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Minimal Open Strings

Kazuo Hosomichi

TL;DR

The paper addresses open-string dynamics in minimal string theories by linking a two-matrix-model formulation to a worldsheet Liouville-based construction. It maps simple boundary-changing operators between the matrix model and worldsheet frameworks, and reveals a novel linear relation among FZZT branes alongside a straightforward realization of the boundary ground ring, enabling recursion for higher-point open-string amplitudes. The spectral-curve and disk-amplitude structure provide a concrete bridge between matrix-model resolvents and Liouville correlators, clarifying how open-string data are encoded in D-brane moduli. Overall, the work strengthens the matrix-model / worldsheet correspondence in non-critical strings and furnishes practical tools for perturbative open-string calculations through linear relations and ground-ring actions.

Abstract

We study FZZT-branes and open string amplitudes in (p,q) minimal string theory. We focus on the simplest boundary changing operators in two-matrix models, and identify the corresponding operators in worldsheet theory through the comparison of amplitudes. Along the way, we find a novel linear relation among FZZT boundary states in minimal string theory. We also show that the boundary ground ring is realized on physical open string operators in a very simple manner, and discuss its use for perturbative computation of higher open string amplitudes.

Minimal Open Strings

TL;DR

The paper addresses open-string dynamics in minimal string theories by linking a two-matrix-model formulation to a worldsheet Liouville-based construction. It maps simple boundary-changing operators between the matrix model and worldsheet frameworks, and reveals a novel linear relation among FZZT branes alongside a straightforward realization of the boundary ground ring, enabling recursion for higher-point open-string amplitudes. The spectral-curve and disk-amplitude structure provide a concrete bridge between matrix-model resolvents and Liouville correlators, clarifying how open-string data are encoded in D-brane moduli. Overall, the work strengthens the matrix-model / worldsheet correspondence in non-critical strings and furnishes practical tools for perturbative open-string calculations through linear relations and ground-ring actions.

Abstract

We study FZZT-branes and open string amplitudes in (p,q) minimal string theory. We focus on the simplest boundary changing operators in two-matrix models, and identify the corresponding operators in worldsheet theory through the comparison of amplitudes. Along the way, we find a novel linear relation among FZZT boundary states in minimal string theory. We also show that the boundary ground ring is realized on physical open string operators in a very simple manner, and discuss its use for perturbative computation of higher open string amplitudes.

Paper Structure

This paper contains 17 sections, 55 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Matrix Model
  3. Spectral curve.
  4. Continuum limit.
  5. Some disk amplitudes
  6. Worldsheet Theory
  7. Liouville theory with boundary
  8. Degenerate operators.
  9. The second Liouville theory.
  10. FZZT-branes
  11. Some disk amplitudes
  12. Amplitudes of ${\cal B}$.
  13. Amplitudes of ${\cal T}$.
  14. New linear relation among D-branes.
  15. Boundary ground ring
  16. ...and 2 more sections

Figures (1)

  • Figure 1: The oblique lines form the spectral curve for the two-matrix model realizing $(p,q)=(8,7)$ minimal string. The curve covers the $x$-plane 7 times and $y$-plane 8 times. The white dot is an FZZT-brane $\vert \theta;3,3\rangle$ which decomposes into nine elementary FZZT-branes described by black dots.