Haagerup symmetry in a rank-six lattice conformal field theory

Introduction

Symmetries of a two-dimensional conformal field theory can be described by topological lines, which can be moved across a correlation function until they meet a field insertion. Joining two such lines gives their fusion product. Ordinary group symmetries give invertible lines, whereas a general rational theory can also have noninvertible lines. Fixing a rational chiral algebra and restricting to lines that preserve its left and right actions gives a finite category of lines and junctions. In the TFT construction, these lines are described by bimodules over the algebra specifying the full theory [17, 16]. For a simple such algebra, they form a fusion category.

The Haagerup fusion categories arise from finite-index subfactors and provide a test for realizing categorical symmetries in conformal field theory [AHKZ26, Section II]. In $\mathcal{H}_3$, the noninvertible simple objects have fusion dimension $\frac{3+\sqrt{13}}{2}$, whereas a simple line implementing a group element has dimension one. The question is to construct a full theory whose lines contain one of these categories, and to identify its fields and the action of the lines. Albert, Honda, Kaidi and Zheng proposed Haagerup symmetries in the $(\mathfrak{e}_8)_1$ theory and suggested related theories at central charges two and six, supported by modular bootstrap calculations [AHKZ26, Section III pp. 4–5]. We construct a theory at central charge six from a Haagerup vertex operator algebra at central charge eight. Its full bulk space has a unique vacuum: the space of states of weights $(0,0)$ is one-dimensional. The theory is the diagonal theory of an explicit rank-six lattice, and it carries all three Haagerup categories through different conformal subalgebras.

The Haagerup lines preserve a conformal subalgebra $\Omega$ on the left and right, but not the maximal lattice algebra, whose representation category is pointed (Remark 5.1). The main step is therefore the determination of this inclusion, including the multiplication of its extension algebra. The relation between conformal extensions and categorical symmetries is discussed in [25 Sections 7.1–7.4].

We call a vertex operator algebra $U$ a Haagerup algebra if it is positive and strongly rational, has central charge eight and admits a braided equivalence $\operatorname{Rep}(U) \simeq Z(\mathcal{H}_3)$. Here $Z(\mathcal{H}_3)$ is the Drinfeld center of the Haagerup fusion category $\mathcal{H}_3$. We recall the definitions and conventions in Section 2. The construction uses the rank-two associated lattice $L$ of $U$ and the common Heisenberg commutant $\Omega$ of its two lattice realizations [2 Section 12]. A holomorphic extension of $U$ places $L$ primitively in $E_8$. Taking its orthogonal complement produces a conformal inclusion $\Omega\subset V_K$. The extension algebra of this inclusion is a Haagerup Lagrangian algebra tensored with the unit of the remaining pointed factor. This factorization gives the Haagerup lines at central charge six.

3. For each $i$, the diagonal theory of $V_K$, regarded as a theory over $\iota_i(\Omega) \otimes\overline{\iota_i(\Omega)}$, has central charges $c_L = c_R = 6$, a unique vacuum and a category of topological lines preserving this chiral algebra that contains a full fusion subcategory equivalent to $\mathcal{H}_i$. The three embeddings are pairwise non-conjugate, even up to automorphisms of $\Omega$.

4. There are exactly five conformal extensions of $\Omega$ with positive strongly rational target and global module dimension 39, up to isomorphisms fixing $\Omega$. Every target is isomorphic to $V_K$. There are exactly four $\operatorname{Aut}(V_K)$-orbits of conformal subalgebras isomorphic to $\Omega$: the three pure classes above and one mixed class.

For each $i$, the noninvertible simple Haagerup lines do not preserve both chiral actions of any full-rank finite-index lattice subalgebra $V_N \subset V_K$ pointwise (Proposition 5.3).

The equality of the three complement lattices requires proof. The genus of $K$ has two classes, with different root systems (Proposition 6.2), so the discriminant form alone does not determine the lattice. We identify the complement for all three holomorphic extensions, and give its Gram matrix and explicit bases. More generally, every positive strongly rational conformal extension of $\Omega$ with global representation-category dimension 39 has twenty currents (Theorem 6.5). This excludes the other lattice algebra in the genus. The fixed-source classification then gives five extensions (Theorem 6.8). Current Weyl automorphisms identify the two mixed extensions after a change of parametrization of the source, giving the four image classes of Theorem 6.11.

The invariant fields describe the perturbations compatible with the Haagerup lines of both chiralities. They are determined through the representation of the fusion algebra of the lines on the multiplicity spaces of the bulk fields (Lemma 7.1 and Proposition 7.2). For each of the three embeddings, there are eight nonvacuum invariant relevant scalar fields, with scaling dimensions $\frac{4}{3}$, $\frac{24}{13}$ and $\frac{76}{39}$ and multiplicities two, two and four. There is no invariant field of weights $(1,1)$ (Corollary 7.3 and Proposition 7.4). We determine their nonzero relevant cubic couplings and fix their absolute values in unit two-point normalization (Proposition 7.8). The three actions have isomorphic invariant full field algebras and the same normalized genus-zero vacuum correlators (Proposition 7.9).

We also compare the invariant relevant fields with affine coset stresses of $\Omega$. Four relevant fields commute with an affine-coset stress of charge $\frac{18}{5}$, while four have nonzero weight for this stress (Proposition 7.12). A unitary conformal endpoint that retains it as a commuting conformal component has central charge at least $\frac{18}{5}$ (Remark 7.13). This stress is itself the sum of two commuting Virasoro vectors of central charges $\frac{182}{95}$ and $\frac{32}{19}$ [2 Theorem 10.5 p. 30], and the four fields of nonzero weight have nonzero weight for each of them. Using the realization of $\Omega$ with current levels $(1,39)$, we find a second affine-coset stress, of charge $\frac{170}{41}$, which commutes with one pair of the four other relevant fields, and its conjugate by a Weyl involution commutes with the last pair. Thus each of the eight invariant relevant fields commutes with an affine-coset stress of charge at least $\frac{18}{5}$ (Proposition 7.14). Corollary 7.15 shows that finite operator products of either single weak pair have zero coefficients on all six other relevant lowest-weight lines. We do not construct a perturbed theory or an infrared fixed point.

The existence of the initial Haagerup algebra is an input. The explicit algebra $S$ of [2 Theorem 10.10 p. 33] is a Haagerup algebra by [4 Theorem 6.1 p. 25], see also [20] for a computer-assisted proof for an algebra isomorphic to $S$ [2 Theorem 10.24 p. 42], and [18, 29, 9] for the algebras whose currents have levels $(1,39)$. Theorem 1.1 applies to $S$ and its twin, which have the same Heisenberg commutant [2 Theorem 12.3(1) p. 64].

We work in the TFT construction of rational full conformal field theory. Its correlators satisfy covariance and boundary and bulk factorization [FFRS06, Theorems 2.2 p. 7, 2.9 p. 16 and 2.13 p. 19]. The full center and bimodule descriptions are recalled in § 2. The fourth, mixed image class has a pointed bulk algebra with thirteen diagonal summands (Proposition 7.16). Its full defect category and the relation of its lines to the Haagerup categories remain open.

Structure of the paper § 2 recalls the definitions and constructions used below. § 3 describes the associated lattice and proves the positivity of $\Omega$. § 4 identifies the commutant and its extension algebra. § 5 constructs the full theory and its lines, and shows that the noninvertible Haagerup lines preserve no finite-index lattice subalgebra pointwise. § 6 gives the explicit lattice and its genus, and classifies the conformal extensions of global dimension 39 and their images. § 7 determines the invariant fields, their operator products and the affine coset stresses which commute with them.

Preliminaries

Vertex operator algebras and commutants We write $Y(v,z)=\sum_{n\in\mathbb{Z}}v(n)z^{-n-1}$ for the vertex operator of $v$, and $Y(\omega_V,z)=\sum_{n\in\mathbb{Z}}L_V(n)z^{-n-2}$ for the conformal vector of a vertex operator algebra $V$. Its central charge is denoted by $c(V)$. The algebra is of CFT type if $V=\bigoplus_{n\geq0}V_n$ and $V_0=\mathbb{C}1$. We call $V$ strongly rational if it is simple, of CFT type, self-contragredient, rational and $C_2$-cofinite. Self-contragredience means that the graded dual $V'=\bigoplus_n V_n^*$ is isomorphic to $V$ as a $V$-module. Rationality means complete reducibility of admissible modules, whose nonnegative integer grading is compatible with the modes. Here $C_2(V)$ is the span of the vectors $u(-2)v$, and $C_2$-cofiniteness means that $V/C_2(V)$ is finite-dimensional. An irreducible ordinary module has finite-dimensional $L(0)$-eigenspaces and a lowest conformal weight $h$, with weights in $h+\mathbb{N}$. The algebra is positive if every irreducible ordinary module other than $V$ has strictly positive lowest conformal weight. A holomorphic algebra is a strongly rational algebra whose only irreducible ordinary module is itself. We use the module and mode conventions of [24].

A unitary structure consists of a positive definite invariant Hermitian form and a compatible antilinear PCT involution, with $\lVert1\rVert=1$. On a unitary module, $L(n)^*=L(-n)$ on the algebraic direct sum of the conformal-weight spaces. We use complete unitarity in the sense of the companion results quoted in § 3. In particular, all irreducible ordinary modules are unitarizable, and the representation category carries the compatible unitary tensor structure. For the definitions of unitary vertex operator algebras and modules, see [CKLW18, § 5].

An inclusion $W\subset V$ is conformal if it preserves the vacuum and conformal vector. For a vertex subalgebra $W$ with a possibly different conformal vector $\omega_W$, put

Whenever we give this commutant the conformal vector $\omega_V-\omega_W$, we require that this difference be a Virasoro vector belonging to $\operatorname{Com}_V(W)$. Its field then commutes with all fields of $W$, in particular with the Virasoro field of $\omega_W$, and the central charges add to $c(V)$. This condition holds for the Heisenberg and affine Sugawara subalgebras used below, with their standard conformal vectors, by the coset construction [[15]]. A dual pair consists of two such subalgebras, each equal to the commutant of the other.

For a conformal extension $V\subset T$ of strongly rational vertex operator algebras, the restriction of $T$ is a finite direct sum of ordinary $V$-modules. Indeed, rationality gives a direct-sum decomposition, there are only finitely many irreducible $V$-modules, and the multiplicity of a module of lowest weight $h$ is bounded by the finite dimension of $T_h$. Thus this restriction is an object of $\operatorname{Rep}(V)$, as required by the extension correspondence used below [19].

Even lattices For an even positive-definite lattice $L$, put $L'=\{x\in L\otimes_{\mathbb{Z}}\mathbb{R}:\langle x,L\rangle\subset\mathbb{Z}\}$, $A_L=L'/L$ and $q_L(x+L)=\frac{|x|^2}{2}+\mathbb{Z}$. The associated lattice vertex operator algebra $V_L$ has central charge $\operatorname{rank}L$. Its irreducible modules are $V_{x+L}$, with fusion given by addition in $A_L$, and their lowest weights are $\min_{y\in x+L}\frac{|y|^2}{2}$ [14 Chapter 8], [8], see also [24 Theorem 6.5.1 p. 240 and Theorem 6.5.20 p. 259]. We denote its pointed modular category by $\mathcal{C}(L)$. The twist of the object labelled by $x+L$ is $\exp(2\pi i q_L(x+L))$. Even overlattices of $L$ correspond to isotropic subgroups of $(A_L,q_L)$. A primitive sublattice and its orthogonal complement in an even unimodular lattice have anti-isometric discriminant forms [28].

Fusion categories and algebra objects A fusion category is a finite semisimple rigid $\mathbb{C}$-linear tensor category with simple tensor unit. We write $\operatorname{Irr}(\mathcal{C})$ for its simple objects and $\operatorname{FPdim}(X)$ for the Frobenius–Perron dimension. Its global dimension is

In a unitary fusion category the canonical categorical dimensions are positive and equal to these dimensions. An object is invertible if it has a tensor inverse, and a fusion category is pointed if its simple objects are invertible. We write $Z(\mathcal{C})$ for the Drinfeld center, whose objects are objects of $\mathcal{C}$ equipped with a compatible half-braiding [12].

A modular category is a ribbon fusion category with nondegenerate braiding. For a braided category $\mathcal{C}$, the notation $\mathcal{C}^{\mathrm{rev}}$ means that the braiding and twist are reversed. The notation $\mathcal{C}^{\mathrm{op}}$ used for tensor categories below means reversal of the order of the tensor product. The first convention describes the opposite chiral orientation, and the second occurs in the composition of module endofunctors. The tensor product of linear categories is denoted by $\boxtimes$, and $X^\vee$ denotes the dual of $X$.

An algebra object $A$ in a braided fusion category is connected if $\dim\operatorname{Hom}(\mathbb{1},A)=1$, and étale if its multiplication is commutative and separable. Separability means that the multiplication has an $A$-$A$-bimodule section. We denote the category of right $A$-modules by $\mathcal{C}_A$ and its subcategory of local modules by $\mathcal{C}_A^0$. A right module $(M,\rho)$ is local if its action is unchanged by double braiding with $A$, that is, $\rho c_{A,M}c_{M,A}=\rho$. For a connected étale algebra in a nondegenerate braided fusion category,

Such an algebra is Lagrangian if $\operatorname{FPdim}(A)^2=\operatorname{FPdim}(\mathcal{C})$, equivalently if $\mathcal{C}_A^0$ is the category of finite-dimensional vector spaces. For an $A$-module $M$, its module dimension is $\operatorname{FPdim}_{\mathcal{C}_A}(M)=\frac{\operatorname{FPdim}(M)}{\operatorname{FPdim}(A)}$. These facts and extension in stages are recalled in [7 Sections 3 and 3.6]. A connected étale algebra with trivial twist and the VOA extension hypotheses determines a conformal extension, whose ordinary modules are its local modules [19]. We apply this correspondence to the extensions specified in the text.