CONSTRUCTIONS OF VERTEX OPERATOR ALGEBRAS

Alex J. Feingold

Introduction

In these notes are presented the definitions and constructions of the new algebraic structures, vertex operator algebras and superalgebras, which play an important role in mathematics and physics. In mathematics, one cannot avoid them if one is interested in the representation theory of Kac-Moody Lie algebras, the Virasoro algebra or the Moonshine Module for the Monster group [B,FLM]. In physics, one cannot avoid them if one is interested in string theory or conformal field theory (Wess-Zumino-Witten models, minimal models) [BPZ,G]. I also present here the definition of a vertex operator para-algebra, a generalization introduced by myself, Frenkel and Ries in our monograph [FFR], from which most of the material here is taken. (Some is from [FR1].) We believe that para-algebras will be of importance in the study of intertwining vertex operators and their relationship to quantum groups. We have made some first steps in this direction in [FR2].

Primary references for spinor and lattice constructions of the affine Kac-Moody algebras are [F1,KP] and [FK]. See [F2] for the correspondence between the two constructions, and see [FF] for a unified treatment of bosonic and fermionic constructions of all classical affine algebras. The basic references for vertex operator algebras are [B,FLM], but in addition to [FFR] for generalizations one also should mention [T].

Definitions

Lie Algebras.

Let V be a vector space over ℂ, and let

ad : V → End(V)  (1)

be a linear transformation satisfying the Jacobi identity

ad(u)ad(v) − ad(v)ad(u) = ad(ad(u)v)  (2)

for any u, v ∈ V. With the bracket [u, v] = ad(u)v, V becomes a Lie algebra with trivial center if ker(ad) is trivial.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 17B65, 17B67, 81R10; Secondary 17B68, 17A70, 81T40.

Work partially supported by National Science Foundation Grants DMS-8420318 and DMS-8703156, and by the Institute for Advanced Study in Princeton, N. J.

This paper is in final form and no version of it will be submitted for publication elsewhere.

The Cauchy residue formula.

For f(z) ∈ ℂ[z, z^{−1}, (z − z_{0})^{−1}] we have

−Res_{z=∞}f(z) − Res_{z=0}f(z) = Res_{z=z_{0}}f(z).  (3)

Vertex Operator Algebras.

Let V be a vector space and let V[[z, z^{−1}]] denote the vector space of formal Laurent series with coefficients in V. Let

ad_{z} : V → End(V)[[z, z^{−1}]]  (4)

be a linear transformation, which may be viewed as a generating function

ad_{z} = ∑_{n∈ℤ} ad_{(n)}z^{−n−1}  (5)

for the linear transformations

ad_{(n)} : V → End(V),  (6)

such that for u, v ∈ V, we have

ad_{(n)}(u)v = 0 for n sufficiently large.  (7)

For any z_{0} ∈ ℂ^{∗} assume that ad_{z} satisfies the Jacobi Identity

− Res_{z=∞}(ad_{z}(u)ad_{z_{0}}(v)f(z)) − Res_{z=0}(ad_{z_{0}}(v)ad_{z}(u)f(z))

= Res_{z=z_{0}}(ad_{z_{0}}(ad_{z−z_{0}}(u)v)f(z)) (8)

for any f(z) ∈ ℂ[z, z^{−1}, (z−z_{0})^{−1}]. For f(z) = z^{p}(z−z_{0})^{r}, p, r ∈ ℤ, (8) is equivalent to the identities

\begin{matrix}∑_{0≤i∈ℤ} \binom{r}{i}(−1)^{i}(ad_{(m+r−i)}(u)ad_{(n+i)}(v) \\ − (−1)^{r}ad_{(n+r−i)}(v)ad_{(m+i)}(u)) \\ = ∑_{0≤k∈ℤ} \binom{m}{k}ad_{(m+n−k)}(ad_{(r+k)}(u)v) \end{matrix}  (9)

for all m, n, r ∈ ℤ. The finiteness of (9) when applied to any particular vector in V is guaranteed by (7), so (8) is well-defined. We use the notation Y (v, z) for the vertex operator ad_{z}(v) and {v}_{n} for ad_{(n)}(v). We also require that ker(ad_{z}) be trivial. Additional axioms postulate the existence of two distinguished vectors 1, ω ∈ V such that

ad_{z}(1) = Id_{V},  (10)
ad_{z}(ω) = ∑_{n∈ℤ} D(n)z^{−n−2}  (11)

where D(n) = ad_{(n+1)}(ω), n ∈ ℤ, generate the Virasoro algebra

[D(m), D(n)] = (n − m)D(m + n) + \frac{1}{12}(m^{3} − m)δ_{m,−n}c Id_{V},  (12)

the scalar c is called the rank of V, and

[D(−1), ad_{z}(v)] = −(d/dz)ad_{z}(v).  (13)

Note that (9), with m = 0, r = 0 and u = ω, gives

[D(−1), ad_{z}(v)] = ad_{z}(D(−1)v).  (14)

The last axioms require that D(0) defines a ℤ-grading of V,

V = ∐_{n∈ℤ}(V)_{n}  (15)

where

(V)_{n} = {v ∈ V | D(0)v = −nv}  (16)

is finite-dimensional for each n ∈ ℤ and (V)_{n} = {0} for n sufficiently small. Define a vertex operator algebra to be a quadruple (V, ad_{z}, 1, ω) satisfying the above axioms.

Define the weight of a homogeneous vector by the rule

wt(v) = n if v ∈ (V)_{n}.  (17)

Note that (9), with m = 0, r = 1 and u = ω, and (13) imply

wt(ad_{(n)}(v)w) = wt(v) + wt(w) − n − 1.  (18)

With r = 0 in (9) we get the commutator formula

[ad_{(m)}(u), ad_{(n)}(v)] = ∑_{0≤k∈ℤ} \binom{m}{k}ad_{(m+n−k)}(ad_{(k)}(u)v)  (19)

for all m, n ∈ ℤ, and the summation is actually finite.

A module for a Lie algebra.

Let V be a Lie algebra with bracket [u, v] = ad(u)v as in (1)-(2), and let W be a vector space. Then (W, π) is a V-module if

π : V → End(W)  (20)

is a linear transformation satisfying

π(u)π(v) − π(v)π(u) = π(ad(u)v)  (21)

for any u, v ∈ V. We say π is faithful if ker(π) is trivial.

A module for a vertex operator algebra.

Let

π_{z} : V → End(W)[[z, z^{−1}]]  (22)

be a linear transformation, which may be viewed as a generating function

π_{z} = ∑_{n∈ℤ} π_{(n)}z^{−n−1}  (23)

for the linear transformations

π_{(n)} : V → End(W),  (24)

such that for v ∈ V, w ∈ W, we have

π_{(n)}(v)w = 0 for n sufficiently large.  (25)

For any z_{0} ∈ ℂ^{∗} assume that π_{z} satisfies the Jacobi Identity

− Res_{z=∞}(π_{z}(u)π_{z_{0}}(v)f(z)) − Res_{z=0}(π_{z_{0}}(v)π_{z}(u)f(z))

= Res_{z=z_{0}}(π_{z_{0}}(ad_{z−z_{0}}(u)v)f(z)) (26)

for any f(z) ∈ ℂ[z, z^{−1}, (z − z_{0})^{−1}]. This is equivalent to the identities

\begin{matrix}∑_{0≤i∈ℤ} \binom{r}{i}(−1)^{i}(π_{(m+r−i)}(u)π_{(n+i)}(v) \\ − (−1)^{r}π_{(n+r−i)}(v)π_{(m+i)}(u)) \\ = ∑_{0≤k∈ℤ} \binom{m}{k}π_{(m+n−k)}(ad_{(r+k)}(u)v) \end{matrix}  (27)

for all m, n, r ∈ ℤ, which imply

[π_{(m)}(u), π_{(n)}(v)] = ∑_{0≤k∈ℤ} \binom{m}{k}π_{(m+n−k)}(ad_{(k)}(u)v).  (28)

As additional axioms we require the analogs of (10)-(13) with ad_{z} replaced by π_{z} and Id_{V} replaced by Id_{W}. The analog of (14) also follows. The only new feature is that W is not necessarily ℤ-graded, but it is ℂ-graded. In fact, if W is irreducible then it is graded by ℤ + Δ_{W} where Δ_{W} ∈ ℂ, defined mod ℤ, is an important characteristic of the module. Define a V-module to be a pair (W, π_{z}) satisfying these axioms.

The main task in constructing vertex operator algebras and their modules lies in the verification of the Jacobi Identity (8) and (26). We use matrix coefficients which can be defined from nondegenerate Hermitian forms on V and W. It suffices to establish three analytic properties of the matrix coefficients, rationality, permutability and associativity. Let (· , ·) denote a suitable Hermitian form on V.

Rationality.

For u, v, w, w′ ∈ V the series

(ad_{z}(u)ad_{z_{0}}(v)w, w′)  (29)

converges absolutely when |z| > |z_{0}| > 0 to a rational function

f(z, z_{0}) ∈ ℂ[z, z^{−1}, (z − z_{0})^{−1}].

Permutability.

The rational function to which the series

(ad_{z_{0}}(v)ad_{z}(u)w, w′)  (30)

converges absolutely when |z_{0}| > |z| > 0, is the same function f(z, z_{0}).

Associativity.

The series

(ad_{z_{0}}(ad_{z−z_{0}}(u)v)w, w′)  (31)

converges absolutely when |z_{0}| > |z − z_{0}| > 0 to the same function f(z, z_{0}). Analogous definitions of these properties can be made on a module W.

Lie superalgebras.

Let V = V_{0} ⊕ V_{1} be a ℤ_{2}-graded vector space and let |v| = 0 if v ∈ V_{0}, |v| = 1 if v ∈ V_{1}. Let ad : V → End(V) be a linear transformation satisfying the super-Jacobi identity

ad(u)ad(v) − (−1)^{|u| |v|}ad(v)ad(u) = ad(ad(u)v)  (32)

for u and v homogeneous, and

|ad(u)v| = |u| + |v|.  (33)

With the super-bracket [u, v] = ad(u)v, V becomes a Lie superalgebra with trivial center if ker(ad) is trivial.

Vertex operator superalgebras.

Let V = V_{0} ⊕ V_{1} be \frac{1}{2}ℤ-graded in (15), where

V_{0} = ∐_{n∈ℤ}(V)_{n},  V_{1} = ∐_{n∈ℤ+ \frac{1}{2}}(V)_{n},  (34)

and define |v| as above. The only change to (8) is the additional factor (−1)^{|u| |v|} in the Res_{z=0} term, which changes the corresponding term in (9). The left side of (19) becomes a super-bracket. We also require that

|ad_{(n)}(u)v| = |u| + |v|.  (35)

A module for a Lie superalgebra.

A module W for a Lie superalgebra V has a ℤ_{2}-grading W = W_{0} ⊕ W_{1}, the factor (−1)^{|u| |v|} in the second term of (21), and |π(v)w| = |v|+|w| for homogeneous v ∈ V, w ∈ W.

A module for a vertex operator superalgebra.

A vertex operator superalgebra module W = W_{0} ⊕ W_{1} may be ℂ-graded, but for W irreducible, it is graded by \frac{1}{2}ℤ + Δ_{W}. The Jacobi Identity (26) is modified by the additional factor (−1)^{|u| |v|} in the Res_{z=0} term, and we also require |π_{(n)}(v)w| = |v| + |w|.

Vertex operator para-algebras.

Let Γ be a finite abelian group with exponent g. For each γ ∈ Γ, suppose Δ_{γ} ∈ ℚ and, for γ_{1}, γ_{2} ∈ Γ, let

Δ(γ_{1}, γ_{2}) = Δ_{γ_{1}} + Δ_{γ_{2}} − Δ_{γ_{1}+γ_{2}} ∈ \frac{1}{g}ℤ  (36)

be bilinear mod ℤ. Assume that

V = ⨁_{γ∈Γ} V_{γ}  (37)

is a Γ-graded vector space and

ad_{z} : V → End(V)[[z^{1/g}, z^{−1/g}]]  (38)

is a linear transformation with trivial kernel such that for u ∈ V_{γ_{1}}, and v ∈ V_{γ_{2}}, we have

ad_{z}(u)v = ∑_{n∈ℤ+Δ(γ_{1},γ_{2})} ad_{(n)}(u)v z^{−n−1}  (39)

and

ad_{(n)}(u)v ∈ V_{γ_{1}+γ_{2}}  (40)

is zero for n sufficiently large. Suppose that

(a) There are distinguished vectors 1, ω ∈ V_{0} satisfying the usual axioms (10) - (13),

(b) D(0) defines a ℚ-grading on V such that V_{γ} is graded by ℤ + Δ_{γ},

(c) Each graded piece (V_{γ})_{n} is finite-dimensional,

(d) (V_{γ})_{n} = {0} for n sufficiently small. The main axiom is the Jacobi Identity

− Res_{z=∞}(ad_{z}(u)ad_{z_{0}}(v)wf(z, z_{0})) − η Res_{z=0}(ad_{z_{0}}(v)ad_{z}(u)wf(z, z_{0}))

= Res_{z=z_{0}}(ad_{z_{0}}(ad_{z−z_{0}}(u)v)wf(z, z_{0})) (41)

for u ∈ V_{γ_{1}}, v ∈ V_{γ_{2}}, w ∈ V_{γ_{3}},

f(z, z_{0}) ∈ z^{Δ(γ_{1},γ_{3})} z_{0}^{Δ(γ_{2},γ_{3})} (z − z_{0})^{Δ(γ_{1},γ_{2})}ℂ[z, z^{−1}, z_{0}, z_{0}^{−1}, (z − z_{0})^{−1}] (42)

where η = η(γ_{1}, γ_{2}) ∈ ℂ^{∗} is a bilinear function on Γ × Γ. Bilinearity implies that η(γ_{1}, γ_{2}) is a root of unity and η is a 2-cocycle representing a cohomology class in H^{2}(Γ, ℤ_{g}). This class is restricted by the further assumption that η e^{πiΔ} is skew-symmetric, that is,

η(γ_{1}, γ_{2}) e^{πiΔ(γ_{1},γ_{2})} = η(γ_{2}, γ_{1})^{−1} e^{−πiΔ(γ_{2},γ_{1})}.  (43)

We must be precise about how the three residues in (41) are to be computed. Let

p ∈ ℤ + Δ(γ_{1}, γ_{2}) and m ∈ ℤ + Δ(γ_{1}, γ_{3}).  (44)

In the first term of (41) we expand (z − z_{0})^{p} as

z^{p} (1 − \frac{z_{0}}{z} )^{p} = z^{p} ∑_{0≤k∈ℤ} \binom{p}{k}(−1)^{k} (\frac{z_{0}}{z} )^{k}  (45)

and find −Res_{z=∞} to be the coefficient of z^{−1}. In the second term we expand (z − z_{0})^{p} as

e^{πip}z_{0}^{p} (1 − \frac{z}{z_{0}} )^{p} = e^{πip}z_{0}^{p} ∑_{0≤k∈ℤ} \binom{p}{k}(−1)^{k} (\frac{z}{z_{0}} )^{k}  (46)

and find Res_{z=0} to be the coefficient of z^{−1}. In the third term we let ζ = z − z_{0} and expand z^{m} = (z_{0} + ζ)^{m} as

z_{0}^{m} (1 + \frac{ζ}{z_{0}} )^{m} = z_{0}^{m} ∑_{0≤k∈ℤ} \binom{m}{k} (\frac{ζ}{z_{0}} )^{k}  (47)

and find Res_{z=z_{0}} to be the coefficient of ζ^{−1}. The Jacobi Identity is equivalent to the identities

\begin{matrix}∑_{0≤k∈ℤ} \binom{p}{k}(−1)^{k}(ad_{(m+p−k)}(u)ad_{(n+k)}(v) \\ − η(γ_{1}, γ_{2})e^{πip}ad_{(n+p−k)}(v)ad_{(m+k)}(u))w \\ = ∑_{0≤k∈ℤ} \binom{m}{k}ad_{(m+n−k)}(ad_{(p+k)}(u)v)w \end{matrix}  (48)

for all m ∈ ℤ + Δ(γ_{1}, γ_{3}), n ∈ ℤ + Δ(γ_{2}, γ_{3}), p ∈ ℤ + Δ(γ_{1}, γ_{2}). We then say that (V, ad_{z}, 1, ω, Γ, Δ, η) is a vertex operator para-algebra.

Discussion.

Let us explain the meaning of the bilinearity of η and the skew-symmetry of ηe^{πiΔ} in our definition. In the examples we consider, we prove the Jacobi Identity by proving rationality, permutability and associativity theorems. These properties apply to matrix coefficients multiplied by algebraic functions chosen so that the results are rational functions. When the permutability theorem is applied to do a transposition twice, the rational function should remain unchanged. Along with (46), this requires the skew-symmetry condition on ηe^{πiΔ}. Applying the permutability and associativity theorems to rationalized matrix coefficients involving more than two vertex operators, the same considerations yield the bilinearity condition on η. In other approaches one considers the analytic continuations of the matrix coefficients, compares the branches in a common domain, and choices can lead to representations of the braid group.

A vertex operator superalgebra is the case when Γ = ℤ_{2} with Δ_{0} = 0, Δ_{1} = \frac{1}{2} and η(γ_{1}, γ_{2}) = −1 if γ_{1} = γ_{2} = 1, η(γ_{1}, γ_{2}) = 1 otherwise.

The Fermionic (Spinor) Construction

Spinor Construction of affine Kac-Moody algebra D_{l}^{(1)}.

Let g be a finite dimensional Lie algebra of type D_{l}. The homogeneously graded affine Lie algebra ĝ associated with g is defined to be

ĝ = g ⊗ ℂ[t, t^{−1}] ⊕ ℂc ⊕ ℂd

where c is central and d = t(d/dt). For x ∈ g, n ∈ ℤ, we write x(n) = x ⊗ t^{n}. Then the brackets in ĝ are

\begin{aligned}[x(m), y(n)] & = [x, y](m + n) + mδ_{m,−n}⟨x, y⟩c, \\ [d, x(m)] & = mx(m). \end{aligned}

The spinor construction of ĝ begins with a nondegenerate symmetric bilinear form ⟨ , ⟩ on a vector space A ≅ ℂ^{2l} such that A = A^{+} ⊕ A^{−} is a polarization into maximally isotropic subspaces A^{±} ≅ ℂ^{l}. Let {a_{1}, …, a_{l}} be a basis of A^{+} and let {a_{1}^{∗}, …, a_{l}^{∗}} be a basis of A^{−} such that ⟨a_{i}, a_{j}^{∗}⟩ = δ_{ij}. We call {a_{1}, …, a_{l}, a_{1}^{∗}, …, a_{l}^{∗}} a “canonical” basis of A. Let

A(ℤ) = A ⊗ ℂ[t, t^{−1}] and A(ℤ + \frac{1}{2}) = A ⊗ t^{1/2}ℂ[t, t^{−1}].  (1)

For Z = ℤ or ℤ + \frac{1}{2} denote a ⊗ t^{n} by a(n) for a ∈ A, n ∈ Z, and let

ι = \{ \begin{matrix}+1 & if Z = ℤ \\ −1 & Z = ℤ + \frac{1}{2}. \end{matrix}  (2)

Extend the form to A(Z) by setting

⟨a(m), b(n)⟩ = ⟨a, b⟩δ_{m,−n}.  (3)

Then A(Z) = A(Z)^{+} ⊕ A(Z)^{−} is a polarization with respect to the extended form, where

\begin{aligned}A(ℤ)^{±} & = A^{±} ⊕ (A ⊗ t^{±1}ℂ[t^{±1}]), \\ A(ℤ + \frac{1}{2})^{±} & = A ⊗ t^{±1/2}ℂ[t^{±1}]. \end{aligned}  (4)

Define an associative algebra Cliff(Z) with unit element 1, generators from A(Z) and relations

a(m)b(n) + b(n)a(m) = ⟨a(m), b(n)⟩1 = ⟨a, b⟩δ_{m,−n}1.  (5)

VERTEX OPERATOR ALGEBRAS 9

Let ℑ(Z) be the left ideal in Cliff(Z) generated by A(Z)^{+}. Then

CM(Z) = Cliff(Z)/ℑ(Z)  (6)

is an irreducible left Cliff(Z)-module having a “vacuum vector”

vac(Z) = 1 + ℑ(Z)  (7)

satisfying A(Z)^{+} ·vac(Z) = 0 and 1·vac(Z) = vac(Z). The subalgebra of Cliff(Z) generated by 1 and A(Z)^{−} is the exterior algebra ∧A(Z)^{−}, and

CM(Z) = (∧A(Z)^{−}) · vac(Z).  (8)

Therefore, CM(Z) is spanned by vectors of the form

a_{1}(−m_{1}) … a_{r}(−m_{r})vac(Z)  (9)

for a_{i}(−m_{i}) ∈ A(Z)^{−}, 1 ≤ i ≤ r. We have the decomposition

CM(Z) = CM(Z)^{0} ⊕ CM(Z)^{1}  (10)

into the even and odd parity subspaces, CM(Z)^{0} = (∧^{even}A(Z)^{−}) · vac(Z),

CM(Z)^{1} = (∧^{odd}A(Z)^{−}) · vac(Z). (11)

Let ζ be a formal complex variable and for a ∈ A define the generating function

a(ζ) = ∑_{n∈Z} a(n)ζ^{−n}.  (12)

For a(m), b(n) ∈ A(Z) define the “fermionic normal ordering”

∘ ∘a(m)b(n)∘ ∘ = \{ \begin{matrix}a(m)b(n) & if n > m \\ \frac{1}{2}(a(m)b(n) − b(n)a(m)) & if n = m \\ −b(n)a(m) & if n < m. \end{matrix}  (13)

Then the generating function

∘ ∘a(ζ)b(ζ)∘ ∘ = ∑_{k∈ℤ} (∑_{n∈Z} ∘ ∘a(n)b(k − n)∘ ∘) ζ^{−k}  (14)

defines the homogeneous components

∘ ∘a(ζ)b(ζ)∘ ∘_{k} = ∑_{n∈Z} ∘ ∘a(n)b(k − n)∘ ∘ for k ∈ ℤ  (15)

which are well-defined operators on CM(Z). Define

D_{i}^{Z}(ζ) = −\frac{1 + ι}{16} + ∑_{k∈ℤ} ∑_{n∈Z}(n − \frac{1}{2}k) ∘ ∘a_{i}(n)a_{i}^{∗}(k − n)∘ ∘ζ^{−k}  (16)

for 1 ≤ i ≤ l, and define

D^{Z}(ζ) = ∑_{1≤i≤l} D_{i}^{Z}(ζ).  (17)

Then for k ∈ ℤ we have the homogeneous components

D^{Z}(k) = ∑_{1≤i≤l} ∑_{n∈Z}(n − \frac{1}{2}k) ∘ ∘a_{i}(n)a_{i}^{∗}(k − n)∘ ∘ for k ≠ 0,  (18)

and

D^{Z}(0) = −\frac{l(1 + ι)}{16} + ∑_{1≤i≤l} ∑_{n∈Z} n ∘ ∘a_{i}(n)a_{i}^{∗}(−n)∘ ∘.  (19)

These are well-defined operators on CM(Z), independent of the choice of canonical basis. Let Z′ = \frac{1}{2}ℤ − Z and for a ∈ A, n ∈ Z, define a(n) to act on CM(Z′) as zero. Then the operators

D(m) = D^{ℤ}(m) + D^{ℤ+ \frac{1}{2}} (m), m ∈ ℤ,  (20)

are well-defined on CM(ℤ) ⊕ CM(ℤ + \frac{1}{2}) and D(m) = D^{Z}(m) on CM(Z).

Theorem. For Z = ℤ or ℤ + \frac{1}{2} the operators (15) along with the operator (19) and the identity operator represent the affine algebra ô(2l) on CM(Z). For l > 1, CM(Z) = CM(Z)^{0} ⊕CM(Z)^{1} is the decomposition of CM(Z) into two irreducible ô(2l)-modules. The operators (20) and the identity operator represent the Virasoro algebra Vir on CM(ℤ) ⊕ CM(ℤ + \frac{1}{2}).

Choose a ℂ-antilinear involution ϑ : A → A such that for a, b ∈ A we have

\begin{matrix}⟨ϑa, ϑb⟩ = \overline{⟨a, b⟩}, \\ 0 < ⟨a, ϑa⟩ ∈ ℝ if a ≠ 0, \\ A^{−} = ϑ(A^{+}). \end{matrix}  (21)

For a, b ∈ A define a form

(a, b) = ⟨a, ϑb⟩  (22)

and note that it is positive Hermitian because of (21). Extend this form to A(Z)^{−} by

(a(m), b(n)) = δ_{m,n}(a, b),  (23)

and further extend it to CM(Z) by

(a_{1}(m_{1}) … a_{r}(m_{r})vac(Z), b_{1}(n_{1}) … b_{s}(n_{s})vac(Z))

= δ_{r,s} det[(a_{i}(m_{i}), b_{j}(n_{j}))]

= δ_{r,s} ∑_{σ∈S_{r}} sgn(σ) ∏_{1≤k≤r} (a_{k}(m_{k}), b_{σ(k)}(n_{σ(k)})),

(24)

giving a positive Hermitian form. If L is any linear operator on CM(Z), denote by L^{∗} the adjoint of L with respect to this form.

Proposition. Let ( , ) denote the positive Hermitian form on CM(Z) defined above. Then we have

(a) a(m)^{∗} = (ϑa)(−m) for a(m) ∈ A(Z),

(b) (∘ ∘a(ζ)b(ζ)∘ ∘_{k})^{∗} = ∘ ∘(ϑb)(ζ)(ϑa)(ζ)∘ ∘_{−k} for a, b ∈ A, k ∈ ℤ,

(c) D^{Z}(m)^{∗} = D^{Z}(−m) for m ∈ ℤ.

The eigenspaces of D(0) provide CM(Z) with a grading as follows. Let

(CM(Z))_{n} = {u ∈ CM(Z) | D(0)u = −nu}  (25)

be the subspace of vectors of weight n, and write wt(u) for the weight of u. Then we have

D(0)vac(Z) = −\frac{l(1 + ι)}{16}vac(Z),  (26)

and for u = a_{1}(−m_{1}) … a_{r}(−m_{r})vac(Z) we have

D(0)u = − (m_{1} + … + m_{r} + \frac{l(1 + ι)}{16}) u.  (27)

Then CM(Z) is the direct sum of the eigenspaces (25), and the \frac{1}{2}ℤ-grading of CM(ℤ + \frac{1}{2}) begins at 0 and the ℤ-grading of CM(ℤ) begins at l/8. For W = CM(Z)^{α}, α = 0, 1, we define Δ_{W} to be to be the minimal value of wt(w) for w ∈ W, so that W is ℤ+Δ_{W}-graded. Since D(0) is self-adjoint, the eigenspaces for distinct eigenvalues are orthogonal with respect to the Hermitian form on CM(Z). If A(m) is any of the operators a(m) ∈ Cliff(Z), x(m) ∈ ô(2l) or D(m) ∈ Vir, then

A(m) : (CM(Z))_{n} → (CM(Z))_{n−m}.  (28)

Let V_{0} = CM(ℤ + \frac{1}{2})^{0}, V_{2} = CM(ℤ + \frac{1}{2})^{1},

V_{1} = CM(ℤ)^{0}, V_{3} = CM(ℤ)^{1}, (29)

V = V_{0} ⊕ V_{1} ⊕ V_{2} ⊕ V_{3},  (30)
V = V_{0} ⊕ V_{2},  (31)

and extend the Hermitian form ( , ) to V so that CM(ℤ + \frac{1}{2}) and CM(ℤ) are orthogonal. D(0) provides a grading of V_{i}, 0 ≤ i ≤ 3, into mutually orthogonal eigenspaces

V_{i} = ⨁_{0≤k∈ℤ+Δ_{i}}(V_{i})_{k}  (32)

where we define Δ_{i} = Δ_{V_{i}} so that Δ_{0} = 0, Δ_{2} = \frac{1}{2}, and Δ_{1} = Δ_{3} = l/8.

Spinor construction of a vertex operator superalgebra and module.

Define generalized vertex operators Y (v, ζ) on CM(ℤ + \frac{1}{2}) and on CM(ℤ) for any v ∈ V. The definition of Y (v, ζ) on CM(ℤ) is more complicated than it is on CM(ℤ + \frac{1}{2}), but there is a common part which we denote by Ȳ (v, ζ). With vac = vac(ℤ+ \frac{1}{2}) define Ȳ (vac, ζ) = 1 (the identity operator) on V, and for a ∈ A, 0 ≤ n ∈ ℤ, on CM(Z) let

Ȳ (a(−n − \frac{1}{2})vac, ζ) = n!^{−1}(d/dζ)^{n}(ζ^{−1/2}a(ζ))  (33)

where

a(ζ) = ∑_{m∈Z} a(m)ζ^{−m}.  (34)

Define the notations a^{(n)}(ζ), a^{(n)}(ζ)^{±} and a^{(n)}(ζ)^{0} by

a^{(n)}(ζ) = Ȳ (a(−n − \frac{1}{2})vac, ζ) = ∑_{m∈Z}\binom{−m − \frac{1}{2}}{n}a(m)ζ^{−m−n− \frac{1}{2}} ,  (35)

a^{(n)}(ζ)^{+} = ∑_{0<m∈Z} \binom{−m − \frac{1}{2}}{n}a(m)ζ^{−m−n− \frac{1}{2}} ,  (36)

a^{(n)}(ζ)^{−} = ∑_{0>m∈Z} \binom{−m − \frac{1}{2}}{n}a(m)ζ^{−m−n− \frac{1}{2}} ,  (37)

a^{(n)}(ζ)^{0} = \binom{−\frac{1}{2}}{n}a(0)ζ^{−n− \frac{1}{2}} if Z = ℤ,  (38)

so that

a^{(n)}(ζ) = \{ \begin{matrix}a^{(n)}(ζ)^{+} + a^{(n)}(ζ)^{−} & on CM(ℤ + \frac{1}{2}) \\ a^{(n)}(ζ)^{+} + a^{(n)}(ζ)^{0} + a^{(n)}(ζ)^{−} & on CM(ℤ). \end{matrix}  (39)

For a_{1}, …, a_{r} ∈ A, n_{1}, …, n_{r} ∈ ℤ + \frac{1}{2} extend the definition of fermionic normal ordering (13) to any product of Clifford generators

∘ ∘a_{1}(n_{1}) … a_{r}(n_{r})∘ ∘ = sgn(σ)a_{σ1}(n_{σ1}) … a_{σr}(n_{σr}),  (40)

where σ ∈ .S_{r} is any permutation such that n_{σ1} ≤ … ≤ n_{σr}. The Clifford relations (5) show that this is well-defined and agrees with (13) for r = 2. For n_{1}, …, n_{r} ∈ ℤ, we modify the definition in (40) in case some n_{i} = 0. We want to have

∘ ∘a_{1}(n_{1}) … a_{r}(n_{r})∘ ∘ = sgn(σ)∘ ∘a_{σ1}(n_{σ1}) … a_{σr}(n_{σr})∘ ∘  (41)

for any permutation σ ∈ S_{r}. Make the definition when n_{1} ≤ … ≤ n_{r} and then define the other cases by (41). If n_{1} ≤ … ≤ n_{r} and each 0 ≠ n_{i} ∈ ℤ, let

∘ ∘a_{1}(n_{1}) … a_{r}(n_{r})∘ ∘ = a_{1}(n_{1}) … a_{r}(n_{r}),  (42)

but if n_{i−1} < 0 = n_{i} = n_{i+1} = … = n_{i+s−1} < n_{i+s} for some s ≥ 1, then define

\begin{matrix}∘ ∘a_{1}(n_{1}) … a_{r}(n_{r})∘ ∘ = \\ a_{1}(n_{1}) … a_{i−1}(n_{i−1})(∘ ∘a_{i}(0) … a_{i+s−1}(0)∘ ∘)a_{i+s}(n_{i+s}) … a_{r}(n_{r}), \end{matrix}  (43)

where for any a_{1}, …, a_{k} ∈ A,

∘ ∘a_{1}(0) … a_{k}(0)∘ ∘ = k!^{−1} ∑_{σ∈S_{k}} sgn(σ)a_{σ1}(0) … a_{σk}(0).  (44)

For 0 ≤ n_{1}, …, n_{r} ∈ ℤ, and for elements of the form

v = a_{1}(−n_{1} − \frac{1}{2}) … a_{r}(−n_{r} − \frac{1}{2})vac ∈ (V)_{n},  (45)

n = wt(v) = n_{1} + … + n_{r} + \frac{1}{2}r, on CM(Z) we define

\begin{aligned}Ȳ (v, ζ) & = ∘ ∘Ȳ (a_{1}(−n_{1} − \frac{1}{2})vac, ζ) … Ȳ (a_{r}(−n_{r} − \frac{1}{2})vac, ζ)∘ ∘ \\ & = ∘ ∘a_{1}^{(n_{1})}(ζ) … a_{r}^{(n_{r})}(ζ)∘ ∘. \end{aligned}  (46)

Then on CM(ℤ + \frac{1}{2}) we have

Ȳ (v, ζ) = ∑ sgn(σ)a_{σ1}^{(n_{σ1})}(ζ)^{±} … a_{σr}^{(n_{σr})}(ζ)^{±}  (47)

where the summation is over the 2^{r} terms obtained by splitting each factor as in the first line of (39) and in each term permuting the resulting factors by some σ ∈ S_{r} so that all ^{−} factors are to the left of all ^{+} factors. Any two ^{−} factors anticommute, and any two ^{+} factors anticommute. On CM(ℤ) we have

Ȳ (v, ζ) = ∑ sgn(σ)a_{σ1}^{(n_{σ1})}(ζ)^{−} … (∘ ∘a_{σi}^{(n_{σi})}(ζ)^{0} … a_{σj}^{(n_{σj})}(ζ)^{0}∘ ∘) … a_{σr}^{(n_{σr})}(ζ)^{+} (48)

where the summation is over the 3^{r} terms obtained by splitting each factor as in the second line of (39) and in each term permuting the resulting factors by some σ ∈ S_{r} so that all ^{−} factors are to the left and all ^{+} factors are to the right. Extend the definitions of Ȳ (v, ζ) to any v ∈ CM(ℤ + \frac{1}{2}) by linearity. For v ∈ (V)_{k} define the homogeneous components Ȳ_{m}(v) of Ȳ (v, ζ) and the notation {v̄}_{n} by

Ȳ (v, ζ) = ∑_{n∈ \frac{1}{2} ℤ} Ȳ_{n+1−wt(v)}(v)ζ^{−n−1} = ∑_{n∈ \frac{1}{2} ℤ}\{v̄\}_{n}ζ^{−n−1}.  (49)

On CM(ℤ + \frac{1}{2}) the coefficient of ζ^{−n−1} is zero unless n ∈ ℤ. On CM(ℤ) the coefficient of ζ^{−n−1} is zero unless n ∈ ℤ for v ∈ V_{0}, but for v ∈ V_{2}, it is zero unless n ∈ ℤ + \frac{1}{2}. The components Ȳ_{m}(v) are well-defined operators on V such that

wt(Ȳ_{m}(v)w) = wt(w) − m.  (50)

Proposition. For v ∈ V we have

(a) lim_{ζ→0} Ȳ (v, ζ)vac = v,

(b) Ȳ (D(−1)v, ζ) = −(d/dζ)Ȳ (v, ζ) on V,

(c) [D(0), Ȳ (v, ζ)] = −ζ(d/dζ)Ȳ (v, ζ) + Ȳ (D(0)v, ζ) on V.

We can write the generating function of the Virasoro operators on CM(ℤ + \frac{1}{2}) as follows. Letting

ω = D(−2)vac = −\frac{1}{2} ∑_{1≤i≤l}(a_{i}^{∗}(−\frac{3}{2})a_{i}(−\frac{1}{2}) + a_{i}(−\frac{3}{2})a_{i}^{∗}(−\frac{1}{2}))vac  (51)

we find that

Ȳ (ω, ζ) = ζ^{−2}D(ζ)  (52)

on V.

Computing Ȳ (ω, ζ) on CM(ℤ) we find that the homogeneous components agree with the definitions of the Virasoro operators in (18)-(19), except that the scalar term −l/8 is missing from the operator which should be giving D(0). So we want

Ȳ (ω, ζ) − \frac{1}{8}ζ^{−2}l = ζ^{−2}D(ζ)  (53)

to be the generating function of the Virasoro operators on CM(ℤ). The vertex operators Y (v, ζ) which we will define later incorporate this scalar term on CM(ℤ).

Definition. For 0 ≤ m, n ∈ ℤ, let

C_{mn} = \frac{1}{2} \frac{m − n}{m + n + 1}\binom{−\frac{1}{2}}{m}\binom{−\frac{1}{2}}{n},  (54)

and note that

(m + 1)C_{(m+1)n} + (n + 1)C_{m(n+1)} = −(m + n + 1)C_{mn}.  (55)

Lemma. For 0 ≤ k, r, t ∈ ℤ the coefficients defined in (54) satisfy

∑_{0≤m≤k} \binom{m + r}{r}\binom{k − m + t}{t}C_{(m+r)(k−m+t)} = \binom{−r − t − 1}{k}C_{rt}.

We now define a generating function Δ(ζ) of quadratic operators on CM(ℤ + \frac{1}{2}) which allows us to define the operators Y (v, ζ) we want.

Definition. For {a_{1}, …, a_{l}, a_{1}^{∗}, …, a_{l}^{∗}} a canonical basis of A, define

Δ(ζ) = ∑_{1≤i≤l} ∑_{0≤m,n∈ℤ} C_{mn}a_{i}(m + \frac{1}{2})a_{i}^{∗}(n + \frac{1}{2})ζ^{−m−n−1}  (56)

on CM(ℤ + \frac{1}{2}). Since Δ(ζ) is quadratic, it preserves V_{0} and V_{2}.

Definition. For v ∈ CM(ℤ + \frac{1}{2}), let

Y (v, ζ) = \{ \begin{matrix}Ȳ (v, ζ) & on CM(ℤ + \frac{1}{2}) \\ Ȳ (exp(Δ(ζ))v, ζ) & on CM(ℤ). \end{matrix}  (57)

For v ∈ (V)_{k}, define the homogeneous components Y_{m}(v) of Y (v, ζ) and the notation {v}_{n} by

Y (v, ζ) = ∑_{n∈ \frac{1}{2} ℤ} Y_{n+1−wt(v)}(v)ζ^{−n−1} = ∑_{n∈ \frac{1}{2} ℤ}\{v\}_{n}ζ^{−n−1}.  (58)

Note that

Y (D(−2)vac, ζ) = ζ^{−2}D(ζ)  (59)

includes the correction term −l/8 to the operator D(0) on CM(ℤ).

Theorem. (V, Y ( , ζ), vac, ω) is a vertex operator superalgebra. (CM(ℤ), Y ( , ζ)) is a module for that vertex operator superalgebra.

The Bosonic (Lattice) Construction

Bosonic Construction of affine Kac-Moody algebras.

Let Q be a rank l root lattice of type A, D or E, and let P be the corresponding weight lattice containing Q. They come equiped with a positive definite symmetric bilinear form ⟨·, ·⟩ such that

⟨α, α⟩ ∈ 2ℤ and ⟨λ, α⟩ ∈ ℤ  (1)

for α ∈ Q, λ ∈ P. Let ℂ[P] be the group algebra with basis {e^{μ} | μ ∈ P } and product e^{λ}e^{μ} = e^{λ+μ}. Let

P = Q_{0} ∪ Q_{1} ∪ … ∪ Q_{t}  (2)

be the coset decomposition of P with respect to Q = Q_{0} and let {0, …, t} be given the group structure of P/Q. Let h = P ⊗_{ℤ} ℂ and extend the symmetric form to h.

Define the infinite dimensional Heisenberg algebra,

ĥ = h ⊗ ℂ[t, t^{−1}] ⊕ ℂc  (3)

with brackets

[h_{1}(m), h_{2}(n)] = mδ_{m,−n}⟨h_{1}, h_{2}⟩c,  [c, h_{1}(m)] = 0.  (4)

Then

ĥ^{−} = h ⊗ t^{−1}ℂ[t^{−1}]  (5)

is an abelian subalgebra of ĥ, and S(ĥ^{−}) is the symmetric algebra of polynomials in ĥ^{−}. Define

V = S(ĥ^{−}) ⊗ ℂ[P],  (6)

and the subspaces

V_{k} = S(ĥ^{−}) ⊗ ℂ[Q_{k}], 0 ≤ k ≤ t.  (7)

Then ĥ acts on V as follows.

For 0 < m ∈ ℤ, h ∈ h, u ∈ S(ĥ^{−}), μ ∈ P, we have

\begin{aligned}h(m) · (u ⊗ e^{μ}) & = m(∂_{h(−m)}u) ⊗ e^{μ}, \\ h(−m) · (u ⊗ e^{μ}) & = (h(−m)u) ⊗ e^{μ}, \\ h(0) · (u ⊗ e^{μ}) & = u ⊗ (∂_{h}e^{μ}) = μ(h)u ⊗ e^{μ}, \\ c · (u ⊗ e^{μ}) & = u ⊗ e^{μ}. \end{aligned}  (8)

For any two Heisenberg generators h_{i}(m), h_{j}(n) ∈ ĥ we define their “bosonic normal ordering” to be

\begin{aligned}: h_{i}(m)h_{j}(n): & = \{ \begin{matrix}h_{i}(m)h_{j}(n) & if n ≥ m \\ h_{j}(n)h_{i}(m) & if n < m \end{matrix} \\ & =: h_{j}(n)h_{i}(m): . \end{aligned}  (9)

For {h_{1}, …, h_{l}} an orthonormal basis of h and m ∈ ℤ, let

D(m) = −\frac{1}{2} ∑_{1≤i≤l} ∑_{n∈ℤ}: h_{i}(n)h_{i}(m − n): .  (10)

Then D(m) is a well-defined operator on V, independent of the choice of orthonormal basis. For m, n ∈ ℤ, h ∈ h, we have

[D(m), h(n)] = nh(n + m)  (11)

and

[D(m), D(n)] = (n − m)D(m + n) + \frac{1}{12}(m^{3} − m)δ_{m,−n} l,  (12)

so the operators D(m), m ∈ ℤ, and the identity operator represent Vir on V. Any vector in V can be written as a linear combination of vectors of the form

v = u ⊗ e^{λ} = h_{i_{1}}(−m_{1}) … h_{i_{r}}(−m_{r})1 ⊗ e^{λ}  (13)

with 0 ≤ r, 0 < m_{1} ≤ m_{2} ≤ … ≤ m_{r}, 1 ≤ i_{j} ≤ l, λ ∈ P. For such a vector we have

D(0)(v) = − (\frac{1}{2}⟨λ, λ⟩ + ∑_{1≤j≤r} m_{j}) (v ⊗ e^{λ}),  (14)

and

wt(v) = (\frac{1}{2}⟨λ, λ⟩ + ∑_{1≤j≤r} m_{j})  (15)

provides a ℤ + Δ_{k}-grading on V_{k}, where Δ_{k} = \frac{1}{2}⟨λ, λ⟩ for λ ∈ Q_{k} is determined mod ℤ. For each λ ∈ P, S(ĥ^{−}) ⊗ ℂe^{μ}, is an irreducible ĥ-module.

Let [ϵ] ∈ H^{2}(P, ℂ^{∗}) be a cohomology class represented by a 2-cocycle ϵ such that ϵ(0, 0) = 1, whose restriction to Q is the unique class in H^{2}(Q, {±1}) satisfying

ϵ(α, β)ϵ(β, α)^{−1} = (−1)^{⟨α,β⟩} for α, β ∈ Q,  (16)

Define an operator ϵ_{λ} : V → V for λ ∈ P by

ϵ_{λ}(u ⊗ e^{μ}) = ϵ(λ, μ)u ⊗ e^{μ}.  (17)

Let ζ be a formal complex variable, let v = 1 ⊗ e^{λ}. Then

\begin{matrix}Y (v, ζ) = & exp ( ∑_{m≥1} λ(−m)\frac{ζ^{m}}{m} ) e^{λ}ζ^{λ(0)} exp (− ∑_{m≥1} λ(m)\frac{ζ^{−m}}{m} ) ϵ_{λ} \\ = & ∑ Y_{n}(v)ζ^{−n−wt(v)} \end{matrix}  (18)

defines the homogeneous components Y_{n}(v). The first and last exponentials act on S(ĥ^{−}). The middle operators act on ℂ[P] by

e^{λ}ζ^{λ(0)} · e^{μ} = ζ^{⟨λ,μ⟩}e^{λ+μ}.  (19)

These components are well-defined operators on V. If λ ∈ Q = Q_{0} then the spaces V_{k} are each preserved and the summation is over n ∈ ℤ.

Theorem. The operators Y_{n}(1 ⊗ e^{α}) for α ∈ Q with ⟨α, α⟩ = 2, n ∈ ℤ, along with D(0) and the operators representing the Heisenberg algebra ĥ represent the affine Kac-Moody algebra ĝ on V. For l > 1, each subspace V_{k} is an irreducible ĝ−module. The operators D(m), m ∈ ℤ, along with the identity operator, represent the Virasoro algebra Vir.

We can define a positive Hermitian form on V and we can determine the Hermitian adjoints of the operators we have acting on V. If h = ∑_{1≤i≤l}a_{i}α_{i} ∈ h then

we define a ℂ-antilinear involution by

h̄ = ∑_{1≤i≤l} ā_{i}α_{i}.  (20)

For h_{1}, h_{2} ∈ h define a positive Hermitian form

(h_{1}, h_{2}) = ⟨h_{1}, h̄_{2}⟩.  (21)

Extend this form to ĥ^{−} by

(h_{1}(m), h_{2}(n)) = −mδ_{m,n}(h_{1}, h_{2}),  (22)

and further extend it to S(ĥ^{−}) by

\begin{matrix}(h_{i_{1}}(m_{1}) … h_{i_{r}}(m_{r}), h_{j_{1}}(n_{1}) … h_{j_{s}}(n_{s})) = \\ δ_{r,s}P erm[(h_{i_{k}}(m_{k}), h_{j_{p}}(n_{p})] = \\ δ_{r,s} ∑_{σ∈S_{r}} ∏_{1≤k≤r} (h_{i_{k}}(m_{k}), h_{j_{σ(k)}}(n_{σ(k)})) , \end{matrix}  (23)

giving a positive Hermitian form. Using the form

(e^{λ}, e^{μ}) = δ_{λ,μ}  (24)

on ℂ[P], we finally obtain a positive Hermitian form on V by letting

(u_{1} ⊗ e^{λ}, u_{2} ⊗ e^{μ}) = (u_{1}, u_{2})δ_{λ,μ}.  (25)

If L is any linear operator on V then denote by L^{∗} the adjoint of L with respect to this form.

Proposition. Let ( , ) be the positive Hermitian form on V defined above. Choose a cocycle ϵ that takes values in T = {z ∈ ℂ | |z| = 1}. Then we have

(a) h(m)^{∗} = h̄(−m) for h(m) ∈ ĥ,

(b) Y_{n}(λ)^{∗} = ϵ(−λ, λ)^{−1}Y_{−n}(−λ) for λ ∈ P,

(c) D(m)^{∗} = D(−m) for m ∈ ℤ.

The eigenspaces of D(0),

(V)_{n} = {v ∈ V | D(0)u = −nu},  (26)

provide V with a ℚ-grading, and (V)_{n} = 0 for n < 0.

Lattice construction of a vertex operator superalgebra and module.

Let L be an integral lattice of rank l with positive definite bilinear form ⟨ , ⟩, and let h = L ⊗_{ℤ} ℂ. We have the coset decomposition

L = L_{0} ∪ L_{1},  (27)

where for i = 0, 1,

L_{i} = {λ ∈ L | ⟨λ, λ⟩ ∈ 2ℤ + i}.  (28)

Let

V = S(ĥ^{−}) ⊗ ℂ[L],  (29)

so we have the corresponding decomposition

V = V_{0} ⊕ V_{1},  (30)

where

V_{i} = S(ĥ^{−}) ⊗ ℂ[L_{i}].  (31)

We choose a 2-cocycle ϵ : L × L → {±1} representing the unique class in H^{2}(L, ℤ_{2}) satisfying

ϵ(λ, μ)ϵ(μ, λ)^{−1} = (−1)^{⟨λ,μ⟩+⟨λ,λ⟩⟨μ,μ⟩}.  (32)

With {h_{1}, …, h_{l}} an orthonormal basis of h, we have the Virasoro operators D(m), m ∈ ℤ, on V as in (10). Then these operators satisfy (11) and (12), and D(0) provides a ℤ-grading on V_{0} and a ℤ + \frac{1}{2}-grading on V_{1}. For λ ∈ L we have the generating function Y (λ, ζ) defined in (18), whose homogeneous components Y_{n}(λ), n ∈ ℤ, are well-defined operators on V.

We now define vertex operator Y (v, ζ) for any v ∈ V. For h ∈ h and λ ∈ L, define

Y (h(−n − 1) ⊗ e^{0}, ζ) = n!^{−1}(d/dζ)^{n}(ζ^{−1}h(ζ))  (33)

for 0 ≤ n ∈ ℤ, where

h(ζ) = ∑_{n∈ℤ} h(n)ζ^{−n}.  (34)

Note that

Y (1 ⊗ e^{0}, ζ) = 1.  (35)

We extend the definition of bosonic normal ordering (9) to any product of Heisenberg generators,

: h_{1}(n_{1}) … h_{r}(n_{r}): = h_{σ1}(n_{σ1}) … h_{σr}(n_{σr}),  (36)

where σ ∈ S_{r} is any permutation such that n_{σ1} ≤ … ≤ n_{σr}. The Heisenberg commutation relations (4) show that this is well-defined, agrees with (9) for r = 2, and is equivalent to putting all h(n) with n < 0 to the left of those with n ≥ 0. Note that e^{λ} commutes with h(n) for n ≠ 0. Define the bosonic normal ordering of any product involving operators e^{λ}, ζ^{μ(0)}, and h(n) to be the rearranged product with the h(n) operators normally ordered as above, and e^{λ} to the left of all ζ^{μ(0)} and all h(0). For elements of the form

v = h_{1}(−n_{1}) … h_{r}(−n_{r}) ⊗ e^{λ} ∈ (V)_{n},  (37)

n = n_{1} + … + n_{r} + \frac{1}{2}⟨λ, λ⟩, define

Y (v, ζ) = : Y (h_{1}(−n_{1}) ⊗ e^{0}, ζ) … Y (h_{r}(−n_{r}) ⊗ e^{0}, ζ)Y (1 ⊗ e^{λ}, ζ): (38)

and we extend this definition linearly to Y (v, ζ) for any v ∈ V. For v ∈ V with wt(v) = m, the homogeneous components Y_{n}(v) of Y (v, ζ) are defined by

Y (v, ζ) = ∑_{n∈Z} Y_{n}(v)ζ^{−n−wt(v)},  (39)

where Z = ℤ for v ∈ V_{0} and Z ∈ ℤ + \frac{1}{2} for v ∈ V_{1}. We extend this definition to arbitrary v ∈ V by linearity. It is easy to see that the components Y_{n}(v) are well-defined operators on V sending (V)_{r} to (V)_{r−n}.

We can write the generating function of the Virasoro operators,

D(ζ) = −\frac{1}{2} ∑_{1≤i≤l}: h_{i}(ζ)^{2}: = ∑_{n∈ℤ} D(n)ζ^{−n}  (40)

in terms of Y (v, ζ) for a certain v ∈ V with wt(v) = 2. Letting

ω = D(−2)(1 ⊗ e^{0}) = −\frac{1}{2} ∑_{1≤i≤l} h_{i}(−1)^{2} ⊗ e^{0}  (41)

for {h_{1}, …, h_{l}} an orthonormal basis of h, we have

Y (ω, ζ) = ζ^{−2}D(ζ).  (42)

We say that a lattice L equiped with a symmetric bilinear form ⟨ , ⟩ is orthonormal when L has an orthonormal ℤ-basis with respect to that form.

Theorem. (V, Y ( , ζ), 1 ⊗ e^{0}, ω) is a vertex operator superalgebra. If L is an orthonormal lattice then L_{0} is a root lattice of type D_{l} and L is contained in the corresponding weight lattice P with index 2. The above construction with P yields a vertex operator superalgebra module on the other coset of L in P.

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Department of Mathematical Sciences, The State University of New York, Binghamton, New York 13902-6000

E-mail address: alex@math.binghamton.edu