Simple toroidal vertex algebras and their irreducible modules.

In this paper, we continue the study on toroidal vertex algebras initiated in [15] , to study concrete toroidal vertex algebras associated to toroidal Lie algebra L r ( g ˆ ) = g ˆ ⊗ L r , where g ˆ is an untwisted affine Lie algebra and L r = C [ t 1 ± 1 , … , t r ± 1 ] . We first construct an ( r + 1 ) -toroidal vertex algebra V ( T , 0 ) and show that the category of restricted L r ( g ˆ ) -modules is canonically isomorphic to that of V ( T , 0 ) -modules. Let c denote the standard central element of g ˆ and set S c = U ( L r ( C c ) ) . We furthermore study a distinguished subalgebra of V ( T , 0 ) , denoted by V ( S c , 0 ) . We show that (graded) simple quotient toroidal vertex algebras of V ( S c , 0 ) are parametrized by a Z r -graded ring homomorphism ψ : S c → L r such that Im ψ is a Z r -graded simple S c -module. Denote by L ( ψ , 0 ) the simple quotient ( r + 1 ) -toroidal vertex algebra of V ( S c , 0 ) associated to ψ . We determine for which ψ , L ( ψ , 0 ) is an integrable L r ( g ˆ ) -module and we then classify irreducible L ( ψ , 0 ) -modules for such a ψ . For our need, we also obtain various general results. [ABSTRACT FROM AUTHOR]