Vertex Algebras and Algebraic Curves

Nonsemisimple Fusion Algebras and the Verlinde Formula

“…The relevant first principles are the properly formulated axioms of chiral conformal field theory. The situation is thus reminiscent of the one with the ordinary (semisimple) Verlinde formula, whose proof could be attacked only after those axioms had been formulated [22] (see also [23,24]) for rational conformal field theory. In the semisimple case, the structure constants are expressed through the defining data of the representation category, which is a modular tensor category, and thus through the matrices of the basic B and F operations of [22] as…”

Nonrational Conformal Field Theory

“…There is a canonical Lie algebra U ′ (V ) which can be attached to a vertex algebra V , see [FBZ,Section 4.1]. U ′ (V ) is generated from the expansion coefficients A n introduced in (2.4).…”

“…Representations M of the vertex algebra V must in particular be representations of the Lie-algebra U ′ (V ) generated from the coefficients A n , see [FBZ,Section 5] for more details.…”

“…, z n , respectively. It can then be shown that the spaces F (X, R) and F (X ′ , R ′ ) are canonically isomorphic [FBZ,Theorem 10.3.1]. The isomorphism is defined by demanding that…”

The PBW Filtration, Demazure Modules and Toroidal Current Algebras

“…These series are also called fields. They play a crucial role in the theory of vertex operator algebras (see [15,22,3]). We will need the field which corresponds to the highest root θ of g. Namely, let e θ ∈ g be a highest weight vector in the adjoint representation.…”