Vertex operator algebra

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 related notion of vertex algebra was introduced by Richard Borcherds in 1986, motivated by a construction of an infinite-dimensional Lie algebra due to Igor Frenkel. In the course of this construction, one employs a Fock space that admits an action of vertex operators attached to lattice vectors. Borcherds formulated the notion of vertex algebra by axiomatizing the relations between the lattice vertex operators, producing an algebraic structure that allows one to construct new Lie algebras by following Frenkel's method.

The notion of vertex operator algebra was introduced as a modification of the notion of vertex algebra, by Frenkel, James Lepowsky, and Arne Meurman in 1988, as part of their project to construct the moonshine module. They observed that many vertex algebras that appear in nature have a useful additional structure (an action of the Virasoro algebra), and satisfy a bounded-below property with respect to an energy operator. Motivated by this observation, they added the Virasoro action and bounded-below property as axioms.

We now have post-hoc motivation for these notions from physics, together with several interpretations of the axioms that were not initially known. Physically, the vertex operators arising from holomorphic field insertions at points (i.e., vertices) in two dimensional conformal field theory admit operator product expansions when insertions collide, and these satisfy precisely the relations specified in the definition of vertex operator algebra. Indeed, the axioms of a vertex operator algebra are a formal algebraic interpretation of what physicists call chiral algebras, or "algebras of chiral symmetries", where these symmetries describe the Ward identities satisfied by a given conformal field theory, including conformal invariance. Other formulations of the vertex algebra axioms include Borcherds's later work on singular commutative rings, algebras over certain operads on curves introduced by Huang, Kriz, and others, and D-module-theoretic objects called chiral algebras introduced by Alexander Beilinson and Vladimir Drinfeld. While related, these chiral algebras are not precisely the same as the objects with the same name that physicists use.

Important basic examples of vertex operator algebras include lattice VOAs (modeling lattice conformal field theories), VOAs given by representations of affine Kac–Moody algebras (from the WZW model), the Virasoro VOAs (i.e., VOAs corresponding to representations of the Virasoro algebra) and the moonshine module V, which is distinguished by its monster symmetry. More sophisticated examples such as affine W-algebras and the chiral de Rham complex on a complex manifold arise in geometric representation theory and mathematical physics.

The Locality axiom has several equivalent formulations in the literature, e.g., Frenkel-Lepowsky-Meurman introduced the Jacobi identity:

Borcherds[1] initially used the following two identities: for any vectors u, v, and w, and integers m and n we have

He later gave a more expansive version that is equivalent but easier to use: for any vectors u, v, and w, and integers m, n, and q we have

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.

Any finite-dimensional vertex algebra is commutative. In particular, even the smallest examples of noncommutative vertex algebras require significant introduction.

For a vertex operator algebra, the other Virasoro operators satisfy similar properties:

A basic example of a noncommutative vertex algebra is the rank 1 free boson, also called the Heisenberg vertex operator algebra. It is "generated" by a single vector b, in the sense that by applying the coefficients of the field b(z) = Y(b,z) to the vector 1, we obtain a spanning set. The underlying vector space is the infinite-variable polynomial ring C[x1,x2,...], where for positive n, the coefficient b–n of Y(b,z) acts as multiplication by xn, and bn acts as n times the partial derivative in xn. The action of b0 is multiplication by zero, producing the "momentum zero" Fock representation V0 of the Heisenberg Lie algebra (generated by bn for integers n, with commutation relations [bn,bm]=n δn,–m), i.e., induced by the trivial representation of the subalgebra spanned by bn, n ≥ 0.

The Fock space V0 can be made into a vertex algebra by the following reconstruction:

where :..: denotes normal ordering (i.e. moving all derivatives in x to the right). The vertex operators may also be written as a functional of a multivariable function f as:

if we understand that each term in the expansion of f is normal ordered.

The rank n free boson is given by taking an n-fold tensor product of the rank 1 free boson. 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.

Virasoro vertex operator algebras are important for two reasons: First, the conformal element in a vertex operator algebra canonically induces a homomorphism from a Virasoro vertex operator algebra, so they play a universal role in the theory. Second, they are intimately connected to the theory of unitary representations of the Virasoro algebra, and these play a major role in conformal field theory. In particular, the unitary Virasoro minimal models are simple quotients of these vertex algebras, and their tensor products provide a way to combinatorially construct more complicated vertex operator algebras.

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–1z as n ranges over integers greater than 1. The module then has partition function

This space has a vertex operator algebra structure, where the vertex operators are defined by:

Given a vertex algebra homomorphism from a Virasoro vertex algebra of central charge c to any other vertex algebra, the vertex operator attached to the image of ω automatically satisfies the Virasoro relations, i.e., the image of ω is a conformal vector. Conversely, any conformal vector in a vertex algebra induces a distinguished vertex algebra homomorphism from some Virasoro vertex operator algebra.

The Virasoro vertex operator algebras are simple, except when c has the form 1–6(pq)2/pq for coprime integers p,q strictly greater than 1 – this follows from Kac's determinant formula. In these exceptional cases, one has a unique maximal ideal, and the corresponding quotient is called a minimal model. When p = q+1, the vertex algebras are unitary representations of Virasoro, and their modules are known as discrete series representations. They play an important role in conformal field theory in part because they are unusually tractable, and for small p, they correspond to well-known statistical mechanics systems at criticality, e.g., the Ising model, the tri-critical Ising model, the three-state Potts model, etc. By work of Weiqang Wang[2] concerning fusion rules, we have a full description of the tensor categories of unitary minimal models. For example, when c=1/2 (Ising), there are three irreducible modules with lowest L0-weight 0, 1/2, and 1/16, and its fusion ring is Z[x,y]/(x2–1, y2x–1, xyy).

By replacing the Heisenberg Lie algebra with an untwisted affine Kac–Moody Lie algebra (i.e., the universal central extension of the loop algebra on a finite-dimensional simple Lie algebra), one may construct the vacuum representation in much the same way as the free boson vertex algebra is constructed. Here, WZW refers to the Wess–Zumino–Witten model, which produces the anomaly that is interpreted as the central extension.

By choosing a basis J a of the finite type Lie algebra, one may form a basis of the affine Lie algebra using J an = J a 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] For any choice of dual bases J a, Ja with respect to the level 1 inner product, the conformal element is

This construction can be altered to work for the rank 1 free boson. In fact, the Virasoro vectors form a one-parameter family ωs = 1/2 x12 + s x2, endowing the resulting vertex operator algebras with central charge 1−12s2. When s = 0, we have the following formula for the graded dimension:

This is known as the generating function for partitions, and is also written as q1/24 times the weight −1/2 modular form 1/η (the Dedekind eta function). The rank n free boson then has an n parameter family of Virasoro vectors, and when those parameters are zero, the character is qn/24 times the weight −n/2 modular form ηn.

Much like ordinary rings, vertex algebras admit a notion of module, or representation. Modules play an important role in conformal field theory, where they are often called sectors. 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.

Given a vertex algebra V with multiplication Y, a V-module is a vector space M equipped with an action YM: VMM((z)), satisfying the following conditions:

The modules of a vertex algebra form an abelian category. When working with vertex operator algebras, the previous definition is given the name "weak module", and V-modules are required to satisfy the additional condition that L0 acts semisimply with finite-dimensional eigenspaces and eigenvalues bounded below in each coset of Z. Work of Huang, Lepowsky, Miyamoto, and Zhang has shown at various levels of generality that modules of a vertex operator algebra admit a fusion tensor product operation, and form a braided tensor category.

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 of SL2(Z). In particular, if a VOA is holomorphic, i.e., its representation category is equivalent to that of vector spaces, then its partition function is SL2(Z)-invariant up to a constant. Huang showed that the category of modules of a regular VOA is a modular tensor category, and its fusion rules satisfy the Verlinde formula.

To connect with our first example, the irreducible modules of the rank 1 free boson are given by Fock spaces Vλ with some fixed momentum λ, i.e., induced representations of the Heisenberg Lie algebra, where the element b0 acts by scalar multiplication by λ. The space can be written as C[x1,x2,...]vλ, where vλ is a distinguished ground-state vector. The 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 bCn 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. For an automorphism σ of order N, the action has the form VMM((z1/N)), with the following monodromy condition: if uV satisfies σ u = exp(2πik/N)u, then un = 0 unless n satisfies n+k/NZ (there is some disagreement about signs among specialists). Geometrically, twisted modules can be attached to branch points on an algebraic curve with a ramified Galois cover. In the conformal field theory literature, twisted modules are called twisted sectors, and are intimately connected with string theory on orbifolds.

The lattice vertex algebra construction was the original motivation for defining vertex algebras. It is constructed by taking a sum of irreducible modules for the free boson corresponding to lattice vectors, and defining a multiplication operation by specifying intertwining operators between them. That is, if Λ is an even lattice, the lattice vertex algebra VΛ decomposes into free bosonic modules as:

Lattice vertex algebras are canonically attached to double covers of even integral lattices, rather than the lattices themselves. While each such lattice has a unique lattice vertex algebra up to isomorphism, the vertex algebra construction is not functorial, because lattice automorphisms have an ambiguity in lifting.[1]

The double covers in question are uniquely determined up to isomorphism by the following rule: elements have the form ±eα for lattice vectors α ∈ Λ (i.e., there is a map to Λ sending eα to α that forgets signs), and multiplication satisfies the relations eαeβ = (–1)(α,β)eβeα. Another way to describe this is that given an even lattice Λ, there is a unique (up to coboundary) normalised cocycle ε(α, β) with values ±1 such that (−1)(α,β) = ε(α, β) ε(β, α), where the normalization condition is that ε(α, 0) = ε(0, α) = 1 for all α ∈ Λ. This cocycle induces a central extension of Λ by a group of order 2, and we obtain a twisted group ring Cε[Λ] with basis eα (α ∈ Λ), and multiplication rule eαeβ = ε(α, β)eα+β – the cocycle condition on ε ensures associativity of the ring.[3]

The vertex operator attached to lowest weight vector vλ in the Fock space Vλ is

where zλ is a shorthand for the linear map that takes any element of the α-Fock space Vα to the monomial z(λ,α). The vertex operators for other elements of the Fock space are then determined by reconstruction.

As in the case of the free boson, one has a choice of conformal vector, given by an element s of the vector space Λ ⊗ C, but the condition that the extra Fock spaces have integer L0 eigenvalues constrains the choice of s: for an orthonormal basis xi, the vector 1/2 xi,12 + s2 must satisfy (s, λ) ∈ Z for all λ ∈ Λ, i.e., s lies in the dual lattice.

If the even lattice Λ is generated by its "root vectors" (those satisfying (α, α)=2), and any two root vectors are joined by a chain of root vectors with consecutive inner products non-zero then the vertex operator algebra is the unique simple quotient of the vacuum module of the affine Kac–Moody algebra of the corresponding simply laced simple Lie algebra at level one. This is known as the Frenkel–Kac (or FrenkelKacSegal) construction, and is based on the earlier construction by Sergio Fubini and Gabriele Veneziano of the tachyonic vertex operator in the dual resonance model. 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. In particular, one obtains a construction of all ADE type Lie groups directly from their root lattices. And this is commonly considered the simplest way to construct the 248 dimensional group E8.[3][4]

One of the simplest examples is the vertex operator superalgebra generated by a single free fermion ψ. 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. One may also describe it as a spin representation of the Clifford algebra on the quadratic space t1/2C[t,t−1](dt)1/2 with residue pairing. The vertex operator superalgebra is holomorphic, in the sense that all modules are direct sums of itself, i.e., the module category is equivalent to the category of vector spaces.

The tensor square of the free fermion is called the free charged fermion, and by boson–fermion correspondence, it is isomorphic to the lattice vertex superalgebra attached to the odd lattice Z.[3] This correspondence has been used by Date-Jimbo-Kashiwara-Miwa to construct soliton solutions to the KP hierarchy of nonlinear PDEs.

The Virasoro algebra has some supersymmetric extensions that naturally appear in superconformal field theory and superstring theory. The N=1, 2, and 4 superconformal algebras are of particular importance.

Infinitesimal holomorphic superconformal transformations of a supercurve (with one even local coordinate z and N odd local coordinates θ1,...,θN) are generated by the coefficients of a super-stress–energy tensor T(z, θ1, ..., θN).

When N=1, T has odd part given by a Virasoro field L(z), and even part given by a field

By examining the symmetry of the operator products, one finds that there are two possibilities for the field G: the indices n are either all integers, yielding the Ramond algebra, or all half-integers, yielding the Neveu–Schwarz algebra. These algebras have unitary discrete series representations at central charge

and unitary representations for all c greater than 3/2, with lowest weight h only constrained by h≥ 0 for Neveu–Schwarz and hc/24 for Ramond.

An N=1 superconformal vector in a vertex operator algebra V of central charge c is an odd element τ ∈ V of weight 3/2, such that

G−1/2τ = ω, and the coefficients of G(z) yield an action of the N=1 Neveu–Schwarz algebra at central charge c.

For N=2 supersymmetry, one obtains even fields L(z) and J(z), and odd fields G+(z) and G(z). The field J(z) generates an action of the Heisenberg algebras (described by physicists as a U(1) current). There are both Ramond and Neveu–Schwarz N=2 superconformal algebras, depending on whether the indexing on the G fields is integral or half-integral. 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. The unitary representations are given by discrete series with central charge c = 3-6/m for integers m at least 3, and a continuum of lowest weights for c > 3.

An N=2 superconformal structure on a vertex operator algebra is a pair of odd elements τ+, τ of weight 3/2, and an even element µ of weight 1 such that τ± generate G±(z), and µ generates J(z).

For N=3 and 4, unitary representations only have central charges in a discrete family, with c=3k/2 and 6k, respectively, as k ranges over positive integers.