Constraints for toroidal cosmology

We discuss the constraints for a universe with the spatial topology of a ({ital d}{minus}1)-dimensional torus and metric {ital G}{sub {ital i}{ital j}}. We argue that if dynamical duality exists it should lead to an O({ital d}{minus}1,{ital d}{minus}1,{ital Z})-invariant metric on superspace. We give an example of an indefinite O({ital d}{minus}1,{ital d}{minus}1,{ital Z})-invariant metric on the superspace ({ital G}{sub {ital i}{ital j}},{ital B}{sub {ital i}{ital j}}) where {ital B}{sub {ital i}{ital j}} is the antisymmetric tensor field. On the hypersurface {ital B}{sub {ital i}{ital j}}=0 duality represents a parity time-reversal transformation of superspace.