STRINGS WITH DISCRETE TARGET SPACE

We investigate the field theory of strings having as a target space an arbitrary discrete one-dimensional manifold. The existence of the continuum limit is guaranteed if the target space is a Dynkin diagram of a simply laced Lie algebra or its affine extension. In this case the theory can be mapped onto the theory of strings embedded in the infinite discrete line Z which is the target space of the SOS model. On the regular lattice this mapping is known as Coulomb gas picture. Introducing a quantum string field PHI(x)(l) depending on the position x and the length l of the closed string, we give a formal definition of the string field theory in terms of a functional integral. The classical string background is found as a solution of the saddle-point equation which is equivalent to the loop equation we have previously considered. The continuum limit exists in the vicinity of the singular points of this equation. We show that for given target space there are many ways to achieve the continuum limit; they are related to the multicritical points of the ensemble of surfaces without embedding. Once the classical background is known, the amplitudes involving propagation of strings can be evaluated by perturbative expansion around the saddle point of the functional integral. For example, the partition function of the non-interacting closed string (toroidal world sheet) is the contribution of the gaussian fluctuations of the string field. The vertices in the corresponding Feynman diagram technique are constructed as the loop amplitudes in a random matrix model with suitably chosen potential.