Permutation orbifolds of the Heisenberg vertex algebra H(3)

Milas, Antun; Penn, Michael; Shao, Hanbo

doi:10.1063/1.5045164

We study the S₃-orbifold of a rank three Heisenberg vertex algebra in terms of generators and relations. By using invariant theory, we prove that the orbifold algebra has a minimal strong generating set of vectors whose conformal weights are 1, 2, 3, 4, 5, 6² (two generators of degree 6). The structure of the cyclic $Z_{3}$ -orbifold is determined by similar methods. We also study characters of modules for the orbifold algebra.

I. INTRODUCTION

For every vertex operator algebra V, the n-fold tensor product $V^{\otimes^{n}}$ has a natural vertex operator algebra structure. The symmetric group S_n acts on V(n) ≔ V^⊗n by permuting tensor factors and thus S_n ⊂ Aut(V(n)). Denote by $V {(n)}^{S_{n}}$ the fixed point vertex operator subalgebra also called the S_n-orbifold of V(n). It is an open problem to classify irreducible modules of $V {(n)}^{S_{n}}$ although it is widely believed that every such module should come from a g-twisted V^⊗n-module for some g ∈ S_n. When it comes to the inner structure of $V {(n)}^{S_{n}}$ (e.g., a strong system of generators), very little is known. Even for the Heisenberg orbifold $H {(n)}^{S_{n}}$ ⁠, where $H$ is the rank one Heisenberg vertex algebra, this problem seems quite difficult.

Finite and permutation orbifolds have been extensively studied both in physics and mathematics literature. According to Ref. 5, the earliest study of permutation orbifolds seems to be Ref. 17. The first systematic construction of cyclic orbifolds, including their twisted sectors appeared in Ref. 8. Explicit formulas for characters and modular transformation properties of permutation orbifolds of rational conformal field theories were given in Ref. 4.

In the literature on vertex algebras, the main focus has been on the classification and construction of twisted modules starting with Refs. 15 and 19. Further developments include “Quantum Galois” theory developed in Ref. 11, work on twisted sectors of permutation orbifolds⁶ (see also Ref. 7), and tensor category structure for general G-orbifolds of rational vertex algebras^13,18. There are also numerous papers on orbifold vertex algebras for Abelian groups and groups of small order. Structure and representations of 2-permutation orbifolds were subjects of Refs. 1, 2, and 12; see also a more recent work on 3-permutation orbifolds of lattice vertex algebras.¹⁴ The ADE orbifolds of rank one lattice vertex algebras (e.g., Ref. 10) are important for classification of c = 1 rational vertex algebras. Orbifolds of irrational C₂-cofinite vertex algebras have been investigated in Refs. 3, 9, and 2. Linshaw (see Refs. (9) and (20)–(22)) extensively studied orbifolds of “free field” vertex operator (super)algebras using the classical invariant theory²⁶ and its (super) extensions. As a consequence of the main result in Ref. 22, every finite orbifold of an affine vertex algebra is finitely strongly generated. In particular, this implies that $H {(n)}^{S_{n}}$ has a finite strong set of generators. Characters of orbifolds of affine vertex algebras were investigated earlier in Ref. 16.

In this paper, we are concerned with the structure of one of the simplest non-Abelian orbifolds, $H {(3)}^{S_{3}}$ ⁠, where $H$ is the rank one Heisenberg vertex algebra [case $H {(2)}^{S_{2}}$ is well-understood^12,1]. We prove three main results. The first result pertains to generators of $H {(3)}^{S_{3}}$ ⁠. We show in Theorem 4.1 that this vertex algebra is isomorphic to a W-algebra of type (2, 3, 4, 5, 6²) tensored with a rank one Heisenberg vertex algebra. Here, labels 2, 3, 4, 5, 6² indicate that our W-algebra is strongly generated by the Virasoro vector (of degree two) and five primary vectors of degrees 3, 4, 5 and two of degree 6. These five generators are explicitly given in Sec. IV where we denoted them by J₁, J₂, C₁, C₂, and C₃. Our second result is about the cyclic $Z_{3}$ -orbifold of $H (3)$ (see Theorem 5.1 for details).

In Secs. VI and VII, we discuss characters of certain $H {(3)}^{S_{3}}$ -modules and their modular properties. It is expected that many irrational vertex algebras will enjoy modular invariance in a generalized sense involving iterated integrals instead of sums. For the rank n permutation orbifold $H {(n)}^{S_{n}},$ the character of a module M is expected to transform as

c h [M] (- \frac{1}{τ}) = \sum_{i \geq 1}^{n} \int_{R^{i}} S_{M, M_{λ_{i}}} c h [M_{λ_{i}}] (τ) d λ_{i},

(1.1)

where $S_{M, M_{λ_{i}}} \in C$ and $λ_{i} \in R^{i}$ ⁠, 1 ≤ i ≤ n, parametrize certain $H {(n)}^{S_{n}}$ -modules. Our third main result gives strong evidence for this conjecture for n = 3 (see Theorem 7.1).

II. THE S_n-ORBIFOLD OF THE HEISENBERG VERTEX OPERATOR ALGEBRA

Let $H$ denote the rank one Heisenberg vertex operator algebra generated by $α (- 1) 1$ ⁠, with the usual conformal vector (and grading) given by $ω = \frac{1}{2} α^{2} (- 1) 1$ ⁠. Let $H (n) = H^{\otimes^{n}}$ ⁠. For convenience, we suppress the tensor product symbol and let $α_{1} (- 1) 1 ≔ α (- 1) 1 \otimes \dots \otimes 1 \in H (n)$ ⁠, and similarly, we define $α_{i} (- 1) 1$ ⁠, i ≥ 2, such that $H (n) = ⟨ α_{1} (- 1) 1, \dots, α_{n} (- 1) 1 ⟩$ ⁠. We consider the natural action of S_n on $H (n)$ given by

σ \cdot α_{i_{1}} (m_{1}) \dots α_{i_{k}} (m_{k}) 1 = α_{σ (i_{1})} (m_{1}) \dots α_{σ (i_{k})} (m_{k}) 1,

(2.1)

for 1 ≤ i_j ≤ n, m_j < 0, and σ ∈ S_n.

We clearly have a natural linear isomorphism

H (n) ≅ C [x_{i} (m) | 1 \leq i \leq n, m \geq 0]

(2.2)

induced by α_i(−m − 1) ↦ x_i(m) for m ≥ 0. Using the terminology of Ref. 21, we say that $C [x_{i} (m) | 1 \leq i \leq n, m \geq 0]$ is the associated graded algebra of the vertex algebra $H (n)$ ⁠. Further, we may endow the polynomial algebra $C [x_{i} (m) | 1 \leq i \leq n, m \geq 0]$ with the structure of a ∂-ring by defining the map

\begin{aligned} \partial : C [x_{i} (m) | 1 \leq i \leq n, m \geq 0] & \to C [x_{i} (m) | 1 \leq i \leq n, m \geq 0], \\ x_{i} (m) & \mapsto (m + 1) x_{i} (m + 1), \end{aligned}

(2.3)

where the action of ∂ is extended to the whole space via the Leibniz rule. This definition of ∂ is compatible with the translation operator in $H (n)$ given by $T (v) = v_{- 2} 1$ ⁠.

The following lemma is from Ref. 20.

Lemma 2.1.

Let $A$ be a vertex algebra with a $Z_{\geq 0}$ filtration, where $\tilde{A}$ is the associated ∂-ring. If {ã_i|i ∈ I} generates $\tilde{A}$ , then {a_i|i ∈ I} strongly generates $A$ ⁠, where a_i and ã_i are related via the natural linear isomorphism described by the $Z_{\geq 0}$ filtration.

Further, we recall that the invariant ring

C {[x_{1}, \dots, x_{n}]}^{S_{n}}

has a variety of generating sets, including the power sum polynomials

p_{i} = x_{1}^{i} + \dots + x_{n}^{i}, 1 \leq i \leq n .

In addition to this, it is common to study the invariant theory of the ring of infinitely many commuting copies of this polynomial algebra, where we denote by x_i(m) the copy of x_i from the mth copy of the polynomial algebra. A theorem of Weyl²⁶ shows that

C {[x_{i} (m) | 1 \leq i \leq n, m \geq 0]}^{S_{n}}

is generated by the polarizations of these polynomials

q_{k} (m_{1}, \dots, m_{k}) = \sum_{i = 1}^{n} x_{i} (m_{1}) \dots x_{i} (m_{k}),

(2.4)

for 1 ≤ k ≤ n. Now, applying Lemma 2.1, we have an initial strong generating set for the orbifold

H {(n)}^{S_{n}}

given by the vectors

ω_{k} (m_{1}, \dots, m_{k}) = \sum_{i = 1}^{n} α_{i} (- 1 - m_{1}) \dots α_{i} (- 1 - m_{k}) 1,

(2.5)

for 1 ≤ k ≤ n and m_j ≥ 0. It should be noted that the conformal vector of

H (n)

is

\frac{1}{2} ω_{2} (0,0)

⁠, making the orbifold a vertex operator algebra.

III. WARMUP: $H {(2)}^{S_{2}}$

Here, we describe the structure of $H {(2)}^{S_{2}}$ ⁠. This case is well-known, so we only provide a few details. Denote by $M {(1)}^{+} ≔ H {(1)}^{Z_{2}}$ the fixed point subalgebra under the action $α (- 1) 1 \to - α (- 1) 1$ ⁠, where α is the Heisenberg generator [in the physics literature, M(1)⁺ is often denoted by W(2, 4)]. This vertex algebra was thoroughly studied in Ref. 12 and elsewhere. Let

\begin{array}{l} h & = α_{1} (- 1) 1 + α_{2} (- 1) 1, h^{⊥} = α_{1} (- 1) 1 - α_{2} (- 1) 1, \\ ω = \frac{1}{4} α_{1}^{2} (- 1) 1 + \frac{1}{4} α_{2}^{2} (- 1) 1 - \frac{1}{2} α_{1} (- 1) α_{2} (- 1) 1 . \end{array}

Observe that as vertex algebras

H (2) = H {(1)}_{h} \otimes H {(1)}_{h^{⊥}},

where $H {(1)}_{h} = ⟨ h ⟩$ and $H {(1)}_{h^{⊥}} = ⟨ h^{⊥} ⟩$ with conformal vector ω. The nontrivial element of the group S₂ fixes the first tensor factor and

h^{⊥} \to - h^{⊥} .

Thus, we immediately get

H {(2)}^{S_{2}} ≅ H (1) \otimes M {(1)}^{+} .

(3.1)

It is easy to see that

\begin{matrix} H = \frac{1}{2} () α_{1} {(- 1)}^{4} + α_{2} {(- 1)}^{4} - 4 α_{1}^{3} (- 1) α_{2} (- 1) - 4 α_{1} (- 1) α_{2}^{3} (- 1) + 6 α_{1}^{2} (- 1) α_{2}^{2} (- 1) + 3 α_{1}^{2} (- 2) + 3 α_{2}^{2} (- 2) \\ - 6 α_{1} (- 2) α_{2} (- 2) + 2 α_{1} (- 1) α_{2} (- 3) + 2 α_{1} (- 3) α_{2} (- 1) - 2 α_{1} (- 3) α_{1} (- 1) - 2 α_{2} (- 3) α_{2} (- 1))) 1 \end{matrix}

is contained in ⟨h^⊥⟩ and is primary of conformal weight 4 and thus a generator of M(1)⁺.

IV. THE RANK THREE CASE

A. The orbifold $H {(3)}^{S_{3}}$

As described above, we know that the orbifold $H {(3)}^{S_{3}}$ will be strongly generated by the vectors

\begin{aligned} ω_{1} (a) & = \sum_{i = 1}^{3} α_{i} (- 1 - a) 1, \\ ω_{2} (a, b) & = \sum_{i = 1}^{3} α_{i} (- 1 - a) α_{i} (- 1 - b) 1, \\ ω_{3} (a, b, c) & = \sum_{i = 1}^{3} α_{i} (- 1 - a) α_{i} (- 1 - b) α_{i} (- 1 - c) 1 . \end{aligned}

(4.1)

The following change of basis of the generating set will allow for an efficient reduction in the generating set of the corresponding orbifold

\begin{aligned} β_{1} (- 1) 1 & = \frac{1}{\sqrt{3}} (α_{1} (- 1) 1 + α_{2} (- 1) 1 + α_{3} (- 1) 1), \\ β_{2} (- 1) 1 & = \frac{1}{\sqrt{3}} (α_{1} (- 1) 1 + η^{2} α_{2} (- 1) 1 + η α_{3} (- 1) 1), \\ β_{3} (- 1) 1 & = \frac{1}{\sqrt{3}} (α_{1} (- 1) 1 + η α_{2} (- 1) 1 + η^{2} α_{3} (- 1) 1), \end{aligned}

(4.2)

where η is a primitive third root of unity. Using this generating set, we have $σ \cdot β_{1} (- 1) 1 = β_{1} (- 1) 1$ for all σ ∈ S₃. Further, examining the action of the generators of S₃ on the generators of $H (3)$ ⁠, we have

(\begin{matrix} 2 & 3 \end{matrix}) \cdot β_{2} (- 1) 1 = β_{3} (- 1) 1, (\begin{matrix} 2 & 3 \end{matrix}) \cdot β_{3} (- 1) 1 = β_{2} (- 1) 1,

(4.3)

and

(\begin{matrix} 1 & 2 & 3 \end{matrix}) \cdot β_{2} (- 1) 1 = η β_{2} (- 1) 1, (\begin{matrix} 1 & 2 & 3 \end{matrix}) \cdot β_{3} (- 1) 1 = η^{2} β_{3} (- 1) 1 .

(4.4)

From this action, we see that an initial generating set for the orbifold $H {(3)}^{S_{3}}$ may be taken to be

\begin{aligned} ω_{1}^{0} (a) & = β_{1} (- 1 - a) 1, \\ ω_{2}^{0} (a, b) & = β_{2} (- 1 - a) β_{3} (- 1 - b) 1 + β_{3} (- 1 - a) β_{2} (- 1 - b) 1, \\ ω_{3}^{0} (a, b, c) & = β_{2} (- 1 - a) β_{2} (- 1 - b) β_{2} (- 1 - c) 1 + β_{3} (- 1 - a) β_{3} (- 1 - b) β_{3} (- 1 - c) 1, \end{aligned}

(4.5)

for 0 ≤ a ≤ b ≤ c. In fact, we can explicitly write the relation between our original generators (4.1) and our new generators (4.5) as follows:

\begin{aligned} ω_{1} (a) & = \sqrt{3} ω_{1}^{0} (a), \\ ω_{2} (a, b) & = ω_{2}^{0} (a, b) + ω_{1}^{0} {(a)}_{- 1} ω_{1}^{0} (b), \\ ω_{3} (a, b, c) & = \frac{1}{3} ω_{3}^{0} (a, b, c) + \frac{1}{\sqrt{3}} (ω_{1}^{0} {(a)}_{- 1} ω_{2}^{0} (b, c) + ω_{1}^{0} {(b)}_{- 1} ω_{2}^{0} (a, c)) \\ (+ ω_{1}^{0} {(c)}_{- 1} ω_{2}^{0} (a, b) + ω_{1}^{0} {(a)}_{- 1} ω_{1}^{0} {(b)}_{- 1} ω_{1}^{0} (c)) . \end{aligned}

(4.6)

We may make a similar change of variables for the generating set of $C [x_{i} (m) | 1 \leq i \leq 3, m \geq 0]$ by setting

\begin{aligned} y_{1} (m_{1}) & = \frac{1}{\sqrt{3}} (x_{1} (m_{1}) + x_{2} (m_{1}) + x_{3} (m_{1})), \\ y_{2} (m_{2}) & = \frac{1}{\sqrt{3}} (x_{1} (m_{2}) + η^{2} x_{2} (m_{2}) + η x_{3} (m_{2})), \\ y_{3} (m_{3}) & = \frac{1}{\sqrt{3}} (x_{1} (m_{3}) + η x_{2} (m_{3}) + η x_{3} (m_{3})), \end{aligned}

(4.7)

for m_i ≥ 0, and

\begin{aligned} q_{1}^{0} (a) & = y_{1} (a), \\ q_{2}^{0} (a, b) & = y_{2} (a) y_{3} (b) + y_{3} (a) y_{2} (b), \\ q_{3}^{0} (a, b, c) & = y_{2} (a) y_{2} (b) y_{2} (c) + y_{3} (a) y_{3} (b) y_{3} (c), \end{aligned}

(4.8)

for a, b, c ≥ 0.

Using the translation operator, T, restricted to the orbifold $H {(3)}^{S_{3}}$ ⁠, the initial strong generating set (4.5) can be reduced per the following lemma:

Lemma 4.1.

The orbifold

H {(3)}^{S_{3}}

is strongly generated by the vectors

\begin{aligned} ω_{1}^{0} (0), \\ ω_{2}^{0} (0,2 a) & for a \geq 0, \\ ω_{3}^{0} (0, a, b) & for 0 \leq a \leq b . \end{aligned}

(4.9)

Proof.

It is clear that since

ω_{1}^{0} {(a)}_{- 1} 1 = \frac{1}{a!} ω_{1}^{0} {(0)}_{- 1 - a} 1,

(4.10)

the linear portion of the generating set () can be immediately minimized to contain the single weight one element ω₁(0).

Moving on toward the quadratic terms in (), we consider the vector spaces

A_{2} (m) = span {ω_{2}^{0} (a, b) | a + b = m}

(4.11)

and a natural family of subspaces

\partial A_{2} (m) = {v_{- 2} 1 | v \in A_{2} (m)} .

(4.12)

Observe that

\begin{aligned} ω_{2}^{0} {(a, b)}_{- 2} 1 & = Res z^{- 1} \frac{\partial}{\partial z} (Y (β_{2} (- 1 - a) β_{3} (- 1 - b) 1, z) + Y (β_{3} (- 1 - a) β_{2} (- 1 - b) 1, z)) \\ = \frac{1}{a! b!} Res z^{- 1} (\begin{matrix} ◦ \\ ◦ \end{matrix} \frac{\partial^{a}}{{\partial z}^{a}} β_{2} (z) \frac{\partial^{b}}{{\partial z}^{b}} β_{3} (z) \begin{matrix} ◦ \\ ◦ \end{matrix} + \begin{matrix} ◦ \\ ◦ \end{matrix} \frac{\partial^{a}}{{\partial z}^{a}} β_{3} (z) \frac{\partial^{b}}{{\partial z}^{b}} β_{2} (z) \begin{matrix} ◦ \\ ◦ \end{matrix}) \\ = (a + 1) ω_{2}^{0} (a + 1, b) + (b + 1) ω_{2}^{0} (a, b + 1), \end{aligned}

(4.13)

where

β_{i} (z) = Y (β_{i} (- 1) 1, z)

⁠. Also notice that for all a, b ≥ 0, we have

ω_{2}^{0} (a, b) = ω_{2}^{0} (b, a)

⁠. We may take an initial basis for A₂(m) to be the set

B_{2} (m) = \{ω_{2}^{0} (i, m - i) | 0 \leq i \leq ⌊\frac{m}{2}⌋\},

(4.14)

which implies that

dim A_{2} (2 n + 1) = dim A_{2} (2 n) = n + 1 .

(4.15)

Further, the set

{v_{- 2} 1 | v \in B_{2} (m)}

(4.16)

is clearly a basis for ∂A₂(m). It follows that

\begin{aligned} A_{2} (2 n) & = \partial A_{2} (2 n - 1) \oplus C ω_{2}^{0} (0,2 n), \\ A_{2} (2 n + 1) & = \partial A_{2} (2 n), \end{aligned}

(4.17)

and thus by induction, we may take a more convenient basis of A₂(m) to be

\{ω_{2}^{0} {(0,2 i)}_{- 1 - m + 2 i} 1 | 0 \leq i \leq ⌊\frac{m}{2}⌋\} .

(4.18)

Thus, a more efficient set of quadratic generators for the

H {(3)}^{S_{3}}

may be taken to be

ω_{2}^{0} (0,2 a)

for a ≥ 0.

Finally, we consider the cubic generators. Analogous to (), we have

ω_{3}^{0} {(a, b, c)}_{- 2} 1 = (a + 1) ω_{3}^{0} (a + 1, b, c) + (b + 1) ω_{3}^{0} (a, b + 1, c) + (c + 1) ω_{3}^{0} (a, b, c + 1),

(4.19)

which may be used inductively to write all cubic generators in terms of those of the form

ω_{3}^{0} (0, a, b)

with 0 ≤ a ≤ b. Throughout, we use the fact that

ω_{3}^{0} (a, b, c)

is invariant under any permutation of the entries a, b, c.

Remark 4.1.

Using (), we may take

\begin{aligned} ω_{1} (0), \\ ω_{2} (0,2 a) & for a \geq 0, \\ ω_{3} (0, a, b) & for 0 \leq a \leq b, \end{aligned}

(4.20)

as our strong generating set. We take advantage of this translation tool as generators of this form are somewhat more natural to the parent algebra,

H (3)

⁠.

Remark 4.2.

This simplification of the generating set also holds if we consider the ∂-ring associated with

H {(3)}^{S_{3}}

⁠, which has un-reduced generators given by (). We may reduce these to a minimal generating set for

C {[x_{i} (m) | 1 \leq i \leq 3, m \geq 0]}^{S_{3}}

given by

\begin{aligned} q_{1} (0), \\ q_{2} (0,2 a) & for a \geq 0, \\ q_{3} (0, a, b) & for 0 \leq a \leq b . \end{aligned}

(4.21)

We now present the following relations among the generators of $C {[x_{i} (m) | 1 \leq i \leq 3, m \geq 0]}^{S_{3}}$ ⁠.

Lemma 4.2.

For a = (a₁, a₂, a₃, a₄, a₅, a₆) with a₁, a₂, a₃, a₄, a₅, a₆ ≥ 0, we have

\begin{aligned} D_{6}^{C, 1} (a) & = q_{2}^{0} (a_{1}, a_{2}) q_{2}^{0} (a_{3}, a_{4}) q_{2}^{0} (a_{5}, a_{6}) - q_{2}^{0} (a_{1}, a_{2}) q_{2}^{0} (a_{3}, a_{6}) q_{2}^{0} (a_{4}, a_{5}) \\ + q_{2}^{0} (a_{1}, a_{4}) q_{2}^{0} (a_{2}, a_{6}) q_{2}^{0} (a_{3}, a_{5}) - q_{2}^{0} (a_{1}, a_{4}) q_{2}^{0} (a_{2}, a_{3}) q_{2}^{0} (a_{5}, a_{6}) \\ + q_{2}^{0} (a_{1}, a_{5}) q_{2}^{0} (a_{2}, a_{4}) q_{2}^{0} (a_{3}, a_{6}) - q_{2}^{0} (a_{1}, a_{5}) q_{2}^{0} (a_{2}, a_{6}) q_{2}^{0} (a_{3}, a_{4}) \\ + q_{2}^{0} (a_{1}, a_{6}) q_{2}^{0} (a_{2}, a_{3}) q_{2}^{0} (a_{4}, a_{5}) - q_{2}^{0} (a_{1}, a_{6}) q_{2}^{0} (a_{2}, a_{4}) q_{2}^{0} (a_{3}, a_{5}) = 0, \end{aligned}

(4.22)

\begin{aligned} D_{6}^{C, 2} (a) & = q_{3}^{0} (a_{1}, a_{2}, a_{3}) q_{3}^{0} (a_{4}, a_{5}, a_{6}) - q_{3}^{0} (a_{1}, a_{2}, a_{4}) q_{3}^{0} (a_{3}, a_{5}, a_{6}) \\ + \frac{1}{2} q_{2}^{0} (a_{1}, a_{3}) q_{2}^{0} (a_{2}, a_{6}) q_{2}^{0} (a_{4}, a_{5}) - \frac{1}{2} q_{2}^{0} (a_{1}, a_{4}) q_{2}^{0} (a_{2}, a_{5}) q_{2}^{0} (a_{3}, a_{6}) \\ + \frac{1}{2} q_{2}^{0} (a_{1}, a_{5}) q_{2}^{0} (a_{2}, a_{3}) q_{2}^{0} (a_{4}, a_{6}) - \frac{1}{2} q_{2}^{0} (a_{1}, a_{5}) q_{2}^{0} (a_{2}, a_{4}) q_{2}^{0} (a_{3}, a_{6}) = 0, \end{aligned}

(4.23)

and

\begin{aligned} D_{5}^{C} (a) & = q_{2}^{0} (a_{1}, a_{2}) q_{3}^{0} (a_{3}, a_{4}, a_{5}) - q_{2}^{0} (a_{1}, a_{5}) q_{3}^{0} (a_{2}, a_{3}, a_{4}) \\ - q_{2}^{0} (a_{2}, a_{5}) q_{3}^{0} (a_{1}, a_{3}, a_{4}) - q_{2}^{0} (a_{3}, a_{4}) q_{3}^{0} (a_{1}, a_{2}, a_{5}) \\ + q_{2}^{0} (a_{3}, a_{5}) q_{3}^{0} (a_{1}, a_{2}, a_{4}) + q_{2}^{0} (a_{4}, a_{5}) q_{3}^{0} (a_{1}, a_{2}, a_{3}) = 0, \end{aligned}

(4.24)

for a = (a₁, a₂, a₃, a₄, a₅) with a₁, a₂, a₃, a₄, a₅ ≥ 0.

Remark 4.3.

These relations play the role of the determinant (and similar) relations in the invariant theory of classical Lie groups. We do not claim that these two families of expressions generate all relations in this case. In fact, there are such relations at every degree, and since the Frobenius number of 5 and 6 is 19, there are new relations at least up to this degree.

We identify the relations from Lemma 4.2 via the isomorphism () with expressions involving the generators of the orbifold

H {(3)}^{S_{3}}

⁠. For a = (a₁, a₂, a₃, a₄, a₅, a₆) with a₁, a₂, a₃, a₄, a₅, a₆ ≥ 0, set

\begin{aligned} D_{6}^{1} (a) & = ω_{2}^{0} {(a_{1}, a_{2})}_{- 1} ω_{2}^{0} {(a_{3}, a_{4})}_{- 1} ω_{2}^{0} (a_{5}, a_{6}) - ω_{2}^{0} {(a_{1}, a_{2})}_{- 1} ω_{2}^{0} {(a_{3}, a_{6})}_{- 1} ω_{2}^{0} (a_{4}, a_{5}) \\ + ω_{2}^{0} {(a_{1}, a_{4})}_{- 1} ω_{2}^{0} {(a_{2}, a_{6})}_{- 1} ω_{2}^{0} (a_{3}, a_{5}) - ω_{2}^{0} {(a_{1}, a_{4})}_{- 1} ω_{2}^{0} {(a_{2}, a_{3})}_{- 1} ω_{2}^{0} (a_{5}, a_{6}) \\ + ω_{2}^{0} {(a_{1}, a_{5})}_{- 1} ω_{2}^{0} {(a_{2}, a_{4})}_{- 1} ω_{2}^{0} (a_{3}, a_{6}) - ω_{2}^{0} {(a_{1}, a_{5})}_{- 1} ω_{2}^{0} {(a_{2}, a_{6})}_{- 1} ω_{2}^{0} (a_{3}, a_{4}) \\ + ω_{2}^{0} {(a_{1}, a_{6})}_{- 1} ω_{2}^{0} {(a_{2}, a_{3})}_{- 1} ω_{2}^{0} (a_{4}, a_{5}) - ω_{2}^{0} {(a_{1}, a_{6})}_{- 1} ω_{2}^{0} {(a_{2}, a_{4})}_{- 1} ω_{2}^{0} (a_{3}, a_{5}), \end{aligned}

(4.25)

\begin{aligned} D_{6}^{2} (a) & = ω_{3}^{0} {(a_{1}, a_{2}, a_{3})}_{- 1} ω_{3}^{0} (a_{4}, a_{5}, a_{6}) - ω_{3}^{0} {(a_{1}, a_{2}, a_{4})}_{- 1} ω_{3}^{0} (a_{3}, a_{5}, a_{6}) \\ + \frac{1}{2} ω_{2}^{0} {(a_{1}, a_{3})}_{- 1} ω_{2}^{0} {(a_{2}, a_{6})}_{- 1} ω_{2}^{0} (a_{4}, a_{5}) - \frac{1}{2} ω_{2}^{0} {(a_{1}, a_{4})}_{- 1} ω_{2}^{0} {(a_{2}, a_{5})}_{- 1} ω_{2}^{0} (a_{3}, a_{6}) \\ + \frac{1}{2} ω_{2}^{0} {(a_{1}, a_{5})}_{- 1} ω_{2}^{0} {(a_{2}, a_{3})}_{- 1} ω_{2}^{0} (a_{4}, a_{6}) - \frac{1}{2} ω_{2}^{0} {(a_{1}, a_{5})}_{- 1} ω_{2}^{0} {(a_{2}, a_{4})}_{- 1} ω_{2}^{0} (a_{3}, a_{6}), \end{aligned}

(4.26)

and

\begin{aligned} D_{5} (a) & = ω_{2}^{0} {(a_{1}, a_{2})}_{- 1} ω_{3}^{0} (a_{3}, a_{4}, a_{5}) - ω_{2}^{0} {(a_{1}, a_{5})}_{- 1} ω_{3}^{0} (a_{2}, a_{3}, a_{4}) \\ - ω_{2}^{0} {(a_{2}, a_{5})}_{- 1} ω_{3}^{0} (a_{1}, a_{3}, a_{4}) - ω_{2}^{0} {(a_{3}, a_{4})}_{- 1} ω_{3}^{0} (a_{1}, a_{2}, a_{5}) \\ + ω_{2}^{0} {(a_{3}, a_{5})}_{- 1} ω_{3}^{0} (a_{1}, a_{2}, a_{4}) + ω_{2}^{0} {(a_{4}, a_{5})}_{- 1} ω_{3}^{0} (a_{1}, a_{2}, a_{3}), \end{aligned},

(4.27)

for a = (a₁, a₂, a₃, a₄, a₅) with a₁, a₂, a₃, a₄, a₅ ≥ 0.

Observe that for i ∈ {1, 2}, the weight of the expression

D_{6}^{i} (a)

is |a| + 6, while the weight of D₅(a) is |a| + 5, where we take |(u₁, …, u_n)| = u₁ + ⋯ + u_n for any multi-index. By Lemma 4.2 and repeated applications of the weak associativity properties of vertex algebras, along with our linear isomorphism (), we see that for i ∈ {1, 2}, we may rewrite

D_{6}^{i} (a) = D_{6}^{(4)} (a) + D_{6}^{(2)} (a),

(4.28)

where

D_{6}^{(4)} (a) = \sum_{\begin{array}{c} a, b, c, d \geq 0 \\ a + b + c + d = | a | + 2 \end{array}} μ_{a, b, c, d}^{(4)} ω_{2}^{0} {(a, b)}_{- 1} ω_{2}^{0} (c, d)

(4.29)

and

D_{6}^{(2)} (a) = \sum_{\begin{array}{c} a, b \geq 0 \\ a + b = | a | + 4 \end{array}} μ_{a, b}^{(2)} ω_{2}^{0} (a, b),

(4.30)

where

μ_{a, b, c, d}^{(4)}

and

μ_{a, b}^{(2)}

are appropriate constants. Importantly, this expansion has no terms that are “cubic” in the generators

ω_{2}^{0} (a, b)

or “quadratic” in the generators

ω_{3}^{0} (a, b, c)

⁠, each of which would contain a combination of six of the original β_i vectors with i ∈ {2, 3}. We have a similar (and simpler) decomposition

D_{5} (a) = D_{5}^{(3)} (a),

(4.31)

where

D_{5}^{(3)} (a) = \sum_{\begin{array}{c} a, b, c \geq 0 \\ a + b + c = | a | + 2 \end{array}} μ_{a, b, c}^{(3)} ω_{3}^{0} (a, b, c),

(4.32)

where

μ_{a, b, c}^{(3)}

are constants. Again, Lemma 4.2, weak associativity, and () imply that this expansion of will not contain terms that are “products” of the generators

ω_{2}^{0} (a, b)

and

ω_{3}^{0} (a, b, c)

or otherwise a combination of five of the original vectors β₂ and β₃.

Now we are poised to use the expressions $D_{6}^{i} (a)$ and D₅(a) to further reduce the generating set of $H {(3)}^{S_{3}}$ described in Lemma 4.1.

Lemma 4.3.

The full list of quadratic generators described in Lemma 4.1 can be replaced with the set ${ω_{2}^{0} (0,0), ω_{2}^{0} (0,2), ω_{2}^{0} (0,4)}$ ⁠.

Proof.

Theorem 4.7 of Ref. 21 implies that, using only combination of quadratic generators, we may immediately reduce our quadratic generators to the set

{ω_{2}^{0} (0,0), ω_{2}^{0} (0,2), ω_{2}^{0} (0,4), ω_{2}^{0} (0,6), ω_{2}^{0} (0,8)} .

(4.33)

In order to remove the remaining two quadratic generators, we use decoupling relations that involve both quadratic and cubic generators. Using the expression

D_{6}^{1} (0,0,0,0,1,1)

⁠, the decomposition described in ()–() may be used to construct the following equation:

\begin{aligned} ω_{2}^{0} (0,6) & = \frac{143}{742} ω_{2}^{0} {(0,4)}_{- 3} 1 - \frac{81}{371} ω_{2}^{0} {(0,2)}_{- 5} 1 + \frac{743}{1113} ω_{2}^{0} {(0,0)}_{- 7} 1 + \frac{1}{1484} ω_{2}^{0} {(0,0)}_{- 1} ω_{2}^{0} (0,4) \\ - \frac{13}{2968} ω_{2}^{0} {(0,0)}_{- 1} ω_{2}^{0} (1,3) - \frac{1}{4452} ω_{2}^{0} {(0,0)}_{- 1} ω_{2}^{0} (2,2) + \frac{2}{1113} ω_{2}^{0} {(0,1)}_{- 1} ω_{2}^{0} (0,3) \\ - \frac{1}{1113} ω_{2}^{0} {(0,1)}_{- 1} ω_{2}^{0} (1,2) + \frac{1}{2226} ω_{2}^{0} {(0,2)}_{- 1} ω_{2}^{0} (0,2) - \frac{3}{1484} ω_{2}^{0} {(0,2)}_{- 1} ω_{2}^{0} (1,1) \\ - \frac{1}{2968} ω_{2}^{0} {(1,1)}_{- 1} ω_{2}^{0} (1,1) + \frac{1}{4452} ω_{2}^{0} {(0,0)}_{- 1} ω_{2}^{0} {(0,0)}_{- 1} ω_{2}^{0} (0,1) \\ - \frac{1}{8904} ω_{2}^{0} {(0,0)}_{- 1} ω_{2}^{0} {(0,1)}_{- 1} ω_{2}^{0} (0,1) + \frac{1}{4452} ω_{3}^{0} {(0,0,0)}_{- 1} ω_{3}^{0} (0,1,1) \\ - \frac{1}{4452} ω_{3}^{0} {(0,0,1)}_{- 1} ω_{3}^{0} (0,0,1), \end{aligned}

(4.34)

and thus, we may remove

ω_{2}^{0} (0,6)

from (). Using () and

D_{6}^{1} (0,0,0,0,2,2)

⁠, we can similarly write

ω_{2}^{0} (0,8)

as a vertex algebraic polynomial using the quadratic generators

ω_{2}^{0} (0,0)

⁠,

ω_{2}^{0} (0,2)

⁠,

ω_{2}^{0} (0,4)

⁠, along with cubic generators

ω_{3}^{0} (0,0,0)

⁠,

ω_{3}^{0} (0,0,1)

⁠,

ω_{3}^{0} (0,1,1)

⁠,

ω_{3}^{0} (0,0,2)

⁠, and

ω_{3}^{0} (0,2,2)

⁠.

Lemma 4.4.

The full list of cubic generators described in Lemma 4.1 can be replaced with the set

{ω_{3}^{0} (0,0,0), ω_{3}^{0} (0,0,2), ω_{3}^{0} (0,1,2)} .

(4.35)

Moreover, these generators together with

ω_{1}^{0} (0), ω_{2}^{0} (0,0), ω_{2}^{0} (0,2), ω_{2}^{0} (0,4)

form a strong generating set of

H {(3)}^{S_{3}}

⁠.

Proof.

To get started, we notice that

\begin{aligned} ω_{3}^{0} (0,0,1) & = \frac{1}{3} ω_{3}^{0} {(0,0,0)}_{- 2} 1, \\ ω_{3}^{0} (0,1,1) & = - ω_{3}^{0} (0,0,2) + \frac{2}{3} ω_{3}^{0} {(0,0,0)}_{- 3} 1, \\ ω_{3}^{0} (0,0,3) & = - \frac{2}{3} ω_{3}^{0} (0,1,2) + \frac{1}{3} ω_{3}^{0} {(0,0,2)}_{- 2} 1, \end{aligned}

(4.36)

and in other words, all cubic generators of conformal weight 6 or less may be written in terms of the generators

{ω_{3}^{0} (0,0,0), ω_{3}^{0} (0,0,2), ω_{3}^{0} (0,1,2)}

using only the translation operator. Furthermore, we have

\begin{aligned} ω_{3}^{0} (0,1,3) & = - \frac{1}{12} ω_{3}^{0} (0,0,4) - \frac{1}{72} ω_{3}^{0} {(0,1,2)}_{- 2} 1 + \frac{1}{72} ω_{3}^{0} {(0,0,2)}_{- 3} 1, \\ ω_{3}^{0} (0,2,2) & = 3 ω_{3}^{0} (0,0,4) + \frac{4}{3} ω_{3}^{0} {(0,1,2)}_{- 2} 1 - \frac{1}{3} ω_{3}^{0} {(0,0,2)}_{- 3} 1 - \frac{1}{3} ω_{3}^{0} {(0,0,0)}_{- 5} 1, \\ ω_{3}^{0} (0,2,3) & = - \frac{1}{30} ω_{3}^{0} (0,1,4) + \frac{1}{60} ω_{3}^{0} {(0,0,4)}_{- 2} 1 + \frac{1}{35} ω_{3}^{0} {(0,1,2)}_{- 3} 1 - \frac{1}{36} ω_{3}^{0} {(0,0,0)}_{- 6} 1, \\ ω_{3}^{0} (0,3,3) & = \frac{5}{4} ω_{3}^{0} (0,0,6) - \frac{1}{2} ω_{3}^{0} (0,2,4) + \frac{1}{5} ω_{3}^{0} {(0,1,4)}_{- 2} 1 + \frac{1}{20} ω_{3}^{0} {(0,0,4)}_{- 3} 1 \\ + \frac{1}{6} ω_{3}^{0} {(0,1,2)}_{- 4} 1 - \frac{1}{12} ω_{3}^{0} {(0,0,2)}_{- 5} 1 - \frac{1}{4} ω_{3}^{0} {(0,0,0)}_{- 7} 1, \\ ω_{3}^{0} (0,3,4) & = \frac{5}{7} ω_{3}^{0} (0,1,6) + \frac{15}{7} ω_{3}^{0} {(0,0,6)}_{- 2} 1 - \frac{1}{3} ω_{3}^{0} {(0,2,4)}_{- 2} 1 + \frac{2}{3} ω_{3}^{0} {(0,1,4)}_{- 3} 1 \\ + \frac{5}{9} ω_{3}^{0} {(0,1,2)}_{- 5} 1 - \frac{5}{9} ω_{3}^{0} {(0,0,2)}_{- 6} - \frac{10}{9} ω_{3}^{0} {(0,0,0)}_{- 8} 1, \\ ω_{3}^{0} (0,4,4) & = - 21 ω_{3}^{0} (0,0,8) + 4 ω_{3}^{0} (0,2,6) - \frac{8}{7} ω_{3}^{0} {(0,1,6)}_{- 2} 1 + \frac{29}{7} ω_{3}^{0} {(0,0,6)}_{- 3} 1 \\ - \frac{2}{3} ω_{3}^{0} {(0,2,4)}_{- 3} 1 + \frac{8}{5} ω_{3}^{0} {(0,1,4)}_{- 4} 1 - \frac{1}{5} ω_{3}^{0} {(0,0,4)}_{- 5} 1 \\ + \frac{16}{9} ω_{3}^{0} {(0,1,2)}_{- 6} 1 - \frac{7}{3} ω_{3}^{0} {(0,0,2)}_{- 7} 1 - \frac{47}{9} ω_{3}^{0} {(0,0,0)}_{- 9} 1, \end{aligned}

(4.37)

that is, all vectors of the form

ω_{3}^{0} (0, a, b)

with 0 ≤ a ≤ b ≤ 4 can be written, using only the translation operator, in terms of our proposed generating set () with the addition of the vectors

ω_{3}^{0} (0,0,4)

⁠,

ω_{3}^{0} (0,1,4)

⁠,

ω_{3}^{0} (0,2,4)

⁠, and

ω_{3}^{0} (0, c, d)

with 0 ≤ c ≤ d where d ≥ 5. The remainder of our argument will be concerned with eliminating the need for these additional vectors.

Using D₅(0, 0, 0, 1, 1) and the decomposition described in () and (), we have

\begin{aligned} ω_{3}^{0} (0,0,4) & = - \frac{16}{15} ω_{3}^{0} {(0,1,2)}_{- 2} 1 + \frac{4}{15} ω_{3}^{0} {(0,0,2)}_{- 3} 1 + \frac{24}{45} ω_{3}^{0} {(0,0,0)}_{- 5} 1 - \frac{2}{5} ω_{2}^{0} {(0,0)}_{- 1} ω_{3}^{0} (0,1,1) \\ + \frac{4}{5} ω_{2}^{0} {(0,1)}_{- 1} ω_{3}^{0} (0,0,1) - \frac{2}{5} ω_{2}^{0} {(1,1)}_{- 1} ω_{3}^{0} (0,0,0), \end{aligned}

(4.38)

where again this calculation was performed using Ref. 25. Using D₅(0, 0, 1, 1, 1), a similar decoupling equation for

ω_{3}^{0} (0,1,4)

can be found. Furthermore, a linear combination of D₅(0, 0, 0, 1, 3) and D₅(0, 0, 1, 1, 2) leads to the decoupling equation

\begin{aligned} ω_{3}^{0} (0,2,4) & = \frac{1}{18} ω_{2}^{0} {(0,0)}_{- 1} ω_{3}^{0} (1,2,2) + \frac{1}{64} ω_{2}^{0} {(0,0)}_{- 1} ω_{3}^{0} (0,1,3) - \frac{1}{18} ω_{2}^{0} {(0,1)}_{- 1} ω_{3}^{0} (0,1,2) \\ - \frac{1}{64} ω_{2}^{0} {(0,1)}_{- 1} ω_{3}^{0} (0,0,3) - \frac{1}{18} ω_{2}^{0} {(0,2)}_{- 1} ω_{3}^{0} (0,1,1) + \frac{1}{18} ω_{2}^{0} {(1,2)}_{- 1} ω_{3}^{0} (0,0,1) \\ - \frac{1}{64} ω_{2}^{0} {(0,3)}_{- 1} ω_{3}^{0} (0,1,1) + \frac{1}{64} ω_{2}^{0} {(1,3)}_{- 1} ω_{3}^{0} (0,0,1) - \frac{7}{18} ω_{3}^{0} {(0,0,0)}_{- 7} 1 \\ - \frac{59}{144} ω_{3}^{0} {(0,0,2)}_{- 5} 1 + \frac{187}{240} ω_{3}^{0} {(0,0,4)}_{- 3} 1 + \frac{115}{288} ω_{3}^{0} {(0,1,2)}_{- 4} 1 + \frac{37}{60} ω_{3}^{0} {(0,1,4)}_{- 2} 1 . \end{aligned}

(4.39)

Next, for

a \in N

⁠, each of the six expressions D₅(0, 0, 0, 1, a − 3), D₅(0, 0, 0, 2, a − 4), D₅(0, 0, 1, 1, a − 4), D₅(0, 0, 0, 3, a − 5), D₅(0, 0, 1, 2, a − 5), and D₅(0, 1, 1, 1, a − 5) can be expanded as linear combinations of the six vectors

ω_{3}^{0} (0,0, a)

⁠,

ω_{3}^{0} (0,1, a - 1)

⁠,

ω_{3}^{0} (0,2, a - 2)

⁠,

ω_{3}^{0} (0,3, a - 3)

⁠,

ω_{3}^{0} (0,4, a - 4)

⁠, and

ω_{3}^{0} (0,5, a - 5)

along with terms of the form

ω_{3} {(0, m_{1}, m_{2})}_{- 1 - n} 1

where m₁ + m₂ + n = a with n ≥ 1. The linear independence of these can be determined by considering the determinant of the matrix, A, whose (i, j) entry is the coefficient of the ith vector in the jth expression as described above. A lengthy elementary calculation shows that, in the case that a is even

\begin{aligned} det A & = \frac{1}{6} {(a - 4)}^{2} (a - 3) (a - 1) (a + 2) () 5331 a^{6} - 70 325 a^{5} + 314 669 a^{4} - 613 567 a^{3} \\ + 97 384 a^{2} + 1 614 156 a - 1 835 568)), \end{aligned}

(4.40)

and a similar expression exists for odd a. In particular, for a ≥ 5, we can write each of the vectors

ω_{3}^{0} (0,0, a)

⁠,

ω_{3}^{0} (0,1, a - 1)

⁠,

ω_{3}^{0} (0,2, a - 2)

⁠,

ω_{3}^{0} (0,3, a - 3)

⁠,

ω_{3}^{0} (0,4, a - 4)

⁠, and

ω_{3}^{0} (0,5, a - 5)

as a vertex algebraic polynomial in terms of lower weight terms.

Next, for 5 ≤ a < b, D₅(b + 2, a − 1, 0, 0, 0) can be used to construct

\begin{aligned} ω_{3}^{0} (0, a + 1, b) & = \frac{2}{a (a + 1) (b + 1)} ({(- 1)}^{b} - {(- 1)}^{a}) ω_{3}^{0} (0, a - 1, b + 2) \\ + (\binom{a + b + 1}{b}) (\frac{1 + 3 {(- 1)}^{a}}{a + 1} + \frac{1 - 3 {(- 1)}^{b}}{b + 2} + \frac{{(- 1)}^{b} - {(- 1)}^{a}}{a + b + 1}) ω_{3}^{0} (0,0, a + b + 1) \\ + Ψ, \end{aligned}

(4.41)

where Ψ is a vertex algebraic polynomial with terms of the form

ω_{3}^{0} {(0, m_{1}, m_{2})}_{- 1 - n} 1

and

ω_{2}^{0} {(r_{1}, r_{2})}_{- 1} ω_{3}^{0} (0, s_{1}, s_{2})

where m₁ + m₂ + n = a + b + 1 and r₂ + r₂ + s₁ + s₂ = a + b − 1 with n ≥ 1.

Finally, (4.38)–(4.41) provide us with a clear inductive path to eliminating all cubic generators other than $ω_{3}^{0} (0,0,0), ω_{3}^{0} (0,0,2),$ and $ω_{3}^{0} (0,1,2)$ ⁠. This proves the first assertion. To prove the second claim, we use Lemma 4.3, where we reduced all quadratic generators down to $ω_{2}^{0} (0,0), ω_{2}^{0} (0,2), ω_{2}^{0} (0,4)$ ⁠. Although in this reduction, we additionally used cubic generators $ω_{3}^{0} (0,0,1)$ ⁠, $ω_{3}^{0} (0,1,1)$ ⁠, and $ω_{3}^{0} (0,2,2)$ ⁠, we can remove them using the first two relations in (4.36), the second relation in (4.37), and relation (4.38).

Theorem 4.1.

The vertex operator algebra $H {(3)}^{S_{3}}$ is simple of type (1, 2, 3, 4, 5, 6²), i.e., it is strongly generated by seven vectors whose conformal weights are: 1, 2, 3, 4, 5, 6, 6. This generating set is minimal.
$H {(3)}^{S_{3}}$ is isomorphic to $H (1) \otimes W$ ⁠, where W is of type (2, 3, 4, 5, 6²).
$H {(3)}^{S_{3}}$ is not freely generated (by any set of generators).

Proof.

Clearly, by a result from Ref. 11, this vertex algebra is simple. By Lemma 4.4 and earlier discussion, we see that

H {(3)}^{S_{3}}

is strongly generated by vectors of conformal weights: 1 (linear generator), 2, 4, 6 (quadratic generators) and 3, 5, 6 (cubic generators). From the character formula (see Proposition 6.1), we see that

q^{1 / 8} c h [H {(3)}^{S_{3}}] (τ) - \frac{1}{{(q; q)}_{\infty} {(q^{2}; q)}_{\infty} {(q^{3}; q)}_{\infty} {(q^{4}; q)}_{\infty} {(q^{5}; q)}_{\infty} {(q^{6}; q)}_{\infty}^{2}} = O (q^{9}),

where (a; q)_∞ ≔ ∏_i≥0(1 − aqⁱ). An easy analysis shows that dropping one (or more) generators from this generating set would imply that certain graded dimensions of

H {(3)}^{S_{3}}

are strictly bigger than the corresponding graded dimension for the smaller subalgebra. Thus, the proposed set of generators must be a minimal generating set. Clearly, this vertex algebra is not freely generated by these generators due to

q^{1 / 8} c h [H {(3)}^{S_{3}}] (τ) - \frac{1 - q^{9}}{{(q; q)}_{\infty} {(q^{2}; q)}_{\infty} {(q^{3}; q)}_{\infty} {(q^{4}; q)}_{\infty} {(q^{5}; q)}_{\infty} {(q^{6}; q)}_{\infty}^{2}} = O (q^{10}),

which implies that there must be a nontrivial relation at degree 9. In fact, one can quickly argue that this is impossible simply from the fact that

c h [H {(3)}^{S_{3}}]

is modular, which is impossible to achieve with a free generating set.

For (ii), we first observe that ⟨ω₁(0)⟩ is isomorphic to $H (1)$ ⁠. By taking the commutant $W ≔ C o m m (H (1), H {(3)}^{S_{3}})$ ⁠, we get $H (1) \otimes W ≅ H {(3)}^{S_{3}}$ ⁠. Each generator of $H {(3)}^{S_{3}}$ ⁠, except ω₁(0), can be written as a linear combination of elements in $H (1) \otimes W$ with a nonzero component in W [otherwise, it would imply that some of the generators are contained in ⟨ω₁(0)⟩, contradicting the minimality in (i)]. These nonzero components form a generating set of W.

Using (), we may take the original invariants ω₁(0), ω₂(0, 0), ω₂(0, 2), ω₂(0, 4), ω₃(0, 0, 0), ω₃(0, 0, 2), and ω₃(0, 1, 2) as our minimal strong generating set. Furthermore, the following change of variables allows us to express the orbifold in terms of primary generators

\begin{aligned} h & = ω_{1} (0), ω = \frac{1}{2} ω_{2} (0,0), J_{1} = 2 ω_{2} (0,2) - \frac{24}{37} ω_{- 1} ω - \frac{30}{37} ω_{- 3} 1, \\ J_{2} & = 24 ω_{2} (0,4) - \frac{460}{81} ω_{2} {(0,2)}_{- 3} 1 - \frac{80}{27} ω_{- 1} ω_{2} (0,2) + \frac{256}{801} ω_{- 1} ω_{- 1} ω \\ - \frac{364}{2403} ω_{- 2} ω_{- 2} 1 + \frac{3472}{2403} ω_{- 3} ω + \frac{40 744}{2403} ω_{- 5} 1, \\ C_{1} & = ω_{3} (0,0,0) - \frac{1}{3} h_{- 1} h_{- 1} h, \\ C_{2} & = 2 ω_{3} (0,0,2) - \frac{2}{15} ω_{3} {(0,0,0)}_{- 3} 1 - \frac{8}{45} h_{- 3} h_{- 1} h + \frac{2}{15} h_{- 2} h_{- 2} h \\ - \frac{16}{45} ω_{- 1} ω_{3} (0,0,0) + \frac{16}{135} h_{- 1} h_{- 1} h_{- 1} ω, \\ C_{3} & = 2 ω_{3} (0,1,2) + \frac{1}{10} ω_{3} {(0,0,2)}_{- 2} 1 - \frac{6}{25} ω_{3} {(0,0,0)}_{- 4} - \frac{2}{25} ω_{- 1} ω_{3} {(0,0,0)}_{- 2} 1 \\ + \frac{3}{25} ω_{- 2} ω_{3} (0,0,0) - \frac{1}{25} h_{- 1} h_{- 1} h_{- 1} ω_{- 2} 1 + \frac{2}{25} h_{- 2} h_{- 1} h_{- 1} ω + \frac{2}{25} ω_{- 2} ω_{- 2} ω_{- 2} 1 \\ - \frac{4}{25} h_{- 3} h_{- 2} h + \frac{6}{75} h_{- 4} h_{- 1} h . \end{aligned}

(4.42)

V. THE ORBIFOLD $H {(3)}^{Z_{3}}$

We now consider the orbifold of $H (3)$ under the action of the cyclic subgroup $Z_{3} ≅ ⟨(\begin{matrix} 1 & 2 & 3 \end{matrix})⟩ \subset S_{3}$ ⁠. Using a similar strategy to Subsection IV A, we may take

\begin{aligned} ω_{1}^{0} & = β_{1} (- 1) 1, ω_{2,3}^{0} (a, b) = β_{2} (- 1 - a) β_{3} (- 1 - b) 1, \\ ω_{2,2,2}^{0} (a, b, c) & = β_{2} (- 1 - a) β_{2} (- 1 - b) β_{2} (- 1 - c) 1, \\ ω_{3,3,3}^{0} (a, b, c) & = β_{3} (- 1 - a) β_{3} (- 1 - b) β_{3} (- 1 - c) 1, \end{aligned}

(5.1)

as our initial generating set. Analogous to Lemma 4.1, we have the following initial reduction of the generating set:

Lemma 5.1.

The orbifold

H {(3)}^{Z_{3}}

is strongly generated by vectors

\begin{aligned} ω_{1}^{0} (0) & , ω_{2,3}^{0} (0, a) for a \geq 0, \\ ω_{2,2,2}^{0} (0, a, b) & f o r 0 \leq a \leq b, ω_{3,3,3}^{0} (0, a, b) & for 0 \leq a \leq b . \end{aligned}

(5.2)

Proof.

The linear generator is the same vector found in Lemma 4.1 which may be used to produce all vectors of the form $ω_{1}^{0} (a)$ for a ≥ 0 as before.

The quadratic generators will be reduced using an argument similar to the proof of Lemma 4.1, where the main difference is due to the fact that for a ≠ b,

ω_{2,3}^{0} (a, b) \neq ω_{2,3}^{0} (b, a)

⁠. We begin by setting

A (m) = span {ω_{2,3} (a, b) | a + b = m} and \partial A (m) = {v_{- 2} 1 | v \in A (m)}

(5.3)

and notice that for all m ≥ 0, ∂A(m) is a subspace of A(m + 1) of co-dimension 1. In fact, we have

A (m + 1) = \partial A (m) \oplus C ω_{2,3}^{0} (0, m + 1),

(5.4)

from which it follows that the set

{ω_{2,3}^{0} {(0, i)}_{- 1 - m + i} 1}

(5.5)

is a basis of A(m). Our result follows.

Again, the reduction of the cubic generators follows from an analogue of (). In particular,

ω_{i, i, i}^{0} {(a, b, c)}_{- 2} 1 = (a + 1) ω_{i, i, i}^{0} (a + 1, b, c) + (b + 1) ω_{i, i, i}^{0} (a, b + 1, c) + (c + 1) ω_{i, i, i}^{0} (a, b, c + 1),

(5.6)

for i ∈ {1, 2}, may be used inductively to achieve the desired set, where we use the fact that

ω_{i, i, i}^{0} (a, b, c)

is fixed under any permutation of the entries a, b, c.

We now present analogues of Lemmas 4.3 and 4.4 in this setting.

Lemma 5.2.

The full list of quadratic generators described in () can be replaced with

ω_{2,3}^{0} (0,0), ω_{2,3}^{0} (0,1), ω_{2,3}^{0} (0,2), and ω_{2,3}^{0} (0,3) .

(5.7)

Proof.

Our main tool for this argument will be the expression

D_{2} (a) = ω_{2,3}^{0} {(0, a)}_{- 1} ω_{2,3}^{0} (1,1) - ω_{2,3}^{0} {(1, a)}_{- 1} ω_{2,3}^{0} (0,1),

(5.8)

which may be expanded to

D_{2} (a) = (\binom{a + 2}{2}) (\binom{a + 4}{2}) ω_{2,3}^{0} (0, a + 4) + 2 (\binom{a + 3}{3}) ω_{2,3}^{0} (1, a + 3) - {(- 1)}^{a} (a + 1) ω_{2,3}^{0} (a + 3,1) .

(5.9)

An elementary calculation allows us to write

ω_{2,3}^{0} (1, a + 3) = - (a + 4) ω_{2,3}^{0} (0, a + 4) + ω_{2,3}^{0} {(0, a + 3)}_{- 2} 1,

(5.10)

and more generally,

\begin{aligned} ω_{2,3}^{0} (a + 3,1) & = \sum_{j = 0}^{a + 3} {(- 1)}^{a + j + 1} (a - j + 4) ω_{2,3}^{0} {(0, a - j + 4)}_{- 1 - j} 1 \\ = {(- 1)}^{a + 1} (a + 4) ω_{2,3}^{0} (0, a + 4) \\ + \sum_{j = 1}^{a + 3} {(- 1)}^{a + j + 1} (a - j + 4) ω_{2,3}^{0} {(0, a - j + 4)}_{- 1 - j} 1 . \end{aligned}

(5.11)

Now combining ()–(), we have

\begin{aligned} D_{2} (a) & = - \frac{1}{12} (a + 6) (a + 4) (a + 1) (a - 1) ω_{2,3}^{0} (0, a + 4) \\ + 2 (\binom{a + 3}{3}) ω_{2,3}^{0} {(0, a + 3)}_{- 2} 1 \\ + \sum_{j = 1}^{a + 3} {(- 1)}^{j} (a + 1) (a - j + 4) ω_{2,3}^{0} {(0, a - j + 4)}_{- 1 - j} 1, \end{aligned}

(5.12)

which, for a = 0 and a ≥ 2, may be used to solve for

ω_{2,3}^{0} (0, a + 4)

in terms of vectors of lower conformal weight.

Inductively, together with the special case

\begin{aligned} ω_{2,3}^{0} (0,5) & = \frac{1}{7} (ω_{2,3}^{0} {(0,0)}_{- 1} ω_{2,3}^{0} (1,2) - ω_{2,3}^{0} {(1,0)}_{- 1} ω_{2,3}^{0} (0,2)) \\ + \frac{3}{7} ω_{2,3}^{0} {(0,4)}_{- 2} 1 - \frac{3}{7} ω_{2,3}^{0} {(0,3)}_{- 3} 1 + \frac{1}{7} ω_{2,3}^{0} {(0,2)}_{- 4} 1, \end{aligned}

(5.13)

this allows us to remove all but the necessary generators.

Lemma 5.3.

The full list of cubic generators described in () can be replaced with

ω_{2,2,2}^{0} (0,0,0), ω_{2,2,2}^{0} (0,0,2), ω_{3,3,3}^{0} (0,0,0), ω_{3,3,3}^{0} (0,0,2) .

(5.14)

Proof.

We follow a strategy similar to the proof of Lemma 4.4. In this case, our argument is greatly simplified due to the fact that our decoupling relations are simpler and occur at a lower initial conformal weight. We focus on the

ω_{2,2,2}^{0} (0, a, b)

terms, as the

ω_{3,3,3}^{0} (0, a, b)

are similar, starting with the observation that

\begin{aligned} ω_{2,2,2}^{0} (0,0,1) & = \frac{1}{3} ω_{2,2,2}^{0} {(0,0,0)}_{- 2} 1, \\ ω_{2,2,2}^{0} (0,1,1) & = - ω_{2,2,2}^{0} (0,0,2) + \frac{2}{3} ω_{2,2,2}^{0} {(0,0,0)}_{- 3} 1, \end{aligned}

(5.15)

meaning that, by using the translation operator, all cubic generators of weight five or less may be written in terms of ().

The expression

D_{3} (a, b) = ω_{2,3}^{0} {(0, a)}_{- 1} ω_{2,2,2}^{0} (0,0, b) - ω_{2,3}^{0} {(b, a)}_{- 1} ω_{2,2,2}^{0} (0,0,0)

(5.16)

will be our main tool moving forward. A special case of () can be used to construct the equation

ω_{2,2,2}^{0} (0,0, a) = \frac{1}{3 a + 1} (ω_{2,2,2}^{0} {(0,0, a - 1)}_{- 2} 1 + \frac{{(- 1)}^{a}}{a - 2} D_{3} (a - 3,1)),

(5.17)

which can be used inductively to remove all generators of the form

ω_{2,2,2}^{3} (0,0, a)

for a ≥ 3 from the generating set. Furthermore, a more general version yields

ω_{2,2,2}^{0} (0, a, b) = \frac{1}{2} (\frac{2 a + 3 b}{a + b} (\binom{a + b}{a}) ω_{2,2,2}^{0} (0,0, a + b) + \frac{{(- 1)}^{a}}{a - 1} D (a - 2, b)),

(5.18)

which can be used to remove all generators of the form

ω_{2,2,2}^{3} (0, a, b)

⁠, where 2 ≤ a ≤ b. This leaves only the generators of the form

ω_{2,2,2}^{0} (0,1, a)

which can be removed using

ω_{2,2,2}^{0} (0,1, a) = \frac{1}{2} (ω_{2,2,2}^{0} {(0,0, a)}_{- 2} 1 - (a + 1) ω_{2,2,2}^{0} (0,0, a + 1)) .

(5.19)

Theorem 5.1.

The vertex operator algebra $H {(3)}^{Z_{3}}$ is simple of type (1, 2, 3³, 4, 5³), i.e., it is strongly generated by seven vectors whose conformal weights are 1, 2, 3, 3, 3, 4, 5, 5, 5. This generating set is minimal.
$H {(3)}^{Z_{3}}$ is isomorphic to $H (1) \otimes W$ ⁠, where W is of type (2, 3³, 4, 5³).
$H {(3)}^{Z_{3}}$ is not freely generated (by any set of generators).

Proof.

This proof is similar to the proof of Theorem 4.1. In particular, we use the fact that the free $W$ -algebra on a generating set containing the same weight elements with any one removed has certain graded dimensions strictly less than those described by the character of $H {(3)}^{Z_{3}}$ ⁠, (6.1).

Finally, the orbifold

H {(3)}^{Z_{3}}

is described in terms of primary generators with the following vectors:

\begin{aligned} h & = ω_{1}^{0} (0), ω = ω_{2}^{0} (0,0) + \frac{1}{2} ω_{1}^{0} {(0)}_{- 1} ω_{1}^{0} (0), J_{1} = ω_{2}^{0} (0,1) - \frac{1}{2} ω_{- 2} 1 + \frac{1}{2} h_{- 2} h, \\ J_{2} & = 2 ω_{2}^{0} (0,2) - ω_{2}^{0} {(0,1)}_{- 2} 1 + \frac{2}{3} ω_{- 1} ω - \frac{22}{3} h_{- 1} h_{- 1} ω + \frac{77}{9} h_{- 3} h \\ - \frac{83}{12} h_{- 2} h_{- 2} 1, \\ J_{3} & = 6 ω_{2}^{0} (0,3) - \frac{8}{3} ω_{2}^{0} {(0,2)}_{- 2} 1 + 4 h_{- 2} h_{- 1} ω + 8 h_{- 1} h_{- 1} ω_{- 2} 1 \\ - 16 h_{- 3} ω + 8 h_{- 1} ω_{- 3} 1 + 4 h_{- 1}^{4} h - \frac{34}{3} h_{- 2} h_{- 2} h - \frac{40}{3} h_{- 3} h_{- 1} h + 4 h_{- 1}^{4} h \\ + \frac{58}{3} h_{- 3} h_{- 2} 1 - 23 h_{- 4} h + \frac{88}{5} h_{- 5} 1, \\ C_{1}^{(2)} & = ω_{2,2,2}^{0} (0,0,0), C_{2}^{(2)} = ω_{2,2,2}^{0} (0,0,2) - \frac{1}{15} ω_{2,2,2}^{0} {(0,0,0)}_{- 3} 1 - \frac{8}{15} ω_{- 1} ω_{2,2,2}^{0} (0,0,0), \\ C_{1}^{(3)} & = ω_{3,3,3}^{0} (0,0,0), C_{2}^{(3)} = ω_{3,3,3}^{0} (0,0,2) - \frac{1}{15} ω_{3,3,3}^{0} {(0,0,0)}_{- 3} 1 - \frac{8}{15} ω_{- 1} ω_{3,3,3}^{0} (0,0,0) . \end{aligned}

(5.20)

VI. THE CHARACTER OF $H {(n)}^{S_{n}}$

We first derive a general formula for the character of the S_n-orbifold of a vertex algebra. Although formulas of this form have appeared in the literature (e.g., Ref. 4 or Ref. 16), we include the proof for completeness.

Theorem 6.1.

Let V be a VOA and

f^{S_{n}} (q)

be the character of the S_n-fixed point subalgebra

V {(n)}^{S_{n}} \subset V^{\otimes^{n}}

(under the usual action). Then

f^{S_{n}} (q) = \frac{1}{n!} \sum_{g \in S_{n}} {tr}_{V^{\otimes n}} g q^{L (0) - c / 24},

where g acts by permutation of tensor factors.

Proof.

Consider the group algebra

C [S_{n}]

and the idempotent

e = \frac{1}{| S_{n} |} \sum_{g \in S_{n}} g \in C [S_{n}], e^{2} = e,

where |S_n| = n!. The space Im(e) is precisely

V {(n)}^{S_{n}}

⁠. Since the eigenvalues of e are 1 and 0 (and 1 for the S_n-fixed subalgebra), the character can be computed simply by taking the trace of e

{tr}_{V^{\otimes^{n}}} e q^{L (0) - c / 24} = {tr}_{V {(n)}^{S_{n}}} q^{L (0) - c / 24} .

Now we specialize to n = 3 and $V = H (1)$ ⁠.

Proposition 6.1.

c h [H {(3)}^{S_{3}}] (τ) = \frac{q^{- 1 / 8}}{6} (\prod_{n = 1}^{\infty} \frac{1}{{(1 - q^{n})}^{3}} + 3 \prod_{n = 1}^{\infty} \frac{1}{(1 - q^{2 n})} \prod_{n = 1}^{\infty} \frac{1}{(1 - q^{n})} + 2 \prod_{n = 1}^{\infty} \frac{1}{(1 - q^{3 n})}) .

Proof.

It suffices to consider the following elements: g = 1, g = (12), and g = (123). By the previous theorem, we have

c h [H {(3)}^{S_{3}}] (τ) = \frac{1}{6} ({tr}_{H (3)} q^{L (0) - 3 / 24} + 3 \cdot {tr}_{H (3)} (12) q^{L (0) - 3 / 24} + 2 \cdot {tr}_{H (3)} (123) q^{L (0) - 3 / 24}) .

Clearly,

{tr}_{H (3)} q^{L (0) - 3 / 24} = q^{- 1 / 8} \prod_{n = 1}^{\infty} \frac{1}{{(1 - q^{n})}^{3}} .

For tr(12), we compute the trace on the following basis of

H (3)

B = {p_{λ_{1}} \otimes p_{λ_{2}} \otimes p_{λ_{3}}, λ_{i} \in P},

where

P

is the set of partitions, and for

p_{λ_{i}}

⁠, we take obvious monomials of Heisenberg elements. The matrix representation of

(12) \in E n d (H (3))

in this basis has a non-zero entry on the diagonal (=1) if and only if the corresponding basis element is

p_{λ} \otimes p_{λ} \otimes p_{ν} .

Since all p_λ are generated by α(−i), the vectors contributing to the trace are generated by α(−i) ⊗ α(−i) ⊗ α(−j). Thus,

{tr}_{H (3)} (12) q^{L (0) - 3 / 24} = q^{- 1 / 8} \prod_{i = 1}^{\infty} \frac{1}{(1 - q^{2 i})} \prod_{j = 1}^{\infty} \frac{1}{(1 - q^{j})} .

One similarly argues for 3-cycles, so we get

{tr}_{H (3)} (123) q^{L (0) - 3 / 24} = q^{- 1 / 8} \prod_{i = 1}^{\infty} \frac{1}{(1 - q^{3 i})} .

Corollary 6.1.

c h [H {(3)}^{Z_{3}}] (τ) = \frac{q^{- 1 / 8}}{3} (\prod_{n = 1}^{\infty} \frac{1}{{(1 - q^{n})}^{3}} + 2 \prod_{n = 1}^{\infty} \frac{1}{(1 - q^{3 n})}) .

VII. MODULAR INVARIANCE OF $H {(3)}^{S_{3}}$ -CHARACTERS AND QUANTUM DIMENSIONS

By a result from Ref. 11 (especially Sec. VII devoted to dihedral groups), we have a decomposition of $H (3)$ in terms of $H {(3)}^{S_{3}}$ -modules

H (3) = H {(3)}^{S_{3}} \oplus H {(3)}^{S_{3}, s g n} \oplus V (2) \otimes H {(3)}^{S_{3}, s t},

(7.1)

where V(2) is the 2-dimensional standard representation of S₃, and $H {(3)}^{S_{3}, s g n}$ and $H {(3)}^{S_{3}, s t}$ are irreducible $H {(3)}^{S_{3}}$ -modules.

Lemma 7.1.

We have

\begin{array}{l} c h [H {(3)}^{S_{3}, s g n}] (τ) & = \frac{q^{- 1 / 8}}{6} (\prod_{n = 1}^{\infty} \frac{1}{{(1 - q^{n})}^{3}} - 3 \prod_{n = 1}^{\infty} \frac{1}{(1 - q^{2 n})} \prod_{n = 1}^{\infty} \frac{1}{(1 - q^{n})} + 2 \prod_{n = 1}^{\infty} \frac{1}{(1 - q^{3 n})}), \\ c h [H {(3)}^{S_{3}, s t}] (τ) & = \frac{q^{- 1 / 8}}{6} (2 \prod_{n = 1}^{\infty} \frac{1}{{(1 - q^{n})}^{3}} - 2 \prod_{n = 1}^{\infty} \frac{1}{(1 - q^{3 n})}) . \end{array}

Proof.

The character of the sign module¹¹

H {(3)}^{S_{3}, s g n} = {a \in H (3) : σ (a) = sgn (σ) a, σ \in S_{3}}

is computed along the lines of Proposition 6.1 with the difference that in the sign representation 2-cycles act as −1. This sign contributes with a negative sign to the middle terms of the formula.

The second relation follows directly from the first formula, Proposition 6.1, and decomposition (7.1).

Next, we construct several $H {(3)}^{S_{3}}$ -modules coming from ordinary and g-twisted $H (3)$ -modules, g ∈ S₃.

We first look at $H {(3)}^{S_{3}}$ -modules coming from irreducible $H (3)$ -modules (Fock representations). Let $w = (w_{1}, w_{2}, w_{3}) \in C^{3}$ ⁠. We denote by $F_{w_{1}, w_{2}, w_{3}}$ the irreducible $H (3)$ -module with highest weight w such that α_i(0) acts as multiplication by w_i. Similarly, F_w denotes the Fock representation for $H (1)$ with highest weight w.

Interestingly, $F_{w_{1}, w_{2}, w_{3}}$ is generically irreducible as an $H {(3)}^{S_{3}}$ -module.

Lemma 7.2.

Let $w_{1}, w_{2}, w_{3} \in C^{3}$ such that w_i ≠ w_j, i ≠ j. Then the $H (3)$ -module $F_{w_{1}, w_{2}, w_{3}}$ is irreducible as an $H {(3)}^{S_{3}}$ -module.

Proof.

Let V be a simple VOA, g be an automorphism of V of prime order p, and M be a simple V-module such that g ○ M is not isomorphic to M as V-modules. Then according to Theorem 6.1,¹¹ M is a simple V^⟨g⟩-module. To apply this result, we first observe embeddings

H {(3)}^{S_{3}} ↪ H {(3)}^{< (12) >} ↪ H (3) .

Because of w₁ ≠ w₂,

(12) ◦ F_{w_{1}, w_{2}, w_{3}} ≇ F_{w_{1}, w_{2}, w_{3}}

as

H (3)

-modules. Indeed, for

(12) ◦ F_{w_{1}, w_{2}, w_{3}}

⁠, actions of α₁(0) and α₂(0) are switched, so

(12) ◦ F_{w_{1}, w_{2}, w_{3}} ≅ F_{w_{2}, w_{1}, w_{3}}

⁠. Therefore,

F_{w_{1}, w_{2}, w_{3}}

is irreducible as a

G (3) ≔ H {(3)}^{(< 12 >)}

-module. Similarly, we consider

(123) ◦ F_{w_{1}, w_{2}, w_{3}}

and

F_{w_{1}, w_{2}, w_{3}}

as G(3)-modules. Because of w₁ ≠ w₃ and w₃ ≠ w₂, they are not isomorphic as G(3)-modules, and thus

F_{w_{1}, w_{2}, w_{3}}

is irreducible as a G(3)^<(123)> -module. The proof follows.

In the study of characters, we require modular variables so we let q = e^2πiτ, where

τ \in H

(the upper half-plane). Let

η (τ) = q^{1 / 24} {(q; q)}_{\infty},

the Dedekind η-function. Clearly, for every w,

c h [F_{w_{1}, w_{2}, w_{3}}] (τ) = \frac{q^{- 3 / 24 + w_{1}^{2} / 2 + w_{2}^{2} / 2 + w_{3}^{2} / 2}}{{(q; q)}_{\infty}} = \frac{q^{w_{1}^{2} / 2 + w_{2}^{2} / 2 + w_{3}^{2} / 2}}{η {(τ)}^{3}} .

(7.2)

Next we consider g-twisted $H (3)$ -modules, so g is either a 2- or 3-cycle. We may assume, without loss of generality, that g is either (12) or (123).

We first construct a family of θ-twisted

H (3)

-modules, where θ ≔ (12) switches the first two “coordinates.” By using the isomorphism in Sec. III,

H {(2)}^{< (12) >} ≅ H (1) \otimes M {(1)}^{+}

⁠, for every

w_{1}, w_{3} \in C

⁠, we get a θ-twisted

H (3)

-module

F_{w_{1}, w_{3}} (θ) ≔ F_{w_{1}} \otimes M (1) (θ) \otimes F_{w_{3}},

where

M (1) (θ) ≅ C [α (- 1 / 2), α (- 3 / 2), \dots .]

is a

Z_{2}

-twisted

H (1)

-module studied in Ref. 12. This module is of conformal weight

\frac{1}{16}

and its character is easily computed. We quickly get

c h [F_{w_{1}, w_{3}} (θ)] (τ) = \frac{q^{1 / 16 - 3 / 24 + w_{1}^{2} / 2 + w_{3}^{2} / 2}}{{(q^{1 / 2}; q)}_{\infty} {(q; q)}_{\infty} {(q; q)}_{\infty}} = \frac{q^{- 1 / 16 + w_{1}^{2} / 2 + w_{3}^{2} / 2}}{{(q^{1 / 2}; q^{1 / 2})}_{\infty} {(q; q)}_{\infty}}

= \frac{q^{w_{1}^{2} / 2 + w_{3}^{2} / 2}}{η (τ / 2) η (τ)} .

(7.3)

Further, we construct a family of (123)-twisted

H (3)

-modules. Let σ = (123). We first construct a σ-twisted Heisenberg algebra

{\hat{h}}^{σ}

as in Ref. 19. It is isomorphic to

C [t, t^{- 1}] \oplus t^{1 / 3} C [t, t^{- 1}] \oplus t^{2 / 3} C [t, t^{- 1}] \oplus C k

as a graded Lie algebra. Then for every

w \in C

⁠, we have an associated σ-twisted

H (3)

-representation

F_{w} (σ) ≅ U (h_{< 0}^{σ})

⁠. Moreover,

c h [F_{w} (σ)] (τ) = \frac{q^{w^{2} / 2 + h - 1 / 8}}{{(q; q)}_{\infty} {(q^{1 / 3}; q)}_{\infty} {(q^{2 / 3}; q)}_{\infty}} = \frac{q^{w^{2} / 2 + h - 3 / 24}}{{(q^{1 / 3}; q^{1 / 3})}_{\infty}},

where

h = \frac{1}{4 p^{2}} \sum_{i = 1}^{p - 1} i (p - i) r_{i}

⁠, and p is the order of σ and r_i (1 ≤ i ≤ 3) are dimensions of the eigenspaces of σ. Plugging in p = 3 and r_i = 1 into the last formula, we obtain

h = \frac{1}{9}

and thus

c h [F_{w} (σ)] (τ) = \frac{q^{w^{2} / 2 - 1 / 72}}{{(q^{1 / 3}; q^{1 / 3})}_{\infty}} = \frac{q^{w^{2} / 2}}{η (τ / 3)} .

(7.4)

Here, we do not pursue decomposition and irreducibility of $F_{w_{1}, w_{3}} (σ)$ and F_w(σ) as $H {(3)}^{S_{3}}$ -modules. This will be a subject of Ref. 24.

A. Modular invariance

In this part, we obtain a modularity result for the character of $H {(3)}^{S_{3}}$ under the S-transformation, $τ \to - \frac{1}{τ}$ ⁠. This was briefly discussed in the Introduction.

We make use of a well-known modular relation for the Dedekind η-function

η {(- 1 / τ)}^{n} = {(\sqrt{- i τ})}^{n} η {(τ)}^{n}, n \in N .

(7.5)

Let us also recall a higher dimensional Gauss’ integral formula

\int_{R^{n}} q^{w_{1}^{2} / 2 + \dots + w_{n}^{2} / 2} d w = \frac{1}{{(\sqrt{- i τ})}^{n}} .

From the Gauss formula and (7.5), we immediately get the following relations.

Lemma 7.3.

\begin{array}{l} \frac{1}{η {(- 1 / τ)}^{3}} & = \int_{R^{3}} \frac{q^{w_{1}^{2} / 2 + w_{2}^{2} / 2 + w_{3}^{2} / 2}}{η {(τ)}^{3}} d w, w = (w_{1}, w_{2}, w_{3}), \\ \frac{1}{η (- 1 / τ) η (- 2 / τ)} & = \sqrt{2} \int_{R^{2}} \frac{q^{w_{1}^{2} / 2 + w_{3}^{2} / 2}}{η (τ) η (τ / 2)} d w, w = (w_{1}, w_{3}), \\ \frac{1}{η (- 3 / τ)} & = \sqrt{3} \int_{R} \frac{q^{w^{2} / 2}}{η (τ / 3)} d w . \end{array}

Notice that only the numerator of the integrand depends on w. The denominator is placed inside the integral so that it matches the shape of (1.1).

In Proposition 6.1, the character formulas for $F_{w_{1}, w_{2}, w_{3}}$ ⁠, $F_{w_{1}, w_{3}} (θ)$ ⁠, and F_w(σ), that is, formulas (7.2)–(7.4), together with Lemma 7.3 imply the following theorem:

Theorem 7.1.

The character of $H {(3)}^{S_{3}}$ has a modular invariance property, in the sense of (1.1).

B. Quantum dimensions

Let V be a vertex operator algebra. For a V-module M, we define its quantum dimension

q d i m [M] ≔ lim_{t \to 0^{+}} \frac{c h [M] (i t)}{c h [V] (i t)} .

In a situation where V is rational (in the strongest sense), this limit always exists and is nonzero (see Ref. 13). But for general V, this limit can be zero or even +∞.

As our characters can be expressed in terms of η(τ), we will make use of an asymptotic formula

η {(i t)}^{n} \sim (\frac{1}{t^{\frac{n}{2}}}) e^{- \frac{π n}{12 t}}, (t \to 0^{+}) .

(7.6)

Proposition 7.1.

As $H {(3)}^{S_{3}}$ -modules, we have

For any Fock representation $F_{w_{1}, w_{2}, w_{3}}$ ⁠,

q d i m [F_{w_{1}, w_{2}, w_{3}}] = 6 .

(b)
For representations appearing inside $H (3)$ ⁠,

\begin{array}{l} q d i m [H {(3)}^{S_{3}}] = 1, \\ q d i m [H {(3)}^{S_{3}, s g n}] = 1, \\ q d i m [H {(3)}^{S_{3}, s t}] = 2 . \end{array}

(c)
$F_{w_{1}, w_{3}} (θ)$ and F_w(σ) have quantum dimension +∞.

Proof.

We first observe that in the formulas for $c h [H {(3)}^{S_{3}}] (τ)$ ⁠, $c h [H {(3)}^{S_{3}, s g n}] (τ)$ ⁠, and $c h [H {(3)}^{S_{3}, s t}] (τ)$ ⁠, the infinite product $\frac{q^{- 1 / 8}}{{(q; q)}_{\infty}^{3}} = \frac{1}{η {(τ)}^{3}}$ ⁠, multiplied with a constant, dominates the asymptotics t → 0⁺. Therefore in order to compute quantum dimensions for these modules, we only have to compute the ratio of these constants. This implies assertions (a) and (b).

For (c), we need more precise asymptotic behaviors. We have

\begin{array}{l} \frac{1}{η (τ) η (τ / 2)} \sim \frac{t}{\sqrt{2}} e^{\frac{π}{4 t}}, \\ \frac{1}{η (τ / 3)} \sim \frac{t^{1 / 2}}{\sqrt{3}} e^{\frac{π}{4 t}}, \\ \frac{1}{η {(τ)}^{3}} \sim t^{3 / 2} e^{\frac{π}{4 t}} . \end{array}

From these formulas, we get

\frac{c h [) F_{w_{1}, w_{3}} (θ) (i t)}{c h [H {(3)}^{S_{3}}] (i t)} \sim \frac{1}{\sqrt{2} t^{1 / 2}},

\frac{c h [) F_{w} (σ) (i t)}{c h [H {(3)}^{S_{3}}] (i t)} \sim \frac{1}{\sqrt{3} t},

so both ratios have growing terms at t = 0. The proof follows.

We finish with a conjecture which will be addressed in Ref. 24.

Conjecture 1.

Every irreducible (ordinary)

H {(3)}^{S_{3}}

-module M appears in the decomposition of a g-twisted

H (3)

-module, for g ∈ S₃. Moreover,

q d i m (M) \in {1,2,6, + \infty} .

VIII. FUTURE WORK

Remark 8.1.

In order to confirm Conjecture 1, we first have to describe the Zhu algebra of $H {(3)}^{S_{3}}$ ⁠. This will be addressed in Ref. 24.
In our very recent work (Ref. 23), we determine strong minimal generating sets for the permutation orbifold $F {(3)}^{S_{3}}$ ⁠, where $F$ is the free fermion vertex algebra, and for $S F {(3)}^{S_{3}}$ ⁠, where $S F$ is the rank one symplectic fermion vertex superalgebra.⁹ We prove that these vertex algebras are of type $\frac{1}{2}, 2,4, \frac{9}{2}$ and 1², 2, 3³, 4³, 5⁵, 6⁴, respectively.

ACKNOWLEDGMENTS

We thank Drazen Adamovic, Thomas Creutzig, and especially Andrew Linshaw for useful discussions. We also thank the referee for constructive criticism and other valuable comments. Several computations in this paper are performed by the OPE package (Ref. 25) for Mathematica.

The first author was partially supported by the NSF Grant No. DMS-1601070.

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2019

Author(s)

Permutation orbifolds of the Heisenberg vertex algebra $H (3)$

I. INTRODUCTION

II. THE S_n-ORBIFOLD OF THE HEISENBERG VERTEX OPERATOR ALGEBRA

III. WARMUP: $H {(2)}^{S_{2}}$

IV. THE RANK THREE CASE

A. The orbifold $H {(3)}^{S_{3}}$

V. THE ORBIFOLD $H {(3)}^{Z_{3}}$

VI. THE CHARACTER OF $H {(n)}^{S_{n}}$

VII. MODULAR INVARIANCE OF $H {(3)}^{S_{3}}$ -CHARACTERS AND QUANTUM DIMENSIONS

A. Modular invariance

B. Quantum dimensions

VIII. FUTURE WORK

ACKNOWLEDGMENTS

REFERENCES

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Permutation orbifolds of the Heisenberg vertex algebra H(3)

I. INTRODUCTION

II. THE Sn-ORBIFOLD OF THE HEISENBERG VERTEX OPERATOR ALGEBRA

III. WARMUP: H(2)S2

IV. THE RANK THREE CASE

A. The orbifold H(3)S3

V. THE ORBIFOLD H(3)Z3

VI. THE CHARACTER OF H(n)Sn

VII. MODULAR INVARIANCE OF H(3)S3-CHARACTERS AND QUANTUM DIMENSIONS

A. Modular invariance

B. Quantum dimensions

VIII. FUTURE WORK

ACKNOWLEDGMENTS

REFERENCES

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Permutation orbifolds of the Heisenberg vertex algebra $H (3)$

II. THE S_n-ORBIFOLD OF THE HEISENBERG VERTEX OPERATOR ALGEBRA

III. WARMUP: $H {(2)}^{S_{2}}$

A. The orbifold $H {(3)}^{S_{3}}$

V. THE ORBIFOLD $H {(3)}^{Z_{3}}$

VI. THE CHARACTER OF $H {(n)}^{S_{n}}$

VII. MODULAR INVARIANCE OF $H {(3)}^{S_{3}}$ -CHARACTERS AND QUANTUM DIMENSIONS