any weight 1 field whose residue has square zero. The usual method is to tensor with fermions, as one then has a canonical differential. An important special case is quantum Drinfeld–Sokolov reduction applied to affine Kac–Moody algebras to obtain affine W-algebras as degree 0 cohomology. These W algebras also admit constructions as vertex subalgebras of free bosons given by kernels of screening operators. Related algebraic structures If one considers only the singular part of the OPE in a vertex algebra, one arrives at the definition of a Lie conformal algebra. Since one is often only concerned with the singular part of the OPE, this makes Lie conformal algebras a natural object to study. There is a functor from vertex algebras to Lie conformal algebras that forgets the regular part of OPEs, and it has a left adjoint, called the "universal vertex algebra" functor. Vacuum modules of affine Kac–Moody algebras and Virasoro vertex algebras are universal vertex algebras, and in particular, they can be described very concisely once the background theory is developed. There are several generalizations of the notion of vertex algebra in the literature. Some mild generalizations involve a weakening of the locality axiom to allow monodromy, e.g., the abelian intertwining algebras of Dong and Lepowsky. One may view these roughly as vertex algebra objects in a braided tensor category of graded vector spaces, in much the same way that a vertex superalgebra is such an object in the category of super vector spaces. More complicated generalizations relate to q-deformations and representations of quantum groups, such as in work of Frenkel–Reshetikhin, Etingof–Kazhdan, and Li. Beilinson and Drinfeld introduced a sheaf-theoretic notion of chiral algebra that is closely related to the notion of vertex algebra, but is defined without using any visible power series. Given an algebraic curve X, a chiral algebra on X is a DX-module A equipped with a multiplication operation on X×X that satisfies an associativity condition. They also introduced an equivalent notion of factorization algebra that is a system of quasicoherent sheaves on all finite products of the curve, together with a compatibility condition involving pullbacks to the complement of various diagonals. Any translation-equivariant chiral algebra on the affine line can be identified with a vertex algebra by taking the fiber at a point, and there is a natural way to attach a chiral algebra on a smooth algebraic curve to any vertex operator algebra. See