Negative index Jacobi forms and quantum modular forms
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 rank-crank 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
Here, , , and for , is the q-Pochhammer 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 half-integral weight modular forms . 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-. 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 ϕM,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):=ϕ0,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 ϕ1 is related to the famous Andrews-Dyson-Garvan 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 , as postulated by Dyson . In this paper we describe relations between powers of the crank-generating function with certain Appell-Lerch series, giving a new family of partial differential equations (PDEs) generalizing the ‘rank-crank PDE’ of Atkin and Garvan  (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 congruences-relating ranks and cranks, as well as useful relations between the rank and crank moments . We also note the other examples of similar PDEs related to combinatorics have shown up in, for example Section 3.2 of , where the function ϕ1,3(z;τ) is studied in relation to overpartitions.
so that the Fourier coefficients in ζ of ϕ N (z;τ) immediately allow one to compute the multiplicity with which the character of appears. In physics language, such a multiplicity is called the branching function of the coset .
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 , the universal affine vertex algebra of the simple Lie algebra at level . For certain rational admissible levels, is not simple and one instead prefers to study its simple quotient . The characters of irreducible highest-weight modules at admissible level are the sum expansions in special domains of meromorphic Jacobi forms . Understanding these sum expansions is crucial in studying the modular data of the corresponding conformal field theory ,.
Fourthly, the functions ϕ N appear in the denominator identities of affine Lie super algebras . In , the denominator identity of 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 to prove one of our central theorems.
(note that our theta function 𝜗(z;τ) differs from the theta function θ1(z;τ) in the notation of  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 , 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 single-centered black holes and the degeneracies are all interpreted as accounting for two-centered black holes (see ,). In contrast, in the case m>0, the mock part of ψ m corresponds to single-centered black holes and the Appel Lerch sum corresponds to two-centered black hole bound states .
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 well-defined 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  (see also -). Although the definition is not rigorous, Zagier gave a number of motivating examples. Roughly speaking, a weight k quantum modular form is a function for some subset 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 near-modular property for asymptotic expansions defined at every point in a subset of . Moreover, this behavior comes from the ‘leaking’ of modularity properties of a non-holomorphic Eichler integral defined on the lower half plane (see (38)).
Returning to the Fourier coefficients of ϕM,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 , unimodal sequences ,, and stacks . 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:
For any , and with M<N, the functions χ(M,N,r;τ) are mixed partial theta functions.
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).
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 ,,, this settles the final cases of modularity of Kac-Wakimoto characters raised in .
is a modular form of weight (N−1)/2 for Γ(M) with M=N2(N−1)/2. This statement is true, since θ N is the theta function of the lattice . The level of this lattice is M, and the modularity of theta functions of lattices is discussed for example in .
where is the set of positive roots of sℓ(N). Finally, set and let sign(r)=1 if r≥0 and −1 otherwise. Then we have the following.
The proof of the theorem uses the denominator identity of both sℓ(N|1) and as well as Weyl’s character formula for sℓ(N).
The case N=1 follows from the denominator identity of (see Example 1). In this case, the Fourier coefficients relate to the characters of a well-known logarithmic conformal field theory, the -algebra of central charge −2. The modularity of the coefficients has been studied from a different perspective in .
Here and throughout . Note that this gives a description of ϕ3 in terms of Appell-Lerch sums by (8).
Zwegers  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 .
Note that Theorem 3 is more explicit than Zwegers’ rank-crank 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.
It would be interesting to find a Lie theoretic interpretation of the decomposition in Theorem 3.
then the Fourier coefficients of ϕM,N are as follows.
If N>1 is odd, these partial theta functions fit into the pioneering work of Folsom, Ono, and Rhoades  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 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  to study quantum modularity properties (see also ). A key ingredient in our investigation is a beautiful identity of Warnaar  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.
More details about the specific quantum modular properties can be found in the proof of Theorem 5 in Section ‘Quantum modularity of .’
More generally, using Proposition 3 of , our proof of Theorem 5 shows that has modularity properties on all of . For this, we note that although the function is not defined on all of , it has a well-defined asymptotic expansion at all points in . This expansion still agrees with the non-holomorphic Eichler integral on the lower half plane (see Section ‘Proof of Theorem 5’), so one could say that 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 rank-crank PDE in ‘Second viewpoint on the decomposition into partial theta functions’ section. We conclude by describing the quantum modular properties of in ‘Proof of Theorem 5’ section.
Preliminaries on Lie super algebras and character identities
The Lie super algebra sℓ(N+1|1)
In this subsection, the Lie super algebra sℓ(N+1|1) and its root system are defined.
for all α,α′∈Π and α≠α′ in the last equation. The bilinear form (,) on can be extended to an invariant non-degenerate graded symmetric form on sℓ(N+1|1), which we also denote by (,).
The even Weyl group and denominator identity of sℓ(N+1|1)
We now introduce the even Weyl group and the denominator identity of the Lie super algebra sℓ(N+1|1).
Hence, the even Weyl group W ♯ is just the group SN+1 permuting the ε j . Orthonormality of the ε j implies that the even Weyl group preserves the bilinear form (|). Following  we define
We saw that the even Weyl group W ♯ is just SN+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  applied to our situation gives
The denominator identity of the affine Lie super algebra
be the domain of all elements in on which the action of C has positive real part. Let be the field of meromorphic functions on Y and define . Thus, for all y in Y. Any element λ of L′ extends to a linear function on by defining λ(C)=λ(d)=0. In this way rational exponential functions on L′ embed in .
We need Theorem 4.1 of  which states
The Weyl character formula of sℓ(N+1)
Note that this is Weyl’s character formula for irreducible finite-dimensional highest-weight modules. The second equality also holds if we replace λ+ρ0 by z w(λ+ρ0) for any complex number z and any w in W ♯ .
Note that if μ−ρ0 is dominant, then this is just the character of the irreducible highest-weight module of highest-weight μ−ρ0. We now closely follow the argument of the proof of the dimension formula of .
Basic facts on Jacobi forms and quantum modular forms
Here we recall some special Jacobi forms and previous work on Fourier coefficients of Jacobi forms. Jacobi forms are functions from 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 . 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  (80.31) and (80.8)).
where ψ(γ) is the multiplier arising in the transformation law of Dedekind’s eta function.
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 .
We remark that in Proposition 1, are actually modular forms on a slightly larger congruence subgroup, but we have chosen to use Γ1(2N) for ease of exposition.
Here D j is the j th Laurent coefficient of the level (M,N) Kac-Wakimoto character. Thus, we see that our functions ϕM,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  for additional background on quasimodular forms).
A function 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 with coefficients which are holomorphic on . 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  for background on quantum modular forms).
extends to an open subset of and is analytically ‘nice’. Here nice could mean continuous, smooth, real-analytic, 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 ).
All of the quantum modular forms occurring in this paper have cocycles defined on which are real-analytic 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.
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 is essentially a modular form as and also that G(0,2N;τ)=G(0,1;2N τ). It is clear that is quantum modular at any cusps where the eta quotient vanishes, namely, for . For a>0, the situation is more subtle. Folsom, Ono, and Rhoades proved that G(a,b;τ) have the following quantum properties:
For b even, G(a,b;τ) is a strong quantum modular form of weight 1/2 with quantum set .
Although  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 partial theta functions (see Theorem 5).
The Fourier coefficients and partial theta functions of A N
In this section, we prove Theorem 2.
Proof of Theorem 2.
so that in particular for all x in X. We begin with the following crucial lemma.
Letting , we deduce the following.
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 ϕN+1(z;τ).
and the evaluation v(C) follows. All three evaluations v(A),v(B),v(C) are meromorphic functions on so that the result follows with and .
This completes the proof as Corollary 1 and Lemma 3 imply Theorem 2. □
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 ϕM,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’.
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:
as by assumption . By absorbing the constants into the , Proposition 2 follows.
Proof of Theorem 3 for M=0.
Proof of Lemma 6
This is clearly equivalent to the following, where is defined in (19).
The approach taken here is similar to Zwegers’ proof of Lemma 2.1 in , although we give details for the reader’s convenience. Using (20) and (21), it suffices to prove the lemma for . By (22), we may simply choose f0=1 for N=2. Thus, we assume for the remainder of the proof that N≥4.