Liouville field theory on a pseudosphere

TL;DR

This paper analyzes Liouville field theory on a pseudosphere, solving bootstrap equations to identify solutions linked to Virasoro algebra representations, and concludes that the basic solution offers a natural quantization.

Contribution

It provides a comprehensive solution to the bootstrap equations for Liouville theory on a pseudosphere, connecting solutions to Virasoro representations and verifying their consistency.

Findings

Infinite solutions correspond to degenerate Virasoro representations

Only the basic solution aligns with a natural quantization

Boundary and modular bootstrap techniques confirm solution consistency

Abstract

Liouville field theory is considered with boundary conditions corresponding to a quantization of the classical Lobachevskiy plane (i.e. euclidean version of $AdS_2$). We solve the bootstrap equations for the out-vacuum wave function and find an infinite set of solutions. This solutions are in one to one correspondence with the degenerate representations of the Virasoro algebra. Consistency of these solutions is verified by both boundary and modular bootstrap techniques. Perturbative calculations lead to the conclusion that only the ``basic'' solution corresponding to the identity operator provides a ``natural'' quantization of the Lobachevskiy plane.