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In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory.In addition to physical applications, vertex operator algebras have proven useful in purely mathematical contexts such as monstrous moonshine and the geometric Langlands correspondence.
The center of the algebra is generated by , and the algebra is a ... This field, and affine primary fields in general, are sometimes called vertex operators. [3]
A vertex operator algebra is rational if the category of admissible modules is semisimple and there are only finitely many irreducibles. It was conjectured that rationality is equivalent to C 2 -cofiniteness and a stronger condition regularity, however this was disproved in 2007 by Adamovic and Milas who showed that the triplet vertex operator ...
An example of an operator product algebra is the vertex operator algebra. It is currently hoped that operator product algebras can be used to axiomatize all of quantum field theory; they have successfully done so for the conformal field theories, and whether they can be used as a basis for non-perturbative QFT is an open research area.
An algebra is a module, ... Vertex operator algebra; von Neumann algebra; Weyl algebra; Zinbiel algebra; This is a list of fields of algebra. Linear algebra;
From this point of view, operator algebras can be regarded as a generalization of spectral theory of a single operator. In general, operator algebras are non-commutative rings. An operator algebra is typically required to be closed in a specified operator topology inside the whole algebra of continuous linear operators. In particular, it is a ...
All the structural relations of the Kac–Moody and Virasoro algebra can be recovered from these relations, including the Segal–Sugawara construction. Every other integral representation H i at the same level becomes a module for the vertex algebra, in the sense that for each a there is a vertex operator V i (a, z) on H i such that
The quotient algebra T/E is then the algebraic structure or variety. Thus, for example, groups have a signature containing two operators: the multiplication operator m, taking two arguments, and the inverse operator i, taking one argument, and the identity element e, a constant