Orbifolds of symplectic fermion algebras

Abstract:

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. 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