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ON SOME VERTEX ALGEBRAS RELATED TO V(sl(n)) AND THEIR CHARACTERS

Transformation Groups, ISSN: 1531-586X, Vol: 26, Issue: 1, Page: 1-30
2021
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We consider several vertex operator algebras and superalgebras closely related to V(sl(n)) , n ≥ 3 : (a) the parafermionic subalgebra K(sl(n); −1) for which we completely describe its inner structure, (b) the vacuum algebra Ω(V(sl(n))), and (c) an infinite extension U of V(sl(n)) obtained from certain irreducible ordinary modules with integral conformal weights. It turns out that U is isomorphic to the coset vertex algebra psl(n|n)/sl(n), n ≥ 3. We show that V(sl(n)) admits precisely n ordinary irreducible modules, up to isomorphism. This leads to the conjecture that U is quasi-lisse.We present evidence in support of this conjecture: we prove that the (super)character of U is quasimodular of weight one by virtue of being the constant term of a meromorphic Jacobi form of index zero. Explicit formulas and MLDE for characters and supercharacters are given for g = sl(3) and outlined for general n. We present a conjectural family of 2nd order MLDEs for characters of vertex algebras psl(n|n), n ≥ 2. We finish with a theorem pertaining to characters of psl(n|n) and U-modules.

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