Perturbative and Non-Perturbative Aspects of Two-Dimensional String Theory

Abstract

Two-dimensional string theory has been an extremely valuable arena for the study of fundamental aspects of string theory and the understanding of the holographic principle in quantum theories of gravity. The main subject of this dissertation is the holographic duality between the two-dimensional c=1 string theory and the c=1 matrix quantum mechanics. First, we revisit the perturbative S-matrix of c=1 string theory from the worldsheet formalism. We employ recently developed numerical techniques in conformal field theory to calculate the tree level 4-point amplitude and the genus one 2-point reflection amplitude of closed strings, finding agreement with the known results from the dual c=1 matrix quantum mechanics. This analysis clarifies the perturbative dictionary of the holographic duality and provides the first direct evidence for the duality at one-loop order. Second, we develop a worldsheet formalism to compute non-perturbative corrections to closed string scattering amplitudes, which incorporates disconnected diagrams with ZZ-instanton boundary conditions. By matching these instanton contributions to the anticipated results from the dual matrix model, we deduce the precise matrix quantum mechanics dual and promote the c=1 string theory to an exact duality including non-perturbative corrections. Lastly, we augment the c=1 holographic duality to include long strings, whose worldsheet description is a high-energy limit of open strings on top of extended FZZT branes. We show that this long string limit is well-defined and explicitly calculate the tree level amplitudes of (1) a long string decaying by emitting a closed string, and (2) the scattering of a pair of long strings. We find remarkable numerical agreement with computations in the U(N) adjoint and bi-adjoint sectors of the dual matrix quantum mechanics (based on a proposal by Maldacena), thereby providing strong evidence for the extended duality.