First Column Boundary Operator Product Expansion Coefficients

We calculate boundary operator product expansion coefficients for boundary operators in the first column of the Kac table in conformal field theories. For c=0 we give closed form expressions for all such coefficients. Then we generalize to the augmented minimal models, giving explicit expressions for coefficients valid when \phi_{1,2} mediates a change from fixed to free boundary conditions. These quantities are determined by computing an arbitrary four-point correlation function of first column operators. Our calculation first determines the appropriate (non-logarithmic) conformal blocks by using standard null-vector methods. The behavior of these blocks under crossing symmetry then provides a general closed form expression for the desired coefficients, as a product of ratios of gamma functions. This calculation was inspired by the need for several of these coefficients in certain correlation function formulas for critical two-dimensional percolation and the augmented q=2 and q=3 state critical Potts models.