Feynman rules for string field theories with discrete target space

We derive a minimal set of Feynman rules for the loop amplitudes in unitary models of closed strings, whose target space is a simply laced (extended) Dynkin diagram. The string field Feynman graphs are composed of propagators, vertices (including tadpoles) of all topologies, and leg factors for the macroscopic loops. A vertex of given topology factorizes into a fusion coefficient for the matter fields and an intersection number associated with the corresponding punctured surface. As illustration we obtain explicit expressions for the genus-one tadpole and the genus-zero four-loop amplitude. 1. Introduction One of the most important problems in a theory of strings is the construction of the corresponding second quantized theory, i.e., a field theory in the space of loops [1]. A minimal requirement for a string field theory is to give simple rules for the pertirbative expansion, i.e., a prescription how to decompose the integral over world surfaces with different topologies into a sum of Feynman diagrams built from string propagators and vertices. In the last several years the simplest noncritical string theories were solved using large-N techniques in matrix models (see, for example, [2]). A matrix model is essentially a system of free fermions and the closed strings are represented there as collective excitations of fermions. A possible way to derive the genus expansion in the string theory is to reformulate the matrix model in terms of these collective fields 1. Suitable for this purpose are the ADE andˆAˆDˆE matrix models proposed in [3], describing string theories in which the matter degrees of freedom are labeled by the nodes of a Dynkin diagram X[4]. These models were generalized to describe both closed and open strings and reformulated in terms of the collective loop fields in ref. [5]. The resulting string field theory is identical to the one obtained with the loop gas technique in [6], [7]. The world sheet of the string represents a triangulated surface immersed in the graph X. The diagrammatic rules for the interactions of the string fields share some common features with the explicit construction of the interaction in the critical closed string theory [8]. The interaction is described by a nonpolynomial action, with an elementary vertex for every higher genus amplitude. The vertices are essentially the correlation functions for topological gravity and have the geometrical interpretation of punctured surfaces with various topologies localized at a single point x of the target space X. …