# Sander Zwegers Thesis Statements

## Abstract

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.

**Keywords: **modular forms, mock modular forms, Jacobi forms, elliptic orbifolds, Gromov–Witten potentials

## 1. 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 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 , 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

#### Theorem 1.1 —

*[1] Let**a**∈{(3,3,3),(2,4,4),(2,2,2,2)}. Then the functions arising as the various coefficients of W*_{q}*(x,y,z) are, up to rational powers of q, linear combinations of modular forms of weights 0,1/2,3/2,2 with respect to Γ(c) and with certain multiplier systems.*

This theorem is particularly useful as it allows one to extend the potential to a certain global moduli space, and in fact this is the geometric intuition for why such a modularity statement is expected (cf. [2]). Moreover, such modularity results give an efficient way to calculate complete results of the open Gromov–Witten invariants. Lau & Zhou also discussed the case of **a**=(2,3,6) and gave explicit representations for the potential *W*_{q}(*x*,*y*,*z*). As in the discussion following theorem 1.3 of [1], the same heuristic which shows that modularity is ‘expected’ for **a**∈{(3,3,3),(2,4,4)} also predicts that modularity-type properties should hold and which should allow one to extend the potential to a global Kähler moduli space. In particular, from a geometric point of view, the case of **a**=(2,3,6) is very analogous to those cases covered in theorem 1.1 and it is the next simplest test case. In particular, as for **a**∈{(3,3,3),(2,4,4)}, in this case the Seidel Lagrangian can be lifted to a number of copies of the Lagrangian for the elliptic curve of which the orbifold is a quotient. Motivated by these calculations and heuristics, Lau and Zhou asked the following.

#### Question 1.2 (Lau & Zhou) —

What are the modularity properties of the coefficients of *W*_{q}(*x*,*y*,*z*) when **a**=(2,3,6)?

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*^{2πizj} (*z*_{j} ∈ ℂ) for *j*=1,2,3. We note that *F* is an indefinite theta function of type (1,2).

#### Theorem 1.3 —

*For all**z*_{1}, *z*_{2}, *z*_{3} ∈ ℂ *with 0<*Im(*z*_{2}),Im(*z*_{3})<Im(*τ*), *we have that*

1.1

#### Remark —

The right-hand side of (1.1) provides a meromorphic continuation of *F* to ℂ^{3}, and we frequently identify the left-hand side with this meromorphic continuation implicitly.

Our main results can then be stated as follows, where the functions *c*_{y1}, *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**and**. More specifically, we have:*

(i)

*the function c*_{y}*is a cusp form of weight**on*SL_{2}(ℤ)*with multiplier system**;*(ii)

*the function**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**is modular of weight**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.

## 2. Preliminaries

### 2.1. 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*

We recall that *η* is a weight ½ 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:

2.1

Using this transformation, one can also show that the completed function

2.2

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 *k*th Bernoulli number. For *k*≥4, these are modular forms.

In addition to these *q*-series, we also need the *Jacobi theta function*, defined by

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* ∈ ℤ*τ* + ℤ. We also need the following standard formula:

2.3

Moreover, *ϑ* is an important example of a *Jacobi form* (of weight and index ½), 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.

#### Lemma 2.1 —

*For**λ*, *μ* ∈ ℤ *and*, *we have that*

*ϑ*(*z* + *λ**τ* + *μ*) = (−1)^{λ+μ}*q*^{−λ2/2} e^{−2πiλz}*ϑ*(*z*)

2.4

*and*

2.5

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*:

2.6

#### Proof. —

By (7) of [4] and (2.3), we have

2.7

where *ζ*(*s*) denotes the Riemann zeta function. Using this, we directly find that the coefficient of *z*^{1} in −2*πη*^{3}/*ϑ*(*z*) is *π*^{2}*E*_{2}/6. 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 *z*^{1} in −2*πη*^{3}/*ϑ*(*z*) is equal to

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}.

#### Lemma 2.3 —

*For* 0<*y*_{1},*y*_{2}<*v*, *we have*

2.8

*Hence, the identity* (2.8) *provides a meromorphic continuation of the left-hand side for**z*_{1}, *z*_{2} ∈ ℂ∖(ℤ*τ* + ℤ).

#### 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}(2*πiz*_{1},2*πiz*_{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). ▪

### 2.2. The *μ* function and explicit weight ½ 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∖(ℤ*τ* + ℤ) and *τ* ∈ ℍ as

where *ζ*_{j}:=*e*^{2πizj} ( *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 1.4 of [7]), i.e. that

*μ*(*z*_{1}, *z*_{2}) = *μ*(*z*_{2}, *z*_{1}),

2.9

and so, for example, the ‘elliptic’ transformations of *μ* may be summarized by the following identities.

#### Lemma 2.4 —

*For**z*_{1}, *z*_{2} ∈ ℂ∖(ℤ*τ* + ℤ), *we have*

*μ*(*z*_{1} + 1, *z*_{2}) = −*μ*(*z*_{1}, *z*_{2})

2.10

*and*

2.11

We note that the poles of *z*_{j}↦*μ*(*z*_{1},*z*_{2}) are at *z*_{1}, *z*_{2} ∈ ℤ*τ* + 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 transforms like a Jacobi form.

#### Theorem 2.5 —

*The function**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 function is a *harmonic Maass form* of weight ½. This essentially means that in addition to transforming like a modular form of weight ½, 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* 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]:

2.12

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].

### 2.3. 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 [1], where , and writing the resulting coefficients as functions of *τ*, by (3.29) of [1] we have

*W*_{q}(2, 3, 6) = *q*^{1/8}*x*^{2} − *q*^{1/48}*x**y**z* + *c*_{y}(*τ*)*y*^{3} + *c*_{z}(*τ*)*z*^{6} + *c*_{yz2}(*τ*)*y*^{2}*z*^{2} + *c*_{yz4}(*τ*)*y**z*^{4},

2.13

where

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}.

## 3. 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.

### 3.1. 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

where

This sum converges for all *z*_{3} ∈ ℂ∖(ℤ*τ* + ℤ) (as long as *y*_{2}<*v*). Now, for 0<*y*_{2}<*v*, we compute that

Using the easily checked identity

*ρ*(*k*, *ℓ*) = *ρ*(*k* − 1, *ℓ*) + *δ*_{k}

(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

3.1

Next we consider the right-hand side of (1.1) (as a function of *z*_{3}) for *z*_{1} ∉ ℤ*τ* + 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

3.2

Comparing (3.1) and (3.2) then gives

and so the function *f* given by *f*(*z*_{3}):=*ϑ*(*z*_{3}−*z*_{1})( *f*_{L}(*z*_{3})−*f*_{R}(*z*_{3})) satisfies *f*(*z*_{3})=*f*(*z*_{3}+*τ*). Furthermore, we also (trivially) have

*f*_{L}(*z*_{3} + 1) = −*f*_{L}(*z*_{3}), *f*_{R}(*z*_{3} + 1) = −*f*_{R}(*z*_{3}) and *f*(*z*_{3} + 1) = *f*(*z*_{3}).

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 ℤ*τ* + 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*+ℓ=0, which does not occur for *k*>0 and ℓ≥0 or for *k*≤0 and ℓ<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 −(1/2*π*)*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. ▪

### 3.2. 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].

#### Theorem 3.1 (Lau–Zhou) —

*The function c*_{y}*is a cusp form of weight**on* SL_{2}(ℤ) *with multiplier system*.

#### 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

**Sander Pieter Zwegers** (born 16 April 1975 in Oosterhout) is a Dutchmathematician who made a connection between Maass forms and Srinivasa Ramanujan's mock theta functions in 2002.^{[1]} He is currently professor for number theory at the University of Cologne. Zwegers represented the Irish snowboarding team in the pan-European intervarsity games.^{[citation needed]}

## Works[edit]

- Zwegers, S. P. (2001), "Mock θ-functions and real analytic modular forms",
*q-series with applications to combinatorics, number theory, and physics (Urbana, IL, 2000)*(PDF), Contemp. Math.,**291**, Providence, R.I.: American Mathematical Society, pp. 269–277, ISBN 978-0-8218-2746-8, MR 1874536 - Zwegers, S. P. (2002),
*Mock Theta Functions*, Utrecht PhD thesis, ISBN 90-393-3155-3 - Zwegers, S. P. (2008),
*Appell–Lerch sums as mock modular forms*(PDF)^{[permanent dead link]}

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