Summing Planar Open String Loops on a Worldsheet Lattice with Dirichlet and Neumann Boundaries

TL;DR

This paper extends the lightcone worldsheet lattice approach to include D-branes with Dirichlet boundaries, representing multi-loop open string diagrams via an Ising spin system and exploring the resulting phase structure.

Contribution

It introduces a lattice representation for Dirichlet boundary conditions and models multi-loop open string diagrams with an Ising spin system, expanding the worldsheet approach to D-branes.

Findings

Richer phase structure due to interplay of boundary conditions

Representation of Dirichlet conditions on a lattice

Mean field analysis of coupled Ising/coordinate system

Abstract

We extend the lightcone worldsheet lattice approach to string theory, proposed in 1977 by Giles and me, to allow for coincident D-branes. We find a convenient lattice representation of Dirichlet boundary conditions, which the open string coordinates transverse to the D-branes satisfy. We then represent the sum over all planar open string multi-loop diagrams by introducing an Ising spin system on the worldsheet lattice to keep track of the presence or absence of fluctuating boundaries. Finally we discuss a simple mean field treatment of the resulting coupled Ising/coordinate worldsheet system. The interplay between Neumann and Dirichlet boundary conditions leads to a richer phase structure, within this mean field approximation, than that found by Orland for the original system with only Neumann conditions.