Summing over World-sheet Boundaries

The moduli associated with boundaries in a Riemann surface are parametrized by the positions and strengths of electric charges. This suggests a method for summing over orientable Riemann surfaces with Dirichlet boundary conditions on the embedding coordinates. A light-cone parameterization of such boundaries is also discussed. ⋆ email: [email protected] † email: [email protected] The inclusion of boundaries in the sum over world-sheets that defines string perturbation theory alters the properties of string theory rather dramatically. With Neumann boundary conditions on the embedding coordinates Xμ(σ, τ), the resulting theory describes interacting open and closed strings – a boundary representing the trajectory of an open-string end-point. Boundaries with Dirichlet conditions on the embedding coordinates (Xμ(σ, τ) = yμ) also have significant effects—for some discussion see [1] and references therein. From the target space point of view, the Dirichlet boundary is simply a point. This suggests using world-sheet coordinates in which the boundary is mapped to a point; what distinguishes this point is singular behavior of the intrinsic metric gab. Since the world-sheet now appears to be topologically trivial, the sum over arbitrary numbers of Dirichlet boundaries can at least formally be recast as a world-sheet field theory, which is the motivation for this paper. The strategy to be followed is to reexpress the general metric for a surface (of euclidean signature) with an arbitrary number of holes, g̃, as the sum of a metric on a surface with no holes, g, and a bilinear in a vector field living on the surface, g̃αβ = gαβ + AαAβ, (1) (α, β = 1, 2) where A = − ∗ dφ (in components, Aα = −gαβǫ∂γφ) is a 1-form vector field. We will refer to g̃ in the following as the ‘modified’ metric. It bears an obvious resemblence to a metric that would arise in Kaluza–Klein reduction from a three-dimensional theory (in which gα3 = Aα). The inverse modified metric is given by g̃αβ = gαβ −AαAβ(1+AγAγ)−1, where indices are here raised by gαβ (the inverse of g). The magnetic field strength due to the vector field is taken to vanish everywhere except at B arbitrary points, ∗F ≡ ∗dA = −d ∗ dφ = √g∇2φ = B