# Complete Analysis of Extensions of $D(n)_1$ Permutation Orbifolds

## TL;DR

This paper provides a comprehensive analysis of extensions of $D(n)_1$ permutation orbifolds, including new $S$ matrices and fixed point resolution matrices, utilizing triality of SO(8) to address previously unknown cases.

## Contribution

It introduces the full set of $S$ matrices for extensions of $D(n)_1$ permutation orbifolds, especially for integer and half-integer spin spinor currents, expanding prior work.

## Key findings

- Derived $S$ matrices for new orbifold extensions
- Resolved fixed points for specific spinor currents
- Extended understanding of permutation orbifold symmetries

## Abstract

We give the full set of $S$ matrices for extensions of $D(n)_1$ permutation orbifolds, extending our previous work to the yet unknown case of integer spin spinor currents. The main tool is triality of SO(8). We also provide fixed point resolution matrices for spinor currents of $D(n)_1$ permutation orbifolds with $n$ even and not multiple of four, where the spinor currents have half-integer spin.