A Lorentzian construction of timelike Liouville field theory on the cylinder

Timelike Liouville field theory is a candidate model for positive curvature two-dimensional quantum gravity, but a mathematically controlled Lorentzian formulation has remained elusive. In this paper we construct the theory on the cylinder $\mathbb{R}\times \mathbb{S}^1$ in the integer screening sector for a natural algebra of renormalized exponential observables. Starting from a renormalized finite-volume torus regularization, we construct infinite-volume Euclidean correlation functions, prove analytic continuation in the time variables, and identify the resulting Lorentzian boundary values by explicit contour formulas. This yields exact Lorentzian correlators for a natural class of exponential observables. We then prove locality: spacelike separated vertex operators commute in the Lorentzian theory. For smeared observables generated by the integer-charge fields $e^{2nb\phi}$, these Lorentzian expectation values define a vacuum functional on an ordered $*$-algebra and support an AQFT-type quantization without positivity. More precisely, we obtain isotone local algebras, a complete locally convex space $\mathcal H$ with dense algebraic subspace $\mathcal H_0$ carrying a nondegenerate Hermitian form (shown to be indefinite for $b<8^{-1/2}$), a continuous cyclic representation, operator-topologically closed represented local algebras, an action of cylinder translations by continuous linear homeomorphisms, and locality for the represented local net. The construction does not produce a Hilbert space or a Haag-Kastler net of local von Neumann algebras in the usual sense, but it shows that a substantial part of the Euclidean-to-Lorentzian and algebraic reconstruction mechanism survives in this nonpositive setting for timelike Liouville theory on the cylinder.