Convergence and operadic compatibility of bulk and boundary OPEs in two-dimensional conformal field theory

We prove convergence and compatibility of iterated bulk and boundary operator product expansions (OPEs) in two-dimensional conformal field theory with locally $C_1$-cofinite chiral symmetry. For each tree, we give an explicit domain of convergence for the corresponding iterated OPE. These local expansions glue to single-valued real analytic functions on the configuration spaces, which are the correlation functions of the theory. The proof uses an action of the parenthesized permutation-braid operad on $C_1$-cofinite module categories of a vertex operator algebra. This operad models the fundamental groupoid of the two-dimensional Swiss-cheese operad, and under this action the operadic generators correspond to the genus-zero bootstrap equations of boundary CFT.