A CLOSED CONTOUR OF INTEGRATION IN REGGE CALCULUS

The analytic structure of the Regge action on a cone in d dimensions o ver a boundary of arbitrary topology is determined in simplicial minis uperspace. The minisuperspace is defined by the assignment of a single internal edge length to all 1-simplices emanating from the cone verte x, and a single boundary edge length to all 1-simplices lying on the b oundary. The Regge action is analyzed in the space of complex edge len gths, and it is shown that there are three finite branch points in thi s complex plane. A closed contour of integration encircling the branch points is shown to yield a convergent real wave function. This closed contour can be deformed to a steepest descent contour for all sizes o f the bounding universe. In general, the contour yields an oscillating wave function for universes of size greater than a critical value whi ch depends on the topology of the bounding universe. For values less t han the critical value the wave function exhibits exponential behaviou r. It is shown that the critical value is positive for spherical topol ogy in arbitrary dimensions. In three dimensions we compute the critic al value for a boundary universe of arbitrary genus, while in four and five dimensions we study examples of product manifolds and connected sums.