Orbifolds of symplectic fermion algebras

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 $m,n\ge 1$. 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. Finally, we give a complete solution to the Hilbert problem for $\mathcal{A}(n)$; we show that for any reductive group $G$ of automorphisms, $\mathcal{A}(n{)}^G$ is strongly finitely generated.