LOOP SPACE HAMILTONIAN FOR $c \le 1$ OPEN STRINGS

TL;DR

This paper develops a Hamiltonian framework for open and closed strings in a target space with dimension c ≤ 1, using large N matrix models and algebraic constraints to describe string dynamics.

Contribution

It introduces a novel string field Hamiltonian derived from Dyson-Schwinger equations and algebraic constraints in a large N vector+matrix model for c ≤ 1 strings.

Findings

Hamiltonian includes bulk and boundary terms with different scaling

Constraints form decoupled Virasoro and U(1) current algebras

Time parameters relate to fractal geometry of world surfaces

Abstract

We construct a string field Hamiltonian describing the dynamics of open and closed strings with effective target-space dimension $c\le 1 $. In order to do so, we first derive the Dyson-Schwinger equations for the underlying large $N$ vector+matrix model and formulate them as a set of constraints satisfying decoupled Virasoro and U(1) current algebras. The Hamiltonian consists of a bulk and a boundary term having different scaling dimensions. The time parameters corresponding to the two terms are interpreted from the the point of view of the fractal geometry of the world surface.