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 ℙ1 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 Wq(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] Leta∈{(3,3,3),(2,4,4),(2,2,2,2)}. Then the functions arising as the various coefficients of Wq(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 Wq(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 Wq(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:=e2πiτ (τ ∈ ℍ) and ζj:=e2πizj (zj ∈ ℂ) for j=1,2,3. We note that F is an indefinite theta function of type (1,2).

Theorem 1.3 —

For allz1z2z3 ∈ ℂ with 0<Im(z2),Im(z3)<Im(τ), we have that

1.1

Remark —

The right-hand side of (1.1) provides a meromorphic continuation of F to ℂ3, 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 cy1, cyz2 and cyz4 are certain coefficients of Wq(2,3,6) (see (2.13)).

Theorem 1.4 —

The function cyis modular, and cyz2and cyz4have explicit non-holomorphic modular completionsand. More specifically, we have:

  • (i) the function cyis a cusp form of weighton SL2(ℤ) with multiplier system;

  • (ii) the functionis modular (i.e. transforms as a modular form) of weight 2 on SL2(ℤ) with shadow y3/2|η|6; and

  • (iii) the functionis modular of weightand 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 cz 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 SL2(ℤ) (with a multiplier which we denote by νη). We shall also frequently use the quasi-modular Eisenstein seriesE2, which is essentially the logarithmic derivative of η:

As is well known, E2 is not a modular form but has a slightly more complicated modularity property, known as quasi-modularity. Specifically, E2 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 Bk 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

The Jacobi triple product identity is the following product expansion:

where ζ:=e2πiz. In particular, this identity implies that the zeros of zϑ(z;τ), lie exactly at lattice points z ∈ ℤτ + 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 —

Forλμ ∈ ℤ and, we have that

ϑ(zλτμ) = (−1)λ+μqλ2/2 e−2πiλzϑ(z)

2.4

and

2.5

We next prove an identity for E2 in terms of an Appell–Lerch sum, which we need to compute the completion of cyz4. 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 z1 in −2πη3/ϑ(z) is π2E2/6. 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 z1 in −2πη3/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 yj:=Im(zj) for j∈{1,2,3}.

Lemma 2.3 —

For 0<y1,y2<v, we have

2.8

Hence, the identity (2.8) provides a meromorphic continuation of the left-hand side forz1z2 ∈ ℂ∖(ℤτ + 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 −Fτ(2πiz1,2πiz2) (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 z1z2 ∈ ℂ∖(ℤτ + ℤ) and τ ∈ ℍ as

where ζj:=e2πizj ( 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 z1 and z2 (see proposition 1.4 of [7]), i.e. that

μ(z1z2) = μ(z2z1), 

2.9

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

Lemma 2.4 —

Forz1z2 ∈ ℂ∖(ℤτ + ℤ), we have

μ(z1 + 1, z2) = −μ(z1z2)

2.10

and

2.11

We note that the poles of zjμ(z1,z2) are at z1z2 ∈ 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 transforms like a Jacobi form.

Theorem 2.5 —

The functionsatisfies 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 z1 and z2 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 Wq(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

Wq(2, 3, 6) = q1/8x2 − q1/48xyzcy(τ)y3cz(τ)z6cyz2(τ)y2z2cyz4(τ)yz4

2.13

where

and cz is another explicit q-series, which seems to be of a more complicated nature. In §3.2, we determine the modularity properties of cy, cyz2 and cyz4. Firstly, however, we prove theorem 1.3, which we need for our study of cyz4.

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 y3<v, we can use a geometric series expansion to write the left-hand side of (1.1) as

where

This sum converges for all z3 ∈ ℂ∖(ℤτ + ℤ) (as long as y2<v). Now, for 0<y2<v, we compute that

Using the easily checked identity

ρ(k) = ρ(k − 1, ) + δk

(where δk=1 if k=0 and δk=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 z3) for z1 ∉ ℤτ + ℤ and define

Our goal is to show that fR satisfies the same transformation formula as satisfied by fL 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(z3):=ϑ(z3z1)( fL(z3)−fR(z3)) satisfies f(z3)=f(z3+τ). Furthermore, we also (trivially) have

fL(z3 + 1) = −fL(z3),  fR(z3 + 1) = −fR(z3) and f(z3 + 1) = f(z3).

Hence, f is an elliptic function, which we aim to show is identically zero. Both fL and fR are meromorphic functions, which could have simple poles in ℤτ + ℤ, but could not possibly have any other poles. In z3=0, both functions actually do not have a pole: a pole of fL 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 z3(z1)μ(z1,z2)μ(z1,z3) and z3↦(η3ϑ(z2+z3)/ϑ(z2)ϑ(z3))μ(z1,z2+z3) both have a simple pole in z3=0 with residue −(1/2π)μ(z1,z2), so the residue of fR at z3=0 vanishes. Hence f is holomorphic in z3=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 z3=z1, it is identically zero. ▪

3.2. Modularity of cy

In this section, we determine the modularity properties and explicit formulae of the functions described in theorem 1.4. The first function, cy, is essentially a modular form, as shown in (3.42) of [1].

Theorem 3.1 (Lau–Zhou) —

The function cyis a cusp form of weighton SL2(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]

References[edit]

External links[edit]

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