We discuss a $T$-duality transformation for the $c=1/2\mathrm{}$ matrix model for the purpose of studying duality transformations in a possible toy example of nonperturbative frameworks of string theory. Our approach is to first investigate the scaling limit of the Schwinger-Dyson equations and the stochastic Hamiltonian in terms of the dual variables and then compare the results with those using the original spin variables. It is shown that the $c=1/2\mathrm{}$ model in the scaling limit is $T$-duality symmetric in the sphere approximation. In the case of the standard two-matrix model, however, the duality symmetry is violated when the higher-genus effects are taken into account, due to the nonsymmetrical appearence of global ${\mathbf{Z}}_2$ vector fields corresponding to nontrivial homology cycles. Some universal properties of the stochastic Hamiltonians which play an important role in discussing the scaling limit and have been discussed in a previous work by Sugino and Yoneya are refined in both the original and dual formulations. We also report a number of new explicit results for various amplitudes containing macroscopic loop operators.

• Received 30 July 1996

DOI:https://doi.org/10.1103/PhysRevD.55.5083