SPIN DISORDER CORRELATIONS AND DUALITY IN THE C=1/2 STRING/

We use the method of discrete loop equations to calculate exact correl ation functions of spin and disorder operators on the sphere and on th e boundary of a disk in the c = 1/2 string, both in the Ising and dual Ising matrix model formulations. For both the Ising and dual Ising th eories the results on the sphere are in agreement with the KPZ/DDK sca ling predictions based on Liouville theory; the results on the disk ag ree with the scaling predictions of Martinec, Moore, and Seiberg for b oundary operators. The calculation of Ising disorder correlations on t he 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 ter ms 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 the c = 1/2 conformal field theory survives coupling to quantum gravity, imp lying a duality symmetry of the c = 1/2 string even in the presence of boundary operators.