[Paper Review] From Liouville Theory to the Quantum Geometry of Riemann Surfaces

J. Teschner|ArXiv.org|Aug 5, 2003

History and Theory of Mathematics3 references16 citations

TL;DR

This paper proposes a geometric interpretation of full (non-chiral) correlation functions in quantum Liouville theory as arising from the quantization of Teichmüller spaces of Riemann surfaces. By identifying Liouville conformal blocks with eigenfunctions of geodesic length operators in quantum Teichmüller theory, it establishes a direct link between conformal field theory amplitudes and the quantum geometry of moduli spaces, with explicit asymptotic behavior near degeneration matching the expected quantum operators via operator ordering constraints.

ABSTRACT

The aim of this note is to propose an interpretation for the full (non-chiral) correlation functions of the Liouville conformal field theory within the context of the quantization of spaces of Riemann surfaces.

Motivation & Objective

• To establish a geometric interpretation of full (non-chiral) correlation functions in quantum Liouville theory within the framework of quantum Teichmüller theory.
• To relate the conformal blocks of Liouville theory to eigenfunctions of geodesic length operators on Riemann surfaces.
• To provide a quantum-geometric foundation for two-dimensional quantum gravity by connecting Liouville theory to the quantization of moduli spaces of Riemann surfaces.
• To conjecture a measure for the scalar product in the quantum Teichmüller space that ensures orthogonality of eigenstates corresponding to different geodesic lengths.

Proposed method

• The paper uses holomorphically factorized representations of Liouville correlation functions via conformal blocks, expressed as integrals over intermediate representations with a specific measure.
• It identifies the conformal blocks as solutions to the conformal Ward identities, which encode invariance under holomorphic vector fields on punctured Riemann surfaces.
• Near degeneration of a Riemann surface (as |q|→0), the asymptotic behavior of conformal blocks is matched to eigenfunctions of the geodesic length operator in the q-representation.
• The operator ordering for the length operator is fixed by requiring consistency with the monodromy condition around q=0, leading to a specific form of the differential operator b²∂/∂q.
• The paper proposes a measure ν(x, x̄) on the upper half-plane (via q = e^{2πix}) such that the inner product of eigenfunctions yields a delta function, ensuring orthogonality.
• It conjectures that the Liouville conformal blocks are the unique eigenfunctions compatible with both the monodromy condition and the general operator ordering ansatz for the length operator.

Experimental results

Research questions

• RQ1 Can the full non-chiral correlation functions of Liouville theory be interpreted as matrix elements in the quantum Teichmüller space of Riemann surfaces?
• RQ2 Are the Liouville conformal blocks eigenfunctions of the geodesic length operator in quantum Teichmüller theory, and if so, what is the correct operator ordering?
• RQ3 What is the appropriate measure for the inner product in the quantum Teichmüller space that ensures orthogonality of states with different geodesic lengths?
• RQ4 How does the asymptotic behavior of Liouville conformal blocks near q=0 relate to the quantum geometry of degenerating Riemann surfaces?
• RQ5 Can the relation between the accessory parameter C_q and the quantum length operator be consistently quantized, and does it reproduce the correct monodromy behavior?

Key findings

• The Liouville conformal blocks are shown to match the asymptotic behavior of eigenfunctions of the geodesic length operator in the q-representation, with wavefunctions ψ_l(q) = q^{p² - Q²/4}, where p = l/(4πb).
• The operator ordering for the length operator is uniquely fixed by the monodromy condition around q=0, leading to the differential operator b²∂/∂q.
• The relation qC_q(q, q̄) = (l/4π)² - 1/4 is quantized as q∂/∂q = (l̂_γ/(4πb))² - Q²/4, establishing a direct link between classical accessory parameters and quantum length operators.
• The proposed measure ν(x, x̄) for the inner product ensures that the norm of eigenstates ψ_l(q) yields a delta function δ(l - l'), provided ν depends only on the imaginary part of x.
• The conformal blocks are identified as the unique solutions compatible with both the monodromy condition and the general operator ordering ansatz for the length operator.
• The paper conjectures that the full Liouville correlation functions arise from the inner product of these eigenfunctions in the quantum Teichmüller space, with the measure m(S) in the integral representation corresponding to the quantum measure on moduli space.