Hyperbolizations of affine Lie algebras]Hyperbolization of affine Lie algebras

Kaiwen Sun

Department of Physics and Astronomy, Uppsala University, 75120, Uppsala, SwedenDepartment of Mathematics, Uppsala University, 75106, Uppsala, Sweden

Haowu Wang

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, China

Brandon Williams

Lehrstuhl A für Mathematik, RWTH Aachen, 52056 Aachen, Germany

[2020]11F27, 11F50, 11F55, 17B22, 17B65, 17B69

In 1983, Feingold and Frenkel posed a question about possible relations between affine Lie algebras, hyperbolic Kac–Moody algebras and Siegel modular forms. In this paper we give an automorphic answer to this question and its generalization. We classify hyperbolic Borcherds–Kac–Moody superalgebras whose super-denominators define reflective automorphic products of singular weight on lattices of type $2U\oplus L$. As a consequence, we prove that there are exactly $81$ affine Lie algebras $\hat{\mathfrak{g}}$ which have nice extensions to hyperbolic BKM superalgebras for which the leading Fourier–Jacobi coefficients of super-denominators coincide with the denominators of $\hat{\mathfrak{g}}$. We find that $69$ of them appear in Schellekens’ list of semi-simple $V_1$ structures of holomorphic CFT of central charge $24$, while $8$ of them correspond to the $\mathcal{N}=1$ structures of holomorphic SCFT of central charge $12$ composed of $24$ chiral fermions. The last $4$ cases are related to exceptional modular invariants from nontrivial automorphisms of fusion algebras. This clarifies the relationship of affine Lie algebras, vertex algebras and hyperbolic BKM superalgebras at the level of modular forms.

§ INTRODUCTION

Affine Lie algebras are the simplest class of infinite dimensional Kac–Moody Lie algebras, and they have numerous connections with other areas of mathematics and theoretical physics. The next simplest class of Kac–Moody algebras after the affine algebras are the hyperbolic Lie algebras. In 1983, as an extension of the affine Lie algebra of type $A_1$, Feingold and Frenkel [Feingold and Frenkel, 1983] investigated the hyperbolic Kac–Moody algebra with Cartan matrix

\left( \begin{array}{ccc}
2 & -2 & 0 \\
-2 & 2 & -1 \\
0 & -1 & 2
\end{array} \right)

and found a connection with the Igusa cusp form [Igusa, 1962] of weight $35$ on $\Sp_2(\ZZ)$. This suggests exploring more general relationships between hyperbolic Kac–Moody Lie algebras and modular forms of several variables.

In 1988, Borcherds [Borcherds, 1988] introduced generalized Kac–Moody algebras, which are now usually called Borcherds–Kac–Moody or simply BKM algebras. These infinite-dimensional Lie algebras are also defined in terms of Chevalley–Serre generators and relations that are encoded in a generalized Cartan matrix, and they differ from Kac–Moody algebra mainly by allowing the diagonal entries of the Cartan matrix to be non-positive. In other words, simple roots are allowed to be imaginary, whereas simple roots in a Kac–Moody algebra are always real.
The best-known example of a BKM algebra is the monster Lie algebra. In 1992, Borcherds [Borcherds, 1992] constructed this algebra as the BRST cohomology  related to the monster vertex algebra [Borcherds, 1986, Frenkel et al., 1988] by means of the no-ghost theorem from string theory. By considering the action of the monster group on the denominator identity of the monster Lie algebra, Borcherds proved the celebrated monstrous moonshine conjecture. Furthermore, he observed that the denominator functions of some BKM algebras are modular forms on orthogonal groups of signature $(l,2)$. In 1995 and 1998 Borcherds [Borcherds, 1995, Borcherds, 1998] developed the theory of singular theta lift to construct modular forms for arithmetic subgroups of $\Orth(l,2)$ which have infinite product expansions. These are called automorphic products, or Borcherds products, and they are natural candidates for the denominator functions of BKM algebras. Similarly to affine Lie algebras, BKM algebras and automorphic products also have many applications in physics. For example, Harvey and Moore [Harvey and Moore, 1996, Harvey and Moore, 1998] proposed that BKM algebras should play as the underlying organizing structure of BPS states in string compactifications; in particular, the denominators of BKM algebras might be generating functions for BPS states.

In 1996 Gritsenko and Nikulin [Gritsenko and Nikulin, 1996] constructed an automorphic correction of the rank-three hyperbolic Lie algebra considered previously by Feingold and Frenkel. More precisely, they extended this hyperbolic Lie algebra to a (hyperbolic) BKM algebra by adding infinitely many imaginary simple roots, such that the denominator of the resulting BKM algebra is exactly the Igusa cusp form of weight $35$. Later, they constructed automorphic corrections of other hyperbolic Lie algebras in a series of papers [Gritsenko and Nikulin, 1997, Gritsenko and Nikulin, 1998, Gritsenko and Nikulin, 1998, Gritsenko and Nikulin, 2002, Gritsenko and Nikulin, 2018]. A common feature of these corrections is that the resulting BKM algbera only has finitely many real simple roots, a Weyl chamber of finite volume and a Weyl vector of negative norm.  Moreover, its denominator usually defines a cuspidal automorphic product on $\Orth(l,2)$. These corrections extend affine Lie algebras in a nice way, but they have several features that are not preferred from our perspective:

(1) for the automorphic correction $\mathfrak{G}$ of an affine Lie algebra $\hat{\mathfrak{g}}$, the multiplicity of an imaginary root of $\hat{\mathfrak{g}}$ in $\mathfrak{G}$ is strictly greater than its multiplicity in $\hat{\mathfrak{g}}$;

(2) the set of imaginary simple roots is very complicated, although the set of real simple roots is easy to describe;

(3) the role of the vertex algebra is unclear.

In this paper we extend affine Lie algebras to (hyperbolic) BKM algebras in a different way. Certain affine Kac–Moody algebras $\hat{\mathfrak{g}}$ will be extended to BKM algebras $\mathcal{G}_\mathfrak{g}$ which have infinitely many real simple roots and satisfy:

(a)  for any root $\alpha$ of $\hat{\mathfrak{g}}$, the root multiplicities of $\alpha$ in $\hat{\mathfrak{g}}$ and $\mathcal{G}_\mathfrak{g}$ are the same;

(b) the imaginary simple roots of $\mathcal{G}_\mathfrak{g}$ are negative integral multiples of the Weyl vector;

(c) the Lie algebras $\hat{\mathfrak{g}}$ and $\mathcal{G}_\mathfrak{g}$ are closely related to some exceptional vertex algebras.

Our main results are about the classification and construction of such extensions, which are connected with various types of modular forms. In the rest of the introduction, we will explain the setup and state the main theorems.

§.§ Denominators of affine Lie algebras and Jacobi forms

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra of rank $r$ and let $\Delta^+_\mathfrak{g}$ be a set of positive roots. The product side of the denominator identity of the associated affine Lie algebra $\hat{\mathfrak{g}}$ is the holomorphic function

\vartheta_{\mathfrak{g}}(\tau,\mathfrak{z})=\eta(\tau)^r \prod_{\alpha\in \Delta_\mathfrak{g}^+}\frac{\vartheta(\tau, \latt{\alpha,\mathfrak{z}})}{\eta(\tau)},

where $\eta$ and $\vartheta$ are the Dedekind eta function and the odd Jacobi theta function, respectively:

\begin{align*}
\eta(\tau)&=q^{\frac{1}{24}}\prod_{n=1}^\infty(1-q^n), \quad \tau\in \HH, \; q=e^{2\pi i\tau},\\
\vartheta(\tau,z)&=-q^{\frac{1}{8}}\zeta^{-\frac{1}{2}}\prod_{n=1}^\infty(1-q^{n-1}\zeta)(1-q^n\zeta^{-1})(1-q^n), \quad z\in \CC, \; \zeta= e^{2\pi iz}.
\end{align*}

The function $\vartheta_{\mathfrak{g}}$ has nice modular properties and is an example of a Jacobi form in the sense of Eichler and Zagier [Eichler and Zagier, 1985].  Let $L$ be an even positive definite lattice. A Jacobi form of integral weight $k$ and index $L$ is a holomorphic function $\varphi : \HH \times (L \otimes \CC) \rightarrow \CC$ that is modular under $\SL_2(\ZZ)$ and doubly quasi-periodic, namely

\begin{align*}
\varphi \left( \frac{a\tau +b}{c\tau + d},\frac{\mathfrak{z}}{c\tau + d}
\right)& = (c\tau + d)^k
\exp{\left(i \pi t \frac{c(\mathfrak{z},\mathfrak{z})}{c
\tau + d}\right)} \varphi ( \tau, \mathfrak{z} ), \quad A=\begin{psmallmatrix}
a & b \\
c & d
\end{psmallmatrix} \in \SL_2(\ZZ),\\
\varphi (\tau, \mathfrak{z}+ x \tau + y)&=
\exp{\big(-i \pi t \big( (x,x)\tau +2(x,\mathfrak{z})\big)\big)}
\varphi (\tau, \mathfrak{z} ), \quad x,y\in L,
\end{align*}

and whose Fourier expansion satisfies a boundary condition. The function $\vartheta_{\mathfrak{g}}$ is a Jacobi form of weight $r/2$ and index $P_\mathfrak{g}^\vee(h_\mathfrak{g}^\vee)$ for some character, where $P_\mathfrak{g}^\vee$ is the dual of the root lattice and $h_\mathfrak{g}^\vee$ is the dual Coxeter number. Note that Jacobi forms defined by an expression similar to $\vartheta_{\mathfrak{g}}$ are called theta blocks following Gritsenko–Skoruppa–Zagier [Gritsenko et al., 2019].

§.§ Automorphic products of singular weight

A modular form of integral weight $k$ and trivial character for an arithmetic subgroup $\Gamma<\Orth(l,2)$ is a holomorphic function on the associated type IV symmetric domain which satisfies

\begin{align*}
F(t\mathcal{Z})&=t^{-k} F(\mathcal{Z}), \quad t\in \CC^\times,\\
F(g\mathcal{Z})&=F(\mathcal{Z}),   \qquad g \in \Gamma.
\end{align*}

Let $M$ be an even lattice of signature $(l,2)$.  The input into Borcherds' theta lift is a vector-valued modular form of weight $1-l/2$ with integral Fourier expansion for the Weil representation of $\SL_2(\ZZ)$ attached to the discriminant form $M'/M$, and the output is a meromorphic modular form for a certain subgroup of $\Orth(M)$ which has an infinite product expansion at any $0$-dimensional cusp and whose divisors are linear combinations of hyperplanes.

Since the denominators of affine Lie algebras satisfy modularity, it is natural to focus on hyperbolic BKM algebras whose denominators are modular. Let $\mathcal{G}$ be a BKM algebra whose denominator function defines the Fourier expansion of an $\Orth(l,2)$-modular form $F$ at some $0$-dimensional cusp.  Since $F$ has an infinite product expansion, by Bruinier's converse theorem [Bruinier, 2002, Bruinier, 2014] one expects that it can be constructed by the Borcherds lift. In the case, the roots and their multiplicities are encoded in the Fourier expansion of the input. When $F$ has singular weight, i.e. weight $l/2-1$, the Fourier expansion is supported only on isotropic vectors, which often forces the imaginary simple roots of $\mathcal{G}$ to be negative integral multiples of the Weyl vector. Moreover, it is conjectured in this particular case that $\mathcal{G}$ can be constructed as the BRST cohomology related to some vertex algebra, similarly to the monster algebra. This suggests focusing on BKM algebras whose denominators are automorphic products of singular weight. The zeros of $F$ that contain the cusp are actually hyperplanes orthogonal to real roots of $\mathcal{G}$, hence $F$ is anti-invariant under the reflections through these hyperplanes. It is natural to expect that $F$ is anti-invariant under all reflections associated with zeros of $F$. This has been proven by the last two named authors [Wang and Williams, 2023]. It follows that $F$ is a reflective modular form.

A non-constant modular form on $\Gamma < \Orth(M)$ is called reflective if it vanishes only on mirrors of reflections fixing the lattice $M$. Reflective modular forms were introduced in 1998 by Borcherds [Borcherds, 1995, Borcherds, 1998] and Gritsenko–Nikulin [Gritsenko and Nikulin, 1998], and their classification has been an active project for the past thirty years (see [Gritsenko and Nikulin, 2002, Barnard, 2003, Scheithauer, 2006, Dittmann et al., 2015, Scheithauer, 2017, Ma, 2017, Ma, 2018, Dittmann, 2019, Opitz and Schwagenscheidt, 2019, Wang, 2021, Wang, 2021, Wang, 2022, Driscoll-Spittler, 2022, Wang, 2023, Wang, 2023]), because they have nice applications to hyperbolic reflection groups [Borcherds, 1998, Borcherds, 2000, Gritsenko and Nikulin, 1998, Gritsenko et al., 2007, Gritsenko and Hulek, 2014], algebraic geometry [Borcherds, 1996, Borcherds et al., 1998, Gritsenko et al., 2007, Ma, 2018, Gritsenko, 2018] and free algebras of modular forms [Wang, 2021, Wang and Williams, 2023] in addition to infinite-dimensional Lie algebras.

§.§ Main results

BKM algebras whose denominator functions are reflective automorphic products of singular weight are exceptional. The main examples are the fake monster algebra [Borcherds, 1990] and their twists by the Conway group $\mathrm{Co}_0$ [Borcherds, 1992, Scheithauer, 2004, Wang and Williams, 2022]. There are conjecturally only finitely many such algebras and constructing and classifying them is an open problem.  Many partial results have been proved towards such a classification [Borcherds, 1990, Borcherds, 1992, Niemann, 2002, Barnard, 2003, Scheithauer, 2000, Scheithauer, 2004, Scheithauer, 2006, Scheithauer, 2017, Dittmann, 2019, Wang and Williams, 2022, Driscoll-Spittler, 2022]. In this paper we contribute some new results in this direction.

Let $U$ be an even unimodular lattice of signature $(1,1)$ and let $L$ be an even, positive-definite lattice. The input of the Borcherds lift on $2U\oplus L$ can be identified with Jacobi forms of weight $0$ and index $L$.
We will identify affine Lie algebras, which naturally extends to BKM algebras or superalgebras whose denominators or super-denominators are reflective Borcherds product of singular weight on lattices of type $2U\oplus L$. The setting is inspired by the following result.

Let $F$ be a reflective Borcherds product of singular weight on $2U\oplus L$ whose Jacobi form input has Fourier expansion

\phi(\tau,\mathfrak{z}) = \sum_{n\in \ZZ} \sum_{\ell \in L'} f(n,\ell) q^n \zeta^\ell

satisfying that $f(0,\ell)\geq 0$ for all $\ell\in L'$.
If $L$ is the Leech lattice, then $F$ is the denominator of the fake monster algebra and $\phi$ is the full character of the Leech lattice vertex operator algebra. Otherwise, the set

\mathcal{R}=\{ \ell \in L' : \; \ell\neq 0, \; f(0,\ell) \neq 0 \}

determines a finite-dimensional semi-simple Lie algebra

\mathfrak{g}=\bigoplus_{j=1}^s\mathfrak{g}_{j,k_j}

with the same rank as $L$ such that the identity

\begin{equation}\label{MEQ}
C:= \frac{\dim \mathfrak{g}}{24} - a = \frac{h_j^\vee}{k_j}
\end{equation}

holds for any $1\leq j\leq s$ and such that the leading Fourier–Jacobi coefficient of $F$ at the $1$-dimensional cusp determined by $2U$ coincides with the denominator of the associated affine Lie algebra $\hat{\mathfrak{g}}$. For any $1\leq j\leq s$, $\mathfrak{g}_j$ is a simple ideal of $\mathfrak{g}$, $k_j$ is a positive integer indicating the level of $\mathfrak{g}_j$ and $h_j^\vee$ is the dual Coxeter number of $\mathfrak{g}_j$. The number $a$ equals $f(-1,0)$, which has to be $0$ or $1$. When $a=0$, $k_j>1$ for any $1\leq j\leq s$. The cases $a=0$ and $a=1$ are called symmetric and anti-symmetric, respectively.

If the $q^0$-term of $\phi$ has negative Fourier coefficients, then $\mathcal{R}$ determines a semi-simple Lie superalgebra and the corresponding BKM superalgebra will have odd real roots. We will consider this case in a separate paper.

If the Fourier expansion of $F$ defines the (super)-denominator of a BKM (super)-algebra $\mathcal{G}$, then the (super)-denominator has the form

e^\rho \prod_{\alpha>0} \big( 1 - e^{-\alpha} \big)^{f(nm,\ell)},

where $\rho$ is the Weyl vector of $F$, and where $(n,\ell,m)$ are coordinates of positive roots $\alpha \in U\oplus L'$ with $n\in\ZZ$, $m\in \NN$, $\ell\in L'$ and $\alpha^2=\ell^2-2nm$. The above $\hat{\mathfrak{g}}$ is embedded into $\mathcal{G}$ as the sum of the root spaces associated with roots of type $\pm (n,\ell, 0)$. In this way, $\mathcal{G}$ can be regarded as a graded module over $\hat{\mathfrak{g}}$ graded by $m\in \NN$. This leads us to define $\mathcal{G}$ as a hyperbolization of $\hat{\mathfrak{g}}$ and $F$ as a hyperbolization of the denominator of $\hat{\mathfrak{g}}$.  It turns out that there are only $81$ affine Lie algebras with a hyperbolization:

There are $81$ possibilities for the semi-simple Lie algebra $\mathfrak{g}$ in Theorem <ref> and they fall into three categories:

* $69$ make up Schellekens' list of semi-simple $V_1$ structures of holomorphic vertex operator algebras of central charge $24$;

* $8$ are the $\mathcal{N}=1$ structures of holomorphic vertex operator superalgebras $F_{24}$ of central charge $12$ composed of $24$ fermions;

* The remaining $4$ cases $A_{1,16}$, $A_{1,8}^2$, $A_{1,4}^4$ and $A_{2,9}$ possess an exceptional modular invariant that comes from a nontrivial automorphism of the fusion algebra.

Case (1) is anti-symmetric, while Cases (2) and (3) are symmetric. Schellekens' list [Schellekens, 1993] was established using the solutions of Equation (<ref>) with $a=1$. Holomorphic vertex operator superalgebras of central charge $12$ were classified by Creutzig, Duncan
and Riedler [Creutzig et al., 2018], and the $\mathcal{N}=1$ structures of $F_{24}$ were determined in [Harrison et al., 2021], corresponding to solutions of Equation (<ref>) with $a=0$ and $C=1$. The exceptional modular invariants mentioned in (3) were discovered around the 1990s by Moore and Seiberg [Moore and Seiberg, 1989], Verstegen [Verstegen, 1990] and Gannon [Gannon, 1995]. The $4$ exotic cases satisfy Equation (<ref>) with $a=0$ and $C<1$.

We now present hyperbolizations of these affine Lie algebras.

For any $\mathfrak{g}$ in Theorem <ref> there exists a singular-weight reflective Borcherds product $\Psi_\mathfrak{g}$ on some lattice $2U\oplus L_\mathfrak{g}$ whose leading Fourier–Jacobi coefficient equals the denominator of $\hat{\mathfrak{g}}$. Moreover, the Jacobi form input $\phi_\mathfrak{g}$ can be expressed as a $\ZZ$-linear combination of full characters of the affine vertex operator algebra generated by $\hat{\mathfrak{g}}$.

The construction will be briefly summarized here. If $\mathfrak{g}$ is in Schellekens' list, $L_\mathfrak{g}$ is the orbit lattice in Höhn's construction [Höhn, 2017] of the holomorphic VOA of central charge $24$ with $V_1=\mathfrak{g}$, and $\phi_\mathfrak{g}$ is the full character of the VOA. If $\mathfrak{g}$ is of symmetric type, $L_\mathfrak{g}$ is the maximal even sublattice of the coweight lattice of $\mathfrak{g}$. If $\mathfrak{g}$ defines an $\mathcal{N}=1$ structure of $F_{24}$, then the Jacobi form input can be expressed in terms of characters of $F_{24}$ as

\phi_{\mathfrak{g}}=(\chi_{\mathrm{NS}} - \chi_{\widetilde{\mathrm{NS}}} - \chi_{\mathrm{R}})/2.

Finally, we will explain the relation between Jacobi form inputs and exceptional modular invariants for the remaining four $\mathfrak{g}$. The Jacobi form input for $\mathfrak{g}=A_{1,16}$ can be written in terms of affine characters as

\phi_{A_{1,16}}=\chi^{A_{1,16}}_{2,\frac19}+\chi^{A_{1,16}}_{14,\frac{28}{9}}-\chi^{A_{1,16}}_{8,\frac{10}{9}},

and we find that the difference between the simple current modular invariant and the exceptional modular invariant [Moore and Seiberg, 1989] is given by $|\phi_{A_{1,16}}|^2$. Similar relations hold for the other three $\mathfrak{g}$. Note that these $L_\mathfrak{g}$ are chosen so that the resulting BKM superalgebra has root lattice $U\oplus L_\mathfrak{g}'$.

Clearly, the Borcherds products $\Psi_\mathfrak{g}$ in Theorem <ref> are closely related to vertex algebras. We therefore expect that the BRST cohomology related to these vertex algebras define the BKM (super)-algebras with $\Psi_\mathfrak{g}$ as the (super)-denominators. This type of realization has been achieved in [Borcherds, 1990, Höhn and Scheithauer, 2003, Creutzig et al., 2007, Höhn and Scheithauer, 2014, Möller, 2021, Driscoll-Spittler, 2022, Harrison et al., 2021] under some technical assumptions for $\mathfrak{g}$ from Schellekens' list and the $\mathcal{N}=1$ structure of $F_{24}$.  However, such a realization is completely open for $\mathfrak{g}$ related to the four exceptional modular invariants.

Affine Lie algebras, vertex algebras and BKM (super)-algebras are therefore closely connected from the point of view of the attached modular forms. The connections are illustrated in  Figure <ref>.

[node distance=3cm, auto]
(A) Affine Lie algebra;
(AA) [right of=A] $ $;
(B) [right of=AA] Vertex algebra;
(AAA) [right of=B] $ $;
(C) [right of=AAA] BKM superalgebra;
(D) [below of=A] Theta block;
(E) [below of=B] Weight $0$ Jacobi form;
(F) [below of=C] Borcherds product;
[->] (A) to node  (B);
[->] (B) to node BRST (C);
[->] (D) to node  (E);
[->] (E) to node Lift (F);
[->] (A) to node Denominator (D);
[->] (B) to node Character (E);
[->] (C) to node Denominator (F);
[->, bend left] (F) to node Leading FJ coefficient (D);
[->, bend left] (A) to node Hyperbolization (C);
[blue, ->] (A) to node Character (E);

Hyperbolization of affine Lie algebras

§.§ Outline of the proof

The proof of Theorem <ref> relies on some previous results. We know from [Wang and Williams, 2023] that singular-weight reflective Borcherds products have only simple zeros and are anti-invariant under reflections associated with their zeros. Therefore, the nonzero coefficients $f(0,\ell)$ of the Jacobi form input have to be $1$ if $\ell \neq 0$, and $f(0,0)$ equals the rank of $\mathfrak{g}$ by singular-weight. The theorem follows by applying an argument used by the second named author [Wang, 2023, Wang, 2021, Wang, 2023] to classify reflective modular forms. Theorem <ref> further shows that the central charge of the affine vertex operator algebra generated by $\hat{\mathfrak{g}}$ is

c_\mathfrak{g}= \frac{24(C+a)}{C+1}.

In particular, $c_\mathfrak{g}=24$ if $a=1$, and $c_\mathfrak{g}=12$ if $a=0$ and $C=1$. This motivates the three groupings in Theorem <ref>.

To prove Theorem <ref> we first solve equations of type (<ref>).
In the anti-symmetric case, Equation (<ref>) was first derived by Schellekens [Schellekens, 1993] in the context of conformal field theories. Schellekens found $221$ solutions to this equation and eliminated $152$ of them to arrive at his list. We also have the same extra solutions to eliminate, but we have to use a completely different approach. In the symmetric case, there are $17$ solutions to Equation (<ref>) and we have to rule out $5$ of them. In the setting of Theorem <ref>, we can prove that $L(C)$ is an integral lattice and that $L$ is bounded by

\bQ < L < \bP,

where $\bQ$ and $\bP$ are the coroot lattice and coweight lattice of $\mathfrak{g}$, respectively. On one hand, for every extraneous $\mathfrak{g}$, we will be able to find an even overlattice $K$ of $L$ for which there is no reflective Borcherds product on $2U\oplus K$ satisfying certain constraints. On the other hand, we prove that if $2U\oplus L$ has a singular-weight reflective Borcherds product then $2U\oplus K$ has also a reflective Borcherds product of the same type. Taken together, this allows us to rule out all $157$ extraneous solutions of Equation (<ref>).

We will now sketch the proof of Theorem <ref>, beginning with the anti-symmetric case. Let $V$ be a holomorphic VOA of central charge $24$ with semi-simple $V_1=\mathfrak{g}$. The full character $\chi_V$ of $V$ is known to be a Jacobi form of weight $0$ and lattice index $\bQ$ with non-negative integral Fourier expansion [Zhu, 1996, Miyamoto, 2000, Krauel and Mason, 2012]. This immediately implies that the singular theta lift of $\chi_V$, denoted $\Borch(\chi_V)$, defines a holomorphic Borcherds product of singular weight on $2U\oplus \bQ$. It remains to find an extension $L$ of $\bQ$ for which $\Borch(\chi_V)$ is reflective on $2U\oplus L$. We recognize that $L$ should be the Höhn' orbit lattice $L_\mathfrak{g}$. Höhn [Höhn, 2017, Lam, 2019, Betsumiya et al., 2023] proposed a construction of $V$ as the simple current extension of the tensor product of the lattice VOA $V_{L_\mathfrak{g}}$ and a certain VOA $W_{\mathfrak{g}}$ of central charge $24-\rank(\mathfrak{g})$ with trivial weight-one subspace. This construction corresponds to the theta decomposition of $\chi_V$ as a Jacobi form of index $L_\mathfrak{g}$. It is possible to prove directly that $\Borch(\chi_V)$ is reflective on $2U\oplus L_\mathfrak{g}$ by computing the Fourier expansion of $\chi_V$; however, this is only feasible for certain specific $\mathfrak{g}$. We carry out the calculation for $\mathfrak{g}=B_{12,2}$, $A_{2,2}F_{4,6}$ and $C_{4,10}$, in which case $[L_\mathfrak{g}:\bQ]\leq 2$.

To complete the proof for the remaining $\mathfrak{g}$, we relate $\Borch(\chi_V)$ to a twisted denominator of the fake monster algebra. By [Höhn, 2017, Lam, 2019], there exists a conjugacy class $[g]$ of the Conway group $\mathrm{Co}_0$ such that $W_\mathfrak{g}$ is isomorphic to the orbifold  $V_{\Lambda_g}^{\hat{g}}$, where $\Lambda_g$ is the coinvariant sublattice of the Leech lattice $\Lambda$. Moreover, $\Borch(\chi_V)$ for distinct $V$ define the same modular form on type IV symmetric domain if $V_1=\mathfrak{g}$ correspond to the same $[g]$. Combining this fact with our previous calculation of $\chi_V$, we prove that $\Borch(\chi_V)$ is reflective if $V_1$ corresponds to a class $[g]$ whose order is distinct from its level. The last two named authors [Wang and Williams, 2022] proved that the $g$-twisted denominator of the fake monster algebra defines a reflective Borcherds product $\Phi_g$ of singular weight on $U(n_g)\oplus U\oplus \Lambda^g$ if $g$ has the same level and order $n_g$, where $\Lambda^g$ is the sublattice of $\Lambda$ fixed by $g$. We identify $\Borch(\chi_V)$ with the associated $\Phi_g$ and thus prove that $\Borch(\chi_V)$ is reflective if $V_1$ corresponds to some $[g]$ with the equal level and order. As a by-product, we find that the BKM algebra constructed as the BRST cohomology related to $V$ and the $g$-twist of the fake monster algebra are isomorphic if $g$ has equal order and level, but they are not isomorphic if the order and level of $g$ are distinct.

Then we consider the symmetric case with $C=1$. For any $\mathcal{N}=1$ structure of $F_{24}$, we show that there exists a $\mathrm{Co}_0$-conjugacy class $[g]$ with level equal to its order such that the Borcherds product $\Borch(\phi_\mathfrak{g})$ is the $g$-twisted denominator of the fake monster algebra. This is proven by identifying their Jacobi form inputs, and relies on the construction of these $g$-twisted denominators as Gritsenko (additive) lifts due to Dittmann and the second named author [Dittmann and Wang, 2022]. We then confirm that $\Borch(\phi_\mathfrak{g})$ is a singular-weight reflective Borcherds product and that the BKM superalgebra constructed in [Harrison et al., 2021] as the BRST cohomology related to $F_{24}$ is isomorphic to the $g$-twist of the fake monster algebra.

In the final case, we determine the Jacobi form inputs in terms of affine characters and prove that their singular theta lifts are reflective by direct calculation.

§.§ Outline of the paper

In Section <ref>, we quickly introduce Jacobi forms, affine Kac–Moody algebras, reflective modular forms, automorphic products and Borcherds–Kac–Moody algebras. We also fix some notation that will be used frequently later on.

In Section <ref> we define the hyperbolization of an affine Lie algebra and explain its motivation.

In Section <ref> we present the proof of Theorem <ref> and the solutions of Equation (<ref>).

In Section <ref> we state the full version of the main results.

In Section <ref>, we first introduce holomorphic VOA of central charge $24$ and Höhn's construction, as well as their relations with twisted denominators of the fake monster algebra. We then construct hyperbolizations of the affine Lie algebras on Schellekens' list.

In Section <ref>, we begin by introducing holomorphic SVOA of central charge $12$ and type $F_{24}$, and then construct hyperbolizations for the eight $\mathcal{N}=1$ structures of $F_{24}$.  As an application, we construct many exceptional modular invariants by considering the conformal embedding from the affine VOA generated by the $\mathcal{N}=1$ structure to $F_{24}$ and considering an automorphism of the $D_{12,1}$ fusion algebra.

In Section <ref>, we construct hyperbolizations of the remaining affine Lie algebras and explain their connection with some  exceptional modular invariants.

Combining Sections <ref>-<ref> completes the proof of Theorem <ref>.

In Section <ref>, we present a uniform construction of the $12$ symmetric singular-weight reflective Borcherds products as additive lifts.

In Section <ref>, we compute the Fourier expansions of these singular-weight reflective Borcherds products at the $0$-dimensional cusp determined by one copy of $U$.

Section <ref> is devoted to the proof of the anti-symmetric case of Theorem <ref>.

Section <ref> contains the proof of the symmetric case of Theorem <ref>.

In Section <ref> we give an application of our main results. We use the pull-back to construct a new infinite series of anti-symmetric Siegel paramodular forms of weight $3$.

In Section <ref> we raise some related questions and conjectures.

This article ends with several long tables.

KS thanks Thomas Creutzig, Zhihao Duan, Kimyeong Lee and Hiraku Nakajima for useful discussions. HW thanks Valery Gritsenko, Xingjun Lin and Nils Scheithauer for fruitful discussions. KS and HW thank the hospitality of IBS Center for Geometry and Physics where part of this work was done. KS would also like to thank KIAS (grant QP081001, QP081002), MPIM Bonn, Inter-University Centre in Dubrovnik and Isaac Newton Institute for Mathematical Sciences in Cambridge (during the program Black holes: bridges between number theory and holographic quantum information, by EPSRC grant no EP/R014604/1) for support and hospitality where work on this paper was undertaken.

§ PRELIMINARIES

In this section, we will first review Jacobi forms and denominator identities of affine Lie algebras. We then define reflective modular forms on orthogonal groups $\Orth(n,2)$ and review Borcherds' theory of automorphic products, as well as the denominator identities of Borcherds–Kac–Moody algebras. This background is necessary to state and prove the main theorems.

§.§ Jacobi forms of lattice index

Let $\ZZ$ and $\NN$ denote the sets of integers and non-negative integers, respectively. Let $L$ be an even integral positive-definite lattice with bilinear form $(-,-)$ and dual lattice

L'=\{ v\in L\otimes \QQ : \; (x,y)\in \ZZ, \; \text{for all $y\in L$}  \}.

For any nonzero integer $a$, the lattice with abelian group $L$ and bilinear form $a(-,-)$ is denoted by $L(a)$. Let $\HH$ be the complex upper half plane. The Dedekind eta function

\eta(\tau)=q^{1/24}\prod_{j=1}^\infty(1-q^j), \quad q=e^{2\pi i\tau}, \; \tau\in \HH

is a modular form of weight $\frac{1}{2}$ on $\SL_2(\ZZ)$ with a multiplier system of order $24$. We denote this multiplier system by $\upsilon_{\eta}$.

Let $D$ be a positive integer, $k\in\frac{1}{2}\ZZ$ and $t\in \frac{1}{2}\NN$. A holomorphic function $\varphi : \HH \times (L \otimes \CC) \rightarrow \CC$ is
called a weakly holomorphic Jacobi form of weight $k$, index $t$ and character (or multiplier system) $\upsilon_\eta^D$ for $L$
if it satisfies

\begin{align*}
\varphi \left( \frac{a\tau +b}{c\tau + d},\frac{\mathfrak{z}}{c\tau + d}
\right)& = \upsilon_\eta^D(A) (c\tau + d)^k
\exp{\left(i \pi t \frac{c(\mathfrak{z},\mathfrak{z})}{c
\tau + d}\right)} \varphi ( \tau, \mathfrak{z} ),\\
\varphi (\tau, \mathfrak{z}+ x \tau + y)&=
(-1)^{t\big((x,x)+(y,y)\big)}\exp{\big(-i \pi t \big( (x,x)\tau +2(x,\mathfrak{z})\big)\big)}
\varphi (\tau, \mathfrak{z} ),
\end{align*}

for $A=\begin{psmallmatrix}
a & b \\
c & d
\end{psmallmatrix} \in \SL_2(\ZZ)$ and $x,y\in L$,
and if its Fourier expansion  takes the form

\begin{equation*}
\varphi ( \tau, \mathfrak{z} )=\sum_{ \substack{-\infty \ll n\in  \ZZ -\frac{D}{24} \\ \ell \in \frac{1}{2}L' } }f(n,\ell)q^n\zeta^\ell, \quad \zeta^\ell=e^{2\pi i (\ell,
\mathfrak{z})},
\end{equation*}

where $n\gg -\infty$ means that $n$ is bounded from below.
If $f(n,\ell) = 0$ whenever $n < 0$,
then $\varphi$ is called a weak Jacobi form.  If $ f(n,\ell) = 0 $ whenever
$ 2nt - (\ell,\ell) < 0 $,
then $\varphi$ is called a holomorphic
Jacobi form.

We denote the vector spaces of weakly holomorphic, weak and holomorphic Jacobi forms of weight $k$, index $t$ and character $\upsilon_\eta^D$ for $L$ by

J^{!}_{k,L,t}(\upsilon_\eta^D) \supset J^{\w}_{k,L,t}(\upsilon_\eta^D) \supset J_{k,L,t}(\upsilon_\eta^D),

respectively. We simply write $J^{!}_{k,L,t}\supset J^{\w}_{k,L,t} \supset  J_{k,L,t}$ if the character is trivial.
The spaces $J_{k, t}$ of classical Jacobi forms introduced by Eichler–Zagier [Eichler and Zagier, 1985]
are simply $J_{k,A_1, t}$ for the lattice $A_1=\ZZ$ with $(x,x)=2x^2$. If $\varphi \in J^{!}_{k,L,t}(\upsilon_\eta^D)$ is nonzero for some $t>0$ then $L(t)$ is necessarily an integral lattice [Cléry and Gritsenko, 2013]. Jacobi forms of index $t$ for $L$ are also called Jacobi forms of (lattice) index $L(t)$ and we sometimes write

J^!_{k,L(t)}=J^!_{k,L,t}, \quad J^{\w}_{k,L(t)}=J^{\w}_{k,L,t}, \quad J_{k,L(t)}=J_{k,L,t}.

We will use the following Hecke operators to construct Jacobi forms later.

Let $\varphi \in J_{k,L,t}^!(\upsilon_{\eta}^D)$. Assume that $k\in \ZZ$ and $D$ is an even divisor of $24$. If $Q=24/D$ is odd, we further assume that $t\in \ZZ$. Then for any positive integer $m$ coprime to $Q$ we have

\varphi \lvert_{k,t}T_{-}^{(Q)}(m)(\tau,\mathfrak{z})=m^{-1}\sum_{\substack{ad=m,\,a>0\\ 0\leq b <d}}a^k \upsilon_{\eta}^D(\sigma_a)\varphi\left(\frac{a\tau+bQ}{d},a\mathfrak{z}\right)\in J_{k,L,mt}^!(\upsilon_{\eta}^{D x}),

where $x,y\in \ZZ$ such that $mx+Qy=1$,

$$\sigma_a= \left(\begin{array}{cc}
dx+Qdxy & -Qy \\
Qy & a
\end{array}  \right) \in \SL_2(\ZZ).$$

Moreover, the Fourier coefficients $f_m(-,-)$ of $\varphi \lvert_{k,t}T_{-}^{(Q)}(m)$ are linear combinations of the Fourier coefficients $f(-,-)$ of $\varphi$, i.e.

f_m(n,\ell)=\sum_{a \mid (n,\ell,m)}a^{k-1} \upsilon_{\eta}^D(\sigma_a)f\left( \frac{nm}{a^2},\frac{\ell}{a}\right),

where $a\! \mid\! (n,\ell,m)$ means that $a\! \mid\! nQ$, $a^{-1}\ell\in \frac{1}{2}L'$ and $a\! \mid\! m$.

§.§ Affine Kac–Moody algebras and theta blocks

We review untwisted affine Kac–Moody algebras following [Kac, 1990] and identify their denominators as important examples of Jacobi forms.

Let $\mathfrak{g}$ be a simple Lie algebra of rank $r$ with Cartan subalgebra $\mathfrak{h}$ and root system $\Delta_\mathfrak{g}$. We fix a set of simple roots $\{ \alpha_1, ..., \alpha_r \} \subset \mathfrak{h}^*$ and denote the set of positive roots by $\Delta_\mathfrak{g}^+$ and the highest root by $\theta$. Note that $\dim \mathfrak{g} = r + |\Delta_\mathfrak{g}|$. The invariant symmetric bilinear form $\latt{-,-}$ on $\mathfrak{h}^*$ is normalized such that long roots have (square) norm two; in particular, $\latt{\theta, \theta}=2$. We identify $\mathfrak{h}$ with $\mathfrak{h}^*$ and define the coroot of a root $\alpha$ as

\alpha^\vee=2\alpha/\latt{\alpha, \alpha}.

The fundamental weights $w_i\in \mathfrak{h}^*$ are defined by

\latt{w_i, \alpha_i^\vee} = \delta_{i,j}, \quad 1\leq i,j \leq r,

where $\delta_{i,j}=1$ if $i=j$ and $0$ otherwise.
The Weyl vector $\rho_\mathfrak{g}$ is defined as

\rho_\mathfrak{g}=\frac{1}{2}\sum_{\alpha\in \Delta_\mathfrak{g}^+} \alpha = \sum_{j=1}^r w_j.

Let $Q_\mathfrak{g}$ be the rational lattice generated by the roots of $\mathfrak{g}$ and let $Q_\mathfrak{g}^\vee$ be the even integral lattice generated by the coroots, or equivalently by the long roots of $\mathfrak{g}$. The weight lattice $P_\mathfrak{g}$, generated by the fundamental weights, is the dual of the coroot lattice:

P_\mathfrak{g}=(Q_\mathfrak{g}^\vee)'=\{ x\in Q_\mathfrak{g}\otimes\QQ : \latt{x, \alpha^\vee}\in \ZZ, \; \alpha\in \Delta_\mathfrak{g}  \}\supset Q_\mathfrak{g}.

The coweight lattice $P_\mathfrak{g}^\vee$ is the dual of the root lattice:

P_\mathfrak{g}^\vee=Q_\mathfrak{g}'=\{ x\in Q_\mathfrak{g}\otimes\QQ : \latt{x, \alpha}\in \ZZ, \; \alpha\in \Delta_\mathfrak{g}  \}\supset Q_\mathfrak{g}^\vee.

The reflection associated with a root $\alpha$ is defined as

\sigma_{\alpha}(x)= x - \latt{x,\alpha^\vee}\alpha, \quad x\in Q_\mathfrak{g}\otimes \RR.

The reflections associated with simple roots generate the Weyl group $W_\mathfrak{g}$.
The coroot of $\theta$ can be written as an $\NN$-linear combination of the coroots of simple roots,

\theta = \theta^\vee = \sum_{j=1}^r a_j^\vee \alpha_j^\vee.

The integers $a_j^\vee$ are called comarks. The number

h_\mathfrak{g}^\vee=1+\sum_{j=1}^r a_j^\vee = \frac{1}{r}\sum_{\alpha\in \Delta_\mathfrak{g}^+} \latt{\alpha, \alpha}

is the dual Coxeter number and it satisfies the identity

\sum_{\alpha\in \Delta_\mathfrak{g}^+} \latt{\alpha, \mathfrak{z}}^2 = h_\mathfrak{g}^\vee\latt{\mathfrak{z},\mathfrak{z}}, \quad \mathfrak{z}\in Q_\mathfrak{g}\otimes\CC.

The above identity implies that the rescaled lattice $P_\mathfrak{g}^\vee(h_\mathfrak{g}^\vee)$ is integral.

The classification of irreducible root systems into types $A_n$ for $n\geq 1$, $B_n$ for $n\geq 2$, $C_n$ for $n\geq 3$, $D_n$ for $n\geq 4$, and the exceptional systems $E_6$, $E_7$, $E_8$, $F_4$ and $G_2$ is well-known. We use the same symbol to stand for the corresponding root lattice with its normalized bilinear form.  Some useful data is summarized in Table <ref> for convenience; for the coordinates of the fundamental weights and the values of the comarks, see [Bourbaki, 1981].

\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline
\Delta_\mathfrak{g} & A_n & B_n & C_n & D_n & E_6 & E_7 & E_8 & G_2 & F_4  \\
\hline
|\Delta_\mathfrak{g}| & n(n+1) & 2n^2 & 2n^2 & 2n(n-1) & 72 & 126 & 240 & 12 & 48 \\
\hline
h^\vee_{\mathfrak{g}} & n+1 & 2n-1 & n+1 & 2(n-1)& 12 & 18 & 30 & 4 & 9 \\
\hline
Q^\vee_\mathfrak{g} & A_n & D_n & nA_1 & D_n & E_6 & E_7 & E_8 & A_2 & D_4 \\
\hline
P^\vee_\mathfrak{g} & A_n' & \ZZ^n & D_n'(2) & D_n' & E_6' & E_7' & E_8 & A_2 & D_4 \\
\hline
\end{array}
\]

Data related to the irreducible root systems

The untwisted affine Kac–Moody algebra $\hat{\mathfrak{g}}$ is an extension of $\mathfrak{g}$ defined by

\hat{\mathfrak{g}} = \CC[t,t^{-1}]\otimes \mathfrak{g} \oplus \CC K \oplus \CC d,

where $K$ is a central element and $d$ is a derivative. The algebra $\hat{\mathfrak{g}}$ is an infinite-dimensional Lie algebra with affine Cartan subalgebra $\hat{\mathfrak{h}}=\mathfrak{h}\oplus \CC K \oplus \CC d$. We write $\hat{\mathfrak{h}}^*=\mathfrak{h}^*\oplus \CC \hat{w}_0 \oplus \CC \delta$ with

\begin{align*}
&\hat{w}_0(\mathfrak{h}\oplus \CC d)=0, \qquad\hat{w}_0(K)=1,\\
&\delta(\mathfrak{h}\oplus \CC K)=0, \qquad\ \delta(d)=1.
\end{align*}

We embed $\mathfrak{h}^*$ into $\hat{\mathfrak{h}}^*$ and define

\alpha_0=\delta-\theta \quad \text{and} \quad \alpha_0^\vee=K-\theta^\vee.

Then $\{\alpha_0, \alpha_1,\cdots, \alpha_r\}$ is a set of simple roots of $\hat{\mathfrak{g}}$ and $\{\alpha_0^\vee, \alpha_1^\vee,\cdots, \alpha_r^\vee\}$ is the corresponding set of coroots. Setting

\hat{w}_i = w_i + a_i^\vee \hat{w}_0, \quad 1\leq i\leq r,

$\{ \hat{w}_0, \hat{w}_1,\cdots ,\hat{w}_r \}$ is a set of fundamental weights of $\hat{\mathfrak{g}}$. The Weyl vector of $\hat{\mathfrak{g}}$ is defined by

\hat{\rho}_\mathfrak{g}= \rho_\mathfrak{g} + h_\mathfrak{g}^\vee \hat{w}_0 - (\dim \mathfrak{g} / 24)\delta,

such that the norm of $\hat{\rho}_\mathfrak{g}$ is zero; indeed, by the “strange formula” of Freudenthal–de Vries,

h_\mathfrak{g}^\vee \cdot \dim \mathfrak{g} =12 \latt{\rho_\mathfrak{g},\rho_\mathfrak{g}}.

The integrable highest weight representations of $\hat{\mathfrak{g}}$ are indexed by dominant integral weights, which are elements of

\hat{P}_+ = \sum_{j=0}^r \NN \hat{w}_j + \CC \delta.

The level of $\lambda = \sum_{j=0}^r x_j \hat{w}_j + c\delta \in \hat{P}_+$ is the integer

\lambda(K) = \lambda_0 + \sum_{j=1}^r x_j a_j^\vee.

The character of the irreducible highest weight module labeled by a weight $\lambda \in \hat{P}_+$ of level $k$ is given by the Weyl–Kac character formula,

\begin{equation}
\chi_\lambda^{\hat{\mathfrak{g}}} = \frac{\sum_{\sigma\in W_{\hat{\mathfrak{g}}}} (-1)^{\ell(\sigma)} e^{\sigma(\lambda+\rho_{\hat{\mathfrak{g}}})} }{e^{\rho_{\hat{\mathfrak{g}}}} \prod_{\alpha \in \Delta_{\hat{\mathfrak{g}}}^+}(1-e^{-\alpha})^{\mathrm{mult}(\alpha)}},
\end{equation}

where $W_{\hat{\mathfrak{g}}}$ is the Weyl group of $\hat{\mathfrak{g}}$, a semi-direct product of $W_\mathfrak{g}$ by a certain group of translations, $\ell(\sigma)$ is the length of $\sigma$, $\Delta_{\hat{\mathfrak{g}}}^+$ is the set of positive roots of $\hat{\mathfrak{g}}$, and $\mathrm{mult}(\alpha)$ is the multiplicity of $\alpha$, i.e. the dimension of the root space $\hat{\mathfrak{g}}_\alpha$. When $\lambda=0$, we have $\chi_\lambda^{\hat{\mathfrak{g}}}=1$, which gives the Macdonald–Weyl denominator identity:

\begin{equation}
e^{\rho_{\hat{\mathfrak{g}}}} \prod_{\alpha \in \Delta_{\hat{\mathfrak{g}}}^+}(1-e^{-\alpha})^{\mathrm{mult}(\alpha)} = \sum_{\sigma\in W_{\hat{\mathfrak{g}}}} (-1)^{\ell(\sigma)}e^{\sigma(\rho_{\hat{\mathfrak{g}}})}.
\end{equation}

The algebra $\hat{\mathfrak{g}}$ has real roots (with norm $>0$) and imaginary roots (with norm $= 0$). The set of positive real roots is

\Delta_{\hat{\mathfrak{g}}}^{+,\mathrm{re}} = \{ \alpha + n\delta : 0 < n \in \ZZ, \alpha \in \Delta_\mathfrak{g} \} \cup \Delta_\mathfrak{g}^+.

The set of positive imaginary roots is

\Delta_{\hat{\mathfrak{g}}}^{+,\mathrm{im}} = \{ n\delta : 0< n \in \ZZ \}.

Every real root has multiplicity one, but every imaginary root has multiplicity $r$.

Let $(\tau, \mathfrak{z})\in \HH \times \mathfrak{h}$. By interpreting the formal variable $e^{\alpha}$ as $e^{2\pi i \alpha(\mathfrak{z}-\tau d)}$, the character $\chi_\lambda^{\hat{\mathfrak{g}}}(\tau, \mathfrak{z})$ defines a holomorphic function on $\HH\times \CC^r$. The product side of the denominator identity can be written as a theta block (introduced by Gritsenko–Skoruppa–Zagier [Gritsenko et al., 2019])

\begin{equation}
\vartheta_{\mathfrak{g}}(\tau,\mathfrak{z})=\eta(\tau)^r \prod_{\alpha\in \Delta_\mathfrak{g}^+}\frac{\vartheta(\tau, \latt{\alpha,\mathfrak{z}})}{\eta(\tau)},
\end{equation}

where the odd Jacobi theta function

\vartheta(\tau,z)=\sum_{n\in\ZZ}\left(\frac{-4}{n}\right)q^{\frac{n^2}{8}}\zeta^{\frac{n}{2}} =-q^{\frac{1}{8}}\zeta^{-\frac{1}{2}}\prod_{n=1}^\infty(1-q^{n-1}\zeta)(1-q^n\zeta^{-1})(1-q^n), \quad z\in \CC, \; \zeta= e^{2\pi iz}

is the denominator of $\chi_\lambda^{\hat{\mathfrak{g}}}$ for the $A_1$ Lie algebra $\mathfrak{g}$. Note that $\vartheta_\mathfrak{g}$ is a holomorphic Jacobi form of (singular) weight ${r}/{2}$ and lattice index $P_\mathfrak{g}^\vee(h_\mathfrak{g}^\vee)$ with multiplier system $\upsilon_\eta^{\dim \mathfrak{g}}$.  By [Kac, 1990], all characters $\chi_\lambda^{\hat{\mathfrak{g}}}$ of fixed level $k$ form a vector-valued weakly holomorphic Jacobi form of weight $0$ and lattice index $Q_\mathfrak{g}^\vee(k)$ which is invariant under the action of the Weyl group $W_\mathfrak{g}$ on  $\mathfrak{z}$. We refer to <cit.> for the precise transformation laws with respect to the generators of $\SL_2(\ZZ)$.

Let $k$ be a positive integer. The irreducible $\hat{\mathfrak{g}}$-module $L_{\hat{\mathfrak{g}}}(k,0)$ associated with $k\hat{w}_0$ has a canonical structure as a simple rational vertex operator algebra. This is called the affine VOA generated by $\hat{\mathfrak{g}}$ at level $k$. Such VOAs are physically realized as the well-known Wess–Zumino–Witten (WZW) models, which are nonlinear sigma models describing mapping fields from Riemann surfaces to Lie group manifolds. The central charge of $L_{\hat{\mathfrak{g}}}(k,0)$ is

\begin{equation}\label{cc}
c_\mathfrak{g}=\frac{k \dim \mathfrak{g}}{k + h_\mathfrak{g}^\vee}.
\end{equation}

The irreducible modules of $L_{\hat{\mathfrak{g}}}(k,0)$ are the irreducible $\hat{\mathfrak{g}}$-modules $L_{\hat{\mathfrak{g}}}(k,\lambda)$ associated with level $k$ dominant integral weights $k\hat{w}_0+\lambda \in \hat{P}_+$ for any $\lambda\in \sum_{j=1}^r \NN w_j$ that satisfies

\latt{\lambda, \theta^\vee}=\sum_{j=1}^r a_j^\vee \lambda_j\leq k, \quad \text{where} \quad \lambda=\sum_{j=1}^r \lambda_j w_j.

The conformal weight of $L_{\hat{\mathfrak{g}}}(k,\lambda)$ is

\begin{equation}\label{conformalw}
h_\lambda=\frac{\latt{\lambda, \lambda + 2\rho_\mathfrak{g}}}{2(k+h^\vee_\mathfrak{g})}.
\end{equation}

The full character of $L_{\hat{\mathfrak{g}}}(k,\lambda)$ is actually $\chi_{k\hat{w}_0+\lambda}^{\hat{\mathfrak{g}}}(\tau,\mathfrak{z})$ as defined above. When the root system of $\mathfrak{g}$ is $R_r$ as in Table <ref> and $\lambda=\sum_{j=1}^r \lambda_j w_j$, we often write $\chi_{k\hat{w}_0+\lambda}^{\hat{\mathfrak{g}}}$ as

\begin{equation}\label{eq:symbol-char}
\chi^{R_{r,k}}_{\lambda_1\cdots\lambda_r,h_\lambda}.
\end{equation}

Let $\mathfrak{g}=\oplus_{j=1}^s \mathfrak{g}_j$ be a semi-simple Lie algebra. We will often have to associate to each simple ideal $\mathfrak{g}_j$ a positive integer $k_j$ called the level. In this case we indicate the levels by writing $\mathfrak{g}=\oplus_{j=1}^s \mathfrak{g}_{j,k_j}$. For simplicity we write

h_j^\vee = h_{\mathfrak{g_j}}^\vee, \quad Q_j = Q_{\mathfrak{g}_j}, \quad Q_j^\vee = Q_{\mathfrak{g}_j}^\vee, \quad P_j = P_{\mathfrak{g}_j}, \quad P_j^\vee = P_{\mathfrak{g}_j}^\vee.

We further fix two lattices

\mathbf{Q}_\mathfrak{g} = \bigoplus_{j=1}^s Q_j^\vee(k_j) \quad \text{and} \quad \mathbf{P}_\mathfrak{g} = \bigoplus_{j=1}^s P_j^\vee(k_j).

Let $V_\mathfrak{g}$ denote the affine vertex operator algebra

\bigotimes_{j=1}^s L_{\hat{\mathfrak{g}}_j}(k_j,0).

Clearly, the denominator of $\hat{\mathfrak{g}}$ is given by the theta block

\begin{equation}\label{eq:theta-blcok}
\vartheta_\mathfrak{g}:=\vartheta_{\mathfrak{g}_1}\otimes \vartheta_{\mathfrak{g}_2}\otimes \cdots \otimes \vartheta_{\mathfrak{g}_s}.
\end{equation}

Some remarks on the denominator identities are in order:

* Gritsenko, Skoruppa and Zagier [Gritsenko et al., 2019] found a direct proof of the denominator identity based on their theory of theta blocks.

* The function $\vartheta_\mathfrak{g}(\tau,\mathfrak{z})/\eta(\tau)^{\dim \mathfrak{g}}$ equals the modular Jacobian of the generators of the ring of weak Jacobi forms for $Q_\mathfrak{g}^\vee$ invariant under $W_\mathfrak{g}$ (see [Wang, 2021]).

* If a theta block defines a holomorphic Jacobi form of singular weight, then it has to be the denominator of an affine Lie algebra ([Wang, 2023]). Therefore, there is an one-to-one correspondence between affine Lie algebras and singular-weight theta blocks.

§.§ Reflective modular forms on orthogonal groups

Let $M$ be an even integral lattice of signature $(l,2)$ with $l\geq 3$. We choose one of the two connected components of

\{\mathcal{Z} \in  M\otimes \CC:  (\mathcal{Z}, \mathcal{Z})=0, (\mathcal{Z},\bar{\mathcal{Z}}) < 0\}

and label it $\cA(M)$. The symmetric domain of type IV attached to $M$ is

\cD(M):=\cA(M) / \mathbb{C}^{\times}=\{[\mathcal{Z}] \in  \PP(M\otimes \CC):  \mathcal{Z} \in \cA(M) \}.

Let $\Orth^+(M)$ denote the subgroup of $\Orth(M\otimes\RR)$ which preserves $M$ and $\cA(M)$.  Let $\Gamma$ be a finite-index subgroup of $\Orth^+(M)$. The most important example of $\Gamma$ will be the discriminant kernel

\widetilde{\Orth}^+(M)=\{ g \in \Orth^+(M):\; g(v) - v \in M, \; \text{for all $v\in M'$} \},

where $M'$ is the dual lattice of $M$.

Let $k\in \ZZ$ and $\chi: \Gamma\to \CC^\times$ be a character. A holomorphic function $F: \cA(M)\to \CC$ is called a modular form of weight $k$ and character $\chi$ on $\Gamma$ if it satisfies

\begin{align*}
F(t\mathcal{Z})&=t^{-k}F(\mathcal{Z}), \quad\  \text{for all $t \in \CC^\times$},\\
F(g\mathcal{Z})&=\chi(g)F(\mathcal{Z}), \quad \text{ for all $g\in \Gamma$}.
\end{align*}

Modular forms can be represented by their Fourier expansions. Let $c$ be a primitive isotropic vector of $M$ and choose $c'\in M'$ satisfying $(c,c')=1$. Then $M_{c,c'}=M\cap c^\perp \cap (c')^\perp$ is an even lattice of signature $(l-1,1)$. Around the cusp $c$, one can identify the symmetric domain $\cD(M)$ with a tube domain $\HH_{c,c'}$, a connected component of

\{ Z = X + iY :\; X, Y \in M_{c,c'}\otimes\RR, \; (Y,Y)<0  \}.

This induces an action of $\Orth^+(M)$ on $\HH_{c,c'}$ and an automorphy factor, which allows us to define modular forms of half-integral weight. A modular form of trivial character on $\widetilde{\SO}^+(M)$ can be expanded on $\HH_{c,c'}$ as

F(Z)=\sum_{\lambda \in M_{c,c'}', \; (\lambda, \lambda)\leq 0} c(\lambda) e^{2\pi i(\lambda, Z)}.

Modular forms $F$ on general $\Gamma$ have similar expansions with $M_{c,c'}$ replaced by a finite-index sublattice. If $F$ is nonzero, then either $k=0$ in which case $F$ is constant, or $k\geq l/2-1$. The smallest possible positive weight $l/2-1$ is called the singular weight.  When $F$ has singular weight, its Fourier coefficients $c(\lambda)$ are zero whenever $(\lambda,\lambda)\neq 0$.

Let $\lambda \in M\otimes\QQ$ be a vector of positive norm. The rational quadratic divisor associated with $\lambda$ is

\lambda^\perp=\{ [\mathcal{Z}] \in \cD(M) : (\mathcal{Z}, \lambda)=0 \}.

We define the associated reflection $\sigma_\lambda \in \Orth^+(M\otimes\QQ)$ as

\sigma_\lambda(x)=x-\frac{2(x,\lambda)}{(\lambda,\lambda)}\lambda, \quad x\in M\otimes\QQ.

The divisor $\lambda^\perp$ is called reflective if $\sigma_\lambda$ fixes $M$, i.e. $\sigma_\lambda \in \Orth^+(M)$. If $\lambda$ is a primitive vector in $M'$, then $\lambda^\perp$ is reflective if and only if there exists a positive integer $d$ such that $(\lambda, \lambda)=2/d$ and $d\lambda \in M$. More precisely, the order of $\lambda$ in $M'/M$ is either $d$, or $d/2$ in which case $d/2$ is necessarily even. A non-constant (holomorphic) modular form on $\Gamma$ is called reflective if its zero divisor is a linear combination of reflective rational quadratic divisors.

§.§ Borcherds products

Let $M$ be an even lattice of signature $(b^+,b^-)$ with discriminant form $D_M:=(M'/M, \mathfrak{q})$, where $\mathfrak{q}(x)=(x,x)/2$ is the induced quadratic form. Let $\mathrm{Mp}_2(\mathbb{Z})$ be the metaplectic group, which consists of pairs $A = (A, \phi_A)$, where $A = \begin{psmallmatrix} a & b \\ c & d \end{psmallmatrix} \in \mathrm{SL}_2(\mathbb{Z})$ and $\phi_A$ is a holomorphic square root of $\tau \mapsto c \tau + d$ on $\mathbb{H}$, with the standard generators

T = \big(\begin{psmallmatrix} 1 & 1 \\ 0 & 1 \end{psmallmatrix}, 1\big) \quad  \text{and} \quad S = \big(\begin{psmallmatrix} 0 & -1 \\ 1 & 0 \end{psmallmatrix}, \sqrt{\tau}\big).

The Weil representation $\rho_M$ is the unitary representation of $\mathrm{Mp}_2(\mathbb{Z})$ on the group ring $\mathbb{C}[D_M] = \mathrm{span}(e_x: \, x \in D_M)$ defined by

\rho_M(T) e_x = \mathbf{e}(-\mathfrak{q}(x)) e_x \quad \text{and} \quad \rho_M(S) e_x = \frac{\mathbf{e}( \mathrm{sign}(M) / 8)}{\sqrt{|D_M|}} \sum_{y \in D_M} \mathbf{e}(( x,y)) e_y,

where $\mathbf{e}(t)=e^{2\pi it}$ for $t\in \CC$, and $\mathrm{sign}(M)=b^+ - b^- \mod 8$. The dual representation of $\rho_M$ is the complex conjugate of $\rho_M$; moreover, $\bar{\rho}_M=\rho_{M(-1)}$.

Let $k \in \frac{1}{2}\mathbb{Z}$. A holomorphic function $f : \mathbb{H} \rightarrow \mathbb{C}[D_M]$ is called a weakly holomorphic modular form of weight $k$ if $f$ satisfies

\phi_A(\tau)^{-2k} f(A \cdot \tau) = \rho_M(A) f(\tau), \quad \text{for all $A \in \mathrm{Mp}_2(\mathbb{Z})$,}

and if $f$ is represented by a Fourier series of the form

\begin{equation}
f(\tau) = \sum_{x \in D_M} \sum_{\substack{n \in \mathbb{Z} - \mathfrak{q}(x)\\ n \gg -\infty}} c_x(n) q^n e_x.
\end{equation}

The finite sum $c_x(n)q^n e_x$ with $n<0$ is called the principal part of $f$.
If $f$ is holomorphic at infinity, i.e. its principal part is zero, then it is a holomorphic modular form. Note that $k+\mathrm{sign}(M)/2 \in \ZZ$ if non-zero $f$ exist, and that if $\mathrm{sign}(M)$ is even then $\rho_M$ factors through a representation of $\SL_2(\ZZ)$. We denote the spaces of weakly holomorphic and holomorphic modular forms of weight $k$ for $\rho_M$ respectively by

M_k^!(\rho_M) \supset M_k(\rho_M).

There is a natural homomorphism $\Orth(M) \to \Orth(D_M)$ with kernel $\widetilde{\Orth}(M)$. We define an action of $\Orth(D_M)$ on modular forms $f = \sum c_x(n) q^n e_x$ by

\sigma(f)=\sum c_x(n)q^n e_{\sigma(x)}, \quad \sigma \in \Orth(D_M).

We now assume that $M$ has signature $(l,2)$ with $l\geq 3$.
Let $f$ be a weakly holomorphic modular form of weight $1-l/2$ for $\rho_M$ with integral principal part. The Borcherds singular theta lift [Borcherds, 1995, Borcherds, 1998] produces a meromorphic modular form $\Borch(f)$ of weight $c_0(0)/2$ and some character (or multiplier system) on

\Orth^+(M,f)=\{ \sigma \in \Orth^+(M): \sigma(f)=f\} \supset \widetilde{\Orth}^+(M)

which has an infinite product expansion at any $0$-dimensional cusp involving the Fourier coefficients of $f$ (see below). Moreover, the divisor of $\Borch(f)$ is a linear combination of rational quadratic divisors $\lambda^\perp$, each with multiplicity

\sum_{d=1}^\infty c_{d\lambda}(-d^2\lambda^2/2),

where $\lambda \in M'$ are primitive vectors of positive norm.

There is an identification between modular forms for the Weil representation and Jacobi forms. Let $L$ be an even positive-definite lattice of rank $\rank(L)$. Then one has the isomorphism

\begin{equation}
\begin{split}
M_{k-\frac{1}{2}\rank(L)}^!(\rho_L) &\stackrel{\sim}{\longrightarrow} J_{k,L}^! \\
f(\tau)=\sum_{\gamma\in L'/L} f_\gamma(\tau)e_\gamma &\longmapsto \sum_{\gamma\in L'/L} f_\gamma(\tau) \Theta_{L,\gamma}(\tau,\mathfrak{z}),
\end{split}
\end{equation}

where $\Theta_{L,\gamma}$ is the Jacobi theta function of $L+\gamma$, defined as

\Theta_{L,\gamma}(\tau,\mathfrak{z}) = \sum_{\ell \in L+\gamma} e^{\pi i(\ell,\ell)\tau + 2\pi i(\ell, \mathfrak{z})}, \quad (\tau,\mathfrak{z})\in\HH\times (L\otimes\CC).

This map can be extended to modular forms with characters and it induces an isomorphism between holomorphic modular forms. It follows that a holomorphic Jacobi form of weight $\frac{1}{2}\rank(L)$ and index $L$ is a $\CC$-linear combination of $\Theta_{L,\gamma}$. We call $\frac{1}{2}\rank(L)$ the singular weight for Jacobi forms.

It will be convenient to represent Borcherds products in terms of Jacobi forms. Let $\phi \in J_{0,L,1}^!$ be a weakly holomorphic Jacobi form of trivial character with Fourier expansion

\phi(\tau,\mathfrak{z})=\sum_{n\in \ZZ}\sum_{\ell\in L'}f(n,\ell)q^n\zeta^\ell.

The terms $f(n,\ell)q^n\zeta^\ell$ with $2n-(\ell,\ell)<0$ are called singular Fourier coefficients and they correspond to the principal part of the preimage of $\phi$ under the above isomorphism.

The Fourier expansion of $\phi$ satisfies $f(n,\ell)=f(n,-\ell)$ and $f(n_1,\ell_1)=f(n_2,\ell_2)$ if $2n_1-(\ell_1,\ell_1)=2n_2-(\ell_2,\ell_2)$ and if $\ell_1-\ell_2\in L$. Therefore, any singular Fourier coefficient of $\phi$ already appears as a summand $f(n,\ell)q^n\zeta^\ell$ where $n\leq \hat{\delta}_L$, $\ell \in L'$ and $2n<(\ell,\ell)$, where $\hat{\delta}_L$ is the largest integer less than $\delta_L/2$ and

\begin{equation}
\delta_L := \max\big\{ \min\{(y,y): y\in L + x \} : x \in L' \big\}.
\end{equation}

Let $U$ be a hyperbolic plane, i.e. an even unimodular lattice of signature $(1,1)$. We write

U=\ZZ e + \ZZ f, \quad \text{where} \quad e^2=f^2=0 \quad \text{and} \quad (e,f)=-1.

Let $U_1=\ZZ e_1 +\ZZ f_1$ be a second hyperbolic plane and define $M=U_1\oplus U\oplus L$. We fix coordinates on the tube domain about $U_1$ by writing

\HH(L)=\{ Z= \tau f + \mathfrak{z} +\omega e : \tau, \omega \in \HH,\; \mathfrak{z}\in L\otimes\CC, \; 2\im(\tau)\im(\omega) -(\mathfrak{z},\mathfrak{z})>0  \},

such that

-(\alpha, Z)=n\tau - (\ell, \mathfrak{z}) + m\omega  \quad \text{for} \quad \alpha=ne+\ell+mf\in U\oplus L'.

These coordinates are chosen such that the infinite expansions below match the denominators of Borcherds–Kac–Moody algebras.

We write $v=x_1e_1+xe+\ell+yf+y_1f_1\in U_1\oplus U\oplus L'$ in the coordinate $(x_1,x,\ell,y,y_1)$ and $\alpha=xe+\ell+yf\in U\oplus L'$ in the coordinate $(x,\ell,y)$. In particular, $v^2=(\ell,\ell)-2(xy+x_1y_1)$.

Assume the above $\phi$ has integral singular Fourier coefficients. Then the Borcherds theta lift of $\phi$ is a meromorphic modular form of weight $f(0,0)/2$ on $\widetilde{\Orth}^+(M)$ and it can be expanded on an open subset of the tube domain $\HH(L)$ as

\Borch(\phi)(Z)=q^A \zeta^{\vec{B}} \xi^C\prod_{\substack{n,m\in\ZZ,\; \ell
\in L'\\ (n,\ell,m)>0}}\Big(1-q^n \zeta^{-\ell} \xi^m\Big)^{f(nm,\ell)},

where $q=\exp(2\pi i \tau)$, $\zeta^\ell=\exp(2\pi i (\ell, \mathfrak{z}))$, $\xi=\exp(2\pi i \omega)$, where the positivity condition $(n,\ell,m)>0$ means that either $m>0$, or $m=0$
and $n>0$, or $m=n=0$ and $\ell>0$, and where the Weyl vector $\rho= (-A,\vec{B}, -C)$ of $\Borch(\phi)$ is defined by

A=\frac{1}{24}\sum_{\ell\in L'}f(0,\ell),\quad
\vec{B}=\frac{1}{2}\sum_{\ell>0} f(0,\ell)\ell, \quad C=\frac{1}{2\rank(L)}\sum_{\ell\in L'}f(0,\ell)(\ell,\ell).

The Fourier–Jacobi expansion of $\Borch(\phi)$ on $\HH(L)$ is

\begin{equation}
\Borch(\phi)(Z)=
\biggl(\Theta_{f(0,\ast)}
(\tau,\mathfrak{z})\cdot \xi^C \biggr)
\exp \left(-\sum_{m=1}^\infty \Big(\phi |_0 T_{-}^{(1)}(m)\Big) (\tau, \mathfrak{z}) \cdot \xi^m \right),
\end{equation}

where the leading Fourier–Jacobi coefficient is given by a generalized theta block

\begin{equation*}
\Theta_{f(0,\ast)}(\tau,\mathfrak{z})
=\eta(\tau)^{f(0,0)}\prod_{\ell >0}
\biggl(\frac{\vartheta(\tau,(\ell,\mathfrak{z}))}{\eta(\tau)}
\biggr)^{f(0,\ell)}.
\end{equation*}

We also need the following additive lifts with characters to construct orthogonal modular forms.

Let $D$ be an even divisor of $24$, $k\in\NN$, $t\in \frac{1}{2}\NN$. If $Q=24/D$ is odd, we further assume that $t\in\NN$. Let $\varphi \in J_{k,L,t}(\upsilon_{\eta}^D)$.  Let $G_k(\tau)$ be the normalized Eisenstein series of weight $k$ on $\SL_2(\ZZ)$ whose Fourier coefficient at $q$ is 1. Then the function

\Grit(\varphi)(Z)=f(0,0)G_k(\tau)+\sum_{0<m\in 1+Q\ZZ}\Big(\varphi \lvert_{k,t}T_{-}^{(Q)}(m)\Big)(\tau,\mathfrak{z})\cdot \xi^{m/Q}

is a (holomorphic) modular form of weight $k$ and some character on $\widetilde{\Orth}^+(U_1\oplus U\oplus L(Qt))$. The form $\Grit(\varphi)$ is always invariant under the involution $(\omega,\mathfrak{z},\tau)\mapsto (\tau,\mathfrak{z},\omega)$.

For a given lattice $M$, we will often need to determine all holomorphic, reflective Borcherds products and in some cases prove that no such products exist. To do this, we apply Borcherds's obstruction criterion [Borcherds, 1999] which states that a formal sum
$$\sum_{n < 0} \sum_{x \in D_M} c_x(n) q^n e_x$$
occurs as the principal part of a weakly-holomorphic modular form of weight $\kappa$ for $\rho_M$ if and only if the identity
$$\sum_{n < 0} \sum_{x \in D_M} c_x(n) a_x(-n) = 0$$
holds for every cusp form $\sum_{n > 0} \sum_{x \in D_M} a_x(n) q^n e_x$  of weight $2-\kappa$ for the dual representation $\rho_{M(-1)}$.
We construct a basis of cusp forms for $\rho_{M(-1)}$ following [Williams, 2018] and realize the problem of computing holomorphic reflective products as the problem of enumerating integral lattice points in a polyhedral cone defined by finitely many inequalities (non-negative order along reflective divisors) and finitely many linear equations.
The solution can be conveniently expressed by the notion of a Hilbert basis, i.e. a minimal system of reflective products $F_1,...,F_r$ such that every holomorphic reflective product can be written in the form
$$F = F_1^{n_1} \cdot ... \cdot F_r^{n_r}$$
with non-negative integers $n_1,...,n_r$. Given the polyhedral cone, we used the software Normaliz [16] to find a Hilbert basis.

§.§ Borcherds–Kac–Moody superalgebras

Borcherds–Kac–Moody superalgebras are infinite dimensional Lie superalgebras introduced by Borcherds [Borcherds, 1988] in 1988 which generalize affine Kac–Moody algebras. They can be defined in terms of Chevalley–Serre generators and relations which are encoded in a generalized Cartan matrix. The restrictions on the generalized Cartan matrix are weaker; in particular, simple roots are allowed to have non-positive norm (i.e. they can be imaginary roots). BKM superalgebras also have a character formula for highest-weight modules and a denominator identity. We will review the denominator identity following [Ray, 2006] because it contains valuable information about the roots, root multiplicities and the Weyl group.

Let $\mathcal{G}=\mathcal{G}_0\oplus \mathcal{G}_1$ be a BKM superalgebra with even and odd components $\mathcal{G}_0$ and $\mathcal{G}_1$, respectively. If $\mathcal{G}_1$ is trivial, $\mathcal{G}$ is a BKM algebra.  Let $\mathcal{H}$ be the generalized Cartan subalgebra of $\mathcal{G}$. Let $\Delta^+$ be the set of positive roots of $\mathcal{G}$. We have the root space decomposition

\mathcal{G}=\bigoplus_{\alpha \in \Delta^+} \mathcal{G}_\alpha \oplus \mathcal{H} \oplus \bigoplus_{\alpha \in \Delta^+} \mathcal{G}_{-\alpha}.

For $\alpha \in \Delta^+$, we define

\mathrm{mult}_0(\alpha) = \dim (\mathcal{G}_\alpha\cap \mathcal{G}_0),  \quad \mathrm{mult}_1(\alpha) = \dim (\mathcal{G}_\alpha\cap \mathcal{G}_1), \quad
\mathrm{mult}(\alpha) = \dim (\mathcal{G}_\alpha),

and define the super-multiplicity $s$-$\mathrm{mult}(\alpha)$ as

\mathrm{mult}_0(\alpha) - \mathrm{mult}_1(\alpha).

The sets of positive even roots and odd roots are respectively

\Delta^+_0=\{ \alpha\in \Delta^+: \mathrm{mult}_0(\alpha) >0 \} \quad \text{and} \quad \Delta^+_1=\{ \alpha\in \Delta^+: \mathrm{mult}_1(\alpha) >0 \}.

Let $I$ be a countable set indexing the simple roots $\alpha_i$ and $S$ be the subset of $I$ indexing odd roots. By expanding a root as $\alpha=\sum_{i\in I} k_i \alpha_i$, we define the height and even height of $\alpha$ respectively as

\mathrm{ht}(\alpha) = \sum_{i\in I} k_i \quad \text{and} \quad \mathrm{ht}_0(\alpha) = \sum_{i\in I \backslash S } k_i.

We further define two formal sums

T=e^\rho \sum_{\mu}(-1)^{\mathrm{ht}(\mu)} e^{-\mu}, \quad T'=e^\rho \sum_{\mu}(-1)^{\mathrm{ht}_0(\mu)} e^{-\mu},

where $\rho$ is the Weyl vector of $\mathcal{G}$ and the sums are taken over all sets $\mu$ of distinct pairwise orthogonal imaginary simple roots. Let $W$ be the Weyl group generated by reflections associated with real simple roots. In this paper we assume that $\mathcal{G}$ has no odd real roots. Under this assumption, one has the denominator identity

\begin{equation}
e^\rho \cdot \frac{\prod_{\alpha \in \Delta_0^+}(1-e^{-\alpha})^{\mathrm{mult}_0(\alpha)}}{\prod_{\alpha \in \Delta_1^+}(1+e^{-\alpha})^{\mathrm{mult}_1(\alpha)}} = \sum_{\sigma\in W} \det(\sigma) \sigma(T)
\end{equation}

and the super-denominator identity

\begin{equation}
e^\rho \cdot \frac{\prod_{\alpha \in \Delta_0^+}(1-e^{-\alpha})^{\mathrm{mult}_0(\alpha)}}{\prod_{\alpha \in \Delta_1^+}(1-e^{-\alpha})^{\mathrm{mult}_1(\alpha)}} = \sum_{\sigma\in W} \det(\sigma) \sigma(T').
\end{equation}

An affine Kac–Moody  algebra $\mathcal{G}$ is a BKM algebra with no imaginary simple roots. Therefore, $T=e^{\rho}$, and we recover the denominator identity for affine Kac–Moody algebras.

Following Borcherds, we study BKM superalgebras that have a hyperbolic root lattice (i.e. its signature is of type $(l,1)$) and whose denominator or super-denominator is the Fourier expansion of a holomorphic modular form $F$ on $\Orth(l+1,2)$ at some $0$-dimensional cusp. The product side of the denominator identity suggests that $F$ can usually be constructed as a Borcherds product [Bruinier, 2002, Bruinier, 2014], in which case the roots and their multiplicities are encoded in a weakly holomorphic modular form for the Weil representation of $\SL_2(\ZZ)$. The real roots of $\mathcal{G}$ correspond to the zeros of $F$ through the cusp, so $F$ is locally reflective. Usually, $F$ is also globally reflective.

This is particularly interesting when $F$ is of singular weight. In this case, imaginary simple roots are easy to describe, and in many cases they are just the multiples of the Weyl vector. Moreover, one can write the sum side of the denominator identity explicitly as in [Borcherds, 1998, Scheithauer, 2004, Scheithauer, 2006], and it is expected that such a BKM superalgebra has a natural construction as the BRST cohomology of a suitable vertex algebra. In this paper, we will classify this type of BKM superalgebra under some mild conditions and work out the corresponding vertex algebras. Our classification is closely related to the fake monster algebra and its twists, so we will review that theory as well.

In 1990 Borcherds [Borcherds, 1990] constructed the fake monster algebra $\mathbb{G}$ as the BRST cohomology related to the Leech lattice VOA. The root lattice of the BKM algebra $\mathbb{G}$ is $U\oplus \Lambda$, where $\Lambda$ is the Leech lattice. The Weyl vector is $\rho=(-1,0,0)$ (see Notation <ref>) and the real roots are $\alpha\in U\oplus \Lambda$ with $\alpha^2=2$. The real simple roots are characterized by the equality $(\rho,\alpha)=1$. The imaginary simple roots are $n\rho$ for negative integers $n$, each with multiplicity $24$. The denominator of $\mathbb{G}$ defines a reflective Borcherds product $\Phi_{12}$ of singular weight on $2U\oplus \Lambda$ whose input is the full character of the Leech lattice VOA (see [Borcherds, 1995]). Let $\mathrm{Co}_0=\Orth(\Lambda)$ be the Conway group.  Borcherds [Borcherds, 1992] proved that the denominator identity of $\mathbb{G}$ is a cohomological identity and that $\mathrm{Co}_0$ acts on $\mathbb{G}$ naturally. In this way, he obtained a twisted denominator identity for each conjugacy class $[g]$ of $\mathrm{Co}_0$ and proved that it is the (untwisted) super-denominator identity of a BKM superalgebra, denoted $\mathbb{G}_g$. The root lattice of $\mathbb{G}_g$ is $L_g=U\oplus \Lambda^g$, where

\Lambda^g=\{ v \in \Lambda: g(v)=v \}

is the fixed-point sublattice. The real simple roots are the roots $\alpha$ of $L_g$ satisfying

(\rho, \alpha) = \alpha^2/2

and the imaginary simple roots are $m\rho$ with super-multiplicity $\sum_{k|(m,n_g)}b_k$ for all negative integers $m$, where $n_g$ is the order of $g$ and where $b_k$ describe the cycle shape $\prod_{b_k\neq 0}k^{b_k}$ of $g$. The last two named authors [Wang and Williams, 2022] proved that the twisted denominator function associated with $g$ defines a holomorphic Borcherds product (denoted $\Phi_g$) of singular weight on $U(N_g)\oplus U\oplus \Lambda^g$, where $N_g$ is the level of $g$, and that $\Phi_g$ is reflective on $U(N_g)\oplus U\oplus \Lambda^g$ if $N_g=n_g$, confirming a conjecture of Borcherds <cit.>.

§ THE HYPERBOLIZATION OF AFFINE KAC–MOODY ALGEBRAS

It is well-known how to realize affine Kac–Moody algebras as extensions of semi-simple Lie algebras. In this section, we introduce extensions of
affine Kac–Moody algebras to BKM superalgebras that we call their hyperbolizations.

Let $L$ be an even positive-definite lattice of rank $\rank(L)$, let $U$ and $U_1$ be two hyperbolic planes, and define $M=U_1\oplus U\oplus L$. Let $\mathcal{G}$ be a BKM superalgebra with root lattice $U\oplus L'$ and generalized Cartan subalgebra $\mathcal{H}$ with no odd real roots. We assume that the super-denominator of $\mathcal{G}$ is given by the Fourier expansion of a Borcherds product $F$ for $\widetilde{\Orth}^+(M)$ at the $0$-dimensional cusp determined by $U_1$.

Every divisor $\lambda^\perp$ of $F$ lies in the $\widetilde{\Orth}^+(M)$-orbit of the rational quadratic divisor associated with some primitive vector $\alpha\in U\oplus L'$. If $\alpha^\perp$ appears in the divisor of $F$, then $\alpha$ is a real root. Since we have assumed that $\mathcal{G}$ has no odd real roots, the divisor $\alpha^\perp$ is always a zero and therefore $F$ is holomorphic. Moreover, the associated reflection $\sigma_\alpha$ lies in the Weyl group of $\mathcal{G}$. Therefore, $\sigma_\alpha$ fixes the root lattice $U\oplus L'$, so it also fixes $M'=U_1\oplus U\oplus L'$, which implies that $\sigma_\alpha \in \Orth^+(M')=\Orth^+(M)$ and finally that $F$ is a reflective modular form.

We denote by $\phi$ the Jacobi form input of $F$ and write its expansion at $\infty$ as

\phi(\tau,\mathfrak{z}) = \sum_{n\in\ZZ,\; \ell\in L'} f(n,\ell)q^n \zeta^\ell \in J_{0,L}^!.

By Theorem <ref>, the product side of the denominator identity is (see Notation <ref>)

e^\rho \prod_{\alpha>0}\left(1-e^{-\alpha}\right)^{f(nm,\ell)}, \quad \alpha=(n,\ell,m)\in U\oplus L',\quad \rho=(-A,\vec{B}, -C).

In particular, if $f(nm,\ell)\neq 0$ then $\alpha$ is a root of $\mathcal{G}$ and

\mathrm{mult}_0(\alpha)-\mathrm{mult}_1(\alpha)=f(nm,\ell).

The weight of $F$ is given by $f(0,0)/2$. The root $\alpha$ is positive if and only if either $m>0$, or $m=0$ and $n>0$, or $m=n=0$ and $\ell>0$.

Let $\mathcal{R}_+$ be the set of positive roots of $\mathcal{G}$ of type $(n,\ell,0)$. For any positive integer $n$, $(n,0,0)$ is an imaginary root in $\mathcal{R}_+$. All other roots in $\mathcal{R}_+$ are (even) real and have multiplicity $f(0,\ell)$. Since the multiplicity of a real root is always $1$ (see e.g. <cit.>), we have $f(0,\ell)=1$.

Assume further that none of $(n,0,0)$ is odd. Then the even roots $(n,0,0)$ always have multiplicity $f(0,0)$. Now consider the sum of root spaces

\mathcal{G}^0:=\bigoplus_{\alpha\in \mathcal{R}_+} \mathcal{G}_{\alpha} \oplus \mathcal{H} \oplus \bigoplus_{\alpha\in \mathcal{R}_+} \mathcal{G}_{-\alpha}.

If $F$ is of singular weight, i.e. $f(0,0)=\rank(L)$, then $\mathcal{G}^0$ defines an affine Kac–Moody algebra $\hat{\mathfrak{g}}$ for which $\mathfrak{g}$ is a semi-simple Lie algebra whose positive roots are the roots of type $(0,\ell,0)$ in $\mathcal{R}^+$ (here, we can view $K=\delta=(1,0,0)$ and $d=\hat{w}_0=(0,0,-1)$). The leading Fourier–Jacobi coefficient of $F$ at the $1$-dimensional cusp determined by $U_1\oplus U$ is a holomorphic Jacobi form of singular weight given by the theta block

\eta(\tau)^{\rank(L)} \prod_{(0,\ell,0)\in \mathcal{R}^+} \frac{\vartheta(\tau, (\ell, \mathfrak{z}))}{\eta(\tau)}, \quad \mathfrak{z}\in L\otimes\CC,

which equals the denominator $\vartheta_\mathfrak{g}$ of $\hat{\mathfrak{g}}$. In particular, $\mathcal{G}$ is an extension of $\hat{\mathfrak{g}}$. For $m>0$ we define $\mathcal{G}^m$ to be the sum of root spaces associated with roots of type $\pm (*,*,m)$. Then

\mathcal{G}=\bigoplus_{m=0}^\infty \mathcal{G}^m

is a graded module over $\hat{\mathfrak{g}}=\mathcal{G}^0$. From this point of view, it is natural to regard $\mathcal{G}$ as a hyperbolization of $\hat{\mathfrak{g}}$.

Motivated by the discussion above, we introduce the following definition.

Let $\hat{\mathfrak{g}}$ be an affine Kac–Moody algebra. A BKM superalgebra $\mathcal{G}$ without odd real roots is called a hyperbolization of $\hat{\mathfrak{g}}$ if there exists an even positive-definite lattice $L$ such that the root lattice of $\mathcal{G}$ is $U\oplus L'$ and the super-denominator of $\mathcal{G}$ defines a holomorphic Borcherds product $F$ of (singular) weight $\rank(L)/2$ on $2U\oplus L$ whose leading Fourier–Jacobi coefficient is the denominator $\vartheta_\mathfrak{g}$ of $\hat{\mathfrak{g}}$. We also call $F$ a hyperbolization of $\vartheta_\mathfrak{g}$.

The notion of hyperbolization was first introduced by Gritsenko and the second named author in <cit.>, <cit.> and <cit.>.

In this paper we will classify affine Kac–Moody algebras that admit hyperbolizations and construct a hyperbolization for every such affine Kac–Moody algebra. Moreover, we express every Jacobi form input $\phi$ as a $\ZZ$-linear combination of full characters of an associated vertex operator (super)algebra.

§ ROOT SYSTEMS ASSOCIATED WITH SINGULAR-WEIGHT REFLECTIVE MODULAR FORMS

In this section, we classify reflective automorphic products of singular weight on lattices of type $2U\oplus L$ whose Jacobi form inputs have non-negative $q^0$-terms. Although Jacobi form inputs that have negative coefficients in their $q^0$-terms can lift to holomorphic Borcherds products, the corresponding BKM superalgebras in these cases will have odd real roots.

By an argument similar to <cit.> and <cit.>, we will show that any such Borcherds product belongs to a root system satisfying certain strong restrictions. These restrictions yield a finite list of root systems and a finite list of candidates for affine Kac–Moody algebras that admit hyperbolizations.

Let $L$ be an even positive-definite lattice of rank $\rank(L)$ and let $U$ be a hyperbolic plane. Suppose there is a reflective Borcherds product $F$ of singular weight on $\widetilde{\Orth}^+(2U\oplus L)$ whose Jacobi form input has non-negative $q^0$-term. Then either $(a)$ or $(b)$ below holds:

(a) The lattice $L$ is the Leech lattice, $F$ is the denominator $\Phi_{12}$ of the fake monster algebra, and the Jacobi form input is the full character of the Leech lattice VOA.

(b) There exists a semi-simple Lie algebra $\mathfrak{g}=\oplus_{j=1}^s\mathfrak{g}_{j,k_j}$ of rank $\rank(L)$ satisfying restrictions $(1)$ or $(1')$ and $(2)$ and $(3)$, where $\mathfrak{g}_{j}$ are simple Lie algebras and $k_j$ are positive integers for $1\leq j\leq s$. We refer to Notation <ref> for the symbols below. Let $\lambda\in U$ with $\lambda^2=2$.

(1) If $F$ vanishes on $\lambda^\perp$, then

\begin{equation}\label{antiC}
C:=\frac{\dim \mathfrak{g}}{24} -1 = \frac{h_j^\vee}{k_j}, \quad \text{for $1\leq j\leq s$}.
\end{equation}

($1'$) If $F$ does not vanish on $\lambda^\perp$, then

\begin{equation}\label{symC}
C:=\frac{\dim \mathfrak{g}}{24} = \frac{h_j^\vee}{k_j} \quad \text{and} \quad k_j>1, \quad \text{for $1\leq j\leq s$}.
\end{equation}

(2) The rescaled lattice $L(C)$ is integral and $L$ is bounded by

\begin{equation}
\mathbf{Q}_\mathfrak{g}<L< \mathbf{P}_\mathfrak{g}.
\end{equation}

(3) The leading Fourier–Jacobi coefficient of $F$ at the $1$-dimensional cusp determined by $2U$ coincides with the denominator of the affine Kac–Moody algebra $\hat{\mathfrak{g}}$.

Let $c_\mathfrak{g}$ denote the central charge of the affine VOA generated by $\hat{\mathfrak{g}}=\oplus_{j=1}^s\hat{\mathfrak{g}}_{j,k_j}$. Then $c_\mathfrak{g}$ is always $24$ in case $(1)$ and in case $(1')$ we have

\begin{equation}
c_\mathfrak{g} = \frac{24C}{C+1}.
\end{equation}

We call case $(1)$ (resp. $(1')$) the anti-symmetric (resp. symmetric) case, since $F$ is anti-invariant (resp. invariant) under the involution $(\omega,\mathfrak{z},\tau)\mapsto (\tau,\mathfrak{z},\omega)$.

This is a more precise version of <cit.> in the specific case of singular weight. We provide a detailed proof here as it combines the proof of <cit.> with some recent results proved in [Wang and Williams, 2023], and because it explains the construction of the Lie algebra $\mathfrak{g}$.

Let $\phi\in J_{0,L}^!$ be the Jacobi form input of $F$, and write its Fourier expansion as

\phi(\tau,\mathfrak{z}) = \sum_{n\in \ZZ,\; \ell\in L'} f(n,\ell) q^n \zeta^\ell.

By assumption, $f(0,\ell)$ are non-negative integers.
Each vector $v=(n,\ell,1) \in U\oplus L'$ is primitive in $M:=2U\oplus L$ and satisfies $v^2=(\ell,\ell)-2n$. The multiplicity of $v^\perp$ in the divisor of $F$ is

\sum_{d=1}^\infty f(d^2n, d\ell).

By <cit.>, the Fourier expansion of $F$ begins with

\phi(\tau,\mathfrak{z}) = f(-1,0)q^{-1} + \sum_{\ell \in L'}f(0,\ell)\zeta^\ell + O(q).

Since $F$ is of singular weight, $f(0,0)=\rank(L)$. We know from <cit.> that a holomorphic Borcherds product of singular weight has only simple zeros. Therefore, $f(-1,0)$ and $f(0,\ell)$ can only be $0$ or $1$. By definition, $f(-1,0)=1$ in the anti-symmetric case and $f(-1,0)=0$ in the symmetric case. We define

\mathcal{R}=\{ \ell \in L'\backslash \{0\} : f(0,\ell)=1 \}.

By <cit.>, we have

\begin{align}\label{eq:A}
\frac{|\mathcal{R}|+\rank(L)}{24} -f(-1,0) &= \frac{1}{2\rank(L)}\sum_{\ell \in \mathcal{R}} (\ell, \ell) =:C, \\
\label{eq:B}
\sum_{\ell \in \mathcal{R}} (\ell, \mathfrak{z})^2 &= 2C (\mathfrak{z}, \mathfrak{z}).
\end{align}

If $\mathcal{R}$ is empty, then $C=0$, $f(-1,0)=1$ and $\rank(L)=24$. In this case, <cit.> shows that $M$ is isomorphic to the even unimodular lattice of signature $(26,2)$, which then forces $L$ to be the Leech lattice $\Lambda$ and $F$ to be the Borcherds form $\Phi_{12}$ (see <cit.>). The Jacobi form input of $\Phi_{12}$ is given by

\Theta_{\Lambda,0}(\tau,\mathfrak{z})/\eta^{24}(\tau)=q^{-1}+24+O(q),

which equals the full character of the Leech lattice VOA.

Now assume that $\mathcal{R}$ is non-empty. Equation (<ref>) implies that $C> 0$ and that $L\otimes\QQ$ is spanned by $\mathcal{R}$ over $\QQ$. We will now show that $\mathcal{R}$ is a rescaled root system. Let $\ell_1, \ell_2 \in \mathcal{R}$. Then $-\ell_1 \in \mathcal{R}$ because $f(0,\ell)=f(0,-\ell)$. For any integer $a>1$, $a\ell_1 \not\in \mathcal{R}$, as otherwise $(0,\ell_1,1)^\perp$ would have multiplicity at least $2$ in the divisor of $F$, contradicting the fact that all zeros of $F$ are simple. Since $F$ is reflective, the reflection $\sigma_{(0,0,\ell_1,1,0)}$ fixes $M$ (see Notation <ref>). From

\sigma_{(0,0,\ell_1,1,0)}\big((0,0,\ell_2,1,0)\big)= \big(0,0,\sigma_{\ell_1}(\ell_2),1-2(\ell_1,\ell_2)/(\ell_1,\ell_1),0\big)=:\lambda \in M

we obtain

\sigma_{\ell_1}(\ell_2)\in L' \quad \text{and} \quad 2(\ell_1,\ell_2)/(\ell_1,\ell_1) \in \ZZ.

We know from <cit.> that $F$ is anti-invariant under $\sigma_{(0,0,\ell_1,1,0)}$ and therefore vanishes on $\lambda^\perp$. Since $(0,0,\sigma_{\ell_1}(\ell_2),1,0)$ is primitive in $M'$, by the Eichler criterion (see e.g. <cit.>) there exists $g\in \widetilde{\Orth}^+(M)$ such that $g(\lambda)=(0,0,\sigma_{\ell_1}(\ell_2),1,0)$. Therefore, $F$ also vanishes on $(0,0,\sigma_{\ell_1}(\ell_2),1,0)^\perp$, from which it follows that $f(0,\sigma_{\ell_1}(\ell_2))=1$ and that $\sigma_{\ell_1}(\ell_2)\in \mathcal{R}$. This proves that $\mathcal{R}$ is a (rescaled) root system.

Now we can decompose $\mathcal{R}$ into rescaled irreducible root systems

\mathcal{R}=\bigoplus_{j=1}^s \mathcal{R}_j(a_j),

where $\mathcal{R}_j$ are standard irreducible root systems and $a_j$ are rational numbers by which the (squared) norms of the roots in $\mathcal{R}_j$ are rescaled. Since $(0,0,\ell,1,0)^\perp$ is reflective for any $\ell \in \mathcal{R}$, there exists a positive integer $d$ such that $\ell^2=2/d$ and $d \ell \in L$. It follows that $a_j=1/k_j$ for some positive integers $k_j$. In the symmetric case, there are no $2$-roots in $\mathcal{R}$, so every $k_j$ is larger than $1$. Let $\mathfrak{g}_{j,k_j}$ be the simple Lie algebra of type $\mathcal{R}_j$ with level $k_j$. We see from (<ref>) and (<ref>) that the semi-simple Lie algebra $\mathfrak{g}=\oplus_{j=1}^s \mathfrak{g}_{j,k_j}$ satisfies condition $(1)$ or $(1')$ of the theorem.

By <cit.>, $L(C)$ is an integral lattice. The set $\mathcal{R} = \oplus_{j=1}^s \mathcal{R}_j(1/k_j)$ generates a sublattice of $L'$ over $\ZZ$, from which it follows that

\bigoplus_{j=1}^s Q_j(1/k_j) < L' \quad \text{and therefore}  \quad L<\bigoplus_{j=1}^s P_j^\vee(k_j).

The vectors $\ord(\ell) \ell$ for $\ell \in \mathcal{R}$ generate a sublattice of $L$ over $\ZZ$, where $\ord(\ell)$ is the order of $\ell$ in $L'/L$. It is easy to check that this lattice contains the lattice generated by long roots of $\oplus_{j=1}^s \mathcal{R}_j(k_j)$, and therefore $\oplus_{j=1}^s Q_j^\vee(k_j) < L$. This proves claim (2).

Finally, let $\Delta_j^+$ be the set of positive roots of $\mathcal{R}_j$ and let $\latt{-,-}$ be the normalized bilinear form on $\mathcal{R}_j$ that was fixed in Section <ref>. For $\alpha \in \Delta_j^+$, we have $\alpha / k_j \in \mathcal{R} \subsetneq L'$ and
$$(\alpha/k_j, \mathfrak{z})=k_j\latt{\alpha/k_j, \mathfrak{z}}=\latt{\alpha, \mathfrak{z}}.$$

Therefore, the leading Fourier–Jacobi coefficient of $F$ is a Jacobi form of lattice index $L(C)$ given by

\eta(\tau)^{\rank(L)} \prod_{0<\ell \in \mathcal{R}} \frac{\vartheta(\tau, (\ell, \mathfrak{z}))}{\eta(\tau)} = \prod_{j=1}^s \Big( \eta(\tau)^{\rank(\mathfrak{g}_{j})} \prod_{\alpha \in \Delta_j^+}\frac{\vartheta(\tau, \latt{\alpha, \mathfrak{z}})}{\eta(\tau)} \Big),

which is actually the denominator of $\hat{\mathfrak{g}}$. Note that

\bigoplus_{j=1}^s Q^\vee_j(h_j^\vee)<L(C)< \bigoplus_{j=1}^s P^\vee_j(h_j^\vee),

which matches the index $L(C)$ of the above leading Fourier–Jacobi coefficient of $F$.

The last assertion follows from the central charge formula

\begin{align*}
c_\mathfrak{g} = \sum_{j=1}^s \frac{k_j \dim \mathfrak{g}_{j}}{ k_j+h_j^\vee } = \sum_{j=1}^s \frac{\dim \mathfrak{g}_{j}}{1 + C} = \frac{\dim \mathfrak{g}}{1+C}
\end{align*}

and because $\dim \mathfrak{g} = 24(C+1)$ in the anti-symmetric case and $\dim \mathfrak{g} = 24C$ in the symmetric case.

In the above theorem, when $\mathcal{R}_j$ is of type $E_8$, $F_4$ or $G_2$, we have $Q_j^\vee = P_j^\vee$. Let $J$ denote the subset of such $j$. Then we have (see <cit.>)

L=K\oplus \bigoplus_{j\in J}P_j^\vee(k_j), \quad \text{where} \quad \bigoplus_{j\neq J}Q_j^\vee(k_j) < K < \bigoplus_{j\neq J}P_j^\vee(k_j).

Equation (<ref>) has $221$ solutions, and they are listed in Table <ref>. Equation (<ref>) has $17$ solutions, and they are listed in Table <ref>.

Equation (<ref>) was first derived by Schellekens [Schellekens, 1993] in 1993 in the context of holomorphic vertex operator algebras of central charge $24$, and he determined the solutions that are listed in Table <ref>. The solutions of Equation (<ref>) can be found in a similar way.

§ THE CLASSIFICATION OF AFFINE LIE ALGEBRAS WITH HYPERBOLIZATIONS

In this section we state the main theorems and explain the ideas behind the proofs.

Our first main result is the classification of root systems associated with reflective Borcherds products of singular weight on lattices of type $2U\oplus L$, which was introduced in Theorem <ref>.

Suppose $2U\oplus L$ has a reflective Borcherds product of singular weight whose Jacobi form input has non-negative $q^0$-term. Then the associated semi-simple Lie algebra $\mathfrak{g}$ defined in Theorem <ref> is one of the $81$ Lie algebras colored blue in Tables <ref> and <ref>.

To prove this, we have to rule out most of the solutions of Equation (<ref>) and Equation (<ref>). There are $238$ root systems in Tables <ref> and <ref> altogether. A root system will be called extraneous if it is not one of the $81$ root systems described in Theorem <ref>; there are $152$ extraneous root systems of anti-symmetric type and $5$ extraneous root systems of symmetric type.

Let $\mathfrak{g}$ be a semi-simple Lie algebra associated with a singular-weight reflective Borcherds product on $2U\oplus L$ that is extraneous. By Theorem <ref> (2), the underlying lattice $L$ satisfies the bounds

\bQ < L < \bP.

We will determine an explicit even overlattice $K$ of $L$ with the property that any singular-weight reflective Borcherds product on $2U\oplus L$ lifts to a reflective Borcherds product satisfying certain conditions on $2U\oplus K$, but for which no such reflective products on $2U\oplus K$ exist. This leads to a contradiction. The complete proof appears in Section <ref> for anti-symmetric root systems and in Section <ref> for symmetric root systems.

The second main result is the construction of hyperbolizations for the $81$ semi-simple Lie algebras in the theorem above.

For each semi-simple Lie algebra $\mathfrak{g}$ in Theorem <ref>, there exists an even positive definite lattice $L_{\mathfrak{g}}$ with the same rank as $\mathfrak{g}$ that satisfies the following conditions:

* There is a singular-weight reflective Borcherds product $\Psi_{\mathfrak{g}}$ on $2U\oplus L_{\mathfrak{g}}$;

* The leading Fourier–Jacobi coefficient of $\Psi_{\mathfrak{g}}$ at the $1$-dimensional cusp determined by $2U$ is the denominator of the affine Lie algbera $\hat{\mathfrak{g}}$;

* The Jacobi form input $\phi_\mathfrak{g}$ of $\Psi_{\mathfrak{g}}$ is a $\ZZ$-linear combination of full characters of the affine vertex operator algebra generated by $\hat{\mathfrak{g}}$;

* The lattice generated by $d\rho$ and by those $\lambda=(n,\ell,m)\in U\oplus L_{\mathfrak{g}}'$ for which $\Psi_{\mathfrak{g}}$ vanishes on $\lambda^\perp$ is $U\oplus L_{\mathfrak{g}}'$; that is, $U\oplus L_{\mathfrak{g}}'$ is the root lattice of the BKM (super)algebra with (super)-denominator $\Psi_{\mathfrak{g}}$. Here, $\rho$ is the Weyl vector of $\Psi_{\mathfrak{g}}$ and $d$ is the denominator of $C$ as defined in Theorem <ref>.

* When $\mathfrak{g}$ is of symmetric type, the Borcherds product $\Psi_{\mathfrak{g}}$ coincides with the Gritsenko (additive) lift of the denominator of $\hat{\mathfrak{g}}$.

The above lattices $L_{\mathfrak{g}}$ and Jacobi forms $\phi_\mathfrak{g}$ are constructed as follows.

(a) Let $\mathfrak{g}$ be one of the $69$ semi-simple Lie algebras of anti-symmetric type. In this case, $\mathfrak{g}$ appears as the semi-simple $V_1$ structure of a holomorphic vertex operator algebra $V$ of central charge $24$ in Schellekens’s list. Then $L_\mathfrak{g}$ is the orbit lattice in Höhn's construction of $V$ and $\phi_\mathfrak{g}$ is the unique full character of $V$.

(b) Let $\mathfrak{g}$ be one of the $8$ semi-simple Lie algebras of symmetric type with $C=1$. In this case, $\mathfrak{g}$ appears as the $\mathcal{N}=1$ structure of a holomorphic vertex operator superalgebra of central charge $12$ composed of $24$ chiral fermions. Then $L_\mathfrak{g}$ is the maximal even sublattice of $\bP$, and $\phi_\mathfrak{g}$ can be expressed as a $\ZZ$-linear combination of the full $\mathrm{NS}$-, $\widetilde{\mathrm{NS}}$- and $\mathrm{R}$-characters of the associated SVOA:

\phi_{\mathfrak{g}}=(\chi_{\mathrm{NS}} - \chi_{\widetilde{\mathrm{NS}}} - \chi_{\mathrm{R}})/2.

(c) Let $\mathfrak{g}=A_{1,16}$, $A_{1,8}^2$, $A_{1,4}^4$ or $A_{2,9}$. These semi-simple Lie algebras are of symmetric type with $C<1$. In these cases, the affine Lie algebras $\hat{\mathfrak{g}}$ admit exceptional modular invariants coming from a nontrivial automorphism of the fusion algebra. Then $L_\mathfrak{g}=\bP$, the expressions of $\phi_\mathfrak{g}$ in terms of affine characters are given in Theorem <ref>, and the relationship between $\phi_\mathfrak{g}$ and the exceptional modular invariants is explained in Remark <ref>.

The proof of Theorem <ref> is divided into four sections. In Sections <ref>-<ref> we prove parts (a), (b) and (c) respectively. In Section <ref> we prove property (5). The connections between our construction and the twists of the fake monster algebra are also explained in Sections <ref> and <ref>.

Our last main result is the complete classification of affine Lie algebras with hyperbolizations. This is a direct consequence of Theorem <ref> and Theorem <ref>.

There are exactly $81$ affine Kac–Moody algebras which have a hyperbolization in the sense of Definition <ref>. The $81$ algebras are colored blue in Tables <ref> and <ref>.

Combining Theorem <ref> and Theorem <ref>, we obtain the following constraint on the existence of singular automorphic products:

There are reflective Borcherds products of singular weight on lattices of type $2U\oplus L$ whose input has non-negative principal part if and only if

\rank(L) \in \{1, 2, 4, 6, 8, 10, 12, 16, 24\}.

§ THE ANTISYMMETRIC CASE: HOLOMORPHIC CFT OF CENTRAL CHARGE 24

The $69$ semi-simple Lie algebras $\mathfrak{g}$ in the anti-symmetric case of Theorem <ref> coincide with the semi-simple $V_1$ structures of holomorphic vertex operator algebras (VOA) of central charge $24$ in Schellekens' list [Schellekens, 1993]. In 2017, Höhn [Höhn, 2017] found a uniform construction of the holomorphic VOAs. For each $\mathfrak{g}$, we will take Höhn's orbit lattice $L_\mathfrak{g}$ as the underlying lattice $L$ and construct the reflective Borcherds product of singular weight as the Borcherds lift of the full character of the holomorphic VOA. We also explain the connection between these hyperbolizations and the twisted denominators of the fake monster algebra.

§.§ Holomorphic CFTs of central charge 24 and Schellekens' list

Let $V$ be a holomorphic vertex operator algebra of central charge $24$. The weight-one subspace $V_1$ has a natural Lie algebra structure and by [Schellekens, 1993, Dong and Mason, 2004], $V_1$ is either trivial, abelian of dimension $24$, or semi-simple. In the first case it was conjectured in [Frenkel et al., 1988] that $V$ is isomorphic to the monster VOA, which was  proved under some conditions in [Dong et al., 2007]. In the second case $V$ is isomorphic to the Leech lattice VOA.

We focus on the third case where $V_1$ is a semi-simple Lie algebra $\mathfrak{g}$. Let $(-,-)$ be the unique symmetric, non-degenerate, invariant bilinear form on $V$ normalized such that $(\mathbf{1},\mathbf{1})=-1$, where $\mathbf{1}$ is the vacuum vector. The restriction of $(-,-)$ to a simple ideal $\mathfrak{g}_j$ of $\mathfrak{g}$ satisfies $(-,-)=k_j\latt{-,-}$ for some positive integer $k_j$, where $\latt{-,-}$ is the normalized bilinear form of $\mathfrak{g}_j$ (see [Schellekens, 1993, Dong and Mason, 2006]).  We indicate
these integers by writing

\mathfrak{g}=\mathfrak{g}_{1,k_1}\oplus \cdots \oplus \mathfrak{g}_{s,k_s}.

Then the affine vertex operator algebra

V_\mathfrak{g}\cong L_{\hat{\mathfrak{g}}_1}(k_1,0)\otimes \cdots \otimes L_{\hat{\mathfrak{g}}_s}(k_s,0)

generated by $V_1$ is a full vertex operator subalgebra of $V$. As a $V_\mathfrak{g}$-module, one can decompose $V$ into finitely many irreducible $V_\mathfrak{g}$-modules

\begin{equation}\label{eq:V-decomposition}
V=\bigoplus_{\lambda_1,...,\lambda_s} m_{\lambda_1,...,\lambda_s} L_{\hat{\mathfrak{g}}_1}(k_1,\lambda_1)\otimes \cdots \otimes L_{\hat{\mathfrak{g}}_s}(k_s,\lambda_s),
\end{equation}

where the sum runs over the dominant integral weights $\lambda_j$ of $\mathfrak{g}_j$ that satisfy $\latt{\lambda_j, \theta_j^\vee}\leq k_j$, where $\theta_j$ is the highest root of $\mathfrak{g}_j$.

In 1993 Schellekens [Schellekens, 1993] established the equation (see Notation <ref>)

\frac{\dim \mathfrak{g}}{24} -1 = \frac{h_j^\vee}{k_j}, \quad \text{for $1\leq j\leq s$}

and showed that it has exactly $221$ solutions. By excluding $152$ of them, Schellekens proved that $V_1$ is isomorphic to one of the $69$ semi-simple Lie algebras in <cit.> (named Schellekens's list) and further determined the induced decompositions (<ref>). This result was reproved in [van Ekeren et al., 2020] using similar arguments and in [van Ekeren et al., 2021] by means of the “very strange formula".

Schellekens also conjectured that there exists a unique holomorphic VOA of central charge $24$ with $V_1=\mathfrak{g}$ for every $\mathfrak{g}$ on his list. By the work of many authors over the past three decades, this celebrated conjecture was finally proved (see [Lam, 2011, Lam and Shimakura, 2012, Lam and Shimakura, 2016, Lam and Shimakura, 2016, Lam and Lin, 2020, Sagaki and Shimakura, 2016, van Ekeren et al., 2020] for the existence and [Dong and Mason, 2004, van Ekeren et al., 2020, Lam and Lin, 2020, Kawasetsu et al., 2018, Lam and Shimakura, 2015, Lam and Shimakura, 2019, Lam and Shimakura, 2020, Lam and Shimakura, 2020] for the uniqueness). There are at least three uniform proofs of the conjecture: the Leech lattice method of Höhn ([Höhn, 2017, Lam, 2019, Betsumiya et al., 2023]), the generalized deep hole method of Möller–Scheithauer [Möller and Scheithauer, 2023, Möller and Scheithauer, 2021], and the Niemeier lattice method of Höhn–Möller [Höhn and Möller, 2022].

§.§ Höhn's construction of holomorphic CFT of central charge 24

Our construction of hyperbolizations relies heavily on Höhn's argument, which is reviewed below.

Let $\mathbf{Q}_\mathfrak{g}$ be the rescaled coroot lattice (see Notation <ref>). Then $V$ contains the lattice VOA associated with $\mathbf{Q}_\mathfrak{g}$ denoted $V_{\mathbf{Q}_\mathfrak{g}}$.
If we let $W_\mathfrak{g}=\mathrm{Com}_V(V_{\mathbf{Q}_\mathfrak{g}})$ be the commutant (or centralizer),  then the double commutant

\mathrm{Com}_V(W_\mathfrak{g})=\mathrm{Com}_V(\mathrm{Com}_V(V_{\mathbf{Q}_\mathfrak{g}}))

is a lattice VOA extending $V_{\mathbf{Q}_\mathfrak{g}}$. Therefore, there exists an even positive definite lattice $L_\mathfrak{g}\supset \mathbf{Q}_\mathfrak{g}$ such that $\mathrm{Com}_V(W_\mathfrak{g})$ is isomorphic to the lattice VOA $V_{L_\mathfrak{g}}$. It is well known that $V_{L_\mathfrak{g}}$ has group-like fusion: all irreducible $V_{L_\mathfrak{g}}$-modules are simple current modules. In this case, the set of all irreducible modules $R(V_{L_\mathfrak{g}})$ forms an abelian group with respect to the fusion product, and it also carries the quadratic form $\mathfrak{q}: R(V_{L_\mathfrak{g}})\cong L_\mathfrak{g}'/L_\mathfrak{g} \to \QQ/\ZZ$ defined via

\mathfrak{q}(V_{\alpha + L_\mathfrak{g}}) = \mathrm{wt}(V_{\alpha + L_\mathfrak{g}}) = (\alpha,\alpha)/2 \mod \ZZ,

where $\mathrm{wt}(-)$ denotes the conformal weight of the module. The full character of the irreducible module labeled by $\alpha+L_\mathfrak{g} \in L_\mathfrak{g}'/L_\mathfrak{g}$ can be expressed in terms of the Jacobi theta function and $\eta$-function:

\begin{equation}
\chi_{V_{\alpha+L_\mathfrak{g}}}(\tau, \mathfrak{z}) = \frac{\Theta_{L_\mathfrak{g},\alpha}(\tau,\mathfrak{z})}{\eta(\tau)^{\rank(L_\mathfrak{g})}}, \quad (\tau,\mathfrak{z})\in \HH \times (L_\mathfrak{g}\otimes\CC).
\end{equation}

It is also known that $W_\mathfrak{g}$ is strongly rational and also has group-like fusion (see e.g. [Creutzig et al., 2022, Lin, 2017]). The set $R(W_\mathfrak{g})$ of irreducible modules forms a quadratic space isomorphic to $(R(V_{L_\mathfrak{g}}), -\mathfrak{q})$, where the quadratic form is defined by reducing the conformal weight modulo $\ZZ$. Therefore, $V$ is a simple current extension of $W_\mathfrak{g}\otimes V_{L_\mathfrak{g}}$, i.e. there is an isometry $\iota: L_\mathfrak{g}'/L_\mathfrak{g} \to R(W_\mathfrak{g})$ such that

V \cong \bigoplus_{\alpha + L_\mathfrak{g} \in L_\mathfrak{g}'/L_\mathfrak{g}} W_{\iota(\alpha + L_\mathfrak{g})}\otimes V_{\alpha + L_\mathfrak{g}}.

Note that $V_{L_\mathfrak{g}}$ has central charge $\rank(L_\mathfrak{g})=\rank(V_1)=\rank(\mathfrak{g})$, that $W_\mathfrak{g}$ has central charge $24-\rank(\mathfrak{g})$, and that the weight-one subspace of $W_\mathfrak{g}$ is zero. Höhn computed the lattice $L_\mathfrak{g}$ for each of the $69$ semi-simple $\mathfrak{g}$ and called it the orbit lattice of $\mathfrak{g}$.

Let $\mathfrak{h}$ be the Cartan subalgebra of $\mathfrak{g}$. Recall that the full character of $V$,  defined by

\begin{equation}
\chi_{V}(\tau, \mathfrak{z}) = \mathrm{Tr}_V (e^{2\pi i\mathfrak{z}} q^{L_0-1}), \quad q=e^{2\pi i\tau}, \quad (\tau, \mathfrak{z})\in\HH\times \mathfrak{h},
\end{equation}

is a weakly holomorphic Jacobi form of weight $0$ and lattice index $\mathbf{Q}_\mathfrak{g}$ (with trivial character on $\SL_2(\ZZ)$) (see e.g. [Zhu, 1996, Miyamoto, 2000, Krauel and Mason, 2012]). The above construction implies that $\chi_V$ also defines a weakly holomorphic Jacobi form of weight $0$ and lattice index $L_\mathfrak{g}$. Moreover, the character decomposition

\begin{equation}\label{eq:full-theta}
\chi_V(\tau, \mathfrak{z}) = \sum_{\alpha+L_\mathfrak{g} \in L_\mathfrak{g}'/L_\mathfrak{g}} \chi_{V_{\alpha+L_\mathfrak{g}}}(\tau,\mathfrak{z}) \cdot \chi_{W_{\iota(\alpha+L_\mathfrak{g})}}(\tau)
\end{equation}

determines the theta decomposition of $\chi_V$ as a Jacobi form of lattice index $L_\mathfrak{g}$. In particular, the vector-valued function

\begin{equation}\label{eq:character-orbifold}
f_\mathfrak{g}(\tau):= \sum_{\alpha \in L_\mathfrak{g}'/L_\mathfrak{g}} \eta(\tau)^{-\rank(\mathfrak{g})}\cdot\chi_{W_{\iota(\alpha+L_\mathfrak{g})}}(\tau) e_\alpha
\end{equation}

is a weakly holomorphic modular form of weight $-\rank(\mathfrak{g})/2$ for the Weil representation $\rho_{L_\mathfrak{g}}$.

Höhn described $W_\mathfrak{g}$ in an elegant way. Let $\Lambda$ continue to be the Leech lattice and let

\Lambda_g:=(\Lambda^g)^\perp=\{ x\in \Lambda: (x,y)=0 \quad \text{for all $y\in \Lambda$ satisfying $g(y)=y$} \}

be the coinvariant sublattice for $g\in \mathrm{Co}_0=\Orth(\Lambda)$.
Let $\hat{g}\in \mathrm{Aut}(V_\Lambda)$ be a standard lift of $g$; that is, $\hat{g}$ is a lift of $g$ to the Leech lattice VOA that acts trivially on $\Lambda^g$. Höhn [Höhn, 2017] conjectured that $W_\mathfrak{g}$ is isomorphic to an orbifold VOA $V_{\Lambda_g}^{\hat{g}}$ where $[g]$ is one of $11$ particular conjugacy classes $[g]$ of $\mathrm{Co}_0$ (see Tables <ref> and <ref>). This would imply that $V$ is isomorphic to a simple-current extension of $V_{\Lambda_g}^{\hat{g}}\otimes V_{L_\mathfrak{g}}$. Lam [Lam, 2019] proved that the orbifold $V_{\Lambda_g}^{\hat{g}}$ with $\Lambda^g\neq \{0\}$ has group-like fusion and described the corresponding quadratic form explicitly, and in this way was able to confirm Höhn's conjecture. The simple-current extension of $V_{\Lambda_g}^{\hat{g}}\otimes V_{L_\mathfrak{g}}$ depends on the double cosets

\begin{equation}\label{eq:double-coset}
\Orth(L_\mathfrak{g}) \backslash \Orth(R(W_\mathfrak{g}),-\mathfrak{q}) / \mathrm{Aut}(W_\mathfrak{g}).
\end{equation}

By counting the numbers of the above double cosets  denoted $n(L_\mathfrak{g})$, Betsumiya, Lam and Shimakura [Betsumiya et al., 2023] supplemented Höhn's proof of the uniqueness of holomorphic VOAs of central charge $24$ with semi-simple $V_1$.

We will now describe the $69$ simple current extensions explicitly. The construction involves 11 distinct $\mathrm{Co}_0$-conjugacy classes $[g]$, which are characterized by the property that the discriminant form $(R(V_{\Lambda_g}^{\hat{g}}), -\mathfrak{q})$ of signature $\rank(\Lambda^g) \mod 8$ can be realized by an even positive definite lattice of rank $\rank(\Lambda^g)$.
The orbit lattices $L_\mathfrak{g}$ lie in the $11$ associated genera.

* Let $[g]$ be one of the $8$ conjugacy classes in Table <ref>. For each orbit lattice of genus $[g]$, the number of double cosets (<ref>) is always $1$, so the number of inequivalent simple-current extensions equals to the number of classes in the genus. There are in total $58$ classes (including the Leech lattice) in the $8$ genera, so the $8$ conjugacy classes $[g]$ induce $57$ holomorphic VOAs of central charge $24$ with semi-simple $V_1=\mathfrak{g}$.

* Let $[g]$ be the conjugacy class of cycle shape $2^{12}$.  The associated (lattice) genus has $2$ classes $D_{12}(2)$ and $E_8(2)\oplus D_4(2)$ and the corresponding numbers of double cosets are respectively $6$ and $3$. Therefore, this $[g]$ induces $9$ simple-current extensions.

* Let $[g]$ be the conjugacy class of cycle shape $2^3 6^3$.  The associated (lattice) genus has a unique class and the corresponding number of double cosets is $2$. Therefore, this $[g]$ induces $2$ simple current extensions.

* Let $[g]$ be the conjugacy class of cycle shape $2^2 10^2$. The associated (lattice) genus has a unique class and the corresponding number of double cosets is $1$. Therefore, this $[g]$ induces only $1$ simple current extension.

The relevant data appear in Tables <ref> and <ref>.

§.§ Constructing hyperbolizations

Let $V$ be a holomorphic VOA of central charge $24$ with semi-simple $V_1=\mathfrak{g}$. Let $L_\mathfrak{g}$ be Höhn's orbit lattice of $\mathfrak{g}$ and let $\chi_V$ be the full character of $V$. Then the theta lift $\Borch(\chi_V)$ is a reflective Borcherds product of singular weight on $2U\oplus L_\mathfrak{g}$. Moreover, the leading Fourier–Jacobi coefficient of $\Borch(\chi_V)$ coincides with the denominator of the affine Kac–Moody algebra $\hat{\mathfrak{g}}$.

Let $\Delta_\mathfrak{g}$ be the set of roots of $\mathfrak{g}$.
By definition, the Fourier expansion of $\chi_V$ begins

\chi_V(\tau,\mathfrak{z}) = q^{-1} + \sum_{\alpha\in \Delta_\mathfrak{g}} e^{2\pi i\latt{\alpha, \mathfrak{z}}} + \rank(\mathfrak{g}) + O(q),

and all Fourier coefficients of $\chi_V$ are non-negative integers. It follows from Höhn's construction that $\chi_V$ is a weakly holomorphic Jacobi form of weight $0$ and lattice index $L_\mathfrak{g}$. We conclude immediately that $\Borch(\chi_V)$ is a holomorphic Borcherds product of singular weight on $2U\oplus L_\mathfrak{g}$. It remains to show that $\Borch(\chi_V)$ is reflective.

Suppose first that $\mathfrak{g}$ comes from one of the conjugacy classes $[g]$ in Table <ref>. Then the level and order of $g$ are the same (denoted $n_g$ as in Section <ref>). The orbit lattice $L_\mathfrak{g}$ is determined by

U(n_g) \oplus \Lambda^g \cong U\oplus L_\mathfrak{g}.

As mentioned at the end of Section <ref>, it was proved in <cit.> that the twisted denominator of the fake monster algebra associated with $g$ defines a reflective Borcherds product $\Phi_g$ of singular weight on $U(n_g)\oplus U\oplus \Lambda^g$. Möller [Möller, 2021] calculated the characters of the orbifold VOA $V_{\Lambda_g}^{\hat{g}}$, and by comparing Möller's result and <cit.>, it was also proved <cit.> that the input of $\Phi_g$ equals the full character of $V_{\Lambda_g}^{\hat{g}}$ (up to a factor of $\eta^{-\rank(\mathfrak{g})}$) as a vector-valued modular form. By Höhn's construction, the input of $\Phi_g$ is actually $f_\mathfrak{g}$ as defined in Equation (<ref>). We can then use Equation (<ref>) to see that $\Borch(\chi_V)=\Phi_g$; in particular, $\Borch(\chi_V)$ is reflective.

Otherwise, $\mathfrak{g}$ comes from one of the conjugacy classes $[g]$ in Table <ref>, the level of $g$ is twice its order, and the isomorphism $U(n_g) \oplus \Lambda^g \cong U\oplus L_\mathfrak{g}$ does not hold. In fact, $\Borch(\chi_V)$ is not obviously related to $\Phi_g$ in these cases. We will prove that $\Borch(\chi_V)$ is reflective by directly calculating the input forms $\chi_V$. By Höhn's construction, it is sufficient to do this for any semi-simple $\mathfrak{g}$ from a fixed $[g]$, because the input for any other semi-simple Lie algebra from $[g]$ can be expressed as $\sigma(f_\mathfrak{g})$ for some $\sigma \in \Orth(L_\mathfrak{g}'/L_\mathfrak{g})$. The proof by cases is given in Lemmas <ref>, <ref> and <ref> below.

All products $\Borch(\chi_V)$ for $V_1=\mathfrak{g}$ within a fixed conjugacy class $[g]$ define the same modular form on the type IV symmetric domain of dimension $2+\rank(\mathfrak{g})$ up to automorphism. We denote this modular form by $\Psi_g$. The different products $\Borch(\chi_V)$ can be viewed as the distinct Fourier–Jacobi expansions of $\Psi_g$ at different $1$-dimensional cusps represented by a splitting of two hyperbolic planes. Letting $D_g$ denote the discriminant group of $L_\mathfrak{g}$, the automorphism group $\mathrm{Aut}(V_{\Lambda_g}^{\hat{g}})$ equals the subgroup of $\Orth(D_g)$ that fixes $f_\mathfrak{g}$. If $g$ is in Table <ref>, then

\mathrm{Aut}(V_{\Lambda_g}^{\hat{g}})=\Orth(D_g),

and therefore $\Psi_g$ is modular for the full orthogonal group. This property does not hold if $g$ is in Table <ref>.

By the proof of Theorem <ref>, $\Psi_g=\Phi_g$ for $[g]$ in Table <ref>. The eight $\Phi_g$ were first constructed by Borcherds [Borcherds, 1995, Borcherds, 1998] and Scheithauer in [Scheithauer, 2004, Scheithauer, 2009], but they only considered the Fourier expansions at $0$-dimensional cusps. Gritsenko [Gritsenko, 2012] first calculated the $24$ distinct Fourier–Jacobi expansions of $\Phi_g$ for $g$ of cycle shape $1^{24}$ at the distinct $1$-dimensional cusps, which correspond to the $24$ Niemeier lattices. The Fourier–Jacobi expansions of $\Phi_g$ for $g$ of cycle shapes $1^3 7^3$ and $1^4 5^4$ were first determined in [Gritsenko and Wang, 2019]. We also remark that Scheithauer [Scheithauer, 2015] calculated $\Psi_g$ at $0$-dimensional cusps for $g$ of cycle shape $2^{12}$.

Borcherds [Borcherds, 1990, Borcherds, 1992] proved that the BRST cohomology related to the Leech lattice VOA naturally defines the fake monster algebra. It is natural to expect that the BRST cohomology related to any holomorphic VOA of central charge $24$ defines a BKM algebra whose denominator function is $\Borch(\chi_V)$. This type of construction has been realized for holomorphic vertex operator algebras with $V_1$ structures $A_{1,2}^{16}$, $A_{2,3}^8$, $A_{4,5}^2$, $A_{6,7}$ and $B_{12,2}$ by
Creutzig, Höhn, Klauer and Scheithauer [Höhn and Scheithauer, 2003, Creutzig et al., 2007, Höhn and Scheithauer, 2014]. A uniform construction for all vertex operator algebras was given by Driscoll-Spittler in his thesis [Driscoll-Spittler, 2022] under several technical assumptions. This (conditional) natural construction also implies that $\Borch(\chi_V)$ is reflective on $2U\oplus L_\mathfrak{g}$.

Let $\mathcal{G}_\mathfrak{g}$ be the BKM algebra that arises as the BRST cohomology of a holomorphic VOA with $V_1=\mathfrak{g}$. Let $[g]$ be the $\mathrm{Co}_0$-conjugacy class corresponding to $\mathfrak{g}$. Clearly $\mathcal{G}_\mathfrak{g}$ depends only on $[g]$.

* When $[g]$ is in Table <ref>, we have the two isomorphisms

U(n_g) \oplus \Lambda^g \cong U\oplus L_\mathfrak{g} \quad \text{and} \quad U\oplus \Lambda^g \cong (U \oplus L_\mathfrak{g}')(n_g),

since $(\Lambda^g)'(n_g)\cong \Lambda^g$.
This induces the identifications

\begin{equation}\label{eq:groups}
\begin{split}
&\Orth\!\big(U_1(n_g)\oplus U\oplus \Lambda^g\big) \cong  \Orth\!\big(U_1(n_g)\oplus (U\oplus L_\mathfrak{g}')(n_g)\big) \\
=& \Orth\!\big(U_1\oplus U\oplus L_\mathfrak{g}'\big) = \Orth\!\big(U_1\oplus U\oplus L_\mathfrak{g}\big).
\end{split}
\end{equation}

It follows that the Fourier expansion of $\Borch(\chi_V)$ at $U_1$ equals the Fourier expansion of $\Phi_g$ at $U_1(n_g)$. Recall that the latter is also identical to the denominator of the twist $\mathbb{G}_g$ of the fake monster algebra by $g$ (see Section <ref>). Therefore, the denominator of $\mathcal{G}_\mathfrak{g}$ is the same as the denominator of $\mathbb{G}_g$. The root lattice of
$\mathbb{G}_g$ is always $U\oplus \Lambda^g$. From (<ref>)
we conclude that the root lattice of $\mathcal{G}_\mathfrak{g}$ is $U\oplus L_\mathfrak{g}'$, and moreover that the two BKM algebras $\mathcal{G}_\mathfrak{g}$ and $\mathbb{G}_g$ are isomorphic.

* When $[g]$ is in Table <ref>, $\mathcal{G}_\mathfrak{g}$ and $\mathbb{G}_g$ are not isomorphic because their root lattices are not even isomorphic up to scaling. In addition, the Weyl vector of $\mathbb{G}_g$ lies in the root lattice, but the Weyl vector of $\mathcal{G}_\mathfrak{g}$ does not lie in the root lattice (recall that $C$ is half-integral and the Weyl vector is of type $(-C-1,*,-C)$ in this case). For instance, when $g$ has cycle shape $2^{12}$, the root lattices of $\mathcal{G}_\mathfrak{g}$ and $\mathbb{G}_g$ are respectively

U\oplus D_{12}'(1/2) \quad \text{and} \quad  U\oplus D_{12}^+(2).

In particular, we disagree with <cit.>: the denominator there is not a twisted denominator function of the fake monster algebra. We conjecture that the denominator $\Phi_g$ of $\mathbb{G}_g$ is given by the Fourier expansion of $\Psi_g=\Borch(\chi_V)$ at some other $0$-dimensional cusp.

The $\ZZ$-lattice generated by $d\rho$ and the $\lambda=(n,\ell,m)\in U\oplus L_\mathfrak{g}'$ for which $\Borch(\chi_V)$ vanishes on $\lambda^\perp$ is exactly the dual lattice $U\oplus L_\mathfrak{g}'$. This follows from Remark <ref> and a direct calculation of zero divisors (see Lemmas <ref>, <ref> and <ref> below). Here, $\rho$ is the Weyl vector of $\Borch(\chi_V)$ and $d=1$ or $2$ if the corresponding $[g]$ lies in Table <ref> or Table <ref>, respectively.

Let $[g]$ be a $\mathrm{Co}_0$-conjugacy class of level $N_g$ and order $n_g$. Let $M_g$ be an even lattice of signature $(\rank(\Lambda^g)+2,2)$ whose discriminant form is isomorphic to $(R(V_{\Lambda_g}^{\hat{g}}),-\mathfrak{q})$, as determined by Lam [Lam, 2019]. Let $\Phi_g$ be the $g$-twisted denominator of the fake monster algebra as before, and recall that $\Phi_g$ is a holomorphic Borcherds product of singular weight on $U(N_g)\oplus U\oplus \Lambda^g$ (see <cit.>). We have the following:

* If $N_g=n_g$, then $M_g\cong U(N_g)\oplus U\oplus \Lambda^g$ (see <cit.>) and $\Phi_g$ is reflective on $M_g$ (see <cit.>).

* If $N_g\neq n_g$, then $M_g$ is not isomorphic to $U(N_g)\oplus U\oplus \Lambda^g$ and $\Phi_g$ is not reflective on the lattice $U(N_g)\oplus U\oplus \Lambda^g$. Motivated by Höhn's construction and the discussions above,  we conjecture that the vector-valued characters of $V_{\Lambda_g}^{\hat{g}}$ (divided by $\eta^{\rank(\Lambda^g)}$) can be lifted to a reflective Borcherds product of singular weight on $M_g$. Obviously, the theta lift defines a holomorphic Borcherds product of singular weight on $M_g$. However, we have to compute the conformal weights to confirm that it is reflective.  We further conjecture that the Fourier expansion of this product at a certain $0$-dimensional cusp recovers the $g$-twisted denominator of the fake monster algebra. In other words, $\Phi_g$ defines a reflective Borcherds product on $M_g$, i.e. $M_g$ plays the role of the lattice $\mathbb{M}$ in <cit.>.

We can now complete the proof of Theorem <ref>. By (<ref>), we can express $\chi_V$ as an $\NN$-linear combination of characters of the affine VOA generated by $V_1=\mathfrak{g}$ which have integral conformal weight. Expressions of this type have been determined by Schellekens [Schellekens, 1993]. We will compute the characters $\chi_V$ for $V_1=B_{12,2}$, $F_{4,6}A_{2,2}$ and $C_{4,10}$. In these cases, the index $[L_\mathfrak{g}:\mathbf{Q}_\mathfrak{g}]$ is $1$ or $2$.  To determine the divisor of $\Borch(\chi_V)$, it suffices to compute the singular Fourier coefficients of $\chi_V$, and by Remark <ref> it is enough to consider Fourier coefficients of the form

\begin{equation}\label{eq:check-list}
f(n,\ell)q^n\zeta^\ell, \quad n\leq \hat{\delta}_{L_\mathfrak{g}}, \; \ell \in \mathbf{Q}_\mathfrak{g}', \; 2n<(\ell,\ell).
\end{equation}

Note that the above $f(n,\ell)$ are $0$ or $1$ because $f(n,\ell)\geq 0$ and $\Borch(\chi_V)$ has only simple zeros, hence $\Borch(\chi_V)$ is reflective on $2U\oplus L_\mathfrak{g}$ as soon as we can show that for every nonzero Fourier coefficient of form (<ref>) there exists a positive integer $t$ such that $(\ell,\ell)-2n=2/t$ and $t\ell \in L_\mathfrak{g}$.

In the three lemmas below, we will express the Fourier expansion of $\chi_V$ in terms of Weyl orbits. Suppose $\mathfrak{g}=\mathfrak{g}_{1,k_1}\oplus \mathfrak{g}_{2,k_2}$, where $\mathfrak{g}_{2,k_2}$ may be zero, and let $W_{j}$ be the Weyl group of $\mathfrak{g}_j$. For a dominant integral weight $\lambda_j = \sum_{i} x_i w_i$ of $\mathfrak{g}_j$, we define

O_{\lambda_j, n_j} = \sum_{\ell \in W_{j}\cdot\lambda_j} e^{2\pi i\latt{\ell, \mathfrak{z}}}, \quad \mathfrak{z} \in L_\mathfrak{g}\otimes\CC,

where $n_j=\latt{\lambda_j,\lambda_j}/k_j$. Recall (from Notation <ref>) that

\mathbf{Q}_\mathfrak{g} = Q_1^\vee(k_1)\oplus Q_2^\vee(k_2), \quad  \mathbf{Q}_\mathfrak{g}' = P_1(1/k_1)\oplus P_2(1/k_2)

\mathbf{Q}_\mathfrak{g}< L_\mathfrak{g} < L_\mathfrak{g}' < \mathbf{Q}_\mathfrak{g}'.

When we view $O_{\lambda_j, n_j}$ as part of the Fourier series of $\chi_V$, the vector $\ell \in W_j\cdot \lambda_j \subsetneq \bQ'$ is identified with $\ell / k_j$ and its norm is $(\ell,\ell)=k_j\latt{\ell/k_j,\ell/k_j}=n_j$. Therefore, $q^n \cdot O_{\lambda_1, n_1}\otimes O_{\lambda_2, n_2}$ induces reflective zeros if and only if there exists a positive integer $t$ such that

n_1+n_2-2n=2/t \quad \text{and} \quad t(\lambda_1/k_1 + \lambda_2/k_2)\in L_\mathfrak{g}.

For simplicity, we set

O_{\lambda_1,\lambda_2, n_1+n_2}=O_{\lambda_1,n_1}\otimes O_{\lambda_2,n_2}.

When $V_1=\mathfrak{g}$ is of type $B_{12,2}$, the product $\Borch(\chi_V)$ is reflective on $2U\oplus D_{12}(2)$.

$\chi_V$ can be expressed in terms of characters of the affine VOA generated by $\mathfrak{g}$ (using the notation of (<ref>)) as

\chi_V=\chi^{B_{12,2}}_{0,0}+\chi^{B_{12,2}}_{w_1+w_{12},2}+\chi^{B_{12,2}}_{w_{10},3}+\chi^{B_{12,2}}_{w_5,2}.

In this case, $\mathbf{Q}_\mathfrak{g}=L_\mathfrak{g}=D_{12}(2)$. Note that $\delta_{L}=6$ (see <cit.>), so we only need to calculate  $\chi_V$ up to its $q^2$-term.  We first use SAGE to compute the Fourier coefficients of $\chi^{B_{12,2}}$ as representations of the simple Lie algebra of type $B_{12}$, and then we decompose those representations into Weyl orbits as defined above. We find

\chi_V=q^{-1}+(O_{w_2,1}+O_{w_1,\frac12}+12)+\sum_{i=1}^\infty c_iq^i,

\begin{align*}
&+O_{w_3+w_{12},\frac92}+O_{w_1+w_{6},\frac92}+4 O_{w_8,4}+12 O_{2w_2 ,4}+4 O_{w_1 +w_5,4}+12 O_{w_7,\frac{7}{2}}\\
&+12 O_{w_2+w_{12},\frac{7}{2}}+12 O_{w_1+w_4,\frac{7}{2}}+32 O_{w_6,3}+44 O_{w_1+w_3,3}+90 O_{w_5,\frac{5}{2}}\\
&+90 O_{w_1 +w_{12},\frac{5}{2}}+90 O_{w_1 +w_{2},\frac{5}{2}}+224 O_{w_4,2}+288 O_{2w_1,2}\\
&+520 O_{w_{12},\frac{3}{2}}+520 O_{w_3,\frac{3}{2}}
+1242 O_{w_2,1}
+2535 O_{w_1,\frac{1}{2}}
\end{align*}

The proof follows by verifying that every singular Weyl orbit is reflective. As an example, consider the orbit $q^2\cdot O_{w_9,9/2}$. We write $w_9=(1,1,1,1,1,1,1,1,1,0,0,0)$ as coordinates in the simple roots as in [Bourbaki, 1981]. Then $9/2 - 2\times 2=1/2=2/4$ and $4\cdot(w_9/2)=2w_9 \in D_{12}(2)$, so $q^2\cdot O_{w_9,9/2}$ determines reflective divisors.

When $V_1=\mathfrak{g}$ is of type $A_{2,2}F_{4,6}$, the product $\Borch(\chi_V)$ is reflective on $2U\oplus D_4(6)\oplus A_2(2)$.

We express $\chi_V$ as the linear combination

\begin{align*}
&\Big(\chi^{F_{4,10}}_{0000,0} + \chi^{F_{4,10}}_{0004,2} + \chi^{F_{4,10}}_{0030,3} + \chi^{F_{4,10}}_{1100,2}\Big)\otimes\chi^{A_{2,2}}_{00,0}\\
+&\Big(\chi^{F_{4,10}}_{0101,\frac{26}{15}} + \chi^{F_{4,10}}_{1012,\frac{41}{15}}\Big)\otimes\Big( \chi^{A_{2,2}}_{10,\frac{4}{15} }+ \chi^{A_{2,2}}_{01,\frac{4}{15}}\Big) \\
+&\Big(\chi^{F_{4,10}}_{0003,\frac75} + \chi^{F_{4,10}}_{0006,\frac{17}{5}} + \chi^{F_{4,10}}_{0021,\frac{12}{5}} + \chi^{F_{4,10}}_{2010,\frac{12}{5}}\Big)\otimes\chi^{A_{2,2}}_{11,\frac35}\\
+& \Big(\chi^{F_{4,10}}_{0102,\frac73} + \chi^{F_{4,10}}_{2000,\frac43}\Big)\otimes\Big(\chi^{A_{2,2}}_{02,\frac23} + \chi^{A_{2,2}}_{20,\frac23}\Big).
\end{align*}

Note that

\mathbf{Q}_\mathfrak{g}=L_\mathfrak{g}=A_2(2)\oplus D_4(6).

Since $\delta_L = 22/3$, we only need to calculate  $\chi_V$ up to its $q^3$-term. We write

\chi_V=q^{-1}+ \big(O_{11,0000,1}+O_{00,1000,1}+O_{10,0000,\frac{1}{3}}+O_{01,0000,\frac{1}{3}}+O_{00,0001,\frac{1}{2}}+6\big)+ \sum_{i=1}^\infty c_iq^i

and list the singular Weyl orbits in $c_i$ for $i\leq 3$. There are $17$ orbits with $\mathrm{norm}>2$ in $c_1$:

\begin{align*}
& O_{22,0000,4},O_{03,0000,3},O_{30,0000,3},O_{00,0004,\frac{8}{3}},O_{02,2000,\frac{8}{3}},O_{20,2000,\frac{8}{3}}, \\
& O_{11,0003,\frac{5}{2}},O_{00,1100,\frac{7}{3}},O_{02,0100,\frac{7}{3}}, O_{20,0100,\frac{7}{3}},O_{00,0012,\frac{13}{6}},O_{00,2001,\frac{13}{6}},\\
& O_{01,0101,\frac{13}{6}},O_{02,1001,\frac{13}{6}},O_{10,0101,\frac{13}{6}},O_{11,0011,\frac{13}{6}},O_{20,1001,\frac{13}{6}} .
\end{align*}

There are $26$ orbits with $\mathrm{norm}>4$ in $c_2$:

\begin{align*}
&O_{20,0110,\frac{25}{6}},O_{20,1003,\frac{25}{6}},O_{21,0101,\frac{25}{6}},O_{22,0001,\frac{25}{6}},O_{30,0011,\frac{25}{6}} .
\end{align*}

There are $41$ orbits with $\textrm{norm}>6$ in $c_3$:

\begin{align*}
\end{align*}

We verify that each of these orbits is reflective by direct computation.

When $V_1=\mathfrak{g}$ is of type $C_{4,10}$, the product $\Borch(\chi_V)$ is reflective on $2U\oplus D_{4}(10)$.

We have the expression

\begin{align*}
\chi_V
=\sum_{1}&\Big(\chi^{C_{4,10}}_{0000,0} + \chi^{C_{4,10}}_{0024,4} + \chi^{C_{4,10}}_{0040,2} + \chi^{C_{4,10}}_{0044,6} + \chi^{C_{4,10}}_{00,10,0,8} + \chi^{C_{4,10}}_{0260,5} \\[-2mm]
&+ \chi^{C_{4,10}}_{0321,3}
+\chi^{C_{4,10}}_{0323,5} + \chi^{C_{4,10}}_{0500,2} + \chi^{C_{4,10}}_{0800,4}+ \chi^{C_{4,10}}_{1051,4} + \chi^{C_{4,10}}_{1430,4} \\
&+ \chi^{C_{4,10}}_{1431 ,5}
+\chi^{C_{4,10}}_{2242,6} + \chi^{C_{4,10}}_{3031,3} + \chi^{C_{4,10}}_{4140,4}\Big)+2 \chi^{C_{4,10}}_{ 2222,4},
\end{align*}

where the glue vector $1$ exchanges the affine Dynkin labels $\hat{w}_0,\hat{w}_1,\hat{w}_2,\hat{w}_3,\hat{w}_4$ and $\hat{w}_4,\hat{w}_3,\hat{w}_2,\hat{w}_1,\hat{w}_0$ (such that e.g. $\chi^{C_{4,10}}_{0000,0}$ becomes $\chi^{C_{4,10}}_{000,10,10}$ and $\chi^{C_{4,10}}_{0024,4}$ becomes $\chi^{C_{4,10}}_{2004,3}$; see (<ref>) and the explanations there). In this case, we have

\mathbf{Q}_\mathfrak{g}=\ZZ^4(20) \quad  \text{and}  \quad L_\mathfrak{g}=D_4'(20)\cong D_4(10).

Since $\delta_L = 10$, we only need to calculate  $\chi_V$ up to its $q^4$-term. We write

\chi_V=q^{-1}+\big(O_{2 0 0 0,\frac{1}{5}}+ O_{0 1 0 0,\frac{1}{10}}+4\big)+\sum_{i=1}^\infty c_i q^i

and find that $c_1$ equals

\begin{align*}
&O_{0 5 0 0,\frac{5}{2}}+O_{0 0 4 0,\frac{12}{5}}+O_{4 0 0 2,\frac{12}{5}}+O_{4 0 2 0,\frac{11}{5}}+O_{0 1 2 1,\frac{21}{10}}+O_{1 3 1 0,\frac{21}{10}}+O_{2 1 0 2,\frac{21}{10}}\\
+&O_{4 1 0 1,\frac{21}{10}}+2 O_{0 2 0 2,2}+2 O_{4 2 0 0,2}+2 O_{1 0 1 2,\frac{19}{10}}+2 O_{2 1 2 0,\frac{19}{10}}+2 O_{5 0 1 0,\frac{19}{10}}+2 O_{0 0 0 3,\frac{9}{5}}\\
+&4 O_{0 2 2 0,\frac{9}{5}}+2 O_{2 2 0 1,\frac{9}{5}}+4 O_{6 0 0 0,\frac{9}{5}}+5 O_{0 3 0 1,\frac{17}{10}}+5 O_{1 0 3 0,\frac{17}{10}}+5 O_{2 3 0 0,\frac{17}{10}}+5 O_{3 0 1 1,\frac{17}{10}}\\
+&10 O_{0 4 0 0,\frac{8}{5}}+10 O_{1 1 1 1,\frac{3}{2}}+10 O_{3 1 1 0,\frac{3}{2}}+12 O_{0 0 2 1,\frac{7}{5}}+17 O_{2 0 0 2,\frac{7}{5}}+12 O_{4 0 0 1,\frac{7}{5}}\\
+&20 O_{0 1 0 2,\frac{13}{10}}+20 O_{1 2 1 0,\frac{13}{10}}+20 O_{4 1 0 0,\frac{13}{10}}+32 O_{2 0 2 0,\frac{6}{5}}+38 O_{0 1 2 0,\frac{11}{10}}+38 O_{2 1 0 1,\frac{11}{10}}\\
+&46 O_{0 2 0 11}+56 O_{2 2 0 01}+69 O_{0 3 0 0,\frac{9}{10}}+69 O_{1 0 1 1,\frac{9}{10}}+69 O_{3 0 1 0,\frac{9}{10}}+101 O_{0 0 0 2,\frac{4}{5}}\\
+&101 O_{4 0 0 0,\frac{4}{5}}+120 O_{1 1 1 0,\frac{7}{10}}+168 O_{0 0 2 0,\frac{3}{5}}+148 O_{2 0 0 1,\frac{3}{5}}+205 O_{0 1 0 1,\frac{1}{2}}+205 O_{2 1 0 0,\frac{1}{2}}\\
+&280 O_{0 2 0 0,\frac{2}{5}}+340 O_{1 0 1 0,\frac{3}{10}}+418 O_{0 0 0 1,\frac{1}{5}}+456 O_{2 0 0 0,\frac{1}{5}}+558 O_{0 1 0 0,\frac{1}{10}}+748 .
\end{align*}

There are $10$ orbits with $\textrm{norm}>4$ in $c_2$:

\begin{align*}
&O_{10, 0 0 0,5},O_{2 0 0 4,\frac{21}{5}},O_{4 2 2 0,\frac{21}{5}},O_{0 1 0 4,\frac{41}{10}},O_{0 3 2 1,\frac{41}{10}},\\
&O_{2 3 0 2,\frac{41}{10}},O_{3 0 3 1,\frac{41}{10}},O_{4 3 0 1,\frac{41}{10}},O_{5 0 3 0,\frac{41}{10}},O_{8 1 0 0,\frac{41}{10}}.
\end{align*}

There are $11$ orbits with $\textrm{norm}>6$ in $c_3$:

\begin{align*}
&O_{0 8 0 0,\frac{32}{5}},O_{0 0 2 4,\frac{31}{5}},O_{6 2 0 2,\frac{31}{5}},O_{0 1 0 5,\frac{61}{10}},O_{10, 1 0 0,\frac{61}{10}},O_{1 0 5 1,\frac{61}{10}},\\
&O_{1 4 3 0,\frac{61}{10}},O_{3 4 1 1,\frac{61}{10}},O_{4 1 4 0,\frac{61}{10}},O_{5 0 1 3,\frac{61}{10}},O_{7 0 1 2,\frac{61}{10}}.
\end{align*}

There are $16$ orbits with $\textrm{norm}>8$ in $c_4$:

\begin{align*}
&O_{0 2 6 0,\frac{41}{5}},O_{2 8 0 0,\frac{41}{5}},O_{8 2 2 0,\frac{41}{5}},O_{0 3 2 3,\frac{81}{10}},O_{0 9 0 0,\frac{81}{10}},O_{1 0 7 0,\frac{81}{10}},O_{1 4 3 1,\frac{81}{10}},O_{2 3 0 4,\frac{81}{10}},\\
&O_{3 0 3 3,\frac{81}{10}},O_{3 4 1 2,\frac{81}{10}},O_{4 1 4 1,\frac{81}{10}},O_{5 4 1 1,\frac{81}{10}},O_{6 1 4 0,\frac{81}{10}},O_{7 0 1 3,\frac{81}{10}},O_{8 3 0 1,\frac{81}{10}},O_{9 0 3 0,\frac{81}{10}}.
\end{align*}

To finish the proof, we verify that the required condition on the orders of vectors in the singular Weyl orbits is satisfied. We will work out $q^1 \cdot O_{4002,12/5}$ as an example. The associated dominant weight is

\lambda=4w_1+2w_4=(6,2,2,2)

in coordinates as in [Bourbaki, 1981]. The corresponding vector in $L_\mathfrak{g}'=D_4(1/20)$ is $\lambda / 20$, which has norm $20\times (12/100)=12/5$, and we have $12/5 - 2\times 1 =2/5$. The order of $\lambda/20$ modulo $\mathbf{Q}_\mathfrak{g}=\ZZ^4(20)$ is $10$, but its order modulo $L_\mathfrak{g}=D_4'(20)$ is $5$. Hence the induced zero divisor is reflective on $2U\oplus L_\mathfrak{g}$ although it is not reflective on $2U\oplus \bQ$.

§ THE SYMMETRIC CASE: HOLOMORPHIC SCFTS OF CENTRAL CHARGE 12

In this section, we construct hyperbolizations of affine Kac–Moody algebras $\hat{\mathfrak{g}}$ for the eight $\mathfrak{g}$ of symmetric type with $C=1$ in Theorem <ref>. These semi-simple $\mathfrak{g}$ first appeared in [Dittmann and Wang, 2022], where Dittmann and the second named author proved that the additive lifts of the denominators of these $\hat{\mathfrak{g}}$ are reflective Borcherds products of singular weight on the maximal even sublattices of the (integral) lattices $\mathbf{P}_\mathfrak{g}$. Shortly after [Dittmann and Wang, 2022] appeared on arXiv, Harrison, Paquette, Persson and Volpato [Harrison et al., 2021] described these $\mathfrak{g}$ as the $\mathcal{N}=1$ structures of holomorphic SCFT $F_{24}$ of central charge $12$, and gave a natural construction of a certain BKM superalgebra (denoted $\mathcal{G}_\mathfrak{g}$) as the BRST cohomology  related to $F_{24}$ with a fixed $\mathcal{N}=1$ structure under some technical assumptions. They also determined the super-denominators of these BKM superalgebras.

We will prove that these super-denominators are actually the reflective Borcherds products of singular weight constructed in [Dittmann and Wang, 2022]. We will also describe the connection between these BKM superalgebras and the twists of the fake monster algebra, which resolves some open questions proposed in [Harrison et al., 2021]. Additionally, we express the inputs of these Borcherds products as $\ZZ$-linear combinations of the full characters of the affine VOAs generated by $\mathfrak{g}$, and construct some exceptional modular invariants of $\hat{\mathfrak{g}}$ as an application.

§.§ Holomorphic SCFTs of central charge 12 composed of 24 chiral fermions

Holomorphic vertex operator superalgebras (SVOA) of central charge $12$ were classified by Creutzig, Duncan and Riedler [Creutzig et al., 2018], and they fall into three types: the Conway SCFT $V^{f\natural}$ with trivial weight-$1/2$ subspace, the theory $V^{fE_8}$ of $8$ chiral bosons and $8$ chiral fermions, and the theory $F_{24}$ of $24$ chiral fermions. The hyperbolizations of affine Kac–Moody algebras are related to $F_{24}$. We review the theory of $F_{24}$ following [Harrison et al., 2021].

$F_{24}$ is constructed from the lattice VOA associated with the $D_{12}$ lattice

D_{12}=\left\{ (x_1,x_2,...,x_{12})\in\ZZ^{12} : \quad x_1+x_2+...+x_{12} \in 2\ZZ \right\}.

The lattice VOA $V_{D_{12}}$ has $4$ irreducible modules labeled by the $4$ cosets of $D_{12}'/D_{12}$,

0,\quad  v=(1,0,..,0),\quad  s=\Big(\frac{1}{2},...,\frac{1}{2},\frac{1}{2}\Big) \quad \text{and} \quad c=\Big(\frac{1}{2},...,\frac{1}{2},-\frac{1}{2}\Big).

The four modules $V_0$, $V_v$, $V_s$ and $V_c$ have conformal weights $0$, $1/2$, $3/2$ and $3/2$, respectively. Their full characters can be expressed as the quotients of Jacobi theta functions by $\eta^{12}$:

\chi_*(\tau,z) = \frac{\Theta_{D_{12},*}(\tau, z)}{\eta(\tau)^{12}}, \quad *= 0, v, s, c, \quad z\in D_{12}\otimes\CC.

The sum $V_0\oplus V_v$ is the $\ZZ_2$-graded superspace of $F_{24}$ (i.e. the $\rm NS$ sector in 2d SCFTs). Let $(-1)^F$ be the canonical involution, such that

(-1)^F|_{V_0}=I \quad  \text{and} \quad (-1)^F|_{V_v}=-I.

The SVOA $F_{24}$ has a unique irreducible $(-1)^F$-stable canonically twisted module which is given by $V_s\oplus V_c$ (i.e. the $\rm R$ sector in 2d SCFTs). The weight-$1/2$ subspace $W_{1/2}$ of $F_{24}$ has a natural semi-simple Lie algebra structure, which in physics is called the $\mathcal{N}=1$ superconformal structure. We denote this Lie algebra by $\mathfrak{g}$ and its Cartan subalgebra by $\mathfrak{h}$. For $\mathfrak{z}\in \mathfrak{h}$, the full characters of $F_{24}$ are defined by

\begin{align*}
\chi_{\mathrm{NS}}\big(\tau, \mathfrak{z}\big) &= \mathrm{Tr}_{\mathrm{NS}}\big(q^{L_0-\frac{1}{2}}e^{2\pi i\mathfrak{z}}\big),& \chi_{\widetilde{\mathrm{NS}}}\big(\tau, \mathfrak{z}\big) &= \mathrm{Tr}_{\mathrm{NS}}\big(q^{L_0-\frac{1}{2}}e^{2\pi i\mathfrak{z}}(-1)^F\big),& \\
\chi_{\mathrm{R}}(\tau, \mathfrak{z}) &= \mathrm{Tr}_{\mathrm{R}}\big(q^{L_0-\frac{1}{2}}e^{2\pi i\mathfrak{z}}\big),& \chi_{\widetilde{\mathrm{R}}}(\tau, \mathfrak{z}) &= \mathrm{Tr}_{\mathrm{R}}\big(q^{L_0-\frac{1}{2}}e^{2\pi i\mathfrak{z}}(-1)^F\big).&
\end{align*}

The restriction of the unique normalized invariant bilinear form of $F_{24}$ to a simple ideal $\mathfrak{g}_j$ of $\mathfrak{g}$ satisfies $(-,-)=k_j\latt{-,-}$, where $\latt{-,-}$ is the standard bilinear form of $\mathfrak{g}_j$ as before (see [Harrison et al., 2021]). We indicate these integers by writing

\mathfrak{g}=\mathfrak{g}_{1,k_1}\oplus \cdots \oplus \mathfrak{g}_{s,k_s}.

The semi-simple Lie algebra $\mathfrak{g}$ is of dimension $24$. There are exactly eight distinct possibilities for $\mathfrak{g}$ and they are characterized by the identity (see Notation <ref>)

1=\frac{\dim \mathfrak{g}}{24} = \frac{h_j^\vee}{k_j}, \quad 1\leq j\leq s.

The affine VOA $V_\mathfrak{g}$ generated by $W_{1/2}=\mathfrak{g}$ is a full sub-VOA of $F_{24}$.  Let $\Delta_\mathfrak{g}^+$ denote the set of positive roots of $\mathfrak{g}$. We have the following embedding of integral lattices

\iota_\mathfrak{g}:\quad  \mathbf{P}_\mathfrak{g}:=\bigoplus_{j=1}^s P_{j}^\vee(h_j^\vee) \hookrightarrow \ZZ^{12}, \quad
\mathfrak{z} \mapsto \iota_\mathfrak{g}(\mathfrak{z})=\{\latt{\mathfrak{z},\alpha} \}_{\alpha \in \Delta_\mathfrak{g}^+},

which induces an embedding of the maximal even sublattice $L_\mathfrak{g}$ of $\mathbf{P}_\mathfrak{g}$ into $D_{12}$. From the conformal embedding $V_\mathfrak{g} \hookrightarrow F_{24}$ we deduce that the full characters of $F_{24}$ with $W_{1/2}=\mathfrak{g}$ can be expressed as

\begin{align*}
\chi_{\mathrm{NS}}(\tau, \mathfrak{z}) &= \chi_0 (\tau, \iota_\mathfrak{g}(\mathfrak{z})) + \chi_v (\tau, \iota_\mathfrak{g}(\mathfrak{z})),\\
\chi_{\widetilde{\mathrm{NS}}}(\tau, \mathfrak{z}) &= \chi_0 (\tau, \iota_\mathfrak{g}(\mathfrak{z})) - \chi_v (\tau, \iota_\mathfrak{g}(\mathfrak{z})),\\
\chi_{\mathrm{R}}(\tau, \mathfrak{z}) &= \chi_s (\tau, \iota_\mathfrak{g}(\mathfrak{z})) + \chi_c (\tau, \iota_\mathfrak{g}(\mathfrak{z})),\\
\chi_{\widetilde{\mathrm{R}}}(\tau, \mathfrak{z}) &=\chi_s (\tau, \iota_\mathfrak{g}(\mathfrak{z})) - \chi_c (\tau, \iota_\mathfrak{g}(\mathfrak{z})) \equiv 0,
\end{align*}

where the last equality follows from the fact that $\iota(L_\mathfrak{g}) \hookrightarrow D_{10}$.
It is easy to check that, up to scaling,

\begin{equation}\label{eq:phig}
\begin{split}
\phi_\mathfrak{g}(\tau, \mathfrak{z}) &= \frac{1}{2}\Big( \chi_{\mathrm{NS}}(\tau, \mathfrak{z}) - \chi_{\widetilde{\mathrm{NS}}}(\tau, \mathfrak{z}) - \chi_{\mathrm{R}}(\tau, \mathfrak{z}) \Big) \\
&= \chi_v (\tau, \iota_\mathfrak{g}(\mathfrak{z})) - \chi_c (\tau, \iota_\mathfrak{g}(\mathfrak{z}))   \\
&= \rank(\mathfrak{g}) + \sum_{\alpha \in \Delta_\mathfrak{g}^+}\Big( e^{2\pi i\latt{\alpha, \mathfrak{z}}} + e^{-2\pi i\latt{\alpha, \mathfrak{z}}} \Big) + O(q)
\end{split}
\end{equation}