[Paper Review] Boundary Liouville field theory. 1. Boundary state and boundary two point function

V.A. Fateev, A. B. Zamolodchikov+1 more|arXiv (Cornell University)|Jan 4, 2000

Black Holes and Theoretical PhysicsPhysics and Astronomy2 references257 citations

TL;DR

This paper develops boundary Liouville field theory on the disk, deriving explicit expressions for the expectation value of bulk operators and the boundary two-point function, parameterized by the boundary cosmological constant. It establishes that observables depend on the ratio of bulk to boundary cosmological constants and provides exact results for boundary operators in the boundary sin-Gordon model.

ABSTRACT

Liouville conformal field theory is considered with conformal boundary. There is a family of conformal boundary conditions parameterized by the boundary cosmological constant, so that observables depend on the dimensional ratios of boundary and bulk cosmological constants. The disk geometry is considered. We present an explicit expression for the expectation value of a bulk operator inside the disk and for the two-point function of boundary operators. We comment also on the properties of the degenrate boundary operators. Possible applications and further developments are discussed. In particular, we present exact expectation values of the boundary operators in the boundary sin-Gordon model.

Motivation & Objective

• To formulate boundary Liouville conformal field theory with a family of conformal boundary conditions parameterized by the boundary cosmological constant.
• To derive explicit expressions for the expectation value of bulk operators and the two-point function of boundary operators in disk geometry.
• To analyze the properties of degenerate boundary operators in this framework.
• To explore applications to the boundary sin-Gordon model and provide exact expectation values for boundary operators.

Proposed method

• The study employs conformal field theory techniques on the disk geometry to compute correlation functions involving bulk and boundary operators.
• It introduces a family of conformal boundary conditions labeled by the boundary cosmological constant, which controls the dependence of observables on the ratio of bulk to boundary cosmological constants.
• The derivation uses exact solutions in Liouville theory to compute the expectation value of a bulk operator inside the disk.
• The two-point function of boundary operators is computed using boundary state formalism and conformal invariance.
• The analysis includes a study of degenerate boundary operators through their operator product expansion and null vector conditions.
• Exact results for boundary operators are derived in the context of the boundary sin-Gordon model via duality with Liouville theory.

Experimental results

Research questions

• RQ1 How do boundary conditions in Liouville field theory depend on the boundary cosmological constant, and what is their impact on observables?
• RQ2 What is the explicit form of the expectation value of a bulk operator in the disk geometry with conformal boundary conditions?
• RQ3 What is the structure of the two-point function of boundary operators in boundary Liouville theory?
• RQ4 How do degenerate boundary operators behave in this framework, and what constraints do they impose?
• RQ5 What are the exact expectation values of boundary operators in the boundary sin-Gordon model?

Key findings

• The expectation value of a bulk operator in the disk is derived explicitly, showing dependence on the ratio of bulk and boundary cosmological constants.
• The two-point function of boundary operators is computed in closed form, confirming consistency with conformal symmetry and boundary state formalism.
• Degenerate boundary operators are shown to satisfy specific null vector conditions, leading to constraints on correlation functions.
• The boundary cosmological constant parameterizes a continuous family of conformal boundary conditions, affecting all observables through the dimensionful ratio with the bulk cosmological constant.
• Exact expectation values of boundary operators are obtained in the boundary sin-Gordon model, providing a concrete realization of the framework.