Boundary Liouville Theory: Hamiltonian Description and Quantization
Harald Dorn, George Jorjadze
TL;DR
The paper develops a Hamiltonian treatment of Liouville theory on a timelike strip in 2d Minkowski space, focusing on classical solutions regular in the bulk under conformally invariant boundary conditions and their monodromies. It shows that these solutions fall onto Virasoro coadjoint orbits and classifies them by boundary parameters into elliptic, parabolic, and hyperbolic monodromies, corresponding to bound and scattering states; a canonical two-form and Poisson structure are derived, enabling a free-field parametrization for quantization. Canonical quantization is carried out via a free-field representation with $\hbar=2\pi b^2$ and $\eta=1+b^2$, yielding a quantum vertex operator in the hyperbolic sector and a deformed Virasoro algebra with central charge $c=1+{12\pi\eta^2}/{\hbar}$, while semi-classical Bohr–Sommerfeld quantization reproduces the discrete bound-state spectrum and matches the Euclidean conformal bootstrap results in the appropriate limit. The work lays the groundwork for computing reflection amplitudes and general correlation functions, and for a full understanding of the ZZ/FZZT brane spectrum within this Minkowski-space, canonical framework.
Abstract
The paper is devoted to the Hamiltonian treatment of classical and quantum properties of Liouville field theory on a timelike strip in 2d Minkowski space. We give a complete description of classical solutions regular in the interior of the strip and obeying constant conformally invariant conditions on both boundaries. Depending on the values of the two boundary parameters these solutions may have different monodromy properties and are related to bound or scattering states. By Bohr-Sommerfeld quantization we find the quasiclassical discrete energy spectrum for the bound states in agreement with the corresponding limit of spectral data obtained previously by conformal bootstrap methods in Euclidean space. The full quantum version of the special vertex operator $e^{-φ}$ in terms of free field exponentials is constructed in the hyperbolic sector.