First Column Boundary Operator Product Expansion Coefficients

TL;DR

This paper computes boundary operator product expansion coefficients in conformal field theories, providing explicit formulas for specific models and boundary conditions, crucial for understanding critical phenomena in two-dimensional statistical models.

Contribution

It presents the first closed-form expressions for boundary OPE coefficients in the first column of the Kac table, including for c=0 and augmented minimal models, using null-vector and crossing symmetry methods.

Findings

Explicit formulas for boundary OPE coefficients at c=0.

Generalized coefficients for augmented minimal models.

Application to critical percolation and Potts models.

Abstract

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.