Abstract
In this paper, we consider the Fourier coefficients of a special class of meromorphic Jacobi forms of negative index considered by Kac and Wakimoto. Much recent work has been done on such coefficients in the case of Jacobi forms of positive index, but almost nothing is known for Jacobi forms of negative index. In this paper we show, from two different perspectives, that their Fourier coefficients have a simple decomposition in terms of partial theta functions. The first perspective uses the language of Lie super algebras, and the second applies the theory of elliptic functions. In particular, we find a new infinite family of rankcrank type partial differential equations generalizing the famous example of Atkin and Garvan. We then describe the modularity properties of these coefficients, showing that they are ‘mixed partial theta functions’, along the way determining a new class of quantum modular partial theta functions which is of independent interest. In particular, we settle the final cases of a question of Kac concerning modularity properties of Fourier coefficients of certain Jacobi forms.MSC 11F03; 11F22; 11F37; 11F50
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Background
Since the introduction of the theory of Jacobi forms by Eichler and Zagier [1], connections between Jacobi forms and modulartype objects have been a question of central interest, with applications to many areas including Siegel modular forms, mock modular forms, and Lie theory. In this paper we study the Fourier coefficients of a special family of negative index Jacobi forms. In particular, consider for N\in \mathbb{N},M\in {\mathbb{N}}_{0} the functions
where 𝜗(z;τ) is the usual Jacobi theta function
Here, q:={e}^{2\mathrm{\pi i\tau}}(\tau \in \mathbb{H}), \zeta :={e}^{2\mathrm{\pi iz}}(z\in \u2102), and for n\in {\mathbb{N}}_{0}\cup \left\{\infty \right\}, {(a)}_{n}:={(a;q)}_{n}:=\prod _{j=0}^{n1}\left(1a{q}^{j}\right) is the qPochhammer symbol. The relationship between Jacobi forms and modular forms has appeared in many guises and stems back to important work on holomorphic Jacobi forms, which states that they have theta decompositions relating them to halfintegral weight modular forms [1]. The situation for meromorphic positive index Jacobi forms also well understood; a meromorphic Jacobi form of positive index has Fourier coefficients which are almost mock modular forms, which in turn are holomorphic parts of almost harmonic Maass forms[2][6]. Loosely speaking, almost harmonic weak Maass forms are sums of harmonic weak Maass functions under iterates of the raising operator multiplied by almost holomorphic modular forms. In this paper, we describe new decompositions of the Jacobi forms ϕ_{M,N}(z;τ) which complement this long history of previous work on positive index Jacobi forms in the much more mysterious case of negative index. In addition to being of interest in the subject of general Jacobi forms, here we give further applications of such decompositions, focusing on the special subfamily ϕ_{ N }(τ):=ϕ_{0,N}(τ) as they are of great interest in various areas such as number theory, representation theory, combinatorics, and physics. Here we outline just a few such occurrences.
Firstly, for various choices of N, the functions ϕ_{ N } are of combinatorial interest. In particular, the function ϕ_{1} is related to the famous AndrewsDysonGarvan crank generating function (see (7) and (8)), which was used by Andrews and Garvan to provide a combinatorial explanation for the Ramnaujan congruences for the partition function [7], as postulated by Dyson [8]. In this paper we describe relations between powers of the crankgenerating function with certain AppellLerch series, giving a new family of partial differential equations (PDEs) generalizing the ‘rankcrank PDE’ of Atkin and Garvan [9] (see Theorem 3). This beautiful identity of Atkin and Garvan gives a surprising connection between the rank and crank generating functions which can be used to show various congruencesrelating ranks and cranks, as well as useful relations between the rank and crank moments [9]. We also note the other examples of similar PDEs related to combinatorics have shown up in, for example Section 3.2 of [10], where the function ϕ_{1,3}(z;τ) is studied in relation to overpartitions.
Secondly, the functions ϕ_{ N } contain information about certain affine vertex algebras and their associated affine Lie algebras studied by Kac and Wakimoto [11]. More precisely, let \mathcal{S}(1) be the graded vector space
and \mathcal{S}(N)=\mathcal{S}{(1)}^{\otimes N}. The vector space \mathcal{S}(N) can be given the structure of a vertex algebra, the bosonic β γghost vertex algebra of rank N and central charge −N. The graded character of this vertex algebra has a nice product form in the domain q{}^{\frac{1}{2}}<{\zeta}_{n}<q{}^{\frac{1}{2}}, where {\zeta}_{n}:={e}^{2\mathrm{\pi i}{z}_{n}},
It specializes to ϕ_{ N }(z;τ) for the following choices:
The algebra \mathcal{S}(N) contains as commuting subalgebras the rank 1 Heisenberg vertex algebra \mathcal{\mathscr{H}}(1) and the simple affine vertex algebra of s\ell (N) at level −1, {L}_{1}(s\ell (N)). Note that −1 is not an admissible level in the case of N=2 and that these algebras do not form a mutually commuting pair inside \mathcal{S}(2). However, for N>2 it was shown in [12] that {L}_{1}(s\ell (N)) and \mathcal{\mathscr{H}}(1) form such a mutually commuting pair inside \mathcal{S}(N). The character of the highestweight module {\mathcal{F}}_{\mu}, \mu \in \mathbb{R}, of \mathcal{\mathscr{H}}(1) takes the form
so that the Fourier coefficients in ζ of ϕ_{ N }(z;τ) immediately allow one to compute the multiplicity with which the character of {\mathcal{F}}_{r/\sqrt{N}} appears. In physics language, such a multiplicity is called the branching function of the coset \mathcal{S}(N)/\mathcal{\mathscr{H}}(1).
This leads to the second conformal field theory and vertex algebra importance of decomposing a meromorphic Jacobi form. One of the most interesting classes of vertex algebras is given by {V}_{k}(\mathfrak{\U0001d524}), the universal affine vertex algebra of the simple Lie algebra \mathfrak{\U0001d524} at level k\in \u2102. For certain rational admissible levels, {V}_{k}(\mathfrak{\U0001d524}) is not simple and one instead prefers to study its simple quotient {L}_{k}(\mathfrak{\U0001d524}). The characters of irreducible highestweight modules at admissible level {L}_{k}(\mathfrak{\U0001d524}) are the sum expansions in special domains of meromorphic Jacobi forms [13]. Understanding these sum expansions is crucial in studying the modular data of the corresponding conformal field theory [14],[15].
Fourthly, the functions ϕ_{ N } appear in the denominator identities of affine Lie super algebras [16]. In [14],[15] the denominator identity of \hat{\text{s}\ell}(21) was an essential ingredient to study the relations between characters of admissible level L_{ k }(sℓ(2)), while we use the identities for the family \hat{\text{s}\ell}(N1) to prove one of our central theorems.
Finally, the functions ϕ_{ N } also occur in string theory; we only expound upon one example. The reciprocal of the Igusa cusp form Φ_{10}(Z) (Z\in {\mathbb{H}}_{2}, the Siegel upper half plane of genus (2) arises as the partition function of quarterBPS dyons in the type II compactification on the product of a K_{3} surface and an elliptic curve. Write
For m>0, the Fourier coefficients of the functions ψ_{ m } are the degeneracies of singlecentered black holes and twocentered black holes with total magnetic charge invariant equal to m. This case is studied in pathbreaking work of Dabholkar, Murthy, and Zagier [3] (see also [17] for the appearance of mock modular forms in the context of quantum gravity partition functions and AdS3/CFT2, as well as [18] for a relation between multicentered black holes and mock SiegelNarain theta functions). The coefficient of m=−1 equals
(note that our theta function 𝜗(z;τ) differs from the theta function θ_{1}(z;τ) in the notation of [3] by a factor of i). Analogously to the case of Jacobi forms of positive index, one may view Theorem 3 below as a decomposition giving a ‘polar part’ but no ‘finite part’ as described in [2],[3] and stated in more detail in (24). This is consistent with a string theoretic interpretation of ψ_{−1} in (4) in that there are no singlecentered black holes and the degeneracies are all interpreted as accounting for twocentered black holes (see [19],[20]). In contrast, in the case m>0, the mock part of ψ_{ m } corresponds to singlecentered black holes and the Appel Lerch sum corresponds to twocentered black hole bound states [3].
Returning to the problem of studying ϕ_{M,N}, we define its Fourier coefficients by
and in particular we set χ(N,r;τ):=χ(0,N,r;τ). Note that wallcrossing occurs; the coefficients χ(M,N,r;τ) are only welldefined if we fix a range for z. We show that the Fourier coefficients χ(M,N,r;τ) can be described using partial theta functions (i.e., sums over half a lattice which when summed over a full lattices becomes a theta function), whose modularity properties near the real line we also describe using quantum modular forms. Quantum modular forms were recently defined by Zagier in [21] (see also [22][24]). Although the definition is not rigorous, Zagier gave a number of motivating examples. Roughly speaking, a weight k quantum modular form is a function f:\mathcal{Q}\to \u2102 for some subset \mathcal{Q}\subseteq {\mathbb{P}}_{1}(\mathbb{Q}) such that for any γ in a congruence subgroup Γ, the cocycle f_{ k }(1−γ) extends to an open set of and is ‘nice’ (e.g., continuously differentiable, smooth). In fact, our study of the modularity of the partial theta functions shows that they are what Zagier refers to as strong quantum modular forms, namely, that they have a nearmodular property for asymptotic expansions defined at every point in a subset of {\mathbb{P}}_{1}(\mathbb{Q}). Moreover, this behavior comes from the ‘leaking’ of modularity properties of a nonholomorphic Eichler integral defined on the lower half plane (see (38)).
Returning to the Fourier coefficients of ϕ_{M,N}, we define a mixed partial theta function to be a linear combination of quasimodular forms multiplied with partial theta functions. These functions have known connections to many interesting combinatorial functions, such as concave and convex compositions [25], unimodal sequences [26],[27], and stacks [28]. Throughout, we abuse notation to say that any function is a modular form, partial theta function, mixed partial theta function, etc. if it is equal to such a function up to multiplication by a rational power of q. Our main result is the following:
Theorem 1.
For any N\in \mathbb{N}, M\in 2{\mathbb{N}}_{0} and r\in \mathbb{Z} with M<N, the functions χ(M,N,r;τ) are mixed partial theta functions.
Remarks.

1.
The quasimodular forms appearing in the decomposition of the mixed partial theta functions are canonically determined by the Laurent expansion of the Jacobi form ϕ _{M,N} (see Theorem 4).

2.
Using the techniques of this paper, it is easy to relax the condition on M to allow any natural number less than N; however, we restrict to even M for notational convenience. Together with Theorem 1 and the works in [2],[3],[5], this settles the final cases of modularity of KacWakimoto characters raised in [11].
We first consider the case of ϕ_{ N }(z;τ), which we study from two perspectives. Our first viewpoint describes the Fourier coefficients as derivatives of partial theta functions of a rescaled version of the root lattice
of sℓ(N). Here, the α_{ n } are the simple roots of sℓ(N), which are linear functionals on the Cartan subalgebra \mathfrak{\U0001d525}\cong {\u2102}^{N1} of sℓ(N). The Gram matrix of A_{N−1} is the Cartan matrix of sℓ(N). We denote the bilinear form by () and abbreviate t^{2}:=(tt) for t in A_{N−1}. For r in , we define the subset of \frac{1}{N}{A}_{N1}
and its partial theta function
The e^{t} are functions on the Cartan subalgebra \mathfrak{\U0001d525} defined by e^{t} : u↦e^{t(u)} for u in \mathfrak{\U0001d525}. Its evaluation for u\in \mathfrak{\U0001d525} is then denoted by P_{ r }(u;τ). We call P_{ r } a partial theta function because the theta function obtained by summing over the complete lattice \frac{1}{N}{A}_{N1},
is a modular form of weight (N−1)/2 for Γ(M) with M=N^{2}(N−1)/2. This statement is true, since θ_{ N } is the theta function of the lattice \frac{1}{\sqrt{2M}}{A}_{N1}. The level of this lattice is M, and the modularity of theta functions of lattices is discussed for example in [29].
Further, let ∂ be the differential operator
where {\Delta}_{0}^{+} is the set of positive roots of sℓ(N). Finally, set {d}_{N}:=\prod _{j=1}^{N}j! and let sign(r)=1 if r≥0 and −1 otherwise. Then we have the following.
Theorem 2.
For N≥2, the r th Fourier coefficient of ϕ_{ N }(z;τ) is given by
Remarks.

1.
Using (2) and (3), Theorem 2 implies the character decomposition
\begin{array}{c}\begin{array}{cc}\text{ch}[\phantom{\rule{0.3em}{0ex}}\mathcal{S}(N)](z,\cdots \phantom{\rule{0.3em}{0ex}},z;\tau )& =\sum _{r\in \mathbb{Z}}\text{ch}\left[\phantom{\rule{0.3em}{0ex}}{\mathcal{F}}_{\frac{r}{\sqrt{N}}}(z;\tau )\right]\text{ch}\left[\phantom{\rule{0.3em}{0ex}}{\mathcal{\mathcal{B}}}_{r}\right](\tau ),\phantom{\rule{2em}{0ex}}\end{array}\end{array}
where
The \text{ch}\left[\phantom{\rule{0.3em}{0ex}}{\mathcal{\mathcal{B}}}_{r}\right](\tau ) are then characters of L_{−1}(sℓ(N)).

2.
The proof of the theorem uses the denominator identity of both sℓ(N1) and \hat{s\ell}(N1) as well as Weyl’s character formula for sℓ(N).

3.
The case N=1 follows from the denominator identity of \hat{\mathrm{g}\ell}(11) (see Example 1). In this case, the Fourier coefficients relate to the characters of a wellknown logarithmic conformal field theory, the \mathcal{W}(2,3)algebra of central charge −2. The modularity of the coefficients has been studied from a different perspective in [30].
The second approach is based on a generalization of a deep identity of Atkin and Garvan. To state it, we first recall the rank and crank generating functions (whose combinatorial meanings are not needed in this paper), which arise in many contexts and in particular give combinatorial explanations of Ramnaujan’s congruences (for example, see [7],[8],[31]). Specifically, the generating functions are given as follows:
We also need the normalized versions
Note that ϕ_{ N } is essentially the N th power of {\mathcal{C}}^{\ast} as for N\in \mathbb{N} we have
The simplest case of our decomposition relies on the fact that {\mathcal{C}}^{\ast}, and thus ϕ_{1}, is essentially an AppellLerch sum thanks to the following classical partial fraction expansion (for example, see Theorem 1.4 of [32]).
For the cube of the crank generating function, Atkin and Garvan [9] proved the following rankcrank PDE which is very useful in establishing congruences and relations between the moments of the rank and crank generating functions:
Here and throughout {\mathcal{D}}_{x}:=x\frac{\partial}{\mathrm{\partial x}}. Note that this gives a description of ϕ_{3} in terms of AppellLerch sums by (8).
Zwegers [32] nicely generalized (10) for arbitrary odd powers of the crank generating function using the theory of elliptic forms. For similar results using another clever proof, see also the paper of Chan, Dixit, and Garvan [33].
In this paper, we prove a new family of analogous PDEs which are of independent interest. Moreover, we package Zwegers’ family of PDEs in a way which illuminates their structure coming from negative index Jacobi forms. To describe this, we need the AppellLerch sums
where w:=e^{2πiu}. We note that these AppellLerch sums are similar to the functions f_{ z }(u;τ) considered in Chapter 3 of [6], which transform as a Jacobi form in u and as a ‘mock Jacobi form’ in z. We also require the Laurent coefficients of ϕ_{M,N}(z;τ) at z=0:
Note that only even or odd Laurent coefficients occur, depending on the parity of N, since 𝜗(−z;τ)=−𝜗(z;τ). It is not hard to see that the coefficients D_{ j } are quasimodular forms. Explicitly, they can be computed quickly in terms of the usual Eisenstein series
where {\sigma}_{k}(n):=\sum _{dn}{d}^{k} and B_{ k } is the usual k th Bernoulli number. Specifically, it easily follows from the Jacobi triple product formula that
The following result puts Zwegers family of PDEs as well as our new family of PDEs into a common framework. Setting
we find:
Theorem 3.
For any N\in \mathbb{N}, M\in {\mathbb{N}}_{0}, we have
Remarks.

1.
Note that Theorem 3 is more explicit than Zwegers’ rankcrank type PDEs as it gives the modular coefficients of the PDEs directly from the structure of the Jacobi form ϕ _{M,N}. Chan, Dixit, and Garvan also remarked that it would be interesting to find such an explicit expression for the quasimodular forms in the decomposition in that case.

2.
It would be interesting to find a Lie theoretic interpretation of the decomposition in Theorem 3.
Armed with the decomposition in Theorem 3 into AppellLerch sums, we can easily pick off the Fourier coefficients χ(M,N,r;τ) and write them in terms of the Laurent coefficients of ϕ_{M,N} and certain partial theta functions
Specifically, if we let
then the Fourier coefficients of ϕ_{M,N} are as follows.
Theorem 4.
For any N\in \mathbb{N}, M\in {\mathbb{N}}_{0}, r\in \frac{N}{2}+\mathbb{Z}, 0≤ Im(z)< Im(τ), we have
If N>1 is odd, these partial theta functions fit into the pioneering work of Folsom, Ono, and Rhoades [23] which gives startling relations between the asymptotic expansions of the rank and crank generating functions, generalizing and proving beautiful formulas of Ramanujan. Their work shows that {\Theta}_{\frac{1}{2}}(N,r;\tau ) is a strong quantum modular form for odd N>1. Although their theorem does not directly apply for N=1, in this case we essentially obtain an eta quotient which is trivially a quantum modular form at cusps where it vanishes.
For even N, both the hypergeometric representations used to determine quantum sets and the proof of quantum modularity are not applicable. Here we use the innovative approach of Lawrence and Zagier [34] to study quantum modularity properties (see also [35]). A key ingredient in our investigation is a beautiful identity of Warnaar [36] which relates certain partial and false theta functions (see (35)). Our main result for studying quantum modularity for even N is the following, which gives a new family of quantum modular forms.
Theorem 5.
For any N\in 2\mathbb{N} and r\in \mathbb{Z}, {\Theta}_{\frac{3}{2}}(N,r;\tau ) is a strong quantum modular form with quantum set {\hat{Q}}_{N,r} (defined in (34)) on Γ_{1}(2N), multiplier system χ_{ r } (defined in (23)), and weight \frac{3}{2}.
Remarks.

1.
More details about the specific quantum modular properties can be found in the proof of Theorem 5 in Section ‘Quantum modularity of {\Theta}_{\frac{3}{2}}(N,r;\tau ).’

2.
More generally, using Proposition 3 of [37], our proof of Theorem 5 shows that {\Theta}_{\frac{3}{2}}(N,r;\tau ) has modularity properties on all of . For this, we note that although the function is not defined on all of , it has a welldefined asymptotic expansion at all points in . This expansion still agrees with the nonholomorphic Eichler integral on the lower half plane (see Section ‘Proof of Theorem 5’), so one could say that {\Theta}_{\frac{3}{2}}(N,r;\tau ) is a quantum modular form on if we allow ‘poles’ at certain points in .
The paper is organized as follows. In ‘Preliminaries on Lie super algebras and character identities’ and ‘Basic facts on Jacobi forms and quantum modular forms’ sections, we review the necessary notation and basic objects from Lie theory, Jacobi forms, and quantum modular forms. We give our first proof of the decomposition using Lie theory in ‘The Fourier coefficients and partial theta functions of A_{ N }’ section and our second proof using an analogue of the rankcrank PDE in ‘Second viewpoint on the decomposition into partial theta functions’ section. We conclude by describing the quantum modular properties of {\Theta}_{\frac{1}{2}+\nu}(N,r;\tau ) in ‘Proof of Theorem 5’ section.
Preliminaries on Lie super algebras and character identities
In this section, we recall some known facts of the affine Lie superalgebra \hat{s\ell}(N1), following [16], as well as the finitedimensional Lie algebra sℓ(N) using [38].
The Lie super algebra sℓ(N+11)
In this subsection, the Lie super algebra sℓ(N+11) and its root system are defined.
The even subalgebra of the Lie super algebra sℓ(N+11) is g ℓ(N+1) and the odd part decomposes into the standard representation of the even subalgebra and its conjugate. In order to define the Lie super algebra, it is convenient to first introduce its root system. It lies in the lattice
with bilinear form
Thus, its signature is (N+1,1). The set of roots is Δ=Δ_{0}∪Δ_{1}⊂L_{ N }, where the set of even roots (respective odd roots) is denoted by Δ_{0} (respectively Δ_{1}). They are
It is useful to split these sets into positive and negative subroot spaces, where
A distinguished system of simple positive roots is then chosen to be
The α_{ j } are even roots and β is the only distinguished odd simple root. The inner products of simple positive roots are
Hence, β is an isotropic root. Simple even roots generate the even root lattice
Its dual lattice is
where the inner product of the fundamental weights λ_{ j } with simple roots is (λ_{ j }α_{ k })=δ_{j,k} and (λ_{ j }δ)=0. Roots and weights act on the Cartan subalgebra, which is
with basis {h_{ α }} parameterized by simple positive roots, and {\mathfrak{\U0001d525}}_{0} the Cartan subalgebra of sℓ(N+1). The fundamental weights λ_{ j } are identified with elements of the dual {\mathfrak{\U0001d525}}_{0}^{\ast} of {\mathfrak{\U0001d525}}_{0} via {\lambda}_{j}({h}_{{\alpha}_{k}})={\delta}_{j,k}. A bilinear form (,) on \mathfrak{\U0001d525} is induced from the form on its dual space via
We remark that the Lie superalgebra sℓ(N+11) is then the \mathbb{Z}/2\mathbb{Z}graded algebra generated by \{{h}_{\alpha},{e}_{\alpha}^{\pm}\alpha \in \Pi \} subject to the SerreChevalley relations (14) and the graded Jacobi identity. The parity of h_{ α } and {e}_{{\alpha}_{j}}^{\pm} is even, while the {e}_{\beta}^{\pm} are odd. We denote the graded antisymmetric bracket by [, ]: sℓ(N+11)× sℓ(N+11)→ sℓ(N+11). Then the SerreChevalley relations of the algebra are
for all α,α^{′}∈Π and α≠α^{′} in the last equation. The bilinear form (,) on \mathfrak{\U0001d525} can be extended to an invariant nondegenerate graded symmetric form on sℓ(N+11), which we also denote by (,).
The even Weyl group and denominator identity of sℓ(N+11)
We now introduce the even Weyl group and the denominator identity of the Lie super algebra sℓ(N+11).
For this, we first need to define the Weyl vector ρ. It is the difference of the even Weyl vector ρ_{0} and the odd one ρ_{1}, namely,
The group of even Weyl reflections W^{♯} acts on the dual of the even root lattice, A N′, and is generated by σ_{ j },j=1,⋯,N defined by
This action naturally extends to the lattice L_{ N } via σ_{ j }(δ)=0 and
Hence, the even Weyl group W^{♯} is just the group S_{N+1} permuting the ε_{ j }. Orthonormality of the ε_{ j } implies that the even Weyl group preserves the bilinear form (). Following [16] we define
Definition.
A regular exponential function on A N′ is a finite linear combination of exponentials of the form e^{λ} for λ∈A N′. A rational exponential function is the quotient A/B of two regular exponential functions A and B≠0. The even Weyl group W^{♯} acts on the field of these functions as e^{λ}↦e^{w(λ)} for any w∈W^{♯}. The Weyl denominator of sℓ(N+11) is the rational exponential function
We saw that the even Weyl group W^{♯} is just S_{N+1}, the signum of an element w in W^{♯} is σ(w):=(−1)^{n} if w can be written as a composition of n transpositions. Theorem 2.1 of [16] applied to our situation gives
Lemma 1.
The Weyl denominator of the Lie super algebra sℓ(N+11) is
The denominator identity of the affine Lie super algebra \hat{s\ell}(N+11)
We turn our focus to the affinization of sℓ(N+11) that is
with bracket
for all x,y∈sℓ(N+11) and n,m\in \mathbb{Z}. The Cartan subalgebra extends to its affine counterpart
and the bilinear form extends as
We identify C and d with linear functionals on \mathfrak{\U0001d525} using the bilinear form (,) and extend A N′ to
The bilinear form extends as
The lattice A_{ N }⊂A N′ is then also a sublattice of {\hat{A}}_{N}^{\prime}. The affine Weyl vector is
Note that N is the dual Coxeter number of sℓ(N+11). For α∈A_{ N }, we define
The group of even Weyl translations is {t_{ α }α∈A_{ N }}. Conjugation by a Weyl rotation gives for any w∈W^{♯},α∈A_{ N }
Let
be the domain of all elements in \hat{\mathfrak{\U0001d525}} on which the action of C has positive real part. Let \hat{\mathcal{F}} be the field of meromorphic functions on Y and define \mathfrak{\U0001d52e}:={e}^{C}. Thus, \mathfrak{\U0001d52e}(y)<1 for all y in Y. Any element λ of L^{′} extends to a linear function on {\hat{\mathfrak{\U0001d525}}}^{\ast} by defining λ(C)=λ(d)=0. In this way rational exponential functions on L^{′} embed in \hat{\mathcal{F}}.
Definition.
The denominator of \hat{s\ell}(N+11) is
We need Theorem 4.1 of [16] which states
Lemma 2.
The denominator of \hat{s\ell}(N+11)satisfies
The Weyl character formula of sℓ(N+1)
We also require a wellknown variant of the dimension formula, which itself is a corollary of the famous character formula of Weyl [39]. Let λ=m_{1}λ_{1}+⋯+m_{ N }λ_{ N } be a dominant weight of sℓ(N+1); that is, all m_{ j } are natural numbers. Letting V_{ λ } be the corresponding irreducible highestweight module, then the character formula is in our notation
Since ρ_{1} is W^{♯} invariant, we can replace ρ_{0} by ρ in this formula. Let m\in \mathbb{N} and let v be the linear map from the regular exponential functions on \frac{1}{m}{A}_{N}^{\prime} to the complex numbers defined by v(e^{λ})=1 for every \lambda \in \frac{1}{m}{A}_{N}^{\prime}. Let V_{ λ } be the irreducible finitedimensional highestweight module of highestweight λ. Hence, v(ch[ V_{ λ }]) is just the dimension of this module. The application of v to both nominator and denominator of the character formula (17) vanishes, but the quotient is finite. Using (6), we find [38]
Note that this is Weyl’s character formula for irreducible finitedimensional highestweight modules. The second equality also holds if we replace λ+ρ_{0} by z w(λ+ρ_{0}) for any complex number z and any w in W^{♯}.
Definition.
If m\in \mathbb{N} and μ in \frac{1}{m}{A}_{N}^{\prime}, then v_{ μ } is the rational exponential function
Note that if μ−ρ_{0} is dominant, then this is just the character of the irreducible highestweight module of highestweight μ−ρ_{0}. We now closely follow the argument of the proof of the dimension formula of [38].
Lemma 3.
If m\in \mathbb{N} and μ in \frac{1}{m}{A}_{N}^{\prime}, then
Proof.
Using the explicit description of the positive even roots in (13), it is easy to compute
For an arbitrary weight μ∈A N′, there exists a unique w∈W^{♯} such that w(μ+ρ_{0})−ρ_{0} is dominant. Letting ℓ(w) be the number of positive roots that are mapped to negative ones by w, then (−1)^{ℓ(w)}=σ(w) (see [38]). Then using that the even Weyl group respects the bilinear form.
Basic facts on Jacobi forms and quantum modular forms
Jacobi forms
Here we recall some special Jacobi forms and previous work on Fourier coefficients of Jacobi forms. Jacobi forms are functions from \u2102\times \mathbb{H}\to \u2102 which satisfy both an elliptic and a modular transformation law. For the precise definition and basic facts on Jacobi forms, we refer the reader to [1]. In this paper, we are particularly interested in the classical Jacobi theta function, defined in (1). The following transformation laws are well known (for example, see [40] (80.31) and (80.8)).
Lemma 4.
For \lambda ,\mu \in \mathbb{Z} and \gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \underset{2}{SL}(\mathbb{Z}), we have that
where ψ(γ) is the multiplier arising in the transformation law of Dedekind’s eta function.
We also require the following theta functions of weight \frac{1}{2}+\nu defined for r\in \mathbb{Z}, ν∈{0,1}
We define for convenience the following shifted versions when N\in 2\mathbb{N}
It is trivial to show the following identities:
In Sections ‘Second viewpoint on the decomposition into partial theta functions’ and ‘Proof of Theorem 5,’ we need the following modular transformations, which can be derived as special cases of the transformation formulas for the theta functions of Shimura [41].
Proposition 1.
If N\in 2\mathbb{N} and r\in \mathbb{Z}, then we have:
Moreover, for ν∈{0,1}, {\stackrel{~}{\mathit{\vartheta}}}_{\frac{1}{2}+\nu}(N,r;\tau ) is a modular form of weight \frac{1}{2}+\nu on Γ_{1}(2N) with multiplier
We remark that in Proposition 1, {\mathit{\vartheta}}_{\frac{1}{2}+\nu}(N,r;\tau ) are actually modular forms on a slightly larger congruence subgroup, but we have chosen to use Γ_{1}(2N) for ease of exposition.
We next recall the structure of Fourier coefficients of positive index Jacobi forms for comparison. It is well known that holomorphic Jacobi forms have a theta decomposition involving the functions
The components of this decomposition are classical (vectorvalued) modular forms [1]. The Fourier coefficients of meromorphic Jacobi forms of positive index are also understood. Specifically, in [2], Folsom and the first author, building on illuminating work of Dabholkar, Murthy, and Zagier [3] and Zwegers [6], considered the KacWakimoto character of level (M,N) with M>N, M,N\in 2\mathbb{N}, which essentially corresponds to the meromorphic Jacobi form ϕ_{M,N} (the general case with M>N is considered in [5]). These KacWakimoto characters have a decomposition into a finite and a polar part, where the finite part has a theta decomposition similar to that of holomorphic Jacobi forms (but involving mock modular forms) and where the polar part is
Here D_{ j } is the j th Laurent coefficient of the level (M,N) KacWakimoto character. Thus, we see that our functions ϕ_{M,N} have decompositions which are strikingly similar to the decompositions of positive index Jacobi forms, although in our case there are no associated ‘finite parts’. As mentioned in Remark 1 following Theorem 3, this has an interesting interpretation in physics.
Quantum modular forms
In this section, we recall some definitions and examples of quantum modular forms and describe the quantum sets in Theorem 5. We begin with a few definitions (see [42] for additional background on quasimodular forms).
Definition.
A function f:\mathbb{H}\to \u2102 is an almost holomorphic modular form of weight k on a congruence subgroup Γ if it transforms as a modular form of weight k for Γ and is a polynomial in \frac{1}{Im(\tau )} with coefficients which are holomorphic on \mathbb{H}\cup {\mathbb{P}}_{1}(\mathbb{Q}). Moreover, f is a quasimodular form of weight k if it is the constant term of an almost holomorphic modular form of weight k.
Quantum modular are then defined as follows (see [21] for background on quantum modular forms).
Definition.
For any infinite ‘quantum set’ \mathcal{Q}\subseteq \mathbb{Q}, we say a function f:\mathcal{Q}\to \u2102 is a quantum modular form of weight k on a congruence subgroup Γ if for all γ∈Γ, the cocycle
extends to an open subset of and is analytically ‘nice’. Here nice could mean continuous, smooth, realanalytic, etc. We say that f is a strong quantum modular form if there is a formal power series over attached to each point in with a stronger modularity requirement (see [21]).
Remark.
All of the quantum modular forms occurring in this paper have cocycles defined on which are realanalytic except at one point. Moreover, they have full asymptotic expansions towards rational points in their quantum sets which agree with the asymptotic expansions of mock modular forms defined on the lower half plane.
Especially relevant for us are certain partial theta functions which were shown to be quantum modular forms in recent work of Folsom, Ono, and Rhoades [23], namely,
For any a,b\in \mathbb{Z} with (a,b)=1, a>0, define the following quantum set, where all fractions are assumed to have coprime denominator and numerator throughout
Since for r=\frac{j}{2}\in \frac{1}{2}+\mathbb{Z}
it suffices to study the quantum modular properties of G(a,b;τ). Although a=0 is excluded, it is easy to handle this case directly. Note that G\left(0,1;\frac{\tau}{2}\right) is essentially a modular form as G(0,1;\tau )=\frac{\eta {(\tau )}^{2}}{2\eta (2\tau )}+\frac{1}{2} and also that G(0,2N;τ)=G(0,1;2N τ). It is clear that G\left(0,1;\frac{\tau}{2}\right) is quantum modular at any cusps where the eta quotient vanishes, namely, for \tau \in \left\{\frac{h}{k}\in \mathbb{Q}:k\equiv 1\phantom{\rule{0.3em}{0ex}}(mod\phantom{\rule{0.3em}{0ex}}2)\right\}. For a>0, the situation is more subtle. Folsom, Ono, and Rhoades proved that G(a,b;τ) have the following quantum properties:
Theorem 6([23]).
For b even, G(a,b;τ) is a strong quantum modular form of weight 1/2 with quantum set {\mathcal{Q}}_{a,b}.
Remark.
Although [23] only states the theorem for 0<a<b, an inspection of the proof shows that it is true for general integers (a,b)=1 with a>0 and b are even.
When N is even, we also have the analogous weight \frac{3}{2} partial theta functions {\Theta}_{\frac{3}{2}}(N,r;\tau ) (see Theorem 5).
The Fourier coefficients and partial theta functions of A_{ N }
In this section, we prove Theorem 2.
Proof of Theorem 2.
Define a subdomain of Y
so that in particular \mathfrak{\U0001d52e}(x)<{e}^{(\delta {\epsilon}_{N+1})(x)}<1 for all x in X. We begin with the following crucial lemma.
Lemma 5.
As a function in X, we have
Proof.
Inserting the statement of Lemma 1 into the one of Lemma 2 gives
Using (16) and the bijectivity of the map w:A_{ N }→A_{ N } for every w∈W^{♯}, we get
Let α=m_{1}α_{1}+⋯+m_{ N }α_{ N } be an element of A_{ N }, and set m_{0}:=m_{N+1}:=0. By (15), we have
Hence,
where we used the shorthand notation m=(m_{1},…,m_{ N }) and kept as before m_{0}=m_{N+1}=0. Recall that α_{ n }=ε_{ n }−ε_{n+1}. We split the exponential of the affine Weyl vector as
Note that N d−ρ_{1} is invariant under W^{♯}. Letting q_{ n }:=m_{ n }−m_{n−1}, we then find the identity
Defining the set
we obtain
In the domain X, we can expand in a geometric series to find that {e}^{\hat{\rho}}\hat{R} equals
Since the double sum converges absolutely in the domain X, we can interchange summations. Define
Then
We can express ε_{N+1}−δ in terms of the odd Weyl vector and positive even simple roots:
We see that {\epsilon}_{N+1}\delta \frac{2}{N+1}{\rho}_{1} is in \frac{1}{N+1}{A}_{N}. For (s_{1},…,s_{N+1})∈S, we find that
Combining (25) and (26), we can rewrite
with {t}_{n}:={s}_{n}\frac{r}{N+1}+r{\delta}_{n,N+1}. Then
Here we used that t_{1}+⋯+t_{N+1}=0, which follows from the same property for the s_{ n }. Let q_{ j } be as in the definition of the set S; in particular, we can write q_{ j }=m_{ j }−m_{j−1} with integers m_{ j } for 1≤j≤N, and m_{N+1}=0. Then
Using the sets T_{ r } (5), we finally get
Letting \mathfrak{\U0001d537}={e}^{\frac{2{\rho}_{1}}{N+1}}, we deduce the following.
Corollary 1.
The identity A=B C holds as functions on X, where
Proof.
The corollary follows immediately from Lemma 5 by inserting v_{ t } in the definition of C in (18).
Evaluating the expressions in this equality provides a nice expansion of ϕ_{N+1}(z;τ).
Corollary 2.
Inside the range q<ζ<1, we have
Proof.
The evaluation v maps every regular exponential e^{λ} for \lambda \in \frac{1}{N}{A}_{N1} to 1. The application of v to A and B is finite for \mathfrak{\U0001d52e}(x)<{\mathfrak{\U0001d537}}^{1}(x)<1 and x∈X, and the same is true for C by Lemma 3. The identity (25) implies that v\left({e}^{\delta {\epsilon}_{j}}\right)={e}^{\frac{2{\rho}_{1}}{N}}=\zeta for all j=1,…,N, so that
By Lemma 3,
and the evaluation v(C) follows. All three evaluations v(A),v(B),v(C) are meromorphic functions on \left\{x=2\mathrm{\pi i\tau d}+\frac{4\mathrm{\pi iz}{h}_{{\rho}_{1}}}{N1}:\text{Im}(\tau )>\text{Im}(z)>0\right\} so that the result follows with \zeta ={\mathfrak{\U0001d537}}^{1}(x) and q=\mathfrak{\U0001d52e}(x).
This completes the proof as Corollary 1 and Lemma 3 imply Theorem 2. □
The case N=1 can be proven in a very similar manner using (9), which is the denominator identity of \hat{\mathrm{g}\ell}(11) (see Example 4.1 of [16]).
Example 1.
The Fourier coefficients of ϕ_{1}(z;τ) are given by
Proof.
Suppose q<ζ<1. Expanding (9) in a geometric series and rewriting easily gives the statement.
Second viewpoint on the decomposition into partial theta functions
In this section, we prove Theorem 3 and use it to extract the Fourier coefficients of ϕ_{M,N} in Theorem 4. A key ingredient for the proof of Theorem 3 is the following result whose proof is deferred to Section ‘Proof of Lemma 6’.
Lemma 6.
For N\in \mathbb{N}, there exist meromorphic functions {f}_{j}^{\ast}(\tau ) for 0\le j\le \frac{N1{\delta}_{e}}{2} with {f}_{\frac{N1{\delta}_{e}}{2}}^{\ast}(\tau )\ne 0 such that for all r\in \mathbb{Z}
Proof of Theorem 3 for M=0
The first step in the proof of Theorem 3 is to show the following decomposition for the case when M=0:
Proposition 2.
For N\in \mathbb{N} there exist meromorphic functions g_{ j }(τ)such that
Proof.
We first determine the elliptic transformations of F_{ N }(z;τ) and prove that although this function does not, in general, transform as a negative index Jacobi form, we can ‘correct’ the elliptic transformations to match those of ϕ_{ N }(z;τ). The following periodicity property is evident:
For the elliptic transformation z↦z+τ, a direct calculation gives
Thus, we have the following elliptic transformation formula for the iterated derivative of F_{ N }(z;τ):
We now use the functions {f}_{j}^{\ast} from Lemma 6 to ‘correct’ the elliptic transformation by defining
so that
Thus, P_{ N }(z;τ) satisfies the same elliptic transformations as ϕ_{ N }(z;τ). It also has poles in the same locations and of the same order, namely, poles in \mathrm{\mathbb{Z}\tau}+\mathbb{Z} of order N. Hence, the product
is an entire elliptic function and therefore constant in z. It remains to show that P_{ N }(z;τ)≠0, which we prove by looking at the behavior as z→0. The principal part as z→0 of {\mathcal{D}}_{w}^{j}\left({F}_{N}(z,u;\tau )\right) only comes from the n=0 term in (11), which contributes
where B_{ m }(x) is the usual m th Bernoulli polynomial. Thus, as z→0,
and so
We can then use the wellknown formula
and compare the coefficients of z^{−N} to give
as by assumption {f}_{\frac{N1{\delta}_{e}}{2}}^{\ast}\ne 0. By absorbing the constants into the {f}_{j}^{\ast}, Proposition 2 follows.
Proof of Theorem 3 for M=0.
To finish the proof for M=0, we connect the functions g_{ j } in Proposition 2 to the Laurent coefficients of ϕ_{ N } given in (12) by comparing the principal parts. Namely, using (27), we easily read off:
Proof of Lemma 6
For N odd, Lemma 2.1 of [32] easily gives Lemma 6 by rearranging terms. The condition f_{0}≠0 (in the notation of [32]) is not stated explicitly in the statement; however, the proof shows that one can choose f_{0}=1 in Lemma 2.1 of [32]. Now suppose that N is even. For k\in \mathbb{N}, consider the RamanujanSerre derivative, which raises the weight of a modular form by 2:
and its iterated version starting at weight \frac{3}{2} given by {\mathcal{E}}^{n}:={\mathcal{E}}_{2n\frac{1}{2}}\circ {\mathcal{E}}_{2n\frac{5}{2}}\circ \dots \circ {\mathcal{E}}_{\frac{7}{2}}\circ {\mathcal{E}}_{\frac{3}{2}}. By rearranging, it is enough to show that there are holomorphic functions f_{ j } such that for all r\in \mathbb{Z}
This is clearly equivalent to the following, where {\stackrel{~}{\mathit{\vartheta}}}_{\frac{1}{2}+\nu}(N,r;\tau ) is defined in (19).
Lemma 7.
If N\in 2\mathbb{N}, then there exist meromorphic functions f_{ j }(τ) with {f}_{\frac{N}{2}1}(\tau )\ne 0 such that for all r\in \mathbb{Z}
Proof.
The approach taken here is similar to Zwegers’ proof of Lemma 2.1 in [32], although we give details for the reader’s convenience. Using (20) and (21), it suffices to prove the lemma for 1\le r\le \frac{N}{2}1. By (22), we may simply choose f_{0}=1 for N=2. Thus, we assume for the remainder of the proof that N≥4.
Consider the vectorvalued form
and the matrixvalued form
Using Proposition 1, we see that
Hence,
Here we used the following elementary determinant formula: