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Negative index Jacobi forms and quantum modular forms

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Research

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 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

Background

Since the introduction of the theory of Jacobi forms by Eichler and Zagier [1], connections between Jacobi forms and modular-type 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 , M 0 Open image in new window the functions
ϕ M , N ( z ; τ ) : = 𝜗 z + 1 2 ; τ M 𝜗 ( z ; τ ) N , Open image in new window
where 𝜗(z;τ) is the usual Jacobi theta function
𝜗 ( z ; τ ) : = i ζ 1 2 q 1 8 ( q ) ( ζ ) ζ 1 q . Open image in new window
(1)

Here, q : = e 2 πiτ ( τ ) Open image in new window, ζ : = e 2 πiz ( z ) Open image in new window, and for n 0 { } Open image in new window, ( a ) n : = ( a ; q ) n : = j = 0 n 1 1 a q j Open image in new window 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 [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 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 [7], as postulated by Dyson [8]. 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 [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 congruences-relating 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 ϕ1,3(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 S ( 1 ) Open image in new window be the graded vector space
S ( 1 ) : = Sym n = 0 q n + 1 2 ζℂ ζ 1 , Open image in new window
and S ( N ) = S ( 1 ) N Open image in new window. The vector space S ( N ) Open image in new window can be given the structure of a vertex algebra, the bosonic β γ-ghost 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 | 1 2 < | ζ n | < | q | 1 2 Open image in new window, where ζ n : = e 2 πi z n Open image in new window,
ch [ S ( N ) ] ( z 1 , , z N ; τ ) = n = 1 N q 1 24 ch [ S ( 1 ) ] ( z n ; τ ) = q N 24 n = 1 N 1 ζ n 1 q 1 2 ζ n q 1 2 . Open image in new window
(2)
It specializes to ϕ N (z;τ) for the following choices:
i N ζ N 2 q N 6 q N ch [ S ( N ) ] z τ 2 , , z τ 2 ; τ = ϕ N ( z ; τ ) . Open image in new window
The algebra S ( N ) Open image in new window contains as commuting subalgebras the rank 1 Heisenberg vertex algebra ( 1 ) Open image in new window and the simple affine vertex algebra of sℓ ( N ) Open image in new window at level −1, L 1 ( sℓ ( N ) ) Open image in new window. 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 S ( 2 ) Open image in new window. However, for N>2 it was shown in [12] that L 1 ( sℓ ( N ) ) Open image in new window and ( 1 ) Open image in new window form such a mutually commuting pair inside S ( N ) Open image in new window. The character of the highest-weight module F μ Open image in new window, μ Open image in new window, of ( 1 ) Open image in new window takes the form
ch F μ ( z ; τ ) = ζ N μ q μ 2 2 q 1 24 ( q ) , Open image in new window
(3)

so that the Fourier coefficients in ζ of ϕ N (z;τ) immediately allow one to compute the multiplicity with which the character of F r / N Open image in new window appears. In physics language, such a multiplicity is called the branching function of the coset S ( N ) / ( 1 ) Open image in new window.

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 ( 𝔤 ) Open image in new window, the universal affine vertex algebra of the simple Lie algebra 𝔤 Open image in new window at level k Open image in new window. For certain rational admissible levels, V k ( 𝔤 ) Open image in new window is not simple and one instead prefers to study its simple quotient L k ( 𝔤 ) Open image in new window. The characters of irreducible highest-weight modules at admissible level L k ( 𝔤 ) Open image in new window 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 s ̂ ( 2 | 1 ) Open image in new window was an essential ingredient to study the relations between characters of admissible level L k (s(2)), while we use the identities for the family s ̂ ( N | 1 ) Open image in new window 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 Φ10(Z) ( Z 2 Open image in new window, the Siegel upper half plane of genus (2) arises as the partition function of quarter-BPS dyons in the type II compactification on the product of a K3 surface and an elliptic curve. Write
1 Φ 10 ( Z ) = m = 1 ψ m ( z ; τ ) ρ m Z = τ z z w , ρ = e 2 πiw . Open image in new window
(4)
For m>0, the Fourier coefficients of the functions ψ m are the degeneracies of single-centered black holes and two-centered 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 multi-centered black holes and mock Siegel-Narain theta functions). The coefficient of m=−1 equals
1 η 18 ( τ ) 𝜗 ( z ; τ ) 2 Open image in new window

(note that our theta function 𝜗(z;τ) differs from the theta function θ1(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 single-centered black holes and the degeneracies are all interpreted as accounting for two-centered black holes (see [19],[20]). 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 [3].

Returning to the problem of studying ϕM,N, we define its Fourier coefficients by
ϕ M , N ( z ; τ ) = : r M N 2 + χ ( M , N , r ; τ ) ζ r Open image in new window

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 [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 : Q Open image in new window for some subset Q 1 ( ) Open image in new window 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 1 ( ) Open image in new window. 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 [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 Open image in new window, M 2 0 Open image in new window and r Open image in new window with M<N, the functions χ(M,N,r;τ) are mixed partial theta functions.

Remarks.

  1. 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. 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 Kac-Wakimoto 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
A N 1 : = α 1 α N 1 Open image in new window
of sℓ(N). Here, the α n are the simple roots of sℓ(N), which are linear functionals on the Cartan subalgebra 𝔥 N 1 Open image in new window of sℓ(N). The Gram matrix of AN−1 is the Cartan matrix of sℓ(N). We denote the bilinear form by (|) and abbreviate t2:=(t|t) for t in AN−1. For r in , we define the subset of 1 N A N 1 Open image in new window
T r : = n = 1 N 1 a n α n a n = ( N n ) n 2 rn N + ( N 1 ) m n , m n , m N 1 1 2 r 1 2 0 Open image in new window
(5)
and its partial theta function
P r ( τ ) : = t T r e t q t 2 2 ( N 1 ) . Open image in new window
The e t are functions on the Cartan subalgebra 𝔥 Open image in new window defined by e t : uet(u) for u in 𝔥 Open image in new window. Its evaluation for u 𝔥 Open image in new window 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 1 N A N 1 Open image in new window,
θ N ( τ ) : = t 1 N A N 1 e t ( 0 ) q t 2 2 ( N 1 ) , Open image in new window

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 1 2 M A N 1 Open image in new window. 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
α e λ : = ( λ | α ) e λ and : = α Δ 0 + α Open image in new window
(6)

where Δ 0 + Open image in new window is the set of positive roots of sℓ(N). Finally, set d N : = j = 1 N j ! Open image in new window 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
χ ( N , r ; τ ) = i N sign r N 2 q r 2 2 N η ( τ ) N ( N + 1 ) d N 1 t T r N 2 α Δ 0 + ( t | α ) q t 2 2 ( N 1 ) = i N sign r N 2 q r 2 2 N η ( τ ) N ( N + 1 ) d N 1 P r N 2 ( τ ) e t = 1 t 1 N A N 1 . Open image in new window

Remarks.

  1. 1.
    Using (2) and (3), Theorem 2 implies the character decomposition
    ch [ S ( N ) ] ( z , , z ; τ ) = r ch F r N ( z ; τ ) ch [ r ] ( τ ) , Open image in new window
     
where
ch [ r ] ( τ ) : = sign ( r ) η ( τ ) N 2 1 d N 1 t T r α Δ 0 + ( t | α ) q t 2 2 ( N 1 ) . Open image in new window
The ch [ r ] ( τ ) Open image in new window are then characters of L−1(sℓ(N)).
  1. 2.

    The proof of the theorem uses the denominator identity of both sℓ(N|1) and sℓ ̂ ( N | 1 ) Open image in new window as well as Weyl’s character formula for sℓ(N).

     
  2. 3.

    The case N=1 follows from the denominator identity of g ̂ ( 1 | 1 ) Open image in new window (see Example 1). In this case, the Fourier coefficients relate to the characters of a well-known logarithmic conformal field theory, the W ( 2 , 3 ) Open image in new window-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:
R ( ζ ; q ) : = ( 1 z ) ( q ) n ( 1 ) n q n 2 ( 3 n + 1 ) 1 ζ q n , C ( ζ ; q ) : = ( q ) ( ζq ) ( ζ 1 q ) . Open image in new window
(7)
We also need the normalized versions
R ( ζ ; q ) : = ζ 1 2 q 1 24 R ( ζ ; q ) 1 ζ , C ( ζ ; q ) : = ζ 1 2 q 1 24 C ( ζ ; q ) 1 ζ . Open image in new window
Note that ϕ N is essentially the N th power of C Open image in new window as for N Open image in new window we have
ϕ N ( z ; τ ) = i N η ( τ ) 2 N C ( ζ ; q ) N . Open image in new window
(8)
The simplest case of our decomposition relies on the fact that C Open image in new window, and thus ϕ1, is essentially an Appell-Lerch sum thanks to the following classical partial fraction expansion (for example, see Theorem 1.4 of [32]).
C ( ζ ; q ) = ζ 1 2 η ( τ ) n ( 1 ) n q n ( n + 1 ) 2 1 ζ q n . Open image in new window
(9)
For the cube of the crank generating function, Atkin and Garvan [9] proved the following rank-crank PDE which is very useful in establishing congruences and relations between the moments of the rank and crank generating functions:
2 η ( τ ) 2 C ( ζ ; q ) 3 = 6 D q + D ζ 2 R ( ζ ; q ) . Open image in new window
(10)

Here and throughout D x : = x ∂x Open image in new window. Note that this gives a description of ϕ3 in terms of Appell-Lerch 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 Appell-Lerch sums
F N ( z , u ; τ ) : = ζ N 2 w N 2 n ( w ) Nn q N 2 n ( n + 1 ) 1 ζw q n , Open image in new window
(11)
where w:=e2π i u. We note that these Appell-Lerch 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 ϕM,N(z;τ) at z=0:
ϕ M , N ( z ; τ ) = D N ( τ ) ( 2 πiz ) N + D N 2 ( τ ) ( 2 πiz ) N 2 + + O ( 1 ) . Open image in new window
(12)
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
G k ( τ ) : = B k 2 k + n 1 σ k 1 ( n ) q n , Open image in new window
where σ k ( n ) : = d | n d k Open image in new window and B k is the usual k th Bernoulli number. Specifically, it easily follows from the Jacobi triple product formula that
𝜗 ( z ; τ ) = 2 πz η 3 ( τ ) exp 2 k 1 G 2 k ( τ ) ( 2 πiz ) 2 k ( 2 k ) ! . Open image in new window
The following result puts Zwegers family of PDEs as well as our new family of PDEs into a common framework. Setting
δ e : = 0 if N 2 1 , 1 if N 2 , Open image in new window

we find:

Theorem 3.

For any N Open image in new window, M 0 Open image in new window, we have
ϕ 2 M , N + 2 M ( z ; τ ) = ( 1 ) 1 + δ e j = 0 N 1 δ e 2 + M D 2 j + δ e + 1 ( τ ) ( 2 j + δ e ) ! D w 2 j + δ e F N ( z , u ; τ ) w = 1 . Open image in new window

Remarks.

  1. 1.

    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.

     
  2. 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 Appell-Lerch sums, we can easily pick off the Fourier coefficients χ(M,N,r;τ) and write them in terms of the Laurent coefficients of ϕM,N and certain partial theta functions
Θ 1 2 + δ e ( N , r ; τ ) : = n 0 ( 1 ) Nn n + r N δ e q N 2 n + r N 2 . Open image in new window
Specifically, if we let
ρ ( r ) : = r if r N 2 , N r if r < N 2 , Open image in new window

then the Fourier coefficients of ϕM,N are as follows.

Theorem 4.

For any N Open image in new window, M 0 Open image in new window, r N 2 + Open image in new window, 0≤ Im(z)< Im(τ), we have
χ ( 2 M , N + 2 M , r ; τ ) = ( 1 ) 1 + δ e q r 2 2 N j = 0 N 1 δ e 2 + M D 2 j + δ e + 1 ( τ ) ( 2 j + δ e ) ! N j + δ e 2 j D q j Θ 1 2 + δ e ( N , ρ ( r ) ; τ ) . Open image in new window

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 Θ 1 2 ( N , r ; τ ) Open image in new window 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 2 Open image in new window and r Open image in new window, Θ 3 2 ( N , r ; τ ) Open image in new window is a strong quantum modular form with quantum set Q ̂ N , r Open image in new window (defined in (34)) on Γ1(2N), multiplier system χ r (defined in (23)), and weight 3 2 Open image in new window.

Remarks.

  1. 1.

    More details about the specific quantum modular properties can be found in the proof of Theorem 5 in Section ‘Quantum modularity of Θ 3 2 ( N , r ; τ ) Open image in new window.’

     
  2. 2.

    More generally, using Proposition 3 of [37], our proof of Theorem 5 shows that Θ 3 2 ( N , r ; τ ) Open image in new window 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 Θ 3 2 ( N , r ; τ ) Open image in new window 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 Θ 1 2 + ν ( N , r ; τ ) Open image in new window 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 sℓ ̂ ( N | 1 ) Open image in new window, following [16], as well as the finite-dimensional Lie algebra sℓ(N) using [38].

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.

The even subalgebra of the Lie super algebra sℓ(N+1|1) is g (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
L N : = ε 1 ε N + 1 ℤδ Open image in new window
with bilinear form
ε j | ε k : = δ j , k , ε j | δ : = 0 , δ | δ : = 1 . Open image in new window
Thus, its signature is (N+1,1). The set of roots is Δ=Δ0Δ1L N , where the set of even roots (respective odd roots) is denoted by Δ0 (respectively Δ1). They are
Δ 0 : = ε j ε k | 1 j , k N + 1 , j k , Δ 1 : = ε j δ , δ ε j | 1 j N + 1 . Open image in new window
(13)
It is useful to split these sets into positive and negative subroot spaces, where
Δ 0 + : = ε j ε k | 1 j < k N + 1 , Δ 1 + : = ε j δ | 1 j N + 1 , Δ + : = Δ 0 + Δ 1 + , Δ : = Δ + Δ . Open image in new window
A distinguished system of simple positive roots is then chosen to be
Π : = α j = ε j ε j + 1 , β = ε N + 1 δ | 1 j N . Open image in new window
The α j are even roots and β is the only distinguished odd simple root. The inner products of simple positive roots are
( α j | α k ) = 2 if j = k , 1 if j = k ± 1 , 0 otherwise, ( β | α j ) = δ j , N , ( β | β ) = 0 . Open image in new window
Hence, β is an isotropic root. Simple even roots generate the even root lattice
A N : = α 1 α N . Open image in new window
Its dual lattice is
A N : = λ 1 λ N , Open image in new window
where the inner product of the fundamental weights λ j with simple roots is (λ j |α k )=δj,k and (λ j |δ)=0. Roots and weights act on the Cartan subalgebra, which is
𝔥 : = α Π h α = 𝔥 0 h β , 𝔥 0 : = h α 1 h α N , Open image in new window
with basis {h α } parameterized by simple positive roots, and 𝔥 0 Open image in new window the Cartan subalgebra of sℓ(N+1). The fundamental weights λ j are identified with elements of the dual 𝔥 0 Open image in new window of 𝔥 0 Open image in new window via λ j ( h α k ) = δ j , k Open image in new window. A bilinear form (,) on 𝔥 Open image in new window is induced from the form on its dual space via
h α , h α : = α | α . Open image in new window
We remark that the Lie superalgebra sℓ(N+1|1) is then the / 2 Open image in new window-graded algebra generated by { h α , e α ± | α Π } Open image in new window subject to the Serre-Chevalley relations (14) and the graded Jacobi identity. The parity of h α and e α j ± Open image in new window is even, while the e β ± Open image in new window are odd. We denote the graded anti-symmetric bracket by [, ]: sℓ(N+1|1)× sℓ(N+1|1)→ sℓ(N+1|1). Then the Serre-Chevalley relations of the algebra are
h α , h α = 0 , h α , e α ± = ± α | α e α ± , e α + , e α = δ α , α h α , ad e α ± 1 α | α e α ± = 0 Open image in new window
(14)

for all α,αΠ and αα in the last equation. The bilinear form (,) on 𝔥 Open image in new window 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).

For this, we first need to define the Weyl vector ρ. It is the difference of the even Weyl vector ρ0 and the odd one ρ1, namely,
ρ 0 : = 1 2 α Δ 0 + α = 1 2 j = 1 N + 1 ( N + 2 2 j ) ε j , ρ 1 : = 1 2 α Δ 1 + α = 1 2 ( N + 1 ) δ + j = 1 N + 1 ε j , ρ : = ρ 0 ρ 1 = 1 2 ( N + 1 ) δ + j = 1 N + 1 ( N + 1 2 j ) ε j . Open image in new window
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
σ j : A N A N , σ j ( λ ) : = λ α j | λ α j . Open image in new window
This action naturally extends to the lattice L N via σ j (δ)=0 and
σ j ε k : = ε k ε k | α j α j = ε k δ j , k δ j + 1 , k ε j ε j + 1 = ε j + 1 j = k , ε j j = k 1 , ε k otherwise . Open image in new window

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 [16] we define

Definition.

A regular exponential function on A N′ is a finite linear combination of exponentials of the form e λ for λA 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 ew(λ) for any wW . The Weyl denominator of sℓ(N+1|1) is the rational exponential function
R = α Δ 0 + ( 1 e α ) α Δ 1 + ( 1 + e α ) . Open image in new window

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 [16] applied to our situation gives

Lemma 1.

The Weyl denominator of the Lie super algebra sℓ(N+1|1) is
e ρ R = w W σ ( w ) w e ρ 1 e δ ε N + 1 . Open image in new window

The denominator identity of the affine Lie super algebra sℓ ̂ ( N + 1 | 1 ) Open image in new window

We turn our focus to the affinization of s(N+1|1) that is
s ̂ ( N + 1 | 1 ) : = t , t 1 s ( N + 1 | 1 ) ℂC ℂd Open image in new window
with bracket
t n x , t m y : = t n + m x , y + n δ n + m , 0 x , y C , d , t n x : = n t n x , C , t n x : = C , d : = 0 Open image in new window
for all x,y∈s(N+1|1) and n , m Open image in new window. The Cartan subalgebra extends to its affine counterpart
𝔥 ̂ : = 𝔥 ℂd ℂC Open image in new window
and the bilinear form extends as
t n x , t m y : = ( x , y ) δ n + m , 0 , ( C , d ) : = 1 , ( C , C ) : = ( d , d ) : = C , t n x : = d , t n x : = 0 . Open image in new window
We identify C and d with linear functionals on 𝔥 Open image in new window using the bilinear form (,) and extend A N′ to
A ̂ N : = A N ℤd ℤC. Open image in new window
The bilinear form extends as
( C | d ) : = 1 and ( C | C ) : = ( d | d ) = C | λ = ( d | λ ) = 0 λ A N . Open image in new window
The lattice A N A N′ is then also a sublattice of A ̂ N Open image in new window. The affine Weyl vector is
ρ ̂ : = ρ + Nd. Open image in new window
Note that N is the dual Coxeter number of sℓ(N+1|1). For αA N , we define
t α : A ̂ N A ̂ N , λ λ + ( λ | C ) α ( λ | α ) + 1 2 ( α | α ) ( λ | C ) C. Open image in new window
(15)
The group of even Weyl translations is {t α |αA N }. Conjugation by a Weyl rotation gives for any wW ,αA N
w t α w 1 = t w ( α ) . Open image in new window
(16)
Let
Y : = h 𝔥 ̂ | Re ( C ( h ) ) > 0 Open image in new window

be the domain of all elements in 𝔥 ̂ Open image in new window on which the action of C has positive real part. Let F ̂ Open image in new window be the field of meromorphic functions on Y and define 𝔮 : = e C Open image in new window. Thus, | 𝔮 ( y ) | < 1 Open image in new window for all y in Y. Any element λ of L extends to a linear function on 𝔥 ̂ Open image in new window by defining λ(C)=λ(d)=0. In this way rational exponential functions on L embed in F ̂ Open image in new window.

Definition.

The denominator of sℓ ̂ ( N + 1 | 1 ) Open image in new window is
R ̂ = R j = 1 1 𝔮 j n + 1 α Δ 0 1 𝔮 j e α α Δ 1 1 + 𝔮 j e α 1 . Open image in new window

We need Theorem 4.1 of [16] which states

Lemma 2.

The denominator of sℓ ̂ ( N + 1 | 1 ) Open image in new windowsatisfies
e ρ ̂ R ̂ = α A N t α e ρ ̂ R . Open image in new window

The Weyl character formula of sℓ(N+1)

We also require a well-known variant of the dimension formula, which itself is a corollary of the famous character formula of Weyl [39]. Let λ=m1λ1+⋯+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 highest-weight module, then the character formula is in our notation
ch [ V λ ] = w W σ ( w ) e w ( λ + ρ 0 ) e ρ 0 α Δ 0 + ( 1 e α ) . Open image in new window
(17)
Since ρ1 is W invariant, we can replace ρ0 by ρ in this formula. Let m Open image in new window and let v be the linear map from the regular exponential functions on 1 m A N Open image in new window to the complex numbers defined by v(e λ )=1 for every λ 1 m A N Open image in new window. Let V λ be the irreducible finite-dimensional highest-weight module of highest-weight 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]
v ( ch [ V λ ] ) = v w W σ ( w ) e w ( λ + ρ 0 ) v e ρ 0 α Δ 0 + ( 1 e α ) = α Δ 0 + ( λ + ρ 0 | α ) α Δ 0 + ( ρ 0 | α ) . Open image in new window

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 .

Definition.

If m Open image in new window and μ in 1 m A N Open image in new window, then v μ is the rational exponential function
v μ = w W σ ( w ) e w ( μ ) e ρ 0 α Δ 0 + ( 1 e α ) . Open image in new window
(18)

Note that if μρ0 is dominant, then this is just the character of the irreducible highest-weight module of highest-weight μρ0. We now closely follow the argument of the proof of the dimension formula of [38].

Lemma 3.

If m Open image in new window and μ in 1 m A N Open image in new window, then
v ( v μ ) = 1 d N α Δ 0 + ( μ | α ) = v ( e μ ) d N . Open image in new window

Proof.

Using the explicit description of the positive even roots in (13), it is easy to compute
α Δ 0 + ( ρ 0 | α ) = n = 1 N n ! = d N . Open image in new window
For an arbitrary weight μA N′, there exists a unique wW such that w(μ+ρ0)−ρ0 is dominant. Letting (w) be the number of positive roots that are mapped to negative ones by w, then (−1)(w)=σ(w) (see [38]). Then using that the even Weyl group respects the bilinear form.
v v μ M = v w W σ ( w ) e w μ M e ρ 0 α Δ 0 + ( 1 e α ) = ( 1 ) ( w ) α Δ 0 + w μ M α α Δ 0 + ( ρ 0 | α ) = ( 1 ) ( w ) d N α Δ 0 + μ M w 1 ( α ) = ( 1 ) ( w ) + ( w 1 ) d N α Δ 0 + μ M α = 1 d N α Δ 0 + μ M α = v e μ M d N . Open image in new window

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 × Open image in new window 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 λ , μ Open image in new window and γ = a b c d SL 2 ( ) Open image in new window, we have that
𝜗 z + d ; + b + d = ψ 3 ( γ ) ( + d ) 1 2 e πic z 2 + d 𝜗 ( z ; τ ) , Open image in new window
𝜗 ( z + λτ + μ ; τ ) = ( 1 ) λ + μ q λ 2 2 e 2 πiλz 𝜗 ( z ; τ ) , Open image in new window

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 1 2 + ν Open image in new window defined for r Open image in new window, ν∈{0,1}
𝜗 1 2 + ν ( N , r ; τ ) : = n ( 1 ) nN n + r N 1 2 ν q N 2 n + r N 1 2 2 . Open image in new window
We define for convenience the following shifted versions when N 2 Open image in new window
𝜗 ~ 1 2 + ν ( N , r ; τ ) : = 𝜗 1 2 + ν N , r + N 2 ; τ . Open image in new window
(19)
It is trivial to show the following identities:
𝜗 ~ 1 2 + ν ( N , r + N ; τ ) = 𝜗 ~ 1 2 + ν ( N , r ; τ ) , Open image in new window
(20)
𝜗 ~ 1 2 + ν ( N , r ; τ ) = ( 1 ) ν 𝜗 ~ 1 2 + ν ( N , r ; τ ) , Open image in new window
(21)
𝜗 1 2 ( 1 , 0 ; τ ) = 𝜗 ~ 3 2 ( 2 , r ; τ ) = 0 . Open image in new window
(22)

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 2 Open image in new window and r Open image in new window, then we have:
𝜗 ~ 3 2 N , r ; τ + 1 = e r 2 2 N 𝜗 ~ 3 2 ( N , r ; τ ) , Open image in new window
𝜗 ~ 3 2 N , r ; 1 τ = 2 N ( ) 3 2 k = 1 N 2 1 sin 2 πkr N 𝜗 ~ 3 2 ( N , k ; τ ) . Open image in new window
Moreover, for ν∈{0,1}, 𝜗 ~ 1 2 + ν ( N , r ; τ ) Open image in new window is a modular form of weight 1 2 + ν Open image in new window on Γ1(2N) with multiplier
χ r a b c d : = e r 2 2 N if c = 0 , e b r 2 2 N 2 Nc d if c 0 . Open image in new window
(23)

We remark that in Proposition 1, 𝜗 1 2 + ν ( N , r ; τ ) Open image in new window are actually modular forms on a slightly larger congruence subgroup, but we have chosen to use Γ1(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
𝜗 m , b ( z ; τ ) : = λ λ b ( mod 2 m ) e πi λ 2 τ 2 m + 2 πiλz . Open image in new window
The components of this decomposition are classical (vector-valued) 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 Kac-Wakimoto character of level (M,N) with M>N, M , N 2 Open image in new window, which essentially corresponds to the meromorphic Jacobi form ϕM,N (the general case with M>N is considered in [5]). These Kac-Wakimoto 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
φ P ( z ; τ ) : = j = 1 N 2 D 2 j ( τ ) ( 2 πi ) 2 j 1 ( 2 j 1 ) ! 2 j 1 u 2 j 1 e πi ( M N ) u ζ M N 2 F M N τ 2 u , z τ 2 ; τ u = 0 . Open image in new window
(24)

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 [42] for additional background on quasimodular forms).

Definition.

A function f : Open image in new window 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 1 Im ( τ ) Open image in new window with coefficients which are holomorphic on 1 ( ) Open image in new window. 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’ Q Open image in new window, we say a function f : Q Open image in new window is a quantum modular form of weight k on a congruence subgroup Γ if for all γΓ, the cocycle
r γ ( τ ) : = f | k ( 1 γ ) ( τ ) Open image in new window

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 [21]).

Remark.

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.

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,
G ( a , b ; τ ) : = n 0 ( 1 ) n q n + a b 2 . Open image in new window
For any a , b Open image in new window with (a,b)=1, a>0, define the following quantum set, where all fractions are assumed to have coprime denominator and numerator throughout
Q a , b : = h k : h > 0 , b | 2 h , b h , k a ( mod b ) , k a . Open image in new window
Since for r = j 2 1 2 + Open image in new window
Θ 1 2 ( N , r ; τ ) = G j , 2 N ; 2 , Open image in new window

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 0 , 1 ; τ 2 Open image in new window is essentially a modular form as G ( 0 , 1 ; τ ) = η ( τ ) 2 2 η ( 2 τ ) + 1 2 Open image in new window and also that G(0,2N;τ)=G(0,1;2N τ). It is clear that G 0 , 1 ; τ 2 Open image in new window is quantum modular at any cusps where the eta quotient vanishes, namely, for τ h k : k 1 ( mod 2 ) Open image in new window. 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 Q a , b Open image in new window.

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 3 2 Open image in new window partial theta functions Θ 3 2 ( N , r ; τ ) Open image in new window (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
X : = h Y | Re ( C ( h ) ) > Re ( α ( h ) ) > 0 α Δ + Open image in new window

so that in particular | 𝔮 ( x ) | < | e ( δ ε N + 1 ) ( x ) | < 1 Open image in new window for all x in X. We begin with the following crucial lemma.

Lemma 5.

As a function in X, we have
e ρ ̂ R ̂ = e Nd ρ 1 q ( N + 1 ) ( N + 2 ) 24 r sign ( r ) ( 1 ) r 𝔮 r 2 2 ( N + 1 ) 𝔮 r 2 e r 2 ρ 1 N + 1 t T r w W σ ( w ) w e t 𝔮 t 2 2 N . Open image in new window

Proof.

Inserting the statement of Lemma 1 into the one of Lemma 2 gives
e ρ ̂ R ̂ = α A N t α e ρ ̂ R = α A N t α w W σ ( w ) w e ρ ̂ ( 1 + e δ ε N + 1 ) . Open image in new window
Using (16) and the bijectivity of the map w:A N A N for every wW , we get
e ρ ̂ R ̂ = α A N w W σ ( w ) w t w 1 ( α ) e ρ ̂ ( 1 + e δ ε N + 1 ) = α A N w W σ ( w ) w t α e ρ ̂ ( 1 + e δ ε N + 1 ) . Open image in new window
Let α=m1α1+⋯+m N α N be an element of A N , and set m0:=mN+1:=0. By (15), we have
t α ( ρ ̂ ) = ρ ̂ + n = 1 N + 1 m n + N 2 ( m n m n 1 ) 2 C and t α δ ε N + 1 = δ ε N + 1 m N C. Open image in new window
Hence,
e ρ ̂ R ̂ = m N w W σ ( w ) w e ρ ̂ n = 1 N + 1 e Nm n α n 𝔮 m n + N 2 ( m n m n 1 ) 2 1 + e δ ε N + 1 𝔮 m N 1 , Open image in new window
where we used the short-hand notation m=(m1,…,m N ) and kept as before m0=mN+1=0. Recall that α n =ε n εn+1. We split the exponential of the affine Weyl vector as
e ρ ̂ = e Nd ρ 1 n = 1 N + 1 e ( N + 2 2 n ) 2 ε n . Open image in new window
Note that N dρ1 is invariant under W . Letting q n :=m n mn−1, we then find the identity
n = 1 N + 1 N 2 q n 2 + m n = 1 2 N n = 1 N + 1 N q n + 1 2 + ( 1 n ) 2 ( N + 1 ) ( N + 2 ) 24 . Open image in new window
Defining the set
S : = ( s 1 , , s N + 1 ) 1 2 N + 1 s n = N q n + 1 2 + ( 1 n ) , q n , n = 1 N + 1 q n = 0 , Open image in new window
we obtain
e ρ ̂ R ̂ = e Nd ρ 1 𝔮 ( N + 1 ) ( N + 2 ) 24 s S w W σ ( w ) w n = 1 N + 1 e s n ε n 𝔮 s n 2 2 N 1 + e δ ε N + 1 𝔮 s N + 1 N 1 2 1 . Open image in new window
In the domain X, we can expand in a geometric series to find that e ρ ̂ R ̂ Open image in new window equals
e Nd ρ 1 𝔮 ( N + 1 ) ( N + 2 ) 24 s S s N + 1 1 2 r = 0 ( 1 ) r w W σ ( w ) w e r ( δ ε N + 1 ) 𝔮 r s N + 1 N + 1 2 n = 1 N + 1 e s n ε n 𝔮 s n 2 2 N s S s N + 1 > 1 2 r = 1 ( 1 ) r w W σ ( w ) w e r δ ε N + 1 𝔮 r s N + 1 N + 1 2 n = 1 N + 1 e s n ε n 𝔮 s n 2 2 N . Open image in new window
Since the double sum converges absolutely in the domain X, we can interchange summations. Define
g r : = s S s N + 1 1 2 w W σ ( w ) w e r δ ε N + 1 𝔮 r s N + 1 N + 1 2 n = 1 N + 1 e s n ε n 𝔮 s n 2 2 N if r 0 , s S s N + 1 > 1 2 w W σ ( w ) w e r δ ε N + 1 𝔮 r s N + 1 N + 1 2 n = 1 N + 1 e s n ε n 𝔮 s n 2 2 N if r > 0 . Open image in new window
Then
e ρ ̂ R ̂ = e Nd ρ 1 𝔮 ( N + 1 ) ( N + 2 ) 24 r ( 1 ) r sign ( r ) g r . Open image in new window
We can express εN+1δ in terms of the odd Weyl vector and positive even simple roots:
ε N + 1 δ = δ + 1 N + 1 n = 1 N + 1 ε n + n = 1 N n ( ε n + 1 ε n ) = 2 N + 1 ρ 1 + ε N + 1 1 N + 1 n = 1 N + 1 ε n . Open image in new window
(25)
We see that ε N + 1 δ 2 N + 1 ρ 1 Open image in new window is in 1 N + 1 A N Open image in new window. For (s1,…,sN+1)∈S, we find that
s N + 1 r N + 1 + r 2 + n = 1 N s n r N + 1 2 = N r 2 N + 1 + 2 rs N + 1 + n = 1 N + 1 s n 2 . Open image in new window
(26)
Combining (25) and (26), we can rewrite
e r δ ε N + 1 𝔮 r s N + 1 N + 1 2 n = 1 N + 1 e s n ε n 𝔮 s n 2 2 N = e r 2 ρ 1 N + 1 𝔮 r 2 2 ( N + 1 ) 𝔮 r 2 n = 1 N + 1 e t n ε n 𝔮 t n 2 2 N Open image in new window
with t n : = s n r N + 1 + r δ n , N + 1 Open image in new window. Then
n = 1 N + 1 t n ε n = n = 1 N + 1 j = 1 n t j ( ε j ε j + 1 ) = n = 1 N j = 1 n t j α n . Open image in new window
Here we used that t1+⋯+tN+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 mj−1 with integers m j for 1≤jN, and mN+1=0. Then
j = 1 n t j = rn N + 1 + j = 1 n s j = n ( N n + 1 ) 2 rn N + 1 + j = 1 n Nq j = ( N n + 1 ) n 2 rn N + 1 + Nm n . Open image in new window
Using the sets T r (5), we finally get
g r = e r 2 ρ 1 N + 1 𝔮 r 2 2 ( N + 1 ) 𝔮 r 2 t T r w W σ ( w ) w e t 𝔮 t 2 2 N . Open image in new window

Letting 𝔷 = e 2 ρ 1 N + 1 Open image in new window, we deduce the following.

Corollary 1.

The identity A=B C holds as functions on X, where
A : = j = 1 N + 1 𝔮 1 24 ( e δ ε j ; 𝔮 ) ( e ε j δ 𝔮 ; 𝔮 ) , B : = 𝔮 ( N + 1 ) 2 24 ( 𝔮 ; 𝔮 ) N + 1 α Δ 0 ( e α 𝔮 ; 𝔮 ) ( e α 𝔮 ; 𝔮 ) , C : = r sign ( r ) 𝔷 r 𝔮 r 2 2 ( N + 1 ) 𝔮 r 2 t T r v t 𝔮 t 2 2 N . Open image in new window

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 ϕN+1(z;τ).

Corollary 2.

Inside the range |q|<|ζ|<1, we have
ϕ N ( z ; τ ) = i N d N 1 η ( τ ) N 2 + N r ζ r + N 2 sign ( r ) q r + N 2 2 2 N t T r α Δ 0 + ( t | α ) q t 2 2 ( N 1 ) . Open image in new window

Proof.

The evaluation v maps every regular exponential e λ for λ 1 N A N 1 Open image in new window to 1. The application of v to A and B is finite for | 𝔮 ( x ) | < | 𝔷 1 ( x ) | < 1 Open image in new window and xX, and the same is true for C by Lemma 3. The identity (25) implies that v e δ ε j = e 2 ρ 1 N = ζ Open image in new window for all j=1,…,N, so that
v ( A ) = 𝔮 N 24 𝔷 1 ; 𝔮 N ( 𝔷 𝔮 ; 𝔮 ) N and v ( B ) = 𝔮 N 2 24 ( 𝔮 ; 𝔮 ) N 2 . Open image in new window
By Lemma 3,
v ( v t ) = 1 d N 1 α Δ 0 + ( t | α ) Open image in new window

and the evaluation v(C) follows. All three evaluations v(A),v(B),v(C) are meromorphic functions on x = 2 πiτd + 4 πiz h ρ 1 N 1 : Im ( τ ) > Im ( z ) > 0 Open image in new window so that the result follows with ζ = 𝔷 1 ( x ) Open image in new window and q = 𝔮 ( x ) Open image in new window.

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 g ̂ ( 1 | 1 ) Open image in new window (see Example 4.1 of [16]).

Example 1.

The Fourier coefficients of ϕ1(z;τ) are given by
χ ( 1 , r ; τ ) = i q r 2 2 η ( τ ) 3 m = 0 ( 1 ) m q m + r 1 2 + 1 2 2 2 . Open image in new window

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 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’.

Lemma 6.

For N Open image in new window, there exist meromorphic functions f j ( τ ) Open image in new window for 0 j N 1 δ e 2 Open image in new window with f N 1 δ e 2 ( τ ) 0 Open image in new window such that for all r Open image in new window
j = 0 N 1 δ e 2 f j ( τ ) D q j 𝜗 1 2 + δ e ( N , r ; τ ) = 0 . Open image in new window

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 Open image in new window there exist meromorphic functions g j (τ)such that
ϕ N ( z ; τ ) = j = 0 N 1 δ e 2 g j ( τ ) D w 2 j + δ e F N ( z , u ; τ ) w = 1 . Open image in new window

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:
F N ( z + 1 , u ; τ ) = ( 1 ) N F N ( z , u ; τ ) . Open image in new window
For the elliptic transformation zz+τ, a direct calculation gives
( 1 ) N ζ N q N 2 F N ( z + τ , u ; τ ) F N ( z , u ; τ ) Open image in new window
= r = 0 N 1 ζ r N 2 q 1 2 N r N 2 2 n ( 1 ) Nn w Nn + r N 2 q N 2 n 1 2 + r N 2 . Open image in new window
Thus, we have the following elliptic transformation formula for the iterated derivative of F N (z;τ):
( 1 ) N ζ N q N 2 D w 2 j + δ e ( F N ( z + τ , u ; τ ) ) w = 1 D w 2 j + δ e ( F N ( z , u ; τ ) ) w = 1 Open image in new window
= 2 j N j + δ e r = 0 N 1 ζ r N 2 q 1 2 N r N 2 2 D q j 𝜗 1 2 + δ e ( N , r ; τ ) . Open image in new window
We now use the functions f j Open image in new window from Lemma 6 to ‘correct’ the elliptic transformation by defining
P N ( z ; τ ) : = j = 0 N 1 δ e 2 f j ( τ ) 2 j N j + δ e D w 2 j + δ e ( F N ( z , u ; τ ) ) w = 1 Open image in new window
so that
P N ( z + 1 ; τ ) = ζ N q N 2 P N ( z + τ ; τ ) = ( 1 ) N P N ( z ; τ ) . Open image in new window
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 ℤτ + Open image in new window of order N. Hence, the product
p N ( z ; τ ) : = 𝜗 ( z ; τ ) N P N ( z ; τ ) Open image in new window
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 D w j F N ( z , u ; τ ) Open image in new window only comes from the n=0 term in (11), which contributes
( ζw ) N 2 1 ζw = m 0 B m N 2 ( 2 πi ( u + z ) ) m 1 m ! = 1 2 πi ( u + z ) + O ( 1 ) , Open image in new window
where B m (x) is the usual m th Bernoulli polynomial. Thus, as z→0,
D w j ( F N ( z , u ; τ ) ) w = 1 = ( 1 ) j + 1 j ! ( 2 πiz ) j + 1 + O ( 1 ) Open image in new window
(27)
and so
P N ( z ; τ ) = ( 1 ) N ( N 1 ) ! f N 1 δ e 2 ( τ ) N δ e ( 2 N ) N 1 δ e 2 ( 2 πiz ) N + O z N + 1 . Open image in new window
We can then use the well-known formula
𝜗 ( 0 ; τ ) = 2 πη ( τ ) 3 Open image in new window
and compare the coefficients of zN to give
p N ( z ; τ ) = ( N 1 ) ! ( i ) N f N 1 δ e 2 ( τ ) N δ e ( 2 N ) N 1 δ e 2 0 , Open image in new window

as by assumption f N 1 δ e 2 0 Open image in new window. By absorbing the constants into the f j Open image in new window, 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:
g j ( τ ) = ( 1 ) δ e + 1 D 2 j + δ e + 1 ( τ ) 2 j + δ e ! . Open image in new window

Proof of Lemma 6

For N odd, Lemma 2.1 of [32] easily gives Lemma 6 by rearranging terms. The condition f0≠0 (in the notation of [32]) is not stated explicitly in the statement; however, the proof shows that one can choose f0=1 in Lemma 2.1 of [32]. Now suppose that N is even. For k Open image in new window, consider the Ramanujan-Serre derivative, which raises the weight of a modular form by 2:
E k : = D q k 12 E 2 ( τ ) Open image in new window
and its iterated version starting at weight 3 2 Open image in new window given by E n : = E 2 n 1 2 E 2 n 5 2 E 7 2 E 3 2 Open image in new window. By rearranging, it is enough to show that there are holomorphic functions f j such that for all r Open image in new window
j = 0 N 2 1 f j ( τ ) E j 𝜗 3 2 ( N , r ; τ ) = 0 . Open image in new window

This is clearly equivalent to the following, where 𝜗 ~ 1 2 + ν ( N , r ; τ ) Open image in new window is defined in (19).

Lemma 7.

If N 2 Open image in new window, then there exist meromorphic functions f j (τ) with f N 2 1 ( τ ) 0 Open image in new window such that for all r Open image in new window
j = 0 N 2 1 f j ( τ ) E j 𝜗 ~ 3 2 ( N , r ; τ ) = 0 . Open image in new window
(28)

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 r N 2 1 Open image in new window. By (22), we may simply choose f0=1 for N=2. Thus, we assume for the remainder of the proof that N≥4.

Consider the vector-valued form
𝜗 N ( τ ) : = 𝜗 ~ 3 2 ( N , 1 ; τ ) , , 𝜗 ~ 3 2 N , N 2 1 ; τ T Open image in new window
and the matrix-valued form
T N : = 𝜗 N , E 𝜗 N , , E N 2 2 𝜗 N . Open image in new window
Using Proposition 1, we see that
T N ( τ + 1 ) = diag e j 2 2 N 1 j N 2 1 T N ( τ ) , Open image in new window
T N 1 τ = 2 N ( ) 3 2 sin 2 πkℓ N 1 k , N 2 1 T N ( τ ) diag τ 2 j 2 1 j N 2 1 . Open image in new window
Hence,
det T N ( τ + 1 ) = e ( N 1 ) ( N