The super 𝒲_{1+∞} algebra with integral central charge

Creutzig, Thomas; Linshaw, Andrew

doi:10.1090/S0002-9947-2015-06214-X

The super $\mathcal {W}_{1+\infty }$ algebra with integral central charge
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by Thomas Creutzig and Andrew R. Linshaw PDF

Trans. Amer. Math. Soc. 367 (2015), 5521-5551 Request permission

Abstract:

The Lie superalgebra $\mathcal {S}\mathcal {D}$ of regular differential operators on the super circle has a universal central extension $\widehat {\mathcal {S}\mathcal {D}}$. For each $c\in \mathbb {C}$, the vacuum module $\mathcal {M}_c(\widehat {\mathcal {S}\mathcal {D}})$ of central charge $c$ admits a vertex superalgebra structure, and $\mathcal {M}_c(\widehat {\mathcal {S}\mathcal {D}}) \cong \mathcal {M}_{-c}(\widehat {\mathcal {S}\mathcal {D}})$. The irreducible quotient $\mathcal {V}_c(\widehat {\mathcal {S}\mathcal {D}})$ of the vacuum module is known as the super $\mathcal {W}_{1+\infty }$ algebra. We show that for each integer $n>0$, $\mathcal {V}_{n}(\widehat {\mathcal {S}\mathcal {D}})$ has a minimal strong generating set consisting of $4n$ fields, and we identify it with a $\mathcal {W}$-algebra associated to the purely odd simple root system of $\mathfrak {g} \mathfrak {l}(n|n)$. Finally, we realize $\mathcal {V}_{n}(\widehat {\mathcal {S}\mathcal {D}})$ as the limit of a family of commutant vertex algebras that generically have the same graded character and possess a minimal strong generating set of the same cardinality.

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