Ginsparg, P; Goulian, M; Plesser, MR; Zinn-Justin, J, *(p, q) string actions*,
Nuclear Physics B, vol. 342 no. 3
(1990),
pp. 539-563 [doi] .
**Abstract:**

*We investigate a recently proposed nonperturbative formulation of two-dimensional quantum gravity coupled to (p,q) minimal conformal matter. The coupled differential equations for the partition function summed over topologies are shown to follow from an action principle. The basic action for a (p,q) model takes the general form S(p,q) = ∫Res[Qp/q+ 1 + Σk = 0Σα = 0q - 2t(k), αQk + (α + 1)/q], where Q is a qth-order differential operator and the t(k), α are sources for operator insertions. We illustrate our results with the explicit examples of the Ising (4,3) and tricritical Ising (5,4) models. The action S(p,q) embodies the essential features of the problem (including the relation to generalized KdV hierarchies) in a most compact form. © 1990.*