$S$-matrix in permutation orbifolds

TL;DR

This paper computes the S-matrix for permutation orbifolds of tensor product vertex operator algebras under cyclic permutation groups, advancing understanding of their modular properties.

Contribution

It explicitly determines the S-matrix for the fixed point subalgebra under cyclic permutation, a specific case of permutation orbifolds.

Findings

Explicit S-matrix formula for cyclic permutation orbifolds

Enhanced understanding of modular invariance in permutation orbifolds

Foundation for further study of orbifold conformal field theories

Abstract

For a fixed positive integer $k$, any element $g$ of the permutation group $S_{k}$ acts on the tensor product vertex operator algebra $V^{\otimes k}$ in the obvious way. In this paper, we determine the $S$-matrix of $\left(V^{\otimes k}\right)^{G}$ if $G=\left\langle g\right\rangle $ is the cyclic group generated by $g=\left(1,\ 2,\cdots,k\right).$