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Liouville field theory on a pseudosphere

A. Zamolodchikov, Al. Zamolodchikov

TL;DR

Liouville field theory on the pseudosphere provides a controlled setting to study 2D quantum gravity with boundary of constant negative curvature. The authors solve the out-vacuum bootstrap, modular bootstrap and boundary bootstrap to classify admissible boundary data, revealing an infinite family of out-vacua $U_{m,n}(\alpha)$ tied to degenerate Virasoro representations. The basic $ (m,n)=(1,1) $ vacuum emerges as the perturbatively natural quantization with no nontrivial boundary operators, while other vacua exhibit richer boundary content and modular data consistent with fusion rules. Collectively, these results establish a concrete operator content, fusion structure, and boundary dynamics for Liouville theory on a hyperbolic background, offering insights into 2D quantum gravity and potential AdS2/CFT connections.

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.

Liouville field theory on a pseudosphere

TL;DR

Liouville field theory on the pseudosphere provides a controlled setting to study 2D quantum gravity with boundary of constant negative curvature. The authors solve the out-vacuum bootstrap, modular bootstrap and boundary bootstrap to classify admissible boundary data, revealing an infinite family of out-vacua tied to degenerate Virasoro representations. The basic vacuum emerges as the perturbatively natural quantization with no nontrivial boundary operators, while other vacua exhibit richer boundary content and modular data consistent with fusion rules. Collectively, these results establish a concrete operator content, fusion structure, and boundary dynamics for Liouville theory on a hyperbolic background, offering insights into 2D quantum gravity and potential AdS2/CFT connections.

Abstract

Liouville field theory is considered with boundary conditions corresponding to a quantization of the classical Lobachevskiy plane (i.e. euclidean version of ). 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.

Paper Structure

This paper contains 7 sections, 98 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. One-point bootstrap
  3. The one-point function
  4. Loop perturbation theory
  5. Modular bootstrap
  6. Boundary bootstrap
  7. Discussion

Figures (4)

  • Figure 1: Diagrams contributing to the expectation values $G_{1}$, $G_{2}$ and $G_{3}$ at one- and two-loop order.
  • Figure 2: The annulus with two "boundary conditions" corresponding to the out-vacuum states $(m,n)$ and $(m^{\prime},n^{\prime})$.
  • Figure 3: Normalized two-point function $g_{Q/2,Q/2}(\eta)$ evaluated as the single vacuum block (\ref{['F']}) (solid line) and as the cross-channel integral (\ref{['Ft']}) (circles).
  • Figure 4: Boundary (\ref{['gFF']}) (solid line) and bulk (\ref{['Ft']}) (circles) representations of the normalized two-point function $g_{Q/2,Q/2}(\eta)$ are compared at $b=0.7048\ldots$. Small circles and crosses are respectively the contributions of the first and second terms in eq.(\ref{['gFF']}). The two-point function is almost saturated by the $\psi_{13}$ contribution.