Spin/disorder correlations and duality in the c = 12 string Original Research Article

We use the method of discrete loop equations to calculate exact correlation functions of spin and disorder operators on the sphere and on the boundary of a disk in the c = 12 string, both in the Ising and dual Ising matrix model formulations. For both the Ising and dual Ising theories the results on the sphere are in agreement with the KPZ/DDK scaling predictions based on Liouville theory; the results on the disk agree with the scaling predictions of Martinec, Moore, and Seiberg for boundary operators. The calculation of Ising disorder correlations on the sphere requires the use of boundary variables which have no matrix model analog. A subtlety in the calculation on the disk arises because the expansions of the correlation functions have leading singular terms which are nonuniversal; we show that this issue may be resolved by using separate cosmological constants for each boundary domain. These results give evidence that the Kramers-Wannier duality symmetry of the c = 12 conformal field theory survives coupling to quantum gravity, implying a duality symmetry of the c = 12 string even in the presence of boundary operators.