Vertex operator algebra
In addition to physical applications, vertex operator algebras have proven useful in purely mathematical contexts such as monstrous moonshine and the geometric Langlands correspondence.In the course of this construction, one employs a Fock space that admits an action of vertex operators attached to elements of a lattice.: and satisfies the following properties: A homomorphism of vertex algebras is a map of the underlying vector spaces that respects the additional identity, translation, and multiplication structure.Homomorphisms of vertex operator algebras have "weak" and "strong" forms, depending on whether they respect conformal vectors.For any vector b in n-dimensional space, one has a field b(z) whose coefficients are elements of the rank n Heisenberg algebra, whose commutation relations have an extra inner product term: [bn,cm]=n (b,c) δn,–m.The Virasoro vertex operator algebra is defined as an induced representation of the Virasoro algebra: If we choose a central charge c, there is a unique one-dimensional module for the subalgebra C[z]∂z + K for which K acts by cId, and C[z]∂z acts trivially, and the corresponding induced module is spanned by polynomials in L–n = –z−n–1∂z as n ranges over integers greater than 1.The Virasoro vertex operator algebras are simple, except when c has the form 1–6(p–q)2/pq for coprime integers p,q strictly greater than 1 – this follows from Kac's determinant formula.By work of Weiqang Wang[5] concerning fusion rules, we have a full description of the tensor categories of unitary minimal models.yields a split extension, and the vacuum module is induced from the one-dimensional representation of the latter on which a central basis element acts by some chosen constant called the "level".This matches the loop algebra convention, where levels are discretized by third cohomology of simply connected compact Lie groups.By choosing a basis Ja of the finite type Lie algebra, one may form a basis of the affine Lie algebra using Jan = Ja tn together with a central element K. By reconstruction, we can describe the vertex operators by normal ordered products of derivatives of the fields When the level is non-critical, i.e., the inner product is not minus one half of the Killing form, the vacuum representation has a conformal element, given by the Sugawara construction.A standard assumption in the physics literature is that the full Hilbert space of a conformal field theory decomposes into a sum of tensor products of left-moving and right-moving sectors: That is, a conformal field theory has a vertex operator algebra of left-moving chiral symmetries, a vertex operator algebra of right-moving chiral symmetries, and the sectors moving in a given direction are modules for the corresponding vertex operator algebra.More precisely, they are required to satisfy the additional condition that L0 acts semisimply with finite-dimensional eigenspaces and eigenvalues bounded below in each coset of Z.When the category of V-modules is semisimple with finitely many irreducible objects, the vertex operator algebra V is called rational.Rational vertex operator algebras satisfying an additional finiteness hypothesis (known as Zhu's C2-cofiniteness condition) are known to be particularly well-behaved, and are called regular.For example, Zhu's 1996 modular invariance theorem asserts that the characters of modules of a regular VOA form a vector-valued representation ofThese are used to construct lattice vertex algebras, which as vector spaces are direct sums of Heisenberg modules, when the image ofThe module category is not semisimple, since one may induce a representation of the abelian Lie algebra where b0 acts by a nontrivial Jordan block.For the rank n free boson, one has an irreducible module Vλ for each vector λ in complex n-dimensional space.Each vector b ∈ Cn yields the operator b0, and the Fock space Vλ is distinguished by the property that each such b0 acts as scalar multiplication by the inner product (b, λ).Unlike ordinary rings, vertex algebras admit a notion of twisted module attached to an automorphism.Geometrically, twisted modules can be attached to branch points on an algebraic curve with a ramified Galois cover.Among other features, the zero modes of the vertex operators corresponding to root vectors give a construction of the underlying simple Lie algebra, related to a presentation originally due to Jacques Tits.That is, one forms the direct sum of the Leech lattice VOA with the twisted module, and takes the fixed points under an induced involution.Malikov, Schechtman, and Vaintrob showed that by a method of localization, one may canonically attach a bcβγ (boson–fermion superfield) system to a smooth complex manifold.A prototypical example is the construction of Beem, Leemos, Liendo, Peelaers, Rastelli, and van Rees which associates a vertex algebra to any 4d N=2 superconformal field theory.When the theory admits a weak coupling limit, the vertex algebra has an explicit description as a BRST reduction of a bcβγ system.If in addition there is a Virasoro element ω in the even part of V2, and the usual grading restrictions are satisfied, then V is called a vertex operator superalgebra.As a Virasoro representation, it has central charge 1/2, and decomposes as a direct sum of Ising modules of lowest weight 0 and 1/2.However, the U(1) current gives rise to a one-parameter family of isomorphic superconformal algebras interpolating between Ramond and Neveu–Schwartz, and this deformation of structure is known as spectral flow.