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The Annular Report on Non-Critical String Theory

Emil J. Martinec

TL;DR

This work uses the Liouville annulus partition function with boundary Liouville and matter states to connect non-critical string theories across $c\le 1$ to their matrix-model duals, reproducing macroscopic-loop correlators and leading non-perturbative effects via D-brane (disk) boundary data. It further extends the framework to supercritical strings ($d\ge 25$), where Liouville gravity on the worldsheet realizes 2d de Sitter gravity and annulus amplitudes encode the de Sitter S-matrix between asymptotic boundaries. The analysis clarifies the dual role of matrix eigenvalues as Liouville D-branes and identifies boundary states with D-instanton sectors, while providing a controlled path to de Sitter amplitudes and wavefunctions through boundary-bulk factorization and minisuperspace limits. The results strengthen the Liouville-matrix-model correspondence and illuminate the structure of quantum gravity in two dimensions, including implications for a potential matrix-model description of 2d de Sitter cosmology.

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.

The Annular Report on Non-Critical String Theory

TL;DR

This work uses the Liouville annulus partition function with boundary Liouville and matter states to connect non-critical string theories across to their matrix-model duals, reproducing macroscopic-loop correlators and leading non-perturbative effects via D-brane (disk) boundary data. It further extends the framework to supercritical strings (), where Liouville gravity on the worldsheet realizes 2d de Sitter gravity and annulus amplitudes encode the de Sitter S-matrix between asymptotic boundaries. The analysis clarifies the dual role of matrix eigenvalues as Liouville D-branes and identifies boundary states with D-instanton sectors, while providing a controlled path to de Sitter amplitudes and wavefunctions through boundary-bulk factorization and minisuperspace limits. The results strengthen the Liouville-matrix-model correspondence and illuminate the structure of quantum gravity in two dimensions, including implications for a potential matrix-model description of 2d de Sitter cosmology.

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 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 non-critical string theory. In the context of string theory above the critical dimension, i.e. , 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.

Paper Structure

This paper contains 11 sections, 64 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Minimal models on the annulus
  3. c=1
  4. c$<$1
  5. Branes at infinity
  6. The de Sitter regime
  7. de Sitter Liouville
  8. Liouville theory as a 2d dS CFT
  9. Wavefunctions
  10. Comments
  11. Conventions

Figures (3)

  • Figure 1: Finite cylinder worldsheet for closed string propagation.
  • Figure 2: Conformal diagram for a spacetime which is asymptotically de Sitter at both past and future infinity. The worldline of a hypothetical observer in the spacetime is indicated, together with its past and future horizons. As $\epsilon$ decreases, the observer has access to more of the spacetime.
  • Figure 3: The three classes of solution of de Sitter Liouville theory, corresponding to negative, zero, and positive energy $E_L+{Q^2\over8}={\epsilon^2\over2\beta^2}$ relative to the Casimir shift. The Liouville field has 'wrong sign' kinetic energy (because it is the conformal factor of the metric); the sign of the potential has been flipped here for the purpose of using standard intuition about potential dynamics.