Matrix Models and 2D String Theory

TL;DR

This work surveys how two complementary formalisms—random-matrix models and Liouville/CFT on the worldsheet—realize 2D string theory, highlighting an open/closed string duality in a tractable setting. It demonstrates precise agreement between matrix-model observables (macroscopic loops, density correlators, and eigenvalue dynamics) and Liouville-based continuum computations, notably for disk one-point amplitudes and loop insertions, thereby illuminating tachyon condensation and D-brane interpretations in 2D. The paper also extends the discussion to fermionic (type 0) strings, outlines a rich open problems program (duality maps, gravity, and short-distance structure), and emphasizes the potential of 2D models to yield nonperturbative insights into broader string theory questions.

Abstract

String theory in two-dimensional spacetime illuminates two main threads of recent development in string theory: (1) Open/closed string duality, and (2) Tachyon condensation. In two dimensions, many aspects of these phenomena can be explored in a setting where exact calculations can be performed. These lectures review the basic aspects of this system.

Figures (15)

• Figure 1: Each handle, except the end ones, contributes three closed string propagator tubes to the surface. Each tube has a length and a helical twist angle. The two end handles together contribute only three tubes, and so the number of moduli is $6h-6$.
• Figure 2: The string loop expansion.
• Figure 3: The tachyon background.
• Figure 4: Scattering of a tachyon excitation off the tachyon background.
• Figure 5: (a) Regular polygons for tiling a surface, with dashed red edges; and the dual fatgraph vertices, with solid blue dual edges. (b) A patch of discrete surface tesselated with triangles, and the dual fatgraph.