Orbifolds of symplectic fermion algebras

Understand

We present a systematic study of the orbifolds of the rank $n$ symplectic fermion algebra $\mathcal{A}(n)$, which has full automorphism group $Sp(2n)$.

• First, we show that $\mathcal{A}(n)^{Sp(2n)}$ and $\mathcal{A}(n)^{GL(n)}$ are $\mathcal{W}$-algebras of type $\mathcal{W}(2,4,\dots, 2n)$ and $\mathcal{W}(2,3,\dots, 2n+1)$, respectively.
• Using these results, we find minimal strong finite generating sets for $\mathcal{A}(mn)^{Sp(2n)}$ and $\mathcal{A}(mn)^{GL(n)}$ for all $m,n\geq 1$.
• We compute the characters of the irreducible representations of $\mathcal{A}(mn)^{Sp(2n)\times SO(m)}$ and $\mathcal{A}(mn)^{GL(n)\times GL(m)}$ appearing inside $\mathcal{A}(mn)$, and we express these characters using partial theta functions.