engleski

On some vertex algebras related to $V_{;;;; -1};;;; (sl(n))$ and their characters

We consider several vertex operator algebras and superalgebras closely related to V−1(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−1(sl(n))), and (c) an infinite extension U of V−1(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)1/sl(n)1, n ≥ 3. We show that V−1(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)1, n ≥ 2. We finish with a theorem pertaining to characters of psl(n|n)1 and U-modules.

vertex algebras ; affine Lie algebras ; parafermion algebra ; charactersi

University at Albany, SAD