Matrix Quantum Mechanics and Two-dimensional String Theory in Non-trivial Backgrounds

TL;DR

The thesis demonstrates that Matrix Quantum Mechanics provides a complete, integrable framework for 2D string theory, extending the linear-dilaton background to nontrivial backgrounds via winding and tachyon perturbations. By mapping background deformations to integrable Toda hierarchy flows, it yields exact results for correlation functions, thermodynamics, and the target-space structure, including black-hole-like geometries and FZZ dualities. The work establishes a free-fermion representation of the singlet sector, a Das–Jevicki collective-field picture for tachyon dynamics, and a consistent treatment of winding modes through representations of SU(N), culminating in a unified picture of non-critical string theory in diverse backgrounds. The findings illuminate nonperturbative effects, dualities, and the precise connections between matrix models, 2D gravity, and string theory, with broad implications for solvable models and integrable systems in low dimensions.

Abstract

String theory is the most promising candidate for the theory unifying all interactions including gravity. It has an extremely difficult dynamics. Therefore, it is useful to study some its simplifications. One of them is non-critical string theory which can be defined in low dimensions. A particular interesting case is 2D string theory. On the one hand, it has a very rich structure and, on the other hand, it is solvable. A complete solution of 2D string theory in the simplest linear dilaton background was obtained using its representation as Matrix Quantum Mechanics. This matrix model provides a very powerful technique and reveals the integrability hidden in the usual CFT formulation. This thesis extends the matrix model description of 2D string theory to non-trivial backgrounds. We show how perturbations changing the background are incorporated into Matrix Quantum Mechanics. The perturbations are integrable and governed by Toda Lattice hierarchy. This integrability is used to extract various information about the perturbed system: correlation functions, thermodynamical behaviour, structure of the target space. The results concerning these and some other issues, like non-perturbative effects in non-critical string theory, are presented in the thesis.

Figures (22)

• Figure 1: Scattering amplitudes in dual models.
• Figure 2: Scattering string amplitude can be seen in two ways.
• Figure 3: Open and closed strings.
• Figure 4: Interactions of open and closed strings.
• Figure 5: Chain of dualities relating all superstring theories.