1 Oct 2026) in Section 8, final paragraph titled “More supersymmetry”