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

TL;DR

This paper explores a detailed example of orbifold-string theories, analyzing target space-time properties like dimension, signature, and symmetry, revealing diverse space-time structures including Lorentzian, Euclidean, and null signatures within a single orbifold.

Contribution

It provides a comprehensive analysis of a large example of permutation-type orbifold-string theories, highlighting space-time symmetry enhancements and diverse space-time signatures.

Findings

Orbifold target space-times can be Lorentzian, Euclidean, or null.

Space-time symmetry enhancement matches each cycle's symmetry to its dimension.

Large permutation groups yield varied space-time signatures within a single orbifold.

Abstract

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} $\hat{D}_j(\sigma)$, the \emph{target space-time signature} and the \emph{target space-time symmetry} of each cycle $j$ in each twisted sector $\sigma$. 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}(\sigma)=0), with varying space-time dimensions, signature and symmetry in a single orbifold.