NONPERTURBATIVE 2D GRAVITY, PUNCTURED SPHERES AND THETA-VACUA IN STRING THEORIES

We consider a model of 2D gravity with the coefficient of the Euler ch aracteristic having an imaginary part pi/2. This is equivalent to intr oduce a Theta-vacuum structure in the genus expansion whose effect is to convert the expansion into a series of alternating signs, presumabl y Borel summable. We show that the specific heat of the model has a ph ysical behaviour. It can be represented nonperturbatively as a series in terms of integrals over moduli spaces of punctured spheres and the sum of the series can be rewritten as a unique integral over a suitabl e moduli space of infinitely punctured spheres. This is an explicit re alization a la Friedan-Shenker of 2D quantum gravity. We conjecture th at the expansion in terms of punctures and the genus expansion can be derived using the Duistermaat-Heckman theorem. We briefly analyze expa nsions in terms of punctured spheres also for multicritical models.