Factorization and Topological States in c=1 Matter Coupled to 2-D Gravity

Factorization of the N -point amplitudes in two-dimensional c = 1 quantum gravity is understood in terms of short-distance singularities arising from the operator product expansion of vertex operators after the Liouville zero mode integration. Apart from the tachyon states, there are infinitely many topological states contributing to the intermediate states. It is very important to understand the non-perturbative results of matrix models [1,2] from the viewpoint of the usual continuum approach of the two-dimensional quantum gravity, i.e. the Liouville theory [3-6]. Recently correlation functions on the sphere topology have been computed in the Liouville theory [7-12]. These results are consistent with those of matrix models [13-15]. So far only conformal field theories with central charge c ≤ 1 have been successfully coupled to quantum gravity. The c = 1 case is the richest and the most interesting. It has been observed that this theory can be regarded effectively as a critical string theory in two dimensions, since the Liouville field zero mode provides an additional “time-like” dimension besides the obvious single spatial dimension given by the zero mode of the c = 1 matter [16]. Since there are no transverse directions, the continuous (field) degrees of freedom are exhausted by the tachyon field. In fact, the partition function for the torus topology was computed in the Liouville theory, and was found to give precisely the same partition function as the tachyon field alone [17,18]. However, there are evidences for the existence of other discrete degrees of freedom in the c = 1 quantum gravity. Firstly, the correlation functions obtained in the matrix model exhibit a characteristic singularity structure [14]. In the continuum approach of the Liouville theory, Polyakov has observed that special states with discrete momenta and “energies” can produce such poles, and has called these operators co-dimension two operators [19]. More recently, the two-loop partition function has been computed in a matrix model and evidence has been noted for the occurrence of these topological sates [20]. It is clearly of vital importance to pin down the role played by these topological states as much as possible. In the critical string theory, the particle content of the theory and unitarity has been most clearly revealed through the factorization analysis of scattering amplitudes. On the other hand, the factorization and unitarity of the Liouville theory has not yet been well understood. The purpose of this paper is to understand the factorization of c = 1 quantum gravity in terms of the short-distance singularities arising from the operator