The insertion of boundaries in world-sheets

Abstract

This paper considers the operator that inserts a boundary in a string world-sheet on which the string coordinates satisfy either Neumann or constant Dirichlet boundary conditions. With an arbitrary open-string state attached to the boundary, this describes the vertex coupling an open string to a closed string (the “open-closed string vertex”). A boundary with no open strings attached can be viewed as a vacuum correction to closed-string theory. This factories into two open-closed string vertices joined together by an open-string propagator. BRST-invariant open-closed string vertices of the Neumann and Dirichlet theories are constructed in a “light-cone-like” frame (familiar from some formulations of covariant string field theory) as well as the “vertex operator” frame (in which a general open-string state is represented by a vertex operator attached to the world-sheet boundary). The vertices of the Neumann and Dirichlet theories are related to each other by space-time duality. The insertion of a closed Dirichlet boundary may be expressed in the light-cone-like gauge as an interaction vertex that acts at a fixed “time” and generates an explicit mass term connecting two arbitrary closed-string states. Some examples of how the presence of such boundaries modifies amplitudes for low-lying states are presented.