[Paper Review] Liouville field theory on a pseudosphere

Alexander B. Zamolodchikov, A. B. Zamolodchikov|ArXiv.org|Jan 23, 2001

Black Holes and Theoretical PhysicsPhysics and Astronomy8 references186 citations

TL;DR

This paper investigates Liouville field theory on a pseudosphere, corresponding to the Euclidean $AdS_2$ geometry, by solving the boundary bootstrap equations for the out-vacuum wave function. It finds an infinite family of solutions labeled by degenerate Virasoro representations $(m,n)$, with only the $(1,1)$ solution corresponding to a smooth classical limit and consistent with perturbative quantum gravity, while higher $(1,n)$ states exhibit exponential growth in correlation functions due to negative-dimension boundary operators.

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.

Motivation & Objective

• To formulate a consistent quantum field theory on the pseudosphere, the Euclidean $AdS_2$ geometry, using Liouville field theory.
• To identify which boundary conditions correspond to a natural quantization of the classical Lobachevskiy plane.
• To determine whether multiple solutions exist and which one(s) are physically viable via bootstrap consistency.
• To analyze the behavior of correlation functions in different out-vacua, especially their dependence on geodesic distance.
• To clarify the role of degenerate representations of the Virasoro algebra in defining boundary states on non-compact, negatively curved spaces.

Proposed method

• Solving the boundary bootstrap equations for the out-vacuum wave function in Liouville field theory on the pseudosphere.
• Using the correspondence between degenerate Virasoro representations and solutions labeled by integers $(m,n)$, particularly focusing on $(1,n)$ series.
• Applying both boundary and modular bootstrap techniques to verify consistency of the proposed solutions.
• Computing normalized two-point functions via boundary and bulk channel representations, comparing numerical results.
• Analyzing the behavior of correlation functions in different vacua, especially the exponential growth in $(m,n) \neq (1,1)$ states.
• Using the $\mathcal{F}$-function representation of correlation functions in terms of hypergeometric-type integrals with specific parameters derived from degenerate field dimensions.

Experimental results

Research questions

• RQ1 Which boundary conditions in Liouville field theory on the pseudosphere yield a consistent quantum theory?
• RQ2 Why does only the $(1,1)$ solution correspond to a smooth classical limit despite the existence of an infinite family of solutions?
• RQ3 What is the physical interpretation of the $(1,n)$ solutions with $n>1$, given their exponential growth in correlation functions?
• RQ4 How do degenerate boundary operators with negative dimensions affect the structure of correlation functions in excited vacua?
• RQ5 Can the bootstrap program be completed for all degenerate boundary fields and structure constants in this non-compact, negatively curved setting?

Key findings

• An infinite set of consistent solutions to the boundary bootstrap equations is found, labeled by degenerate Virasoro representations $(m,n)$, each corresponding to a distinct out-vacuum state.
• Only the $(1,1)$ solution exhibits a smooth classical limit and matches the standard perturbative quantum field theory, suggesting it is the 'natural' quantization of the pseudosphere.
• Solutions in the $(1,n)$ series with $n>1$ show exponential growth in two-point functions at large geodesic distances, indicating non-trivial physical behavior.
• The two-point function in the $(1,2)$ vacuum is dominated by the contribution of the $\psi_{13}$ boundary operator with negative dimension $\Delta_{1,3} = Q^2/4 - (b^{-1} + 2b)^2/4$, which drives the exponential growth.
• Numerical comparisons between boundary and bulk channel representations of the two-point function show excellent agreement, validating the bootstrap consistency at $b^2 \approx 0.8086$ and $b \approx 0.7048$, respectively.
• The $(1,1)$ solution is the only one consistent with loop perturbation theory beyond one loop, while higher $(1,n)$ states appear to describe different quantum phases or excited states.