On the modularity of certain functions from the Gromov–Witten theory of elliptic orbifolds

In this paper, we study modularity of several functions which naturally arose in a recent paper of Lau and Zhou on open Gromov–Witten potentials of elliptic orbifolds. They derived a number of examples of indefinite theta functions, and we provide modular completions for several such functions which involve more complicated objects than ordinary modular forms. In particular, we give new closed formulae for special indefinite theta functions of type (1,2) in terms of products of mock modular forms. This formula is also of independent interest.


Introduction and statement of results
In the recent paper [1], Lau & Zhou studied a number of generating functions of importance in Gromov-Witten theory and mirror symmetry, and they showed modularity for several of them. To be more precise, they considered the four elliptic P 1 orbifolds denoted by P 1 a for a ∈ {(3, 3, 3), (2,4,4), (2,3,6), (2, 2, 2, 2)}. In particular, for these choices of a, they explicitly computed the open Gromov-Witten potential W q (x, y, z) of P 1 a , which is in particular a polynomial in x, y, z over the ring of power series in q (where q is interpreted as the Kähler parameter of the orbifold), and which is closely tied with constructions of the associated Landau-Ginzburg mirror. The reader is also referred to [2,3] for related results, as well as to sections 2 and 3 of [1] for the definitions of the relevant geometric objects. Lau & Zhou then proved the following in theorem 1.1 of [1]. Here as usual for c ∈ N We describe our partial answer to question 1.2 in the form of several theorems which give the modular completions of several functions arising in the (2,3,6) case. In each of these cases, we prove modularity by first representing the functions in terms of the μ-function, the Jacobi theta function, and well-known modular forms (see §2 for the definitions).
In order to prove these results, we first establish an identity, which is also of independent interest. To state it, we let with q := e 2πiτ (τ ∈ H) and ζ j := e 2πiz j (z j ∈ C) for j = 1, 2, 3. We note that F is an indefinite theta function of type (1,2).
Our main results can then be stated as follows, where the functions c y 1 , c yz2 and c yz4 are certain coefficients of W q (2, 3, 6) (see (2.13)). Theorem 1.4. The function c y is modular, and c yz2 and c yz4 have explicit non-holomorphic modular completions c yz2 and c yz4 . More specifically, we have: (i) the function c y is a cusp form of weight 3 2 on SL 2 (Z) with multiplier system ν 3 η ; (ii) the function c yz2 is modular (i.e. transforms as a modular form) of weight 2 on SL 2 (Z) with shadow y 3/2 |η| 6 ; and (iii) the function c yz4 is modular of weight 5 2 and is a polynomial of degree 2 in R(0; τ ) over the ring of holomorphic functions on H.

Remarks.
(i) The explicit statements and proofs of the modularity of the functions in theorem 1.4 are given in §3. (ii) Results concerning the modularity properties of these functions could also be proved using work (in progress) of Westerholt-Raum or of Zagier and Zwegers. Moreover, the general shape of the completion of c z should also follow from the same works. We note that the indefinite theta function F we consider here is of a degenerate type and is not representative of the generic case.
Owing to this degeneracy, we are able to express it in terms 'classical' objects, which simply is not possible in the generic case. The paper is organized as follows. In §2, we collect some important facts and definitions from the theory of modular forms, Jacobi forms and mock modular forms, and we define the functions described in theorem 1.4. In §3.1, we prove theorem 1.3. We conclude §3 by giving the explicit statements and proofs comprising theorem 1.4.

Basic modular-type objects
Throughout the paper, we require a few standard examples of modular forms and related objects. Firstly, we recall the Dedekind eta function η(τ ) := q 1/24 We recall that η is a weight 1 2 cusp form on SL 2 (Z) (with a multiplier which we denote by ν η ). We shall also frequently use the quasi-modular Eisenstein series E 2 , which is essentially the logarithmic derivative of η: As is well known, E 2 is not a modular form but has a slightly more complicated modularity property, known as quasi-modularity. Specifically, E 2 is 1-periodic and satisfies the following near-modularity under inversion: Using this transformation, one can also show that the completed function where τ = u + iv, is modular of weight 2. More generally we require the higher weight Eisenstein series, defined by for even natural numbers k by where B k is the kth Bernoulli number. For k ≥ 4, these are modular forms. In addition to these q-series, we also need the Jacobi theta function, defined by ϑ(z; τ ) := n∈1/2+Z q n 2 /2 e 2πi(z+1/2)n .
The Jacobi triple product identity is the following product expansion: where ζ := e 2πiz . In particular, this identity implies that the zeros of z → ϑ(z; τ ), lie exactly at lattice points z ∈ Zτ + Z. We also need the following standard formula: Moreover, ϑ is an important example of a Jacobi form (of weight and index 1 2 ), which essentially means that it satisfies a mixture of transformation laws resembling those of elliptic functions and of modular forms. In particular, we have the following well-known transformation laws. We note that throughout we suppress τ -dependencies whenever they are clear from context.  We next prove an identity for E 2 in terms of an Appell-Lerch sum, which we need to compute the completion of c yz4 . We note in passing that while it is plausible that this identity has been considered before, the authors could not find a specific reference in the literature.

Lemma 2.2. The following holds:
Proof. By (7) of [4] and (2.3), we have where ζ (s) denotes the Riemann zeta function. Using this, we directly find that the coefficient of . We now compare this with the well-known partial fraction expansion of 1/ϑ(z). Namely, a standard application of the Mittag-Leffler theorem gives the following formula (cf. p. 136 of [5] or p. of [6]): An elementary calculation then shows that the coefficient of which, together with the computation above, implies (2.6).
We conclude this subsection by giving an identity for a certain quotient of theta functions in terms of an indefinite theta function which we need for the proof of theorem 1.4. Throughout, we set y j := Im(z j ) for j ∈ {1, 2, 3}.
Proof. In the given range, we may use geometric series to expand: In the notation of theorem of section 3 of [4], this last expression is exactly −F τ (2π iz 1 , 2π iz 2 ) (cf. the first line of the proof of theorem 3 there). The result then follows directly from (vii) of theorem 3 of [4] and (2.3).

The μ function and explicit weight 1 2 mock modular forms
Throughout, we require an important function used in [7] to study several of Ramanujan's mock theta functions. The μ-function is given in terms of an Appell-Lerch series for z 1 , z 2 ∈ C\(Zτ + Z) and τ ∈ H as where ζ j := e 2πiz j ( j = 1, 2). The function μ is a mock Jacobi form, which in particular means that it 'nearly' transforms as a Jacobi form of two variables. It turns out that μ is symmetric in z 1 and z 2 (see proposition We note that the poles of z j → μ(z 1 , z 2 ) are at z 1 , z 2 ∈ Zτ + Z, and by lemma 2.4 and (2.9) the residues are determined by .
The results of [7] give a completion of μ to a (non-holomorphic) Jacobi form. To describe this, we first require the special function R, given by where z = x + iy and E is the entire function Defining the completion Theorem 1.11 of [7] shows that μ transforms like a Jacobi form.
Theorem 2.5. The function μ satisfies the following: The reason that μ is called a mock Jacobi form is closely connected to theorem 2.5. Namely, it follows directly from the theory of Jacobi forms that if z 1 and z 2 are specialized to torsion points, then the completed functionμ is a harmonic Maass form of weight 1 2 . This essentially means that in addition to transforming like a modular form of weight 1 2 , it also satisfies a nice differential equation which in particular implies that it is a real-analytic function. This differential equation can be phrased in terms of an important differential operator in the theory of mock modular forms. Namely, the shadow operator ξ k := 2iv k (∂/∂τ ) maps a harmonic Maass form of weight k to cusp form of weight 2 − k. We are interested in computing the images of certain functions used to prove theorem 1.4 under such operators, and for this, we require the following formula, which follows from lemma 1.8 of [7]: A mock Jacobi form similarly is a holomorphic part of harmonic Maass-Jacobi form. It turns out that μ is essentially the holomoprhic part of a harmonic Maass-Jacobi form [8].

Formulae of Lau & Zhou in the (2, 3, 6) case
To describe the functions occurring in theorem 1.4, we assume throughout that a = (2, 3, 6) and study the function W q (2, 3, 6) defined in [1]. Namely, noting that in the notation of [ and c z is another explicit q-series, which seems to be of a more complicated nature. In §3.2, we determine the modularity properties of c y , c yz2 and c yz4 . Firstly, however, we prove theorem 1.3, which we need for our study of c yz4 .

Statement and proof of theorem 1.4
Before stating the exact formulae and modularity properties of theorem 1.4, we begin with an identity of a special family of indefinite theta functions.

A useful identity for a degenerate type (1, 2) indefinite theta series
In this section, we prove theorem 1.3.
Proof of theorem 1.3. For y 3 < v, we can use a geometric series expansion to write the left-hand side of (1.1) as This sum converges for all z 3 ∈ C\(Zτ + Z) (as long as y 2 < v). Now, for 0 < y 2 < v, we compute that Using the easily checked identity (where δ k = 1 if k = 0 and δ k = 0 otherwise) in the first sum and replacing k by k − 1 in the second, we find where in the second equality we used lemma 2.3. Some rewriting then implies that Next we consider the right-hand side of (1.1) (as a function of z 3 ) for z 1 ∈ Zτ + Z and define Our goal is to show that f R satisfies the same transformation formula as satisfied by f L according to (3.1). This follows from a short calculation using (2.4), (2.9) and (2.11), which yields Comparing (3.1) and (3.2) then gives and so the function f given by f ( . Furthermore, we also (trivially) have Hence, f is an elliptic function, which we aim to show is identically zero. Both f L and f R are meromorphic functions, which could have simple poles in Zτ + Z, but could not possibly have any other poles. In z 3 = 0, both functions actually do not have a pole: a pole of f L has to come from terms in the sum satisfying k + = 0, which does not occur for k > 0 and ≥ 0 or for k ≤ 0 and < 0. The functions z 3 → iϑ(z 1 )μ(z 1 , z 2 )μ(z 1 , z 3 ) and z 3 → (η 3 ϑ(z 2 + z 3 )/ϑ(z 2 )ϑ(z 3 ))μ(z 1 , z 2 + z 3 ) both have a simple pole in z 3 = 0 with residue −(1/2π )μ(z 1 , z 2 ), so the residue of f R at z 3 = 0 vanishes. Hence f is holomorphic in z 3 = 0 and since it is both 1-and τ -periodic it is actually an entire function. By Liouville's theorem, f is then constant, and as it has a zero at z 3 = z 1 , it is identically zero.

Modularity of c y
In this section, we determine the modularity properties and explicit formulae of the functions described in theorem 1.4. The first function, c y , is essentially a modular form, as shown in (3.42) of [1]. 3 2 on SL 2 (Z) with multiplier system ν 3 η .

Theorem 3.1 (Lau-Zhou). The function c y is a cusp form of weight
Remark. Throughout this paper, we slightly abuse terminology and refer to an object as a modular form, cusp form, etc., if it is a rational power of q times such an object.
In fact, theorem 3.1 was shown [1] as a consequence of the following identity.

Modularity of c yz2
The remaining functions in theorem 1.4 are not simply modular forms, but rather mock modular and more complicated modular-type functions. Beginning with the c yz2 case, and defining a natural 'corrected' function by we show the following.
Theorem 3.3. The function c yz2 is modular of weight 2 on SL 2 (Z). In particular, c yz2 is essentially a linear combination of products of mock modular and modular forms, and the image of c yz2 under Remarks.
(i) The 'reason' that c yz2 is actually modular on SL 2 (Z), as opposed to a congruence subgroup, is closely related to the fact that the shadow of R(0) is essentially η 3 , and the fact that the term R(0) is therefore paired with its shadow. (ii) Theorem 3.3 directly gives transformation formulae for the non-completed function c yz2 , and for example can be applied to determine the asymptotic behaviour of the Fourier coefficients of c yz2 . (iii) Using the modularity of E 2 (defined in (2.2)), the last term in the definition of c yz2 could be replaced by a multiple of v −1 . This then yields a function which is modular of weight 2 and which has c yz2 as its 'holomorphic' part.
The modularity of c yz2 follows immediately from theorem 2.5 and from the following identity, where we denote
Deferring the proof of proposition 3.4 to later in this section, we may now prove the modularity of c yz2 .
Proof of theorem 3.3. From proposition 3.4, we find directly that Note that in the last expression, we used the fact that ϑ is an odd function of z. To finish the proof, it suffices to show that F cyz2 transforms like a modular form under inversion. We note that this follows from general facts concerning differential operators acting on Jacobi forms. However, we proceed directly in this case since it is elementary. Namely, using (2.1), lemma 2.1 and theorem 2.5, we compute In the third equality above, we used that the poles of μ only arise from μ, as R does not have any poles. We now turn to the proof of proposition 3.4. We begin by splitting c yz2 = c yz2,1 + c yz2,2 , where c yz2,1 (τ ) := q −1/12 n≥a≥0 (−1) n+a (6n − 2a + 8)q ((n+2)(n+1))/2−a(a+1)/2 , c yz2,2 (τ ) := q −1/12 We first analyse the piece c yz2,1 .
Proof. We begin with the elementary observation that and setting j := n + a + 2 and := n + 1 − a, we rewrite Splitting this sum into two pieces, depending on the parity of j, yields where we shifted → + 1 and used a geometric series expansion. Note that in the second summand of the last expression we can add the term j = 0 freely as it contributes zero overall. We now combine the second piece of each summand in the last formula as This term contributes the following to c(z) − c(−z): where we sent j → −j in the second term. To consider the remaining pieces of c, we need to prove that To see this, we use geometric series expansions to rewrite the second piece of c as and let j → j + 1 in the first sum and switch the roles of and j in the second sum. Differentiating with respect to z and then setting z = 0 gives (1 − E 2 ).
Repeating the calculation for the contribution at −z yields the exact same expression. Hence, we have shown that The proof now follows directly from the definitions of μ and ϑ.

Modularity of c yz4
Define the corrected function c yz4 (τ ) := q 11/48 c yz4 (τ ) + R(0; τ ) − q 1/12 c yz2 (τ ) Then we aim to show the following. Remark. It is interesting to note that there is a certain intertwining between the modularity of the different coefficients of the Gromov-Witten potential, as the function c yz2 naturally arises in considering the completion of c yz4 , and the term c y , which is essentially η 3 also occurs in the completions of c yz2 and c yz4 . It would be interesting to see whether there is a natural geometric explanation for such relations.
In order to prove this theorem, we first express the function c yz4 in terms of the function F studied in §3.1. The definition of F directly implies that c yz4 = −q −11/8 D z,0 (F(3z, 2z, 2z)).
The proof then follows from theorem 1.3.