Operator Approach to Boundary Liouville Theory
Harald Dorn, George Jorjadze
TL;DR
This work develops an operator-based formalism for Boundary Liouville Theory on a Lorentzian strip, constructing the vertex operator V from free-field data and using it to extract the discrete spectrum, reflection amplitude, and correlation functions. By validating the approach against solvable models like the Morse potential, the authors connect BLT data (mass and boundary parameters) to zero-mode dynamics and derive a detailed vacuum-sector spectrum, a representation for the reflection amplitude via Barnes double Gamma functions, and a 1-point function in the ZZ case that matches bootstrap results. The analysis reveals that near critical boundary values BLT can support multiple equidistant spectral series, and it highlights open issues in extending the method to the FZZT case, constructing the full S-matrix, and matching Euclidean bootstrap results in general. Overall, the operator framework provides a concrete bridge between canonical/free-field techniques and the bootstrap approach, offering new avenues to quantify BLT spectra and correlation data while outlining substantial technical challenges remaining for full BLT quantization and S-matrix formulation.
Abstract
We propose new methods for calculation of the discrete spectrum, the reflection amplitude and the correlation functions of boundary Liouville theory on a strip with Lorentzian signature. They are based on the structure of the vertex operator $V=e^{-φ}$ in terms of the asymptotic operators. The methods first are tested for the particle dynamics in the Morse potential, where similar structures appear. Application of our methods to boundary Liouville theory reproduces the known results obtained earlier in the bootstrap approach, but there can arise a certain extension when the boundary parameters are near to critical values. Namely, in this case we have found up to four different equidistant series of discrete spectra, and the reflection amplitude is modified respectively.