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Orbifolds of symplectic fermion algebras. (English) Zbl 1395.17062

Summary: We present a systematic study of the orbifolds of the rank \( n\) symplectic fermion algebra \( \mathcal {A}(n)\), which has full automorphism group \(\mathrm{Sp}(2n)\). First, we show that \( \mathcal {A}(n)^{\mathrm{Sp}(2n)}\) and \( \mathcal {A}(n)^{\mathrm{GL}(n)}\) 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 \( \mathcal {A}(mn)^{\mathrm{Sp}(2n)}\) and \( \mathcal {A}(mn)^{\mathrm{GL}(n)}\) for all \(m\), \(n\geq 1\). We compute the characters of the irreducible representations of \(\mathcal {A}(mn)^{\mathrm{Sp}(2n)\times \mathrm{SO}(m)}\) and \( \mathcal {A}(mn)^{\mathrm{GL}(n)\times \mathrm{GL}(m)}\) 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

MSC:

17B69 Vertex operators; vertex operator algebras and related structures
11F22 Relationship to Lie algebras and finite simple groups
13A50 Actions of groups on commutative rings; invariant theory
17B65 Infinite-dimensional Lie (super)algebras
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