ON SOME VERTEX ALGEBRAS RELATED TO $$ {V}_{-1}\left(\mathfrak{sl}(n)\right) $$ AND THEIR CHARACTERS

Autor: Antun Milas, Dražen Adamović
Rok vydání: 2020
Předmět:
Zdroj: Transformation Groups. 26:1-30
ISSN: 1531-586X
1083-4362
Popis: We consider several vertex operator algebras and superalgebras closely related to $$ {V}_{-1}\left(\mathfrak{sl}(n)\right) $$ , n ≥ 3 : (a) the parafermionic subalgebra K( $$ \mathfrak{sl} $$ (n); −1) for which we completely describe its inner structure, (b) the vacuum algebra Ω(V−1( $$ \mathfrak{sl} $$ (n))), and (c) an infinite extension $$ \mathcal{U} $$ of V−1( $$ \mathfrak{sl} $$ (n)) obtained from certain irreducible ordinary modules with integral conformal weights. It turns out that $$ \mathcal{U} $$ is isomorphic to the coset vertex algebra $$ \mathfrak{psl} $$ (n|n)1/ $$ \mathfrak{sl} $$ (n)1, n ≥ 3. We show that V−1( $$ \mathfrak{sl} $$ (n)) admits precisely n ordinary irreducible modules, up to isomorphism. This leads to the conjecture that $$ \mathcal{U} $$ is quasi-lisse.We present evidence in support of this conjecture: we prove that the (super)character of $$ \mathcal{U} $$ 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 $$ \mathfrak{g} $$ = $$ \mathfrak{sl} $$ (3) and outlined for general n. We present a conjectural family of 2nd order MLDEs for characters of vertex algebras $$ \mathfrak{psl} $$ (n|n)1, n ≥ 2. We finish with a theorem pertaining to characters of $$ \mathfrak{psl} $$ (n|n)1 and $$ \mathcal{U} $$ -modules.
Databáze: OpenAIRE
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