SEWING CONSTRAINTS FOR CONFORMAL FIELD-THEORIES ON SURFACES WITH BOUNDARIES

In a conformal field theory, correlation functions on any Riemann surface are in principle unambiguously defined by sewing together three-point functions on the sphere, provided that the four-point functions on the sphere are crossing symmetric, and the one-point functions on the torus are modular covariant. In this work we extend Sonoda's proof of this result to conformal field theories defined on sur-faces with boundaries. Four additional sewing constraints arise; three on the half-plane and one on the cylinder. These relate the various OPE coefficients in the theory (bulk, boundary, and bulk-boundary) to one another. In rational theories these relations can be expressed in terms of data arising solely within the bulk theory: the matrix S which implements modular transformations on the characters, and the matrices implementing duality transformations on the four-point conformal-block functions. As an example we solve these relations for the boundary and bulk-boundary Structure constants in the Ising model with all possible conformally invariant boundary conditions. The role of the basic sewing constraints in the construction of open string theories is discussed.