The Influence of World-sheet Boundaries on Critical Closed String Theory

This paper considers interactions between closed strings and open strings satisfying either Neumann or constant (point-like) Dirichlet boundary conditions in a BRST formalism in the critical dimension. With Neumann conditions this reproduces the well-known stringy version of the Higgs mechanism. With Dirichlet conditions the open-string states correspond to either auxiliary or Lagrange multiplier targetspace fields and their coupling to the closed-string sector leads to constraints on the closed-string spectrum. ⋆ email: [email protected] Conventional closed-string theory is formulated in terms of a sum over worldsheets with no boundaries. Adding boundaries in the usual manner, i.e. with Neumann boundary conditions on the space-time coordinates, Xμ(σ, τ), leads to a theory with both closed and open strings with free end-points. Adding boundaries with constant Dirichlet conditions leads to a theory with no physical open strings but with a radical modification of the closed-string theory (in which, for example, fixed-angle scattering behaves in a point-like manner). With orientable worldsheets (which is all that is considered here) the boundaries carry quantum numbers of the defining representation of a unitary group, denoted U(m) (or ‘flavour’) in the Neumann case and U(n) (or ‘colour’) in the Dirichlet case. Figure 1. (a) A representation of the interaction of a number of on-shell closed-string states on a world-sheet with the topology of a disk. The thick line indicates the world-sheet boundary. (b) The same process in a configuration in which the boundary represents a closed string disappearing into the vacuum. With Neumann conditions the boundary couples to a linear combination of the dilaton and the trace of the graviton in the cylinder while with Dirichlet conditions only the dilaton couples to the boundary. (c) Illustration of an intermediate open string. With Dirichlet conditions this string has both end-points fixed at the same space-time point. Earlier work (see [1] and references therein) has pointed out certain special features of the Dirichlet theory and its relationship, via space-time duality, to the Neumann theory. A world-sheet with a single boundary (the disk illustrated in fig.1(a)) may be parametrized so that the boundary is the end-state of a cylindrical section of world-sheet (fig.1(b)). The process may then be expressed in terms of the evolution of a closed string coupling to the boundary, which is represented by a state, 〈〈B| (the double ket notation indicates a state defined in the product of the spaces of left and right-moving modes), where 〈〈B|∂nX = 0 in the Neumann theory (the subscript n indicates a derivative normal to the boundary), whereas in the Dirichlet theory 〈〈B|(Xμ−y B) = 0, where the position of each boundary must be integrated (and the total momentum passing through the boundary vanishes as