[Paper Review] The Annular Report on Non-Critical String Theory

Emil J. Martinec|ArXiv.org|May 18, 2003

Black Holes and Theoretical PhysicsPhysics and Astronomy94 references128 citations

TL;DR

This paper establishes a worldsheet formulation of non-critical string theory using Liouville field theory on the annulus, reproducing key matrix model results—specifically the two-loop correlation function and leading non-perturbative instanton effects—by identifying eigenvalue instantons in the matrix model with D-instantons in the worldsheet approach. It further extends this framework to de Sitter gravity in $d \geq 25$ dimensions, showing that annular amplitudes with appropriate boundary conditions yield the $2d$ de Sitter S-matrix and suggesting a dual matrix model description for quantum de Sitter gravity.

ABSTRACT

Recent results on the annulus partition function in Liouville field theory are applied to non-critical string theory, both below and above the critical dimension. Liouville gravity coupled to $c\le 1$ matter has a dual formulation as a matrix model. Two well-known matrix model results are reproduced precisely using the worldsheet formulation: (1) the correlation function of two macroscopic loops, and (2) the leading non-perturbative effects. The latter identifies the eigenvalue instanton amplitudes of the matrix approach with disk instantons of the worldsheet approach, thus demonstrating that the matrix model is the effective dynamics of a D-brane realization of $d\le 1$ non-critical string theory. In the context of string theory above the critical dimension, i.e. $d\ge 25$, Liouville field theory realizes two-dimensional de Sitter gravity on the worldsheet. In this case, appropriate D-brane boundary conditions on the annulus realize the S-matrix for two-dimensional de Sitter gravity.

Motivation & Objective

• To reproduce established matrix model results—specifically the two-macroscopic-loop correlation function and leading non-perturbative effects—in the worldsheet formulation of non-critical string theory using Liouville field theory on the annulus.
• To identify the physical origin of matrix model eigenvalue instantons as D-instantons in the worldsheet approach, thereby confirming a duality between matrix models and D-brane dynamics in $c \leq 1$ string theory.
• To extend the worldsheet approach to supercritical string theories with $d \geq 25$, where Liouville gravity describes 2D de Sitter gravity, and to construct the annular amplitude that realizes the S-matrix for such spacetimes.
• To explore the connection between boundary states in Liouville theory and the UV completion of quantum gravity near cosmological singularities, particularly in the context of big crunch/bang geometries.
• To suggest a potential matrix model dual for 2D de Sitter quantum gravity, inspired by the duality observed in the $c \leq 1$ case and the role of 'S-branes at infinity' in the boundary conditions.

Proposed method

• Uses the annulus partition function in Liouville field theory with boundary cosmological constants to compute amplitudes for non-critical strings below and above the critical dimension.
• Applies boundary states from [24] that are localized in the strong-coupling region ($\varphi \to \infty$) to compute non-perturbative instanton effects, matching matrix model eigenvalue instantons.
• Computes the correlation function of two macroscopic loops via the annular amplitude, using boundary conditions that fix the loop length $\ell = \oint ds\, e^{b\varphi}$, and compares with Moore-Seiberg's matrix model result.
• Analyzes the analytic structure of the annulus amplitude by contour integration, identifying the correct exponential damping behavior that selects the physical reflection amplitude over unphysical sinh terms.
• Uses conformal field theory techniques to compute disk one-point and two-point functions with boundary states, and relates them to the full annular amplitude via factorization.
• Applies Wick rotation ($\phi = i\varphi$) to extend the Liouville theory to supercritical dimensions ($c_{\text{matt}} \geq 25$), mapping classical 2D de Sitter spacetime to the annular worldsheet geometry.

Experimental results

Research questions

• RQ1 How can the annular amplitude in Liouville field theory reproduce the two-loop correlation function of macroscopic loops in $c \leq 1$ non-critical string theory?
• RQ2 What is the worldsheet interpretation of matrix model eigenvalue instantons, and how do they relate to D-brane instantons in Liouville theory?
• RQ3 Can the annular worldsheet amplitude in supercritical Liouville theory ($c_{\text{matt}} \geq 25$) describe the S-matrix of 2D de Sitter gravity?
• RQ4 How are cosmological singularities like big crunches regularized in two-dimensional quantum gravity, and what role does conformal invariance play?
• RQ5 Is there a matrix model dual for 2D de Sitter quantum gravity, analogous to the matrix model duality in $c \leq 1$ string theory?

Key findings

• The annular amplitude in Liouville theory precisely reproduces the two-macroscopic-loop correlation function computed via the matrix model, confirming consistency between worldsheet and matrix model approaches.
• The leading non-perturbative effects in $d < 1$ string theory, previously described as eigenvalue instantons in the matrix model, are identified as D-instantons in the worldsheet formulation, with wavefunctions peaked in the strong-coupling region of Liouville theory.
• The annular amplitude for $c_{\text{matt}} \geq 25$ yields the correct reflection amplitude for 2D de Sitter gravity, with the boundary conditions corresponding to asymptotically de Sitter spacetimes and the Liouville field diverging at the conformal boundary.
• The contour integration of the annular amplitude selects the exponentially damped solution, corresponding to the physical S-matrix amplitude, and not the unphysical sinh terms that would arise from improper contour closure.
• The UV behavior of the theory near cosmological singularities is governed by a specific matter CFT coupled to Liouville gravity, with relevant interactions suppressed at large negative $\phi$, ensuring a well-defined UV completion.
• The results suggest a dual matrix model formulation for 2D de Sitter quantum gravity, with 'S-branes at infinity' in $\phi$-space playing a role analogous to matrix degrees of freedom in the $c \leq 1$ case.