ON THE MODULARITY OF CERTAIN FUNCTIONS FROM THE GROMOV-WITTEN THEORY OF ELLIPTIC ORBIFOLDS

ON THE MODULARITY OF CERTAIN FUNCTIONS FROM THE GROMOV-WITTEN THEORY OF ELLIPTIC ORBIFOLDS KATHRIN BRINGMANN, LARRY ROLEN, AND SANDER ZWEGERS 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 formulas for special indefinite theta functions of type 1, in terms of products of mock modular forms. This formula is also of independent interest. 1. Introduction and statement of results In the recent paper [4], Lau and 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,, 4, 4,, 3, 6,,,, }. 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 [, 3] for related results, as well as to Sections and 3 of [4] for the definitions of the relevant geometric objects. Lau and Zhou then proved the following in Theorem 1.1 of [4]. Here as usual for c N Γc := { M = α β γ δ SL Z; M I mod c }. Theorem 1.1 Lau, Zhou. Let a {3, 3, 3,, 4, 4,,,, }. 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/, 3/, with respect to Γc 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 Date: October 1, 015. The research of the first author was supported by the Alfried Krupp Prize for Young University Teachers of the Krupp foundation and the research leading to these results has received funding from the European Research Council under the European Union s Seventh Framework Programme FP/007-013 / ERC Grant agreement n. 3350 - AQSER. The second author thanks the University of Cologne and the DFG for their generous support via the University of Cologne postdoc grant DFG Grant D-7133-G-403-151001011, funded under the Institutional Strategy of the University of Cologne within the German Excellence Initiative. 1

KATHRIN BRINGMANN, LARRY ROLEN, AND SANDER ZWEGERS statement is expected cf. []. Moreover, such modularity results give an efficient way to calculate complete results of the open Gromov-Witten invariants. Lau and Zhou also discussed the case of a =, 3, 6 and gave explicit representations for the potential W q x, y, z. As in the discussion following Theorem 1.3 of [4], the same heuristic which shows that modularity is expected for a {3, 3, 3,, 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 =, 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,, 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. Lau, Zhou. What are the modularity properties of the coefficients of W q x, y, z when a =, 3, 6? We describe our partial answer to Question 1. in the form of several theorems which give the modular completions of several functions arising in the, 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 Section 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 F z 1, z, z 3 ; τ := q 1 8 ζ 1 1 ζ 1 ζ 1 3 k>0, l,m 0 + k 0, l,m<0 1 k q kk+1 +kl+km+lm ζ k 1 ζ l ζ m 3, with q := e πiτ τ H and ζ j := e πiz j z j C for j = 1,, 3. We note that F is an indefinite theta function of type 1,. Theorem 1.3. For all z 1, z, z 3 C with 0 < Imz, Imz 3 < Imτ, we have that F z 1, z, z 3 ; τ = iϑz 1 ; τµz 1, z ; τµz 1, z 3 ; τ η3 τϑz + z 3 ; τ ϑz ; τϑz 3 ; τ µz 1, z + z 3 ; τ. 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 yz, and c yz4 are certain coefficients of W q, 3, 6 see.13. Theorem 1.4. The function c y is modular, and c yz and c yz4 have explicit non-holomorphic modular completions ĉ yz and ĉ yz4. More specifically, we have: i The function c y is a cusp form of weight 3/ on SL Z with multiplier system ν 3 η. ii The function ĉ yz is modular i.e., transforms as a modular form of weight on SL Z with shadow y 3 η 6. iii The function ĉ yz4 is modular of weight 5/, and is a polynomial of degree in R0; τ over the ring of holomorphic functions on H.

Remarks. MODULARITY OF GROMOV-WITTEN POTENTIALS 3 i The explicit statements and proofs of the modularity of the functions in Theorem 1.4 are given in Section 3. ii Results concerning the modularity properties of these functions could also be proven 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. Due 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 Section, 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 Section 3.1, we prove Theorem 1.3. We conclude Section 3 by giving the explicit statements and proofs comprising Theorem 1.4. Acknowledgements The authors would like to thank Martin Westerholt-Raum and Jie Zhou for useful conversations related to the paper.. Preliminaries.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 ητ := q 1 4 1 q n. n 1 We recall that η is a weight 1/ cusp form on SL Z with a multiplier which we denote by ν η. We shall also frequently use the quasimodular Eisenstein series E, which is essentially the logarithmic derivative of η: E τ := 1 4 dq n. n 1 As is well-known, E is not a modular form, but has a slightly more complicated modularity property, known as quasimodularity. Specifically, E is 1-periodic and satisfies the following near-modularity under inversion: τ E 1 τ d n = E τ + 6 πiτ..1 Using this transformation, one can also show that the completed function Ê τ := E τ 3 πv,.

4 KATHRIN BRINGMANN, LARRY ROLEN, AND SANDER ZWEGERS where τ = u + iv, is modular of weight. More generally we require the higher weight Eisenstein series, defined by for even natural numbers k by E k τ := 1 k d k 1 q n, B k 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 ϑz; τ := e πiz+ n 1. n 1 n 1 +Z q n The Jacobi triple product identity is the following product expansion: ϑz; τ = iq 1 8 ζ 1 1 q n 1 ζq n 1 1 ζ 1 q n, n 1 where ζ := e πiz. In particular, this identity implies that the zeros of z ϑz; τ, lie exactly at lattice points z Zτ + Z. We also need the following standard formula: z [ϑz; τ] z=0 = πη3 τ..3 Moreover, ϑ is an important example of a Jacobi form of weight and index 1/, 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.1. For λ, µ Z and γ = a c d b, we have that and d n ϑz + λτ + µ = 1 λ+µ q λ e πiλz ϑz,.4 z ϑ cτ + d ; γτ = νηγcτ 3 + d 1 πicz e cτ+d ϑz; τ..5 We next prove an identity for E 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.. The following holds: n 0 1 n q nn+3 1 q n = 1 1 E 1..6 Proof. By 7 of [7] and.3, we have πη 3 = z 1 ζne n z n exp,.7 ϑz n n 1

MODULARITY OF GROMOV-WITTEN POTENTIALS 5 where ζs denotes the Riemann zeta function. Using this, we directly find that the coefficient of z 1 in πη 3 /ϑz is π E /6. We now compare this with the well-known partial fraction expansion of 1/ϑz. Namely, a standard application of the Mittag-Leffler theorem gives the following formula cf. page 136 of [6] or page 1 of [5]: η 3 ϑz = iζ 1 n Z 1 n q nn+1. 1 ζq n An elementary calculation then shows that the coefficient of z 1 in πη 3 /ϑz is equal to 4π 1 n q nn+3 1 q n 0 n + 1, 4 which, together with the computation above, implies.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 below. Throughout, we set y j := Imz j for j {1,, 3}. Lemma.3. For 0 < y 1, y < v, we have ζ1 l 1 ζ q = ϑz 1 + z l iη3 ϑz 1 ϑz..8 l Z Hence, the identity.8 provides a meromorphic continuation of the left hand side for z 1, z C \ Zτ + Z. Proof. In the given range, we may use geometric series to expand: ζ1 l 1 ζ q = ζ1ζ l m l q lm = l Z l,m 0 l,m<0 l,m 1 ζ l 1 ζ m ζ1 l ζ m q lm ζ 1 ζ 1 ζ 1 1ζ 1. In the notation of Theorem of Section 3 of [7], this last expression is exactly F τ πiz 1, πiz cf. the first line of the proof of Theorem 3 there. The result then follows directly from vii of Theorem 3 of [7] and.3... The µ function and explicit weight 1/ mock modular forms. Throughout, we require an important function used in [8] 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 C\Zτ + Z and τ H as µz 1, z ; τ := ζ 1 1 ϑz n Z ζ n q nn+1, 1 ζ 1 q n where ζ j := e πiz j j = 1,. 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 see Proposition 1.4 of [8], i.e., that µz 1, z = µz, z 1,.9

6 KATHRIN BRINGMANN, LARRY ROLEN, AND SANDER ZWEGERS and so, for example, the elliptic transformations of µ may be summarized by the following identities. Lemma.4. For z 1, z C \ Zτ + Z, we have µz 1 + 1, z = µz 1, z,.10 µz 1, z + ζ ζ1 1 q 1 µz1 + τ, z = iζ 1 ζ 1 1 q 1 8..11 We note that the poles of z j µz 1, z are at z 1, z Zτ + Z, and by Lemma.4 and.9 the residues are determined by 1 Res z1 =0 µz 1, z = πiϑz. The results of [8] give a completion of µ to a non-holomorphic Jacobi form. To describe this, we first require the special function R, given by Rz; τ := sgnn E n + y v 1 n 1 q n ζ n, v n 1 +Z where z = x + iy and E is the entire function Defining the completion Ez := z 0 e πt dt. µz 1, z := µz 1, z + i Rz 1 z, Theorem 1.11 of [8] shows that µ transforms like a Jacobi form. Theorem.5. The function µ satisfies the following: µz 1 + kτ + l, z + mτ + n = 1 k+l+m+n q k m ζ k m µ z1 cτ + d, z cτ + d ; aτ + b cτ + d = ν 3 1 ζ m k η γcτ + d 1 e πicz 1 z cτ+d µz 1, z for k, l, m, n Z, µz 1, z ; τ for γ = a b c d SL Z. The reason that µ is called a mock Jacobi form is closely connected to Theorem.5. Namely, it follows directly from the theory of Jacobi forms that if z 1 and z are specialized to torsion points, then the completed function µ is a harmonic Maass form of weight 1/. This essentially means that in addition to transforming like a modular form of weight 1/, 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 := iv k τ maps a harmonic Maass form of weight k to cusp form of weight 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 [8]: ξ 1 R0; τ = η 3 τ..1 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 see [1].

MODULARITY OF GROMOV-WITTEN POTENTIALS 7.3. Formulas of Lau and Zhou in the, 3, 6 case. To describe the functions occurring in Theorem 1.4, we assume throughout that a =, 3, 6 and study the function W q, 3, 6 defined in [4]. Namely, noting that in the notation of [4], where q = qd 48, and writing the resulting coefficients as functions of τ, by 3.9 of [4] we have W q, 3, 6 = q 1 8 x q 1 48 xyz + cy τy 3 + c z τz 6 + c yz τy z + c yz4 τyz 4,.13 where c y τ : = q 3 16 1 n+1 n + 1q nn+1 c yz τ : = q 1 1 c yz4 τ : = q 17 48 n 0 n a 0 a,b 0 n a+b, 1 n+a 6n a + 8q n+n+1 aa+1 + n + 4q n+an+1 a, 1 n+a+b 6n a b + 7q n+1n+ aa+1 bb+1, and c z is another explicit q-series, which seems to be of a more complicated nature. In Section 3., we determine the modularity properties of c y, c yz, 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 formulas 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, 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 q 1 8 ζ 1 1 ζ 1 ζ 1 3 1 k q kk+1 +kl ζ1 k ζ l 1 ζ 3 q k+l k>0, l 0 k 0, l<0 = q 1 8 ζ 1 1 ζ 1 ζ 1 3 k,l Z ρk 1, l 1k q kk+1 +kl ζ k 1 ζ l 1 ζ 3 q k+l =: f L z 3, where 1 if k, l 0, ρk, l := 1 if k, l < 0, 0 otherwise.

8 KATHRIN BRINGMANN, LARRY ROLEN, AND SANDER ZWEGERS This sum converges for all z 3 C \ Zτ + Z as long as y < v. Now, for 0 < y < v, we compute that ζ 1 ϑz 1 µz 1, z = ζ 1 ϑz 1 µz, z 1 = k Z = k,l Z = k,l Z 1 k q kk+1 ζ k 1 1 ζ q k ρk, l 1 k q kk+1 +kl ζ1 k ζ l = ρk, l 1 k q kk+1 +kl ζ1 k ζ l 1 ζ 3 q k+l+1 1 ζ 3 q k+l+1 k,l Z ρk, l 1k q kk+1 +kl ζ k 1 ζ l 1 ζ 3 q k+l+1 ζ 3 Using the easily checked identity k,l Z ρk, l = ρk 1, l + δ k ρk, l 1k q kk+3 +1+kl+l ζ k 1 ζ l 1 ζ 3 q k+l+1. 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 ζ 1 ϑz 1 µz 1, z = k,l Z ρk 1, l 1k q kk+1 +kl ζ k 1 ζ l 1 ζ 3 q k+l+1 + l Z + ζ 1 1 ζ 3 k,l Z ρk 1, l 1k q kk+1 +kl ζ k 1 ζ l 1 ζ 3 q k+l ζ l 1 ζ 3 q l+1 = q 3 1 8 ζ 1 ζ 1 ζ 1 3 f L z 3 + τ iζ 1 η 3 ϑz + z 3 ϑz ϑz 3 + q 1 8 ζ 1 1 ζ 1 ζ 1 3 f L z 3, where in the second equality we used Lemma.3. Some rewriting then implies that f L z 3 + q 1 ζ1 ζ3 1 f L z 3 + τ = q 1 1 8 ζ 3 ϑz 1 µz 1, z + iq 1 1 8 ζ 1 ζ 1 1 ζ 1 ζ 1 3 η 3 ϑz + z 3 ϑz ϑz 3. 3.1 Next we consider the right hand side of 1.1 as a function of z 3 for z 1 Zτ + Z, and define f R z 3 := iϑz 1 µz 1, z µz 1, z 3 η3 ϑz + z 3 ϑz ϑz 3 µz 1, z + z 3. 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.4,.9, and.11, which yields f R z 3 + q 1 ζ1 ζ3 1 f R z 3 + τ = q 1 1 8 ζ 3 ϑz 1 µz 1, z + iq 1 1 8 ζ 1 ζ 1 1 ζ 1 ζ 1 3 Comparing 3.1 and 3. then gives f L z 3 f R z 3 = q 1 ζ1 ζ3 1 fl z 3 + τ f R z 3 + τ η 3 ϑz + z 3 ϑz ϑz 3. 3.

MODULARITY OF GROMOV-WITTEN POTENTIALS 9 and so the function f given by fz 3 := ϑz 3 z 1 f L z 3 f R z 3 satisfies fz 3 = fz 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 fz 3 + 1 = fz 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 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 + l = 0, which does not occur for k > 0 and l 0 or for k 0 and l < 0. The functions z 3 iϑz 1 µz 1, z µz 1, z 3 and z 3 η3 ϑz +z 3 µz ϑz ϑz 3 1, z + z 3 both have a simple pole in z 3 = 0 with residue 1 µz π 1, z, 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 since it has a zero at z 3 = z 1, it is identically zero. 3.. Modularity of c y. In this section, we determine the modularity properties and explicit formulas of the functions described in Theorem 1.4. The first function, c y, is essentially a modular form, as shown in 3.4 of [4]. Theorem 3.1 Lau-Zhou. The function c y is a cusp form of weight 3/ on SL Z with multiplier system ν 3 η. 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 [4] as a consequence of the following identity. Lemma 3.. We have that c y τ = q 1 16 η 3 τ. 3.3. Modularity of c yz. 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 yz case, and defining a natural corrected function by we show the following. ĉ yz τ := q 1 1 cyz τ 1 4 + 3 η3 τr0; τ 1 4 E τ, Theorem 3.3. The function ĉ yz is modular of weight on SL Z. In particular, c yz is essentially a linear combination of products of mock modular and modular forms, and the image of ĉ yz under ξ is 3 y 3 ητ 6. Remarks. i The reason that ĉ yz is actually modular on SL Z, as opposed to a congruence subgroup, is closely related to the fact that the shadow of R0 is essentially η 3, and the fact that the term R0 is therefore paired with its shadow.

10 KATHRIN BRINGMANN, LARRY ROLEN, AND SANDER ZWEGERS ii Theorem 3.3 directly gives transformation formulas for the non-completed function c yz, and for example can be applied to determine the asymptotic behavior of the Fourier coefficients of c yz. iii Using the modularity of Ê defined in., the last term in the definition of ĉ yz could be replaced by a multiple of v 1. This then yields a function which is modular of weight and which has c yz as its holomorphic part. The modularity of ĉ yz follows immediately from Theorem.5 and from the following identity, where we denote D z := 1 πi z, D z,0 := D z z=0. Proposition 3.4. We have the following: q 1 1 cyz τ = 1 D z,0 ϑ6z; τµ8z, 6z; τ + ζ4 + 1 1 ζ 8 4 1 E τ. Deferring the proof of Proposition 3.4 to later in this section, we may now prove the modularity of ĉ yz. Proof of Theorem 3.3. From Proposition 3.4, we find directly that ĉ yz = 1 D z,0 ϑ6z µ8z, 6z + ζ4 1 1 ζ 8 3 E, since D z,0 i 4 ϑ6zrz = 3i R0D z,0 ϑz = 3 η3 R0. 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 cyz 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.1, Lemma.1, and Theorem.5, we compute ĉ yz 1 τ = 1 D z,0 τe16z τϑ6zτ; τ µ8zτ, 6zτ; τ + ζ4 1 1 ζ 8 3 τ E τ τ πi 3τ ze16z τϑ6zτ; τ µ8zτ; 6zτ + ζ4 1 ζ τe8πizτ 8 1 e 16πizτ = τ ĉ yz τ + 1 lim z 0 = τ ĉ yz τ 16τ lim zϑ6zτ; τµ8zτ; 6zτ τ z 0 πi = τ ĉ yz τ 16τ z lim τ z 0 1 e 8πizτ πi = τ ĉ yz τ. τ πi In the third equality above, we used that the poles of µ only arise from µ, as R does not have any poles. The claimed formula of the image of ĉ yz under ξ follows directly from.1.

MODULARITY OF GROMOV-WITTEN POTENTIALS 11 We now turn to the proof of Proposition 3.4. We begin by splitting c yz = c yz,1 + c yz,, where c yz,1 τ := q 1 1 1 n+a 6n a + 8q n+n+1 aa+1, c yz, τ := q 1 1 We first analyze the piece c yz,1. n a 0 n a 0 n + 4q n+an+1 a. Lemma 3.5. We have the identity q 1 1 cyz,1 τ = D z,0 ϑ3z; τµ4z, 3z; τ + Proof. We begin with the elementary observation that where q 1 1 cyz,1 τ = D z,0 cz; τ c z; τ, cz; τ := n a 0 ζ + 1 1 ζ 4 1 1 E τ. 1 n+a ζ 3n a+4 q n+n+1 aa+1. Noting that n + n + 1 aa + 1 = 1 n + a + n + 1 a and setting j := n + a + and l := n + 1 a, we rewrite cz = 1 j ζ j+l q jl. j>l 1 l j+1 mod Splitting this sum into pieces, depending on the parity of j, yields cz = ζ j+4l q jl 1 ζ j+4l+1 q j+1l j l 1 = ζj+ q j 1 ζ 4j q j ζj+5 q j+1 1 ζ 4j q jj+1, 1 ζ 4 q j 1 ζ 4 q j+1 j 1 where we shifted l l + 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 j 1 ζ 6j+ q j +j + ζ 6j+5 q j +3j+1 = 1 ζ 4 q j 1 ζ 4 q j+1 j 0 j 1 This term contributes the following to cz c z: j 1 1 j+1 ζ 3j+ q jj+1 1 ζ 4 q j j 1 1 j+1 ζ 3j q jj+1 = 1 ζ 4 q j 1 j+1 ζ 3j+ q jj+1. 1 ζ 4 q j j Z\{0} 1 j+1 ζ 3j+ q jj+1 1 ζ 4 q j,

1 KATHRIN BRINGMANN, LARRY ROLEN, AND SANDER ZWEGERS where we sent j j in the second term. To consider the remaining pieces of c, we need to prove that ζ j+ q j D z,0 1 ζ 4 q ζ j+5 q j+1 = σ j 1 ζ 4 q j+1 1 nq n. j 1 j 0 n 1 To see this, we use geometric series expansions to rewrite the second piece of c as ζ j++4l q j1+l ζ j+5+4l q j+1l+1 = ζ j+4+4l ζ l+5+4j q l+1j+1 j,l 0 j,l 0 j 1 l 0 and let j j + 1 in the first sum and switch the roles of l and j in the second sum. Differentiating with respect to z and then setting z = 0 gives j l + 1q l+1j+1 = n mq mn, j,l 0 m,n 1 m odd where we set m := l + 1 and n := j + 1. Since n mq mn = n mq mn = 0, m,n 1 m even m,n 1 this is equal to m,n 1n mq mn = nq n 1 q = 1 n 4 1 E. n 1 Repeating the calculuation for the contribution at z yields the exact same expression. Hence, we have shown that q 1 1 cyz,1 = D z,0 1 j ζ 3j+ q jj+1 + 1 1 ζ 4 q j 1 1 E. 3.3 j Z\{0} The proof now follows directly from the definitions of µ and ϑ. The following identity was proven in 3.43 of [4]. This, together with Lemma 3.5, completes the proof of Proposition 3.4. Lemma 3.6. We have the identity c yz, τ = q 1 1 6 1 E τ. 3.4. Modularity of c yz4. Define the corrected function ĉ yz4 τ := q 11 48 cyz4 τ + R0; τ q 1 1 c yz τ + 1 6 + 1 1 E τ Then we aim to show the following. Theorem 3.7. The function ĉ yz4 is modular of weight 5/. + 3i 4 R 0; τη 3 τ.

MODULARITY OF GROMOV-WITTEN POTENTIALS 13 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 yz 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 yz and c yz4. It would be interesting to see if 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 Section 3.1. Proposition 3.8. The following identity holds: c yz4 τ = q 11 8 Dz,0 iϑ3z; τµ 3z, z; τ η3 τϑ4z; τ µ3z, 4z; τ. ϑ z; τ Proof. As in the proof of Proposition 3.4, we write q 17 48 cyz4 τ = D z,0 f z; τ f z; τ with f z; τ := 1 n+a+b ζ 3n a b+ 7 n+n+1 q aa+1 bb+1. a,b 0 n a+b Setting N := n a b + 1, we compute f z f z = a,b 0 N 1 + a,b<0 N 0 The definition of F directly implies that The proof then follows from Theorem 1.3. 1N+1 ζ 3N+a+b+ 1 NN+1 q c yz4 = q 11 8 Dz,0 F 3z, z, z. +Na+b+ab. We are now in a position to prove the modularity of ĉ yz4 Proof of Theorem 3.7. We first claim that ĉ yz4 τ = D z,0 iϑ3z; τ µ 3z, z; τ η3 τϑ4z; τ ϑ z; τ For this, we use Proposition 3.8 to compute D z,0 iϑ3z; τ µ 3z, z; τ η3 τϑ4z; τ µ3z, 4z; τ ϑ z; τ =D z,0 ϑ3zµ3z, zrz + i 4 ϑ3zr z + i η 3 ϑ4z ϑ z Rz =D z,0 ζrz 1 ζ + Rzζ n 0 1 n q nn+1 ζ 3n µ3z, 4z; τ. 3.4 q 11 48 cyz4 1 ζ q n + i 4 ϑ3zr z + i η 3 ϑ4zrz. ϑ z

14 KATHRIN BRINGMANN, LARRY ROLEN, AND SANDER ZWEGERS We split this into several pieces, which we denote by ζrz H 1 := D z,0 1 ζ + i η 3 ϑ4zrz, ϑ z H := D z,0 Rz 1 n q nn+1 ζ 3n+1, 1 ζ q n n 0 H = R0 n 0 H 3 := i 4 D z,0 ϑ3zr z. Again using the fact that R is even in particular, that R 0 = 0, we find that 1 n q nn+3 1 q n + 3 n 0 1 n nq nn+1 + 1 q n n 0 1 n q nn+1 1 q n The last sum in the last expression is identically zero, and by Lemma., we have We also directly find that H = R0 1 E 1 + 3R0 n 0 1 n nq nn+1. 3.5 1 q n H 3 = + 3i 4 η3 R 0. 3.6 Finally, we consider the first piece H 1. Using.7 again, we find ζrz 1 ζ + i = 1 + 1 4πi η 3 ϑ4zrz ϑ z 4πiz + πi 6 z + O z R0 + R 0 z + O z 3 1 4z + E ζ + O z 4z 64ζE z 3 + O z 4 R0 + R 0 z + O z 4, and after a short computation using.3 we see that 1 H 1 = R0 1 + 1 6 E. 3.7 Combining 3.5, 3.6, and 3.7 then gives D z,0 iϑ3z; τ µ 3z, z; τ η3 τϑ4z; τ ϑ z; τ = q 11 48 cyz4 + 1 4 R0E + 3R0 n 0 µ3z, 4z; τ 1 n nq nn+1 1 q n + 3i 4 η3 R 0..

MODULARITY OF GROMOV-WITTEN POTENTIALS 15 Using 3.3, Lemma., and Lemma 3.6, we obtain q 1 1 cyz = 1 D 1 n ζ 3n+ q nn+1 z,0 + 1 1 ζ 4 q n 4 1 E = 1 = 3 n 0 n 0 n 0 1 n 3n + q nn+1 1 n q nn+3 1 q n 1 q n 0 n + 1 4 1 E 1 n nq nn+1 + 1 1 q n 3 1 E, which directly implies 3.4. Hence, using.5, 3.4, and Theorem.5, we find that 1 ĉ yz4 = D z,0 iϑ 3z; 1 µ 3z, z; 1 η3 1 τ ϑ 4z; 1 τ τ τ τ ϑ µ 3z, 4z; 1 z; 1 τ τ = iτ 5 ĉyz4 τ + iτ 3 8πiτ lim z 0 z ϑ3zτ; τµ 3zτ, zτ; τ + i η3 τϑ4zτ; τ µ3zτ, 4zτ; τ ϑ zτ; τ, where we used the fact that R does not have a pole, so that all poles of µ come from µ. The inner sum in the limit is essentially just a specialization of F, and by a similar computation of Laurent coefficients as in the proof of Theorem 1.3, it converges to a finite limit as z 0. Hence, the entire limit converges to zero, and so the modularity of ĉ yz4 is proven. References [1] K. Bringmann, M. Raum, and O. Richter, Harmonic Maass-Jacobi forms with singularities and a thetalike decomposition, Transactions of the AMS, accepted for publication. [] C. Cho, H. Hong, S. Kim, and S. Lau, Lagrangian Floer potential oforbifold spheres, preprint 014, arxiv:1403.0990. [3] C. Cho, H. Hong, and S. Lee, Examples of matrix factorizations from SYZ, SIGMA Symmetry Integrability Geom. Methods Appl. 8 01, Paper 053, 4. [4] S. Lau and J. Zhou, Modularity of open Gromov-Witten potentials of elliptic orbifolds, to appear in Communications in Number Theory and Physics. [5] S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988. [6] J. Tannery and J. Molk, Éléments de la Théorie des Fonctions Elliptiques, Vol. III, Gauthier-Villars, Paris, 1896 [reprinted: Chelsea, New York, 197]. [7] D. Zagier, Jacobi theta function and periods of modular forms, Invent. Math. 104 1991, 449 465. [8] S. Zwegers, Mock theta functions, Ph.D. Thesis, Universiteit Utrecht, 00. Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany E-mail address: kbringma@math.uni-koeln.de E-mail address: lrolen@math.uni-koeln.de E-mail address: sander.zwegers@uni-koeln.de