Physics Letters B 268 ( 1991 ) 35-39 North-Holland P H YSIC S k ETTE RS B The double-scaling limit of O (N) vector models and the KP hierarchy Shinsuke Nishigaki and Tamiaki Yoneya Institute of Physics, University of Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan Received 4 June 1991 We show that the known solutions to O(N) symmetric vector models in the double-scaling limit are embedded in the KP hierarchy by a consistent truncation. We also discuss a second-quantizedtopologicalformulation of the theory. 1. In the recent discussion of the double-scaling limit of the matrix model [ 1 ], there are two charac- teristic features; namely, the integrable structure de- scribed by the KP hierarchy [2-5 ] under the flow of the coupling constants, and the equivalence of the theory with the topological formulation of contin- uum 2D gravity [6,7 ]. These features possibilities of, on one hand, a general framework for the description of random geometric systems and, on the other hand, of a new mathematical connection between topology and integrable systems. In a previous work [ 8 ], we have studied the dou- ble-scaling limit of the O(N) vector models corre- sponding to a system of randomly branching chains and found several parallel properties as those in the matrix model. It is of some interest to further pursue and sharpen the similarities. In the present note, we would like to point out that ( 1 ) the chain equation (the counter part of the string equation in the matrix case) and the flow equations of the vector models can be consistently embedded in the KP hierarchy with a certain reduction condition, and that (2) the integral representation of the r-function can be regarded as a second quantized theory of a topological 1D gravity. 2. First we shall briefly review some results con- cerning the vector model shown in a previous paper [ 8 ]. (See also ref. [ 9 ]. ) We make use of the fact that, in the large-Nexpansion of the O(N) symmetric vec- tor integral ZN= J dNq~exp[--flV(~2) ] , (1) each Feynman diagram is dual to a discretized branching world-line of a particle in "zero-dimen- sional spacetime" with a proper weight. The mth critical point is described by the potential satisfying 1-202V'(02) = (¢~ --¢2)m/Oc2m. The double-scal- ing limit is obtained by taking N~ and N/fl--, 1 keeping x- 1/2 N m/°È+~)~ 1 - N / f l ) fixed. Then the linear recursion equation I dO~-~ exp[-flV(O2)+NlnO]=O (2) (for instance, (N/fl)ZN--ZN+z+~ZN+4=O in the m=2 case) reduces to ~l (2x_ d") ~S'~,] Z'(X) = 0 (Z=--2--N/2ZN) . (3) Reflecting the fact that any spin system with suffi- ciently local interaction on one-dimensional lattices is always massive, the critical behaviors of a large class of multi-vector models are proven to fall into those of the one-vector model. Once the content of the scaling operators is known from eq. (3), one can interpolate between different multicritical points by introducing the deformation parameters x, as ~1 We factored out the nonuniversal contribution from the tree graphs. 0370-2693/91/$ 03.50 © 1991 ElsevierSciencePublishers B.V. All rights reserved. 35