THE QUANTUM STRIP - LIOUVILLE THEORY FOR OPEN STRINGS

The quantum group structure of 2D gravity recently put forward by one of us (J.-L. G.) is used to study quantum gravity on the strip. The boundary conditions, previously studied by A. Neveu and this author become easy to implement when one introduces the universal family of chiral operators associated with U(q)(sl(2)). A general formula for inverse powers of the metric-tensor operator is thereby derived. It contains a new universal matrix A, acting in representation-space, which obeys identities involving the R matrix, the Clebsch-Gordon coefficients, and the co-products of U(q)(sl(2)). The physical meaning of these identities is to ensure that these powers of the metric are local and closed by fusion.