CONSTRAINTS FOR TOROIDAL COSMOLOGY

We discuss the constraints for a universe with the spatial topology of a d-1-dimensional torus and metric Gij. We argue that if dynamical duality exists it should lead to an O(d-1,d-1,Z)-invariant metric on superspace. We give an example of an indefinite O(d-1,d-1,Z)-invariant metric on the superspace (Gij,Bij) where Bij is the antisymmetric tensor field. On the hypersurface Bij=0 duality represents a parity time-reversal transformation of superspace. © 1990 The American Physical Society.