$\widehat{sl(2)}$ decomposition of denominator formulae of some BKM Lie superalgebras

We study a family of Siegel modular forms that are constructed using Jacobi forms that arise in Umbral moonshine. All but one of them arise as the Weyl-Kac-Borcherds denominator formula of some Borcherds-Kac-Moody (BKM) Lie superalgebras. These Lie superalgebras have a $\widehat{sl(2)}$ subalgebra which we use to study the Siegel modular forms. We show that the expansion of the Umbral Jacobi forms in terms of $\widehat{sl(2)}$ characters leads to vector-valued modular forms. We obtain closed formulae for these vector-valued modular forms. In the Lie algebraic context, the Fourier coefficients of these vector-valued modular forms are related to multiplicities of roots appearing on the sum side of the Weyl-Kac-Borcherds denominator formulae.


Introduction
The use of automorphic forms in the study of Lie algebras first arose in the work of Macdonald. He associated Jacobi forms with denominator formulae for affine Lie algebras and determined the multiplicity of positive roots using this connection [1,2]. For instance, for sl (2), the product side of the denominator formula is given by 2 s 1/2 ϑ 1 (τ, z) = s 1/2 q 1/8 r 1/2 (1.1) Here we make the identifications: e −α 1 ∼ qr, e −α 2 ∼ r −1 and e −δ = e −α 1 −α 2 ∼ q.
The right hand side of the above equation then reads (with e −̺ ∼ s 1/2 q 1/8 r 1/2 ) where ̺ is the Weyl vector and the set of positive roots L + is This product formula shows that one needs to include imaginary roots, of type mδ, in the the set of positive roots. Borcherds extended this by including situations where imaginary simple roots also appear [3]. This leads to modifications on the sum side of the denominator formulae to account for such simple roots. Today, such Lie algebras are called Borcherds-Kac-Moody (BKM) Lie algebras. Again, automorphic forms play an important role in determining the sum and product side of the denominator formulae which we call the Weyl-Kac-Borcherds denominator formulae. Schematically, one has In the above formula, an automorphic form ∆ is written as a sum and a product. Further, T is the Borcherds correction factor defined in Eq. (B.2), L + corresponds to the set of positive roots and mult(α) is the multiplicity of the root α. The addition of fermionic roots leads to superalgebras and we shall focus on superdenominator formulae where a similar correspondence described above goes through.
This paper studies the connection between five genus two Siegel modular forms (defined in Eq. (2.16)) ∆ k(N ) (Z) , for N = 1, 2, 3, 4, 6 and k(N) = (6/N) − 1 , (1.2) and the BKM Lie superalgebras associated with five rank-three Cartan matrices A (N ) . The Cartan matrices are the inner products of the bosonic real simple roots and are of rank three. The Cartan matrices are given by where λ N is any solution of the quadratic equation For N = 1, 2, 3, the matrices are finite-dimensional with the indices n, m defined modulo 3, 4, 6 respectively. For N = 4, 6, the matrices are infinite-dimensional with m, n ∈ Z. For N = 4, the Cartan matrix has to be obtained as a limit N → 4 leading to a nm = 2 − 4(n − m) 2 . The square of these modular forms appear as the generating function of the refined counting of quarter-BPS states in certain CHL Z N orbifolds of the heterotic string compactified on T 6 [4,5]. Further, the root lattices associated with these Cartan matrices are closely related to the walls of marginal stability where quarter BPS states decay [6][7][8][9][10].
Let g(A (N ) ) denote the Kac-Moody algebra associated with the Cartan matrix A (N ) [11]. For N = 6, the Siegel modular form arises as the Weyl-Kac-Borcherds superdenominator formula for a Borcherds-Kac-Moody (BKM) Lie superalgebra that is a Borcherds extension of g(A (N ) ) by the addition of imaginary simple roots [12,13]. Let us denote this BKM Lie superalgebra by B(A (N ) ). For N = 6, let B(A (6) ) denote an as yet unknown Lie superalgebra whose WKB superdenominator formula is given by ∆ k(6) (Z). All five Siegel modular forms admit a product formula of the form [5] (see Eq. (2.16)) ∆ k(N ) (Z) = e −2πiTr(̺ (N) Z) where L + is the set of positive roots implicitly determined by the product formula and ̺ (N ) is the Weyl vector that satisfies ̺ (N ) , α = −1 for all simple real roots α. The multiplicities m(α) are determined by the Fourier-Jacobi coefficients of a weight zero, index N, Jacobi form that appears in the context of umbral moonshine [14].
These BKM Lie superalgebras for N = 1, 2, 3, 4 naturally fit with the study of Lorentzian Kac-Moody Lie superalgebras associated with rank-three Cartan matrices by Gritsenko and Nikulin [15]. Gritsenko and Nikulin show that there exists no BKM Lie superalgebras associated with hyperbolic root systems with Weyl vector of hyperbolic type. The root lattice associated with A (6) is of this type and hence B(A (6) ) cannot be a BKM Lie superalgebra. Our long-term goal is construct the Lie superalgebra B(A (6) ), if it is exists, whose WKB superdenominator formula is given by the Siegel modular form for N = 6. Clearly some additional inputs beyond the Borcherds correction term is needed. We do not solve this problem here but take the first step towards this by decomposing the Siegel modular forms in terms of a sl(2) subalgebra of g(A (N ) ).
Feingold and Frenkel study a Lorentzian Kac-Moody Lie algebra associated with a rank three Cartan matrix using a sl(2)-subalgebra [16]. Inspired by this, we study all the five modular forms in terms of a sl(2) subalgebra present in all five examples. This enables us to carry out a systematic decomposition of the Seigel modular forms in terms of characters of sl(2) and its Borcherds extension that we denote by B N ( sl (2)) -this is a sub-algebra of B(A (N ) ) for N = 6.
A summary of the main results of this paper is as follows.
1. We decompose the Umbral Jacobi forms in terms of characters of a sl (2) Lie algebra. This leads to vector valued modular forms (vvmfs) with welldefined modular properties that we determine.
2. For N ≤ 4, we show that these vvmfs arise as solutions to a matrix differential equation proposed by Gannon [17]. For N = 6, we obtain a closed formula for the Fourier coefficients of the vvmf using a different method.
3. We re-express the decomposition of the Umbral Jacobi form in terms of B N ( sl(2)) characters and assign weights using roots of B(A (N ) ). Using the covariance properties of the Siegel modular forms, we are able identify Weyl orbits for each of the weights that appear in the Lie algebraic decomposition. This provides a preliminary insight into rewriting the Siegel modular forms as sums of Weyl orbits.
The organisation of the paper is as follows. Following the introductory section, in section 2, we introduce the Lorentzian rank three root lattices, their Weyl group as well as the Siegel modular forms that are potential WKB denominator formula for Lie superalgebras. The Siegel modular forms are constructed using a product formula due to Gritsenko-Nikulin with Umbral Jacobi forms as input. In section 3, we introduce the embedding of sl(2) as a subalgebra of the g(A (N ) ) Lie algebras. This naturally leads to the decomposition of the Umbral Jacobi forms in terms of sl(2) characters whose coefficients form a vector-valued modular form (vvmf). We determine the first few terms in the Fourier expansion of the vvmf on a computer. In section 4, we obtain more explicit formulae for the vvmf using the approach of Gannon by constructing them as solutions to a modular differential equation. This approach did not work of N = 6. In section 4.2, we use another method to get an explicit formula for the vvmf. In section 4.3, we rewrite parts of the Siegel modular form in terms of root vectors that appear in the decomposition of the Umbral Jacobi form. We conclude in section 5 with some remarks. Two appendices provide definitions that are used in the paper. Appendix A provides the background on the various modular forms, Jacobi forms and Siegel modular forms that appear. Appendix B defines the Weyl-Kac-Borcherds superdenominator formula as well as the supercharacter formula for BKM Lie superalgebras.
2 The B(A (N) ) BKM Lie superalgebras 2.1 Root Lattices, hyperbolic polygons and Weyl groups Consider the following two matrices.
Using the above definition, observe that Let X N denote the ordered sequence of distinct matrices α i generated in this fashion. These sequences are periodic for N = 1, 2, 3 and one has We shall call the elements of X N roots in anticipation of the fact that they are indeed the real simple roots of a Lie algebra. The elements of X N are in one-to-one correspondence with edges of a polygon, M N , in the hyperbolic upper half plane with vertices at rational points in the real line which is the boundary of the upper half plane. Let ( b a , d c ) denote adjacent vertices of the hyperbolic polygon. The root corresponding to the edge connecting these two vertices is given by the map [7] b a , d c −→ α = 2bd ad + bc ad + bc 2ac .
We illustrate this for some roots in Figure 1. For N = 6 consider the following additional roots.
For m ∈ Z, define α 2m+a = γ (6) m · α a · (γ (6) ) T m for a = 1, 2. (2.7) Define X 6 = ( α i ) for i ∈ Z. The two (infinite) sets of roots combine to give the hyperbolic polygon M 6 (see Figure 2).  Figure 2: We show some of the roots in X 6 (in blue) and X 6 (in green) which form the edges of the hyperbolic polygon M 6 . The two dark circles indicate limit points.
The operation δ acts on the roots as follows: and α is any root. δ 2 acts as the identity operator on the roots. It acts on the elements of X N as the following involution.
The group Dih(M N ) generated by γ (N ) and δ acts as a dihedral symmetry of the hyperbolic polygon M N . A real symmetric 2 × 2 matrix can be considered as a vector in R 2,1 as follows [7].  . The Cartan matrix for the real simple roots is given by the matrix of inner products of the roots of X N , i.e., A (N ) := (a nm ), with n, m ∈ S N . A closed formula for the Cartan matrices is given by Eq. (1.3). For N = 6, where there are additional roots that appear in X 6 , the following inner products hold α n , α m = α m , α n = α m , α n + 12 . (2.10) The following relation holds This is consistent with the matrices A (N ) having rank three. We will focus on the following four real simple roots (α 0 , α 1 , α 2 , α 3 ) as we will mostly track the Weyl reflections due to these roots. Using this norm, the matrix of inner products of these roots is When N = 1, the first three columns and rows give A (1) as α 0 = α 3 . When N = 2, A trun = A (2) as there are only four real simple roots. In all other cases, we obtain a truncation of the Cartan matrix A (N ) .

The Weyl Group and its extension
Let w i denote the elementary Weyl reflection by the simple root α i ∈ X N and w i the Weyl reflection by the simple root α i ∈ X 6 . Let β be a root. Then Let W = W (A (N ) ) denote the Weyl group generated by elementary Weyl reflections associated with all simple roots in X N . Let us call the group the extended Weyl group. The extended Weyl group is generated by γ (N ) , δ, w 2 .

Root Lattices with Weyl vector
Consider the vectors contained in X N for N ≤ 4. For N = 6, the vectors are given by X N ∪ X N . These generate lattices in R 2,1 . For N ≤ 4, the lattice is given by (2. 13) and for N = 6 (2.14) The L (N ) are all rank-three Lorentzian lattices with lattice Weyl vector The lattice Weyl vector has the following properties: . Thus, the norm is time-like (< 0) for N < 4, light-like (= 0) for N = 4 and space-like (> 0) for N = 6.
2. The inner products of the lattice Weyl vector with real simple roots are: and for N = 6, additionally one has ̺ (6) , α m = +1 ∀ α m ∈ X 6 . The rank-three hyperbolic root lattices L (N ) with lattice Weyl vector ̺ (N ) fit with Nikulin's classification of hyperbolic root systems of rank three [18]. According to that classification, for N ≤ 3, the lattice Weyl vector is of elliptic type, while for N = 4 it is of parabolic type and for N = 6, it is of hyperbolic type. The type is determined by the norm of the lattice Weyl vector.

Construction of the Siegel modular forms
The following theorem enables one to construct Siegel modular forms in the form of a product formula. All the five Siegel modular forms that we study arise in this fashion as we will show. Theorem 2.1 (Gritsenko-Nikulin [13]). Let ψ be a nearly holomorphic Jacobi form of weight 0 and index t with integral Fourier coefficients. ψ(τ, z) = n,ℓ∈Z c(n, ℓ) q n r ℓ , c(n, ℓ) ∈ Z .
Then the product where (n, l, m) > 0 means that: if m > 0, then n ∈ Z and ℓ ∈ Z; if m = 0, then n > 0 and ℓ ∈ Z; if m = n = 0, then ℓ < 0 and defines a meromorphic Siegel modular form of weight k = 1 2 c(0, 0) with respect to Γ + t possibly with character. The character is determined by the zeroth Fourier-Jacobi coefficient (i.e., the coefficient of s C ) of B ψ (Z) which is a Jacobi form of weight k and index C of the Jacobi subgroup of Γ + t .
The five Siegel modular forms of interest are defined as follows. For N = 1, 2, 3, 4, 6, let where ψ 0,N (τ, z) are the Umbral Jacobi forms defined in Eq. (A. 19). Using the Fourier expansion given there, we obtain A = 1/2N, B = C = 1/2 and k(N) = (6/N) − 1. The zeroth Fourier-Jacobi coefficient is The Jacobi forms transform with character v where v η is the character associated with the Dedekind eta function and v H is defined in Eq. (A.27). This implies that the Siegel modular forms have the following character [13]: Further all our Siegel modular forms are symmetric under the operation V N : q ↔ s N .

Properties of the Siegel modular forms:
We can translate the action of the Dihedral and Weyl group on the roots to equivalent actions on Z = τ z z τ ′ ∈ H 2 using the inner product on R 2,1 . One has e −α ←→ e 2πi(α,Z ′ ) = e 2πiTr(α Z) . where All transformations are realised as elements of Γ N and all that one has to do is to compute the character for those elements. A practical method is to directly compare terms in the Fourier expansion of the Siegel form that are related by the generator.
and w 2 • Z acts on H 2 as z → −z and leaves (τ, τ ′ ) invariant. These properties imply that the Siegel modular forms transform covariantly under the extended Weyl group. The proof of these properties is given, for instance, in [4,5,13].

Deconstructing the Denominator Formula
It is known that for N = 1, 2, 3, 4, the Siegel modular forms ∆ k(N ) (Z) are the modular forms associated with the WKB superdenominator formulae for the Borcherds extensions B(A (N ) ) of g(A (N ) ) [4,12,13]. Eq. (2.18) shows that the Siegel modular forms transform as expected from a WKB superdenominator formula. Further, it is easy to see from explicit formulae that one has The simple roots (α 1 , α 2 ) generate a sl (2) sub-algebra. We study this family of Siegel modular forms in terms of this sl (2) sub-algebra as well as a Borcherds extension of it, that we call B N ( sl (2)), obtained by adding imaginary simple roots of the form nδ = n(α 1 + α 2 ) for n ∈ Z >0 .

Defining sl(2)
Let (e, h, f ) be the generators of the sl(2) Lie algebra. The non-zero Lie brackets are: and the (normalised) Killing form is e, f = 1 and h, h = 2. The affine Lie algebra sl(2) is defined by wherek is the central extension and d = −td/dt. The Lie algebra (with x ∈ sl (2)) The Cartan sub-algebra is (h⊗1,k, d) with inner product such that h⊗1, h⊗1 = 2 and k , d = −1.
We would like to embed sl(2) into the Kac-Moody Lie algebra g(A (N ) ) with symmetric Cartan matrix given by A (N ) . Consider the Chevalley generators where (a ij ) is the Cartan matrix for these three roots  The Lie subalgebra of g(A (N ) ) generated by e 1 , f 1 , e 2 , f 2 , h 1 , h 2 and h 3 is isomorphic to sl (2). Following Feingold and Frenkel [16], we make the identification For the Cartan subalgebra of sl(2), using the above identification, we obtain The inverse is (2)) Consider the Borcherds extension of sl(2) obtained by the addition of imaginary simple roots, (δ, 2δ, . . .) with zero norm, each with multiplicity (2k(N) − 1) = (12 − 3N)/N. Let us call this extension B N ( sl (2)). Note that B 4 ( sl(2)) = sl (2) i.e., no imaginary simple roots are added when N = 4.

Characters
As before, let W = W (A (N ) ) denote the Weyl group of g(A (N ) ); this is generated by the elementary reflections w i for all simple roots i ∈ X N . Let W be the subgroup generated by the reflections w 1 , w 2 , corresponding to the simple roots α 1 and α 2 . This is isomorphic to the Weyl group of sl(2), with We will be focusing mostly on the first three terms. The numerator of the Weyl character formula for a highest weight state with weightΛ of B N ( sl(2)) takes the form Note that num(0) is the denominator formula. We have included imaginary simple roots (the Borcherds correction factor defined in Sec. B.2) in the above formula as TΛ.
Using this computation, we can show that forΛ = aδ + bα 2 + cα 3 , one has (with m = (4Nc + 2) and ℓ = (2c − 2b)) After making the following identifications: we obtain the master formula The denominator formula for A (N ) : It is known that for N = 1, 2, 3, 4, the genus two Siegel modular form ∆ k(N ) (Z) arises as the WKB denominator formula for an extension of the Kac-Moody Lie algebra g A (N ) by the addition of imaginary simple roots. For N = 1, 2, 3, 4, 6, it is known that the modular forms admit the following expansion [4,5]: where φ k(N ),1/2 (τ, z) is defined in Eq. (2.17) and ψ 0,N (τ, z) is an Umbral Jacobi form defined in Eq. (A.19)(see [14] for its connection to Umbral Moonshine). It is important to observe that terms in the square brackets, for each power of s, are invariant under the Weyl group W of sl (2). As we will show next, we can expand each of the terms in terms of characters of sl(2) and of B N ( sl (2)). One should keep in mind that the two characters are different. We interpret φ k(N ),1/2 (τ, z) as the WKB denominator formula for the Borcherds extension B N ( sl (2)). Then, using the following result that follows from Eq.

Some examples of interest
Applying the master formula Eq. (3.3) to the real simple roots Λ = α 0 and Λ = α 3 as well as the imaginary simple roots (with zero norm and multiplicity [2k(N) − 1]) Λ = (α 0 + α 1 ) and Λ = (α 2 + α 3 ), we obtain (after dropping a pre-factor of s N that is present in all terms) In the sequel, we will refer to χ α 3 as χ 4N,2 and so on. This extends the labels that we use for sl(2) characters to characters of B N ( sl (2)).

Lie algebra decompositions of Umbral Jacobi Forms
For the cases of interest, we wish to decompose the Umbral Jacobi Forms in terms of characters of sl (2) and B N ( sl (2)). The decomposition takes the form Further, one observes that g j (τ ) = g 2N −j (τ ). This follows from the Z 2 outer automorphism under which α 1 ↔ α 2 and α 0 ↔ α 3 . Thus one has (N + 1) independent functions that we organize into a vector g := (g 1 , g 2 , . . . , g N +1 ) T . Using the modular properties of the normalized characters and the Umbral Jacobi form, we can show that g(τ ) is a weight zero vector-valued modular form (vvmf) with the following modular properties: The above transformations define the matrices T and S.
We also see that the characters χ 16,0 (τ, z) and χ 16,16 (τ, z) do not appear in the above expansion. This is consistent with the fact that there were no imaginary simple roots added to sl(2) in this case. So we do not expect imaginary simple roots (α i + α i+1 ) for i = 0, 1, 2 to be present. This is borne out in the sl(2) decomposition of ψ 0,4 . However, we find a new imaginary simple root of zero norm appearing with character χ 16,8 (τ, z). It is represented by the matrix The imaginary root (α 2 + 6α 1 + α 0 ) with zero norm also appears in the expansion of χ 16,8 (τ, z).

N = 6
As mentioned earlier, we do not have a BKM Lie superalgebra associated with this Cartan matrix. We do know that the Siegel modular form transforms suitably under the extended Weyl group. In particular, one can see that the Umbral Jacobi form is invariant under W and δ. The sl(2) characters may be viewed as providing a basis for expanding the Jacobi form. The Umbral Jacobi form at lambency 7 has the following decomposition in terms of sl(2) characters. We rewrite the above formula in terms of B 6 ( sl(2)) characters using the following definition.

Matrix differential equations and vvmfs
The decomposition of Umbral Jacobi forms in terms of sl(2) characters has given us vvmfs that are weight zero and have multiplier determined by the matrices S and T . We have obtained the first few terms in their Fourier expansions by direct computation. In this section, we will determine them to all orders. For N = 6, we show that they are solutions to a Matrix Differential Equation (MDE) thereby obtaining explicit analytical formulae for the vvmfs. For N = 6, we use a different method to obtain a similar result. Let M ! w (ρ) denote the space of weakly holomorphic vvmf with multiplier ρ of weight w and rank d. Further, let For j = 0, 1, let a j denote the multiplicity of the eigenvalue (−1) j of S and for j = 0, 1, 2 let b j denote the multiplicity of the eigenvalue exp(2πij/3) of U.
Further, let us assume that T is diagonal The exponents λ i are only defined modulo one. In some situations, the exponents can be fixed. Let g(τ ) ∈ M ! w (ρ) have the following Laurent series g(τ ) = q Λ n∈Z a n q n . Proposition 4.1 (Gannon [17]). Let (ρ, w) be admissible and T diagonal. Then, there exists a choice of exponents Λ for which the principal part map P Λ : Using an index theorem argument, Gannon shows that a necessary but not sufficient condition for the bijectivity described above is . In all our examples, we made choices that satisfied the above condition and for N ≤ 4 found choices such that the bijection holds. [17]). Let (ρ, w) be admissible with rank d, T diagonal and Λ be bijective. Further, let

Theorem 4.2 (Theorem 3.3(b) of Gannon
denote the d ×d matrix whose columns are a basis for M ! w (ρ). Then, Ξ(τ ), solves the Matrix Differential Equation (MDE) of the form: where Λ w = Λ − w 12 1 d and χ w = χ + 2w1 d . In all our examples, one column of Ξ(τ ) is obtained from the vvmf that we obtain in Sec. 3.3 from the sl(2) decomposition of the Umbral Jacobi forms. We use the Fourier coefficents of the known vvmf to determine the MDE.

Identifying the MDE for vvmfs of interest
The data entering the MDE of Gannon are the following: 1. The pair (ρ, w), 2. an invertible set of exponents Λ, and 3. the d × d matrix χ defined by For all our examples, the weight w = 0 and the multiplier ρ is known. The unknowns are an invertible Λ and χ. Instead we know a solution to the MDE to any order that we desire. We assume that our solution corresponds to one column of Ξ(τ ) -this leads to a choice of Λ and determines one column of χ.
The first column of Ξ(τ ) is our solution. Thus, we can obtain the q-series for the vvmf associated with A (1) to arbitrary order.
The second column of Ξ(τ ) is our vvmf and is expressed in terms of generalized hypergeometric functions. This is true only for ranks ≤ 3.

N = 3
This is a rank 4 case and hence we do not anticipate that the solution can be expressed in terms of generalized hypergeometric functions. We choose the following exponents: The multiplicity of eigenvalues of S and U are The solution to the matrix DE is the following where the third column is the vvmf of interest. We have checked that column three of the above matrix agrees with expressions for (g 1 , . . . , g 4 ) to O(q 16 ). Thus, even though we do not have simple expression in terms of hypergeometric functions as before, we have identified the MDE that the vvmf satisfies. We can easily solve the recursion relation to obtain the q-series to fairly high orders.

N = 6
We choose the exponents as follows: The multiplicity of eigenvalues of S and U are Hence c ρ,0 = −7/2 = i λ i . However, we have not been able to determine whether the choice of exponents is bijective. The problem is the large number of constants that need to be determined using the data from the known vvmf. Using the action of ∇ i,w for i = 1, 2, 3, we can generate three linear combinations of the solutions. This leaves us with 21 unknown constants and this space is too large for us to solve on a computer. Hence we chose an alternate method to get an all orders formula for the vvmf that we discuss next.

Determining an explicit formula for the N = 6 vvmf
We observe that the theta expansion of the Umbral Jacobi form takes a very simple form after dividing out by a factor of η(τ ).  where M k,m (τ, z) = θ k,m (τ, z) + θ k,−m (τ, z). Kac and Peterson [20, see section 5.5] express the characters of sl(2) in terms of theta functions that appear above. The transformation matrix is given by Hecke modular forms. Explicitly, one has where C (k) λ,n is defined in terms of Hecke indefinite modular forms as follows: We need to express the theta functions in terms of sl (2) characters. This is given by where D 10,λ (τ ) − D (24) 14,λ (τ ) η(τ ) Comparing the above expression with Eq. (3.22), we obtain explicit formulae for (g 1 (τ ), . . . , g 7 (τ )) that agree with the expressions to the order that we have determined them. We thus have obtained explicit formulae for the vvmf associated with N = 6 even though we have not determined the MDE satisfied by the vvmf.

Interpreting the vvmfs
We have seen that the vvmf that we denote by g(τ ) captures the contribution of simple roots. Combining this result with the invariance of the Siegel modular forms under the action of the dihedral group, we obtain formulae that extend our results. The Fourier-Jacobi expansion of the Siegel modular form is compatible with the action of the subgroup w 2 , δ as these are realised as elements of the Jacobi group which preserve the cusp at τ ′ = i∞. The generator γ (N ) does not belong to the Jacobi group. Including its action on the Umbral Jacobi form its decomposition into sl(2) characters enables us to organise the result in terms of orbits of the extended Weyl group. The Siegel modular form ∆ k(N ) (Z) can be written as a sum of terms of the kind that follow from our sl(2) decomposition of the Umbral Jacobi form. where σ N (n) is defined as follows: It is easy to see that σ 4 (n) = 0 for all n > 1 and σ 6 (n) = p(n) where p(n) is the number of partitions of the positive integer n.
3. For N ≤ 4, all other terms correspond to imaginary simple roots and provide Borcherds correction terms. To see how to do this, consider a term of the form The m-th term in the above sum is associated with the sl(2) weight vector (Λ + mδ) with multiplicity b(m). A W -covariant expression that accounts for these roots is 4. The case of N = 6 needs special attention. First, we get new simple real roots that we denoted by α 1 and α 2 . The Siegel modular form is not invariant under Weyl reflections generated by these roots. Further ρ (N ) , α i = +1 and not equal to −1. The multiplicity on the product side is −1 and hence they are fermionic roots. It appears that the term that we obtain arises as follows: (4.23) The above formula is conjectural as we have not checked if the pieces indicated by the ellipsis do appear. We also see that further imaginary roots involving the tilde roots also appear. They are of the form ( α i + α j ) for i, j = 1, 2. These do not appear in the set of positive roots that we obtain from the product formula. This is also true for the weights associated with labels 5 and 7. There is a cancellation of the form 1 − 1 = 0. We can see a similar cancellation in the WKB denominator formula for B 6 ( sl (2)). The root δ = (α 1 +α 2 ) does not appear on the product side. This is because this root appears as a non-simple bosonic imaginary root as well as a fermionic imaginary simple root (with the same weight). This suggests that the positive roots given by the product formula is incomplete and we need to take into account cancellations that occur. Our decomposition in terms of sl(2) characters is able to account for this.

Concluding Remarks
The main result of this paper is a preliminary study of the WKB superdenominator formulae associated with BKM Lie superalgebras using a sl(2) subalgebra (and its Borcherds extension). In the current paper, we have restricted our study to include the first two additional simple real roots (and corresponding imaginary simple roots) that appear in the first Fourier-Jacobi coefficients of the Siegel modular forms. This leads to an interesting connection with vector-valued modular forms associated with some Umbral Jacobi forms. In all cases, we obtained relatively simple formulae for the Fourier coefficients of the vvmfs. These Fourier coefficients correspond to the multiplicities of simple roots, both imaginary and real, of the BKM Lie superalgebras.
The next step would be to carry out a similar decomposition for all Fourier-Jacobi coefficients. The connection with umbral moonshine gives a second formula for the Siegel modular forms. Extending heuristic arguments given in [21] (see also [22]) for Mathieu moonshine to Umbral moonshine, one has The same formula also appears in [13,see Eq. (2.7)]. This formula is very useful in obtaining explicit formulae for higher Fourier-Jacobi coefficients of the ∆ k(N ) (Z). For instance, the second coefficient is given by The above formula has a nice interpretation. Let V N denote a sl(2) module such that (H is the Cartan subalgebra of sl (2)) and the Umbral Jacobi form is equal to supercharacter of V N i.e., where V 0 µ (resp. V 1 µ ) is the bosonic (resp. fermionic) subspace of V N of weight µ. Then, ψ 0,2N (τ, z) is obtained as the supertrace over direct sum of the sl(2) modules: Λ 2 V N and V [2] N . The latter module V [2] N is obtained via the following scaling procedure [23]. The Lie subalgebra sl(2) [2] = sl(2) ⊗ C[t 2 , t −2 ] ⊕ Ck ⊕ C d of sl(2) is in fact isomorphic to sl(2). 4 The sl(2)-module V N is Z + -graded, with the highest weight state being of grade zero and each application of X ⊗ t −m increasing the grade by m. The subspace of V N comprising its graded pieces of even grade is a module for the subalgebra sl (2) [2] ∼ = sl (2). This module is denoted N . It is easy to see that Formulae such as these will enable us to write explicit formulae using the sl(2) decomposition obtained in this paper. This should, in principle, enable us to rewrite the sum side of the WKB denominator formula first in terms of sl (2) representations and then in sums where the covariance under the full Weyl group is manifest. We hope to report on this in the future [24]. In [13], Gritsenko and Nikulin point out that the ∆ k(N ) (Z) for N = 1, 2, 3, 4 are three-dimensional generalizations of the Dedekind eta function. Rankin [25] showed that the weight-twelve modular form Ψ = η(τ ) 24 of Γ 1 satisfies the following nonlinear ODE: (see Zagier [26] for a derivation) Rankin's ODE becomes the Chazy equation: This nonlinear equation satisfies the Painlevé property and connections with integrable systems (see [27,28] and references therein). We have found MDE's for vvmf's associated with the Umbral Jacobi forms. Do all these combine to give a nice three-dimensional modular ODE for the logarithm of the Siegel modular forms? In this context, it is known that the logarithmic derivatives of the genus two theta constants satisfy a system of equations. [29,30]. These methods might help one obtain similar nonlinear modular differential equations for the Siegel modular forms.

A Modular background
In this appendix, we discuss the different kinds of automorphic forms that appear in the paper. In particular, for vector-valued modular forms, we follow the discussion of Gannon [17].

A.1 Basic Group Theory
The paramodular group at paramodular level t that we denote by Γ t is defined as follows (we follow [31] for all definitions) (for t ∈ Z >0 ): When t = 1, then Γ 1 = Sp(4, Z) ≡ Γ (2) is the usual symplectic group. The group Γ + t is generated by V t and its parabolic subgroup The Jacobi group is defined by The embedding of ( a b c d ) ∈ SL(2, Z) in Γ t is given by The above matrix acts on H 2 as with det(CZ + D) = (cτ + d). The Heisenberg group, H(Z), is generated by Sp(4, Z) matrices of the form The above matrix acts on H 2 as (τ, z, σ) −→ τ, z + λτ + µ, τ ′ + λ 2 τ + 2λz + λµ + κ , (A.11) with det(CZ + D) = 1. It is easy to see that Γ J preserves the one-dimensional cusp at Im(τ ′ ) = ∞.

A.2 Modular forms
Definition A.1. A modular form of weight w and character χ : for all γ = ( a b c d ) ∈ Γ (1) . A holomorphic modular form is holomorphic on the extended upper halfplane while a weakly holomorphic modular form is holomorphic on the upper half-plane and meromorphic on the extended upper half-plane. For k ∈ Z >0 , define the Eisenstein series as follows: where q = exp(2πiτ ) and σ s (n) = d|n d s is the divisor function. For k > 1, the Eisenstein series are holomorphic modular forms of weight 2k. For k = 1, it is not a modular form but E * 2 (τ ) = E 2 (τ ) − 3 πIm(τ ) is not holomorphic but is modular of weight 2. The Dedekind eta function is defined by It is a modular form of weight 1 2 of a subgroup of Γ (1) . The q 1/24 implies that under T , it picks up a phase that is a 24-th root of unity. Taking the 24th-power of the Dedekind eta function gives us a modular form of weight 12 called the Discriminant function η(τ ) 24 = q − 24q 2 + 252q 3 + · · · .
The modular J function defined below is a weakly holomorphic modular form of weight zero.

A.4 Modular Differential Operators
Let f be a modular form of weight w and D w denote the modular derivative i.e., This maps a modular form of weight w to a modular form of weight (w + 2). Consider the differential operators that don't change weight. 16) A.6 Classical Theta functions This is a vector-valued Jacobi form of weight half and index k/4. Dividing by η(τ ) makes the weight to zero.
Under the T and S modular transformations, the α k,λ transform as follows: Below we define the normalized sl(2) characters which have nice modular properties.
χ k,λ (τ, z) = θ k+2,λ+1 (τ, z) − θ k+2,−λ−1 (τ, z) θ 2,1 (τ, z) − θ 2,−1 (τ, z) ; for k, λ ∈ Z ≥0 , λ ≤ k . where s = exp(2πiτ ′ ). For each m, φ m (τ, z) is a Jacobi form of weight k and index mt. This can be understood by observing that the cusp at τ ′ = i∞ is preserved by the subgroup Γ J and studying their transformation under this subgroup. We refer to the first non-vanishing term in the above Fourier expansion as the zeroth Fourier-Jacobi coefficient of the Siegel modular form. The character of Siegel modular forms are determined in part by their transformation under the Jacobi group Γ J . Consider the Jacobi form of weight −1 and index 1 2 : This has trivial character under modular transformations and the following character v H ([λ, µ, κ]) = (−1) λ+µ+λµ+κ . (A.27) Multiplying the above Jacobi form by modular form f (τ ) of Γ 1 with character χ leads to another Jacobi form of index half with character (χ × v H ). This data can be obtained from the zeroth Fourier-Jacobi coefficient of the Siegel modular form. We need to determine the character under the involution V t (q ↔ s t ) and [0, 0, κ/t] (for t > 1).

B Supercharacter formula for BKM Lie superalgebras B.1 The superdenominator identity
Let g be a BKM Lie superalgebra. The Weyl-Kac-Borcherds superdenominator identity of g has the form S = P (sum equals product). We describe this in greater detail here, closely following [32]. Let g = g 0 ⊕ g 1 be the decomposition of g into bosonic (even) and fermionic (odd) subspaces. For p = 0, 1, let L + p denote the set of positive roots of bosonic (p = 0) or fermionic (p = 1) type, and let m p (α) = dim(g p ) α denote the multiplicity of the root α in the subspace of appropriate parity.
The product side is given by: To describe the sum side, let α i (i ∈ I) denote the simple roots of g. Let I re = {i ∈ I : α i , α i > 0} and I im = I\I re be the subsets of real and imaginary simple roots. The Weyl group W of g (when g is infinite-dimensional) is the group generated by the simple reflections w α i for i ∈ I re . We can also decompose I = I 0 ∪ I 1 into the disjoint union of simple roots of bosonic and fermionic types.
Consider the set T of all elements µ in the root lattice of g which can be expressed as a finite sum µ = i∈I k i α i satisfying the following conditions: 1. k i is a non-negative integer for all i.
2. k i = 0 for i ∈ I re .
3. If k i and k j are nonzero for some i = j, then α i , α j = 0.
A negative value for m corresponds to isotropic fermionic simple roots. We will encounter such Borcherds extensions of sl (2) i.e., sl(2) with the addition of the imaginary simple roots of the form discussed above.

B.2 The supercharacter formula
More generally, one has the Weyl-Kac-Borcherds formula for the supercharacter of an irreducible integrable highest weight module L(Λ) of g. Here Λ is a dominant integral weight of g, i.e., (Λ, α i ) is a non-negative integer (resp. real number) for i ∈ I re (resp. i ∈ I im ). We define a subset T Λ of T by imposing the following extra condition in addition to (1)-(4) above: 5. k i = 0 if Λ, α i < 0.
The WKB supercharacter formula states that the supercharacter χ Λ of L(Λ) is given by the quotient Sch(L(λ)) : Since L(Λ) is the one-dimensional trivial representation when Λ = 0, this reduces to the superdenominator identity in that case [32].