The Orbifold-String Theories of Permutation-Type: III. Lorentzian and Euclidean Space-Times in a Large Example

To illustrate the general results of the previous paper, we discuss here a large concrete example of the orbifold-string theories of permutation-type. For each of the many subexamples, we focus on evaluation of the \emph{target space-time dimension} $\hat{D}_j(σ)$, the \emph{target space-time signature} and the \emph{target space-time symmetry} of each cycle $j$ in each twisted sector $σ$. We find in particular a gratifying \emph{space-time symmetry enhancement} which naturally matches the space-time symmetry of each cycle to its space-time dimension. Although the orbifolds of $\Z_{2}$-permutation-type are naturally Lorentzian, we find that the target space-times associated to larger permutation groups can be Lorentzian, Euclidean and even null (\hat{D}_{j}(σ)=0), with varying space-time dimensions, signature and symmetry in a single orbifold.

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