Abstract
Using the standard 1/ N expansion, we study O( N) vector models with an arbitrary potential in zero dimensions and we show that a double scaling limit exists as in the case of matrix models. We find in general a hierarchy of critical theories labelled by an integer k. The universal partition function of the k th theory obtained in the double scaling limit is constructed both from the effective action in the double scaling limit and as a solution of a kth order differential equation that follows from the Schwinger-Dyson equations of the theory in the same limit. We also show that the theory possesses the Virasoro symmetry acting on the partition function.
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