[Paper Review] Remarks on Liouville theory with boundary

J. Teschner|ArXiv.org|Sep 18, 2000

Black Holes and Theoretical PhysicsPhysics and Astronomy1 references123 citations

TL;DR

This paper develops the bootstrap program for Liouville field theory with conformally invariant boundary conditions, deriving boundary structure functions via a generalized Cardy condition and linking them to b-Racah-Wigner symbols of $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$. The key result is a precise correspondence between boundary three-point functions and quantum group 6j-symbols, providing a noncompact CFT analog to rational CFT bootstrap methods and extending the Cardy formula to continuous spectra via reflection amplitudes and Plancherel measures.

ABSTRACT

The bootstrap for Liouville theory with conformally invariant boundary conditions will be discussed. After reviewing some results on one- and boundary two-point functions we discuss some analogue of the Cardy condition linking these data. This allows to determine the spectrum of the theory on the strip, and illustrates in what respects the bootstrap for noncompact conformal field theories with boundary is richer than in RCFT. We briefly indicate some connections with $U_q(sl(2,R))$ that should help completing the bootstrap.

Motivation & Objective

• To extend the conformal bootstrap program to noncompact conformal field theories with boundaries, focusing on Liouville theory as a prototype.
• To derive the spectrum and structure functions (one-point, two-point, and three-point functions) for Liouville theory on the strip with boundary conditions.
• To establish a correspondence between boundary structure functions and representation-theoretic data of $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$.
• To generalize the Cardy formula for one-point functions in rational CFT to the continuous spectrum of noncompact CFT.
• To clarify the richer structure of the bootstrap in noncompact CFTs compared to rational CFTs, particularly regarding reflection amplitudes and modular data.

Proposed method

• Constructs the Hilbert space of the theory on the strip as an integral over continuous representations $\mathcal{V}_\alpha$ with $\alpha \in \mathbb{S}^B$, parameterized by boundary parameters $\rho_1, \rho_2$.
• Uses conformal Ward identities and factorization to reduce correlation functions to fundamental structure functions: three-point functions $D$, one-point functions $A(\alpha|\rho)$, boundary two-point functions $N_0$, and bulk-boundary two-point functions $A(\alpha,\beta|\rho)$.
• Imposes a generalized Cardy condition linking one-point functions and boundary two-point functions via the reflection amplitude $R(\alpha)$ and Plancherel measure $\mu(\alpha)$.
• Derives the boundary three-point function $C[\beta_3,\beta_2,\beta_1]^{s_3,s_2,s_1}$ as proportional to b-Racah-Wigner symbols of $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$, with normalization adjusted via Clebsch-Gordan coefficients.
• Fixes normalization by analyzing degenerate representations $\mathcal{V}_{-b}$, leading to finite difference equations for the three-point function.
• Re-expresses the one-point function as $A(\alpha|s) = e^{i\delta(\alpha)} S(\alpha;s)/\sqrt{\mu(\alpha)}$, where $S(\alpha;s)$ is the modular S-matrix and $\mu(\alpha)$ the Plancherel measure, generalizing the Cardy formula.

Experimental results

Research questions

• RQ1 How can the bootstrap program be extended to noncompact CFTs with boundaries, particularly in the context of Liouville theory?
• RQ2 What is the precise relation between boundary structure functions and representation-theoretic data of quantum groups like $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$?
• RQ3 How does the generalized Cardy condition in noncompact CFT differ from the standard Cardy formula in rational CFT?
• RQ4 What role do b-Racah-Wigner symbols play in ensuring associativity and consistency of boundary correlation functions?
• RQ5 How can the one-point function in the presence of boundaries be expressed in terms of modular S-matrix and Plancherel measure, generalizing the rational CFT case?

Key findings

• The boundary three-point function is shown to be proportional to the b-Racah-Wigner symbol $\{\cdots\}'_b$ of $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$, with a normalization shift from standard definitions.
• The associativity of the boundary three-point function is guaranteed by the pentagon identity satisfied by the b-Racah-Wigner symbols.
• The one-point function is expressed as $A(\alpha|s) = e^{i\delta(\alpha)} S(\alpha;s)/\sqrt{\mu(\alpha)}$, where $e^{2i\delta(\alpha)} = R(\alpha)$ is the bulk reflection amplitude and $\mu(\alpha)$ is the Plancherel measure of the quantum group dual.
• The generalized Cardy formula (11) provides a natural extension of the rational CFT one-point function formula to noncompact CFT, with the reflection amplitude replacing the quantum dimension.
• The bootstrap in noncompact CFT is richer than in RCFT: the reflection amplitude is not directly related to modular S-matrix coefficients, and the spectrum is continuous.
• The normalization of boundary fields is fixed by analyzing degenerate representations $\mathcal{V}_{-b}$, leading to finite difference equations that constrain the three-point function.