The string difference equation of the D = 1 matrix model and W1+∞ symmetry of the KP hierarchy

In this paper, the authors give a connection between the D = 1 matrix model and the generalized KP hierarchy. First, the authors find a difference equation satisfied by F, the Legendre transformation of the free energy of the D = 1 matrix model on a circle of radius R. Then the authors show that it is a special case of the difference equation of the generalized KP hierarchy with its zero mode identified with the scaling variable of the D = 1 string theory. The authors propose that the massive D = 1 matrix model is described by the generalized KP hierarchy, which implies the manifest integrability of D = 1 string theory. The authors also show that the (generalized) KP hierarchy has an underlying W1+∞ symmetry. By reduction, we prove that the generalized KdV hierarchy has a subalgebra of the above symmetry which again forms a W1+∞. The authors argue that there are no W constraints in D = 1 string theory, which is in contrast to D < 1 theories, where there are W1+∞ constraints