Constraints for toroidal cosmology

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