Local systems of vertex operators, vertex superalgebras and modules
For an associative algebra \(A\), the representation of \(A\) on a vector space \(M\) is given by an algebra homomorphism from \(A\) to \(\text{End}\,M\). Vertex operator algebras are natural one-variable analogues of commutative associative algebras with an identity element. The author introduces so-called ``local systems of vertex operator algebras'', which are analogues of \(\text{End}\,M\). These local systems form vertex algebras, whose axioms can be obtained from those of vertex operator algebras by dropping the Virasoro relation. Then the representation of a vertex operator algebra \(V\) on a vector space \(M\) is given by a vertex algebra homomorphism from \(V\) to a local system over \(M\).
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