Modularity in Gromov-Witten Theory

Transcription

1 Modularity in Gromov-Witten Theory Jie Zhou Perimeter Institute FRG Workshop, Brandeis Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 1 / 92

2 based on joint works M. Alim, E. Scheidegger, S.-T. Yau, J. Z arxiv: Y. Shen, J. Z arxiv: S.-C. Lau, J. Z arxiv: Y. Ruan, Y. Shen, J. Z work in progress Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 2 / 92

3 Outline 1 Introduction and motivation Gromov-Witten theory and mirror symmetry Exploring modularity 2 Why is modularity expected? Quick introduction to modular forms Elliptic curves Lessons learnt from studying elliptic curves 3 Examples Some non-compact Calabi-Yau 3-folds Open, orbifold Gromov-Witten theory of elliptic orbifold curves Lattice polarized K3 surfaces 4 Conclusions and discussions Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 3 / 92

4 Outline 1 Introduction and motivation Gromov-Witten theory and mirror symmetry Exploring modularity 2 Why is modularity expected? Quick introduction to modular forms Elliptic curves Lessons learnt from studying elliptic curves 3 Examples Some non-compact Calabi-Yau 3-folds Open, orbifold Gromov-Witten theory of elliptic orbifold curves Lattice polarized K3 surfaces 4 Conclusions and discussions Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 4 / 92

5 Introduction Gromov-Witten theory Gromov-Witten theory (topological string A-model) is a theory of counting of holomorphic curves. Given a CY 3-fold Y, N GW d,β = number of holomorphic curves of genus g degree β H 2(Y, Z) Define the generating series to be F g (Y, t) = β H 2 (Y,Z) N GW g,β qβ, q β = e 2πi β ω(t) Here ω(t) = h 1,1 (Y ) i=1 t i ω i, ω i, i = 1, 2 h 1,1 (Y ) are the generators for the Kähler cone of Y. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 5 / 92

6 Introduction Gromov-Witten theory Mathematically, the above generating series is defined using the intersection theory of the moduli space of stable maps F g (Y, t) = e ω(t) g,β β H 2 (Y,Z) ω i1 ω ik g,β = k [M g,k (Y,β)] vir j=1 ev j ω ij Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 6 / 92

7 Introduction Gromov-Witten theory The q parameter is a formal parameter in the formal generating series of GW invariants. Right now we can not really say the generating series is a function since it might be the case that the series is divergent for any nonzero value of q. It is not easy to see whether and how the formal generating series converge. To do that we would have to know the information of the infinite sequence of GW invariants. For some special CY 3-folds, the generating functions F g (Y, t) could be computed by using the localization technique Kontsevich (1994), topological vertex Aganagic, Klemm, Marino & Vafa (2003), etc. For general CY 3-folds, it s extremely difficult to compute them. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 7 / 92

8 Introduction Gromov-Witten theory defined on CY families One can think of putting the CY 3-fold Y in a family p : Y K. Then the quantities F g (Y, t) are formal series defined near the point t = i corresponding to q = 0, called the large volume limit, inside the complexified Kähler cone of Y which is contained in the complexified Kähler moduli space K. The complexified Kähler moduli space K is in general strictly larger than the complexfied Kähler cone. For toric varieties the former is nicely described by a fan (secondary fan) constructed using the toric data and the latter is some cone sitting inside the fan. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 8 / 92

9 Introduction Mirror symmetry The mirror symmetry conjecture says that these generating series are actually global objects defined on the moduli space. More precisely, it says for the family p : Y K, there is another family of CY 3-folds π : X M such that The moduli spaces are isomorphic. The map giving the isomorphism is called the mirror map. Throughout my talk, moduli space means coarse moduli space or the base of the CY family. The quantities F g (Y, t) are essentially identical to their counter parts F (g) (X, t) which are called topological string free energies, where t is now thought of as some coordinate system on the moduli space M via the mirror map. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 9 / 92

10 Introduction Mirror symmetry For genus g = 0, mirror symmetry is initiated by the celebrated work Candelas, de La Ossa, Green & Parkes (1991). Then established by Lian, Liau& Yau (1997), Givental (1997) for a large class of CY 3-fold examples. When g 1, F (g) (X, t) is a (smooth, but non-holomorphic) section of L 2 2g, where L is the Hodge line bundle of the family π : X M.Bershadsky, Cecotti, Ooguri & Vafa (1993) (This puts the story in the language of complex geometry). Furthermore, {F (g) } g 1 are related recursively via the holomorphic anomaly equations. Bershadsky, Cecotti, Ooguri & Vafa (1993) (This allows one to translate the story using only the language of differential equations). Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 10 / 92

11 Introduction Mirror symmetry Thanks to mirror symmetry, one can try to extract Gromov-Witten invariants of Y by studying properties of (the moduli space of) X and by solving F (g) (X, t) from the holomorphic anomaly equations. Techniques are developed by Bershadsky, Cecotti, Ooguri & Vafa (1993), Yamaguchi& Yau (1995), Alim & Länege (2007) to solve the holomorphic anomaly equations (To a large extent, the problem of fixing the integration constant called holomorphic ambiguity in solving the differential equations is still open, though). For example, the F (g) s for the mirror quintic family can be solved up to g = 51 Huang, Klemm& Quackbush (2006). Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 11 / 92

12 Introduction Importance of topological string free energies So far we have seen that the topological string partition functions F (g) are interesting from the perspective of enumerative geometry. These topological quantities also compute physics quantities. For example, according to Bershadsky, Cecotti, Ooguri & Vafa (1993), they compute certain coupling constants in the lower energy effective field theory action. Hence knowing how they behave as functions, rather than formal series, is both interesting and important. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 12 / 92

13 Outline 1 Introduction and motivation Gromov-Witten theory and mirror symmetry Exploring modularity 2 Why is modularity expected? Quick introduction to modular forms Elliptic curves Lessons learnt from studying elliptic curves 3 Examples Some non-compact Calabi-Yau 3-folds Open, orbifold Gromov-Witten theory of elliptic orbifold curves Lattice polarized K3 surfaces 4 Conclusions and discussions Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 13 / 92

14 Motivation Challenges In the A-model the topological string free energies are defined as formal series in the first place. The Kähler moduli space K is usually the extension of the Kähler chambers glued together. Some Kähler chambers represents Kähler cones of some geometries, but not all of them. That is, not every chamber is in the geometric phase. This makes it more difficult to study the global properties of the generating series. In the mirror B-model, the moduli space is a global object and the topological string functions are global non-holomorphic objects (sections of line bundles over a variety). But the (real) analytic properties (e.g, analytic continuation, which is potentially related to duality transformations) of these non-holomorphic objects are not clear. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 14 / 92

15 Motivation Challenges The reason that we care about global properties of these quantities is that in practice we are often interested in the local series expansions of them at different points on the moduli space. For example, at the mirror point of the large volume limit, called large complex structure limit, the Fourier series are related to GW invariants. We usually start from the (perturbative) series expansion at a distinguished point, and want to know the series expansion at any other point that we are interested in on the moduli space. This is essentially analytic continuation. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 15 / 92

16 Motivation Modularity exists in nicest cases In some nicest cases, we can show that for the mirror family π : X M, the base M is a modular curve Γ\H (or more generally, an arithmetic locally symmetric variety of the form Γ\G/K called modular variety). In these cases, the quantities F (g) (X, t) (and hence F g (Y, t)) happen to be the most natural objects defined on the modular curve, that is, modular forms. These are objects enjoying very nice symmetries under the action of SL(2, Z) or its subgroup. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 16 / 92

17 Motivation Why is modularity useful Modularity (the fact that M is a modular variety) gives a natural completion of the Kähler cone into the Kähler moduli space K. Modularity tells how F (g) and thus F g behave globally. Modularity is often related to the arithmeticity, and in particular, integrality of the Fourier development of F g, which in our case gives the Gromov-Witten invariants. Modular transformations is sometimes related to dualities of the physics theory. Also potentially, the re-summation phenomenon arising in the context of modularity (e.g., the shadow in Mock modular forms Dabholkar, Murthy, & Zagier (2012) is related to wall-crossing. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 17 / 92

18 Goal of this talk Exploring modularity through examples In this talk, I shall discuss modularity through a few examples. Explain why the moduli space M is a modular curve for these examples. Discuss modularity of GW theory of some non-compact CY 3-fold families by solving the holomorphic anomaly equations. Alim, Scheidegger, Yau & Zhou (2013), Zhou (2014 thesis) Discuss modularity of genus zero open GW and all genera orbifold GW theory for elliptic orbifold P 1 s. Satake & Takahashi (2011), Shen & Zhou (2014), Lau & Zhou (2014) Discuss modularity of orbifold GW theory for some quotients of K3 surfaces Ruan, Shen & Zhou (work in progress), if time permits. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 18 / 92

19 Relation to previous results Known examples: motivated by physics and proved mathematically Y = elliptic curve Douglas (1993), Rudd (1994), Dijkgraaf (1995), Kaneko & Zagier (1995), Okunkov & Pandharipande (2002)... F 1 (t) = log η(q), q = exp 2πit F 2 (t) 1 = (10E 2(q) 3 6E 2 (q)e 4 (q) 4E 6 (q)) F g (t) is a quasi-modular form of weight 6g 6... Y = elliptic orbifold P 1 Milanov & Ruan (2011), Satake & Takahashi (2011)... Y = a special K3 fibration (STU model), Y = K3 T 2 /Z 2 (FHSV model). IIA HE duality tells that F g (Y ) have nice modular properties. This is generalized gradually and is now known as the KKV conjecture for K3 surfaces Kachru & Vafa (1995), Marino & Moore (1998), Katz, Klemm & Vafa (1999), Klemm & Marino (2005), Maulik & Pandharipande (2006), Pandharipande & Thomas (2014)... Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 19 / 92

20 Outline 1 Introduction and motivation Gromov-Witten theory and mirror symmetry Exploring modularity 2 Why is modularity expected? Quick introduction to modular forms Elliptic curves Lessons learnt from studying elliptic curves 3 Examples Some non-compact Calabi-Yau 3-folds Open, orbifold Gromov-Witten theory of elliptic orbifold curves Lattice polarized K3 surfaces 4 Conclusions and discussions Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 20 / 92

21 Quick introduction to modular forms Definition A function f : H C is modular form of weight k for a congruence subgroup Γ < SL(2, Z) if f ( aτ + b cτ + d ) = (cτ + d)k f (τ), (a, b; c, d) Γ and f is holomorphic, with certain growth condition at the boundary of H. Hence a modular form is a function equivariant under the action of the modular group. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 21 / 92

22 Quick introduction to modular forms Definition Alternatively, we can define the slash operator so that the above condition takes the form of a symmetry condition. Define the slash operator γ, γ SL(2, R) to be the following γ : f f γ (: τ (cτ + d) k f (γτ)) Then f is a modular form for Γ iff f γ = f, γ Γ Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 22 / 92

23 Quick review of modular forms Equivalent definition Equip the quotient Γ\H as the structure of a Riemann surface, where means compactification. Then f is a modular form iff f is section of the line bundle K k 2 Γ\H, where K is a line bundle whose local trivilization can be taken to be dτ, here τ is the natural coordinate on H. The above two definition are the most useful ones. More alternative, yet important, descriptions of modular forms will appear later in my talk. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 23 / 92

24 Quick review of modular forms Quasi-modular forms and almost-holomorphic modular forms Kaneko & Zagier (1995) If we replace the condition f ( aτ + b cτ + d ) = (cτ + d)k f (τ), (a, b; c, d) Γ by f ( aτ + b cτ + d ) = (cτ + d)k f (τ) + k c i (cτ + d) k i f i (τ), i=1 (a, b; c, d) Γ for some f i, i = 1, 2, c... k, then we get quasi-modular forms. If we replace the holomorphicity condition by the real analyticity condition, then we obtain the notion of alomost-holomorphic modular forms. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 24 / 92

25 Quick review of modular forms Example of modular forms Example: Take Γ = SL(2, Z), and define E 2k (τ) = 1 (m,n) Z 2 {(0,0)}. (mτ+n) 2k Then one can check E 2 ( aτ + b cτ + d ) = (cτ + d)2 E 2 (τ) + 12 c(cτ + d) 2πi E 2k ( aτ + b cτ + d ) = (cτ + d)2k E 2k (τ), k 2 1 Im aτ+b = (cτ + d) 2 1 2ic(cτ + d) Imτ cτ+d Hence one can see for example that E 4, E 6 are modular forms E 2 is a quasi-modular form (of course a modular form is also a quasi-modular form) Ê 2 := E 2 3 π 1 Imτ is an almost-holomorphic modular form Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 25 / 92

26 Outline 1 Introduction and motivation Gromov-Witten theory and mirror symmetry Exploring modularity 2 Why is modularity expected? Quick introduction to modular forms Elliptic curves Lessons learnt from studying elliptic curves 3 Examples Some non-compact Calabi-Yau 3-folds Open, orbifold Gromov-Witten theory of elliptic orbifold curves Lattice polarized K3 surfaces 4 Conclusions and discussions Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 26 / 92

27 Elliptic curves Why modularity Let s review the standard picture of mirror symmetry for elliptic curves. Take a genus one Riemann surface C = C/(Z Zτ ), where the lattice Z Zτ gives rise to the complex structure of the Riemann surface C. Take ω 0 = i 2 Imτ dz dz, dz = dx + τ dy. Then the Kähler moduli space is this case is R + ω 0, the complexified Kähler moduli space is thus K = {t = t 1 + it 2 )ω 0 t 2 > 0} = H. While it is trivial to realized the T (: t t + 1) SL(2, Z) transform on a generating series like n a ne 2πint, it is not easy to interpret the S-transform, which is the other generator of the group SL(2, Z), on the generating series. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 27 / 92

28 Elliptic curves Mirror of elliptic curves Now let s go to the mirror side. It is well known that the mirror of an elliptic curve is an elliptic curve. The moduli space complex structures of the mirror curve (with marking=a choice of symplectic basis of H 1 (E, Z)) is parametrized by another copy of H, the mirror map is then the identity map K H, t τ which sends the area of the elliptic curve (C, tω 0 ) to the shape (i.e., the τ-modulus) of the mirror curve. The true moduli space M is the quotient of the moduli space of mirror elliptic curves with markings by SL(2, Z) (the mapping class group) by forgetting about the marking. Hence on the mirror side the map H M = SL(2, Z)\H is obtained by forgetting the extra structure (i.e., marking) carried by the mirror elliptic curve. Hence it is naturally to guess that the extra structure on the B-model should mirror to something in the A-model. And the quotient by the S-transform is, correspondingly, induced by forgetting the extra structure. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 28 / 92

29 Elliptic curves Orbifold construction for mirror of elliptic curves Let s formally apply the orbifold construction for mirror families to the elliptic curve case. For the A-model, take an elliptic curve, say, a Fermat cubic. Then according to the procedure of orbifold construction, the mirror family should be the desingularization of the Z 3 quotient of the Hesse pencil 3 xi 3 z 1 3 i=1 Here the Z 3 action is given by 3 x i = 0 ρ = exp 2πi 3 : [x 1, x 2, x 3 ] [x 1, ρx 2, ρ 2 x 3 ] It is easy to see that the quotient has no singular point, hence no desingularization is actually needed. i=1 Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 29 / 92

30 Elliptic curves mirror map and isogeny Hence the Z 3 quotient of the Hesse pencil π : X M is the mirror family of the family of Fermat cubic curve with varying Kähler structures. (A little work shows that the Z 3 action on a fiber is generated by the translation by 1/3 Z Zτ. The quotient gives a so-called isogeny.) It is a standard fact that the base M as an orbifold (whose underlying space is the P 1 parametrized by z) of the mirror family is a modular curve M = Γ 0 (3)\H where Γ 0 (3) is a nice congruence subgroup of the full modular group SL(2, Z). Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 30 / 92

31 Elliptic curves moduli space as a modular curve Hence we have argued that the correct moduli space K of complexfied kähler structure for the Fermat cubic (as the A-model) should be the modular curve Γ 0 (3)\H according to mirror symmetry. In particular, the modular group that enters the picture is not the full modular group SL(2, Z) but the subgroup Γ 0 (3). Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 31 / 92

32 Elliptic curves Extra structure gives rise to the particular modular group Let s exam what is happening. Recall that we have said For the A-model, take an elliptic curve, say, a Fermat cubic. The word say means that we are not looking at the universal family, but some subfamily. The term Fermat cubic means that the extra structure we put on the genus one Riemann surface is not just the complexfied Kähler structure, but also some complex structure of the Riemann surface such that it is the Fermat cubic in P 2. Forgetting this extra structure in the A-model, the moduli space would change from Γ 0 (3)\H to SL(2, Z)\H. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 32 / 92

33 Elliptic curves Extra structure gives rise to the particular modular group If we shave chosen the elliptic curve to be not the Fermat cubic, but x1 4 + x x 3 2 = 0. Then the naïve application of the orbifold construction would tell that the moduli space for the mirror family (and hence the moduli space in the A-side) is Γ 0 (2)\H. If we have chosen x x x 2 3 = 0, we get the modular curve Γ 0(1 )\H as the moduli space. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 33 / 92

34 Elliptic curves Extra structure gives rise to the particular modular group In retrospect, the reason why the moduli space M = SL(2, Z)\H is as described at the beginning of this section is probably that in the A-model of the elliptic curve, we put no extra structure at all besides the complexified Kähler structure A natural conclusion is that the mirror of the marking (symplectic basis of homology) on the B-side, which makes the moduli space H, should be mirror to some largest structure in the A-side. If we start from the moduli space of complexified Kähler structures with the largest structure and forget about it gradually, the modular curve would change from H to SL(2, Z)\H gradually. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 34 / 92

35 Outline 1 Introduction and motivation Gromov-Witten theory and mirror symmetry Exploring modularity 2 Why is modularity expected? Quick introduction to modular forms Elliptic curves Lessons learnt from studying elliptic curves 3 Examples Some non-compact Calabi-Yau 3-folds Open, orbifold Gromov-Witten theory of elliptic orbifold curves Lattice polarized K3 surfaces 4 Conclusions and discussions Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 35 / 92

36 Lessons learnt from studying elliptic curves Summary of elliptic curves The moduli space involved in the A-model is visible to the extra structure one puts, which is mirror to the corresponding extra structure on the B-model and determines the modular group. In the B-model, one can have some nice subfamilies of elliptic curves so that the bases M are parameterized by modular curves Γ\H. In the examples discussed above, these bases are moduli spaces of complex structures with level structures (which make the modular groups Γ 0 (N) for some N). One can equally consider some other subfamilies corresponding to some other extra structures, e.g., polarization, endormorphism, etc. Then the modular curves would be replaced by the so-called Shimura varieties. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 36 / 92

37 Lessons learnt from studying elliptic curves K3s and higher dimensional CYs Similar story happens for K3s: we have the following generalization elliptic curve: H = SL(2, R)/U(1), Γ < SL(2, Z) H, M = Γ\H K3: D = O + (3, 19)/(O(2) O(1, 19)) +, M = O(Γ 3,19 ) + \D Here D is called a Hermitian symmetric domain, M is called an arithmetic locally symmetric variety. On M the generalization of modular forms, called automorphic forms, can be defined. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 37 / 92

38 Lessons learnt from studying elliptic curves K3s and higher dimensional CYs One can add extra structures in the B-model for K3s, which not only changes the discrete group O(Γ 3,19 ) +, but also D. For example, one can consider the moduli space of CY metrics on K3s, the extra structure would be a Kähler structure compatible with the complex structure. In this talk, we will be mainly interested in moduli space of lattice polarized algebraic K3 surfaces by The extra structure is a lattice M < H 2 (K3, Z). We require the Picard lattice of the K3 contain this lattice. Then one has the decomposition H 2 (K3, Z) = M T. It can be shown that (see e.g., Aspinwall (1996)) the moduli space is M M O(T )\O(2, 20 rankm)/(o(2) O(20 rankm)) Here M is called the Neron-Severi lattice while T is the transcendental lattice. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 38 / 92

39 Lessons learnt from studying elliptic curves K3s and higher dimensional CYs Example: the orbifold construction applied to the Fermat quartic family gives the mirror family for the Fermat quartic. It is a lattice-polarized by M = E 8 E 8 U 4 H 2 (K3, Z) = E 8 E 8 U U U Hence T = U 4, this is a special case studied by Dolgachev (1995). It turns out that M M = Γ0 (2) + \H where Γ 0 (2) + = Γ 0 (2) τ 1 2τ. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 39 / 92

40 Lessons learnt from studying elliptic curves K3s and higher dimensional CYs For CY 3-folds and higher dimensional ones, the natural analogue of D is a complex homogenous space but not a Hermitian symmetry domain, for which the existing theory of automorphic forms do not apply, at least in a straightforward way Griffiths & Schmid (1969). What is worse, in general, it is not easy to establish the isomorphism between the moduli space M and some locally homogenous space Γ\D (due to the lack of a proof of a global Torelli type result.) Nevertheless, there might be some special subfamilies whose bases are modular curves (or nice modular varieties), so there is some hope that modularity can be established for those special families. I will give an example later in this talk. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 40 / 92

41 Lessons learnt from studying elliptic curves Invariant can be obtained by looking at special subfamilies Since we are often interested in extracting invariants of the CY variety, so taking a special family, instead of the universal family, does not hurt and can often help recover the most important information that we want. To illustrate this, a familiar example is as follows. To extract GW invariants of the quintic CY3, one can look at the Dwork pencil of quintics instead of the universal family of quintics. There are some subtitles though, a detailed discussion can be found in e.g., Candelas, De La Ossa, van Geemen & van Straten (2012). In sum, we don t lose too much information by looking at subfamilies instead of the universal family. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 41 / 92

42 Examples Strategy in exploring modularity in Gromov-Witten theory Let s get back to the problem of studying modularity for the Gromov-Witten theory of a CY variety Y. Step 1: Identify the mirror family (mirror B-model) π : X M of the CY family (A-model) p : Y K. Step 2: Establish the isomorphism M = a modular variety. This is the hero in the story. As I said earlier, this can not be true in general. But there are cases this is true. Then nice global properties and symmetries can be obtained for free. Step 3: Prove/check that the generating functions have the modularity predicted in Step 2 (especially the modular group which reflects the extra structure carried by the CY family) by using various ways of identifying modular forms. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 42 / 92

43 Outline 1 Introduction and motivation Gromov-Witten theory and mirror symmetry Exploring modularity 2 Why is modularity expected? Quick introduction to modular forms Elliptic curves Lessons learnt from studying elliptic curves 3 Examples Some non-compact Calabi-Yau 3-folds Open, orbifold Gromov-Witten theory of elliptic orbifold curves Lattice polarized K3 surfaces 4 Conclusions and discussions Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 43 / 92

44 Some non-compact Calabi-Yau 3-folds K P 2: geometry of A-model Consider the example Y = K P 2. The geometry is given by Y = (C 4 Z)/C with the (Cox ring) coordinates are X 1, X 2, X 3, P and the weights of the action of C are 1, 1, 1, 3. One can alternatively use the action by λ U(1) : (X 1, X 2, X 3, P) (λx 1, λx 2, λx 3, λ 3 P) with the moment map given by µ = X X X P 2. Physically Witten (1993), the coordinates parametrize the vacuum field configuration of a 2d N = (2, 2) GLSM. They are the vacuum expectation values of some chiral fields transforming under U(1). The moment map µ = r corresponds to the D-term constraint. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 44 / 92

45 Some non-compact Calabi-Yau 3-folds K P 2: Kähler moduli space The r parameter in the D-term constraint µ r = 0 is the Kähler parameter r = C ω R+, where ω is the Kähler form and C is the curve spanning the Mori cone which is dual to the Kähler cone. The complexfied Kähler cone is parametrized by θ + ir, where θ = C B and B is the B-field. In the GLSM the θ parameter is the θ-angle under the U(1) gauge group action. The complexfied Kähler cone is one-dimensional and is parametrized by α = exp 2πi(θ + ir). Since r 0, this Kähler cone is topologically a hemisphere giving the Calabi-Yau phase. The other hemisphere is the orbifold phase. The Kähler moduli space K is then a copy of P 1 parametrized by α. This is given by the secondary fan and is larger than the complexfied Kähler cone. It can be interpreted as the moduli space of (GIT) stability conditions in this particular example Witten (1993). Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 45 / 92

46 Some non-compact Calabi-Yau 3-folds K P 2: Kähler moduli space As reasoned by Witten (1993) using physics, the singularities on the moduli space should be α = 0, 1,. However, it is not clear from geometry what the orbifold phase means for the geometry K P 2. In particular, the meaning of singularities is obscure by only looking at the A-model geometry. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 46 / 92

47 Some non-compact Calabi-Yau 3-folds K P 2: geometry of B-model We now use the mirror picture to look at the moduli space K. In particular, this helps us understand the (more) mathematical meaning of the singularities more clearly. The Hori-Vafa Hori & Vafa (2000) mirror family π : X M of the geometry p : Y K is the following family where uv H(x, y; α) = 0, ((u, v), (x, y)) C 2 (C ) 2 H(x, y; α) = y 2 (x + 1)y αx 3 Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 47 / 92

48 Some non-compact Calabi-Yau 3-folds K P 2: geometry of mirror curve family The information (e.g, periods, Weil-Petersson metric on the base M) of the mirror family π : X M is fully encoded in the family of elliptic curves π elliptic : E M given by H(x, y; α) = y 2 (x + 1)y αx 3 For example, the Picard-Fuchs equations which are satisfied by the periods are related as follows: L CY 3 = L elliptic θ, θ = α α L elliptic = θ 2 α(θ )(θ ) The singularities of the Picard-Fuchs equation are given by α = 0, 1,. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 48 / 92

49 Some non-compact Calabi-Yau 3-folds K P 2: arithmetic aspects of mirror curve family Now we study modularity of the mirror elliptic curve family π elliptic : E M. It is a standard fact that this family is 3-isogenous to the Hesse pencil π Hesse : E Hesse M x x x 3 3 ( α 27 ) 1 3 x1 x 2 x 3 = 0 That is, the mirror family for the CY 3-fold is the same as the mirror of the Fermat cubic which is discussed earlier. The base is parametrized by the modular curve M = Γ 0 (3)\H Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 49 / 92

50 Some non-compact Calabi-Yau 3-folds K P 2: periods are modular forms The above identification M = Γ 0 (3)\H is the root of modularity Alim, Scheidegger, Yau& Zhou (2013). It implies in particular that the parameter α, the periods, etc, are modular objects. More explicitly, we have the following expression of the parameter α (called Hauptmodul) α(τ) = 3 3 η(3τ) 9 η(τ) η(3τ) 9 η(τ) 3, + η(τ)9 η(3τ) 3 η(τ) = q 1 24 (1 q n ), q = e 2πiτ n The fundamental period (the analytic one at α = 0) annihilated by L elliptic is a Gauss hypergeometric series (and transcendental) in α and is also a modular form 2F 1 ( 1 3, 2 3 ; 1; α) = (3n)! n!(2n)! ( α 27 )n n 2F 1 ( 1 3, 2 3 ; 1; α(τ)) = (33 η(3τ) 12 + η(τ) 12 ) 1 3 η(τ)η(3τ) = Θ A2 (τ) = q m2 +mn+n 2 Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 50 / 92

51 Some non-compact Calabi-Yau 3-folds K P 2: singularities on modular curve and Fricke involution More importantly, the modular curve Γ 0 (3)\H tells the information of the singularities easily. The points α = 0, 1 are the two cusps [τ] = [i ], [0], respectively. The point α = 1 is the unique elliptic fixed point [τ] = [ST 1 (exp 2πi 3 )]. Moreover, the two cusps are related by the Fricke involution W N=3 : τ 1 3τ. It is more natural than the S-transformation in the sense that Γ 0 (3)\H is the moduli space of complex structures of elliptic curves with certain extra structure (in this case the level structure, meaning a cyclic subgroup of order 3 of 3-torsion points on the elliptic curve) and it is the Fricke involution rather than the S-transformation which acts on the extra structure. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 51 / 92

52 Some non-compact Calabi-Yau 3-folds K P 2: singularities on moduli spaces Geometrically, from the perspective of elliptic curve families, the points α = 0, 1 corresponds to singular elliptic curves (with j = ) and should correspond to punctures on the moduli space. The point α = gives an elliptic curve with extra automorphism and hence is an orbifold point on the moduli space. From the perspective of CY 3-fold families, the points α = 0, 1, are the large complex structure limit, conifold point, orbifold point on the moduli space, respectively. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 52 / 92

53 Some non-compact Calabi-Yau 3-folds K P 2: implications of modularity Now let s look at some implications of the fact that the moduli space is a modular curve. According to Bershadsky, Cecotti, Ooguri & Vafa (1993), Yamaguchi& Yau (1995), Alim & Länge (2007), solving the holomorphic anomaly equations is reduced to calculation of Kähler potentials of the Weil-Petersson metric on M. To be a little more precise, F (g) are polynomials of propagators S ij, S i, S, K i and Yukawa couplings or equivalently derivatives of the Kähler potential K i, K ij, K ijk and Yukawa couplings Up to holomorphic ambiguities which presumably can be fixed by boundary ocnditions, the explicit polynomials can be determined recursively in a combinatorial way. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 53 / 92

54 Some non-compact Calabi-Yau 3-folds K P 2: topological string free energies are modular forms Modulo the problem of fixing the holomorphic ambiguities, computing derivatives of the Kähler potential is reduced to calculation of periods. As explained above, periods are modular objects. Straightforward computation Alim, Scheidegger, Yau& Zhou (2013) shows that everything involved in the computation are modular forms. More precisely, almost-holomorphic modular forms (things like Ê 2, E 4, E 6 ), which form a ring that is closed upon taking derivatives. So: topological string free energies are modular forms. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 54 / 92

55 Some non-compact Calabi-Yau 3-folds K P 2: implications of modularity Modular forms for modular groups of small level often have integral Fourier expansions. Assuming mirror symmetry, an easy consequence of modularity is that the GW invariants have certain integrality Zhou (2014 thesis) The Fricke involution acting on the modular curve induces an action on the modular forms. It turns out that the series expansions of F (g) at the two cusps [τ = i ], [τ = 0] or equivalently at the large complex structure limit and at the conifold point, are related by the Fricke involution. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 55 / 92

56 Some non-compact Calabi-Yau 3-folds K P 2: implications of modularity Physically, at the large complex structure limit, F (g) is mirror to GW theory, at the conifold point, F (g) is related to the so-called c = 1 string Ghoshal & Vafa (1995). At the large complex structure limit, we choose a distinguished local coordinate t GW (called flat coordinate), and expand the topological string free energy F g in t GW, then we get F GW g (t GW ). That is, near the large complex structure limit, Fg GW (t GW ) is mirror to the generating series of the GW invariants Fg GW (t GW ) = β N GW g,β qβ Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 56 / 92

57 Some non-compact Calabi-Yau 3-folds K P 2: implications of modularity At the conifold point, we choose another distinguished local coordinate based at this point, denoted by t con, and expand the same function F (g) in terms of t con. Then we get a new function F con g (t con ). That is, near the conifold point, we have (called gap condition) Fg con (t con ) = c tcon 2g 2 + regular terms Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 57 / 92

58 Some non-compact Calabi-Yau 3-folds K P 2: implications of modularity Of course the two functions Fg GW, Fg con are different since their arguments are different. But they are governed by the same function F (g). That is, they are related by analytic continuation. It turns out that the Fricke involution not only acts on the moduli space, but also on the free energy Alim, Scheidegger, Yau & Zhou (2013) in such a way that F con g = Fg GW W3 Here γ is the previously defined slash operator on modular forms and W 3 is the Fricke involution τ 1 3τ. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 58 / 92

59 Some non-compact Calabi-Yau 3-folds Other examples In retrospect, that the geometry K P 2 has something to do with modularity at all is due to the combinatorics of the toric data (e.g, the charge vector is the same as that for the cubic surface in P 2 ). That is, the toric diagram used to define the geometry K P 2 encodes the equivalent information that is used to define the anti-canonical divisor of P 2, which is the Hesse cubic family. This is why the mirror curve family of K P 2 coincides with the mirror elliptic curve family of the Hesse pencil. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 59 / 92

60 Some non-compact Calabi-Yau 3-folds Other examples It is easy to imagine that if one starts with K WP 2 [1,1,2], then the toric data would be the same as the anticanonial divisor of WP 2 [1, 1, 2] which gives rise to the pencil x x x 2 3 z 1 2 x 1 x 2 x 3 = 0. The mirror curve or the CY 3-fold would be the mirror of the above pencil obtained using the orbifold construction. In any case, the moduli space M, that is, the base P 1 of the family parametrized by z, is a modular curve. The same approach of exploring modularity works for some other non-compact CY 3-folds like some local del Pezzo surfaces, as discussed in Alim, Scheidegger, Yau & Zhou (2013). Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 60 / 92

61 Some non-compact Calabi-Yau 3-folds Summary In the non-compact CY 3-fold examples discussed above, modularity is used to: Clarify the meaning of the Kähler moduli space in the A-model Study the sequence of Gromov-Witten invariants Explore the duality of the physics theories Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 61 / 92

62 Some non-compact Calabi-Yau 3-folds Anther example in which modularity gives duality All of the properties, especially integrality and duality, are obtained almost for free once the identification M = Γ\H is established. This phenomenon appears in some other context as well. For example, in the well-studied 4d N = 2 pure supersymmetric gauge theory with G = SU(2) Seiberg & Witten (1994). Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 62 / 92

63 Some non-compact Calabi-Yau 3-folds Anther example in which modularity gives duality If the Seiberg-Witten curve family is taken to be or equivalently z + Λ4 z = 2(x 2 u) y 2 = (x u)(x Λ 2 )(x + Λ 2 ) Then the base parametrized by u is the modular curve Γ Λ 2 0 (4)\H. The singular points u =, Λ 2, Λ 2 corresponds to the cusps τ = i, 0, 1/2. The transformation exchanging the two cusps τ = i, 0 is the Fricke involution W N=4 : τ 1 4τ. This is the electro-magnetic duality in Seiberg-Witten theory. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 63 / 92

64 Some non-compact Calabi-Yau 3-folds Anther example in which modularity gives duality If the Seiberg-Witten curve family is taken to be y 2 = (x u)(x Λ 2 )(x + Λ 2 ) Then the base parametrized by u is the modular curve Γ(2)\H. The Λ 2 singular points u =, Λ 2, Λ 2 corresponds to the cusps τ = i, 0, 1. The transformation exchanging the two cusps τ = i, 0 is the S-transformation τ 1 τ. This is consistent with the previous model since the two modular groups are isomorphic: Γ 0 (4) Γ(2), τ 2τ. Therefore, the Fricke involution on Γ 0 (4)\H is translated to the S-transform on Γ(2)\H. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 64 / 92

65 Outline 1 Introduction and motivation Gromov-Witten theory and mirror symmetry Exploring modularity 2 Why is modularity expected? Quick introduction to modular forms Elliptic curves Lessons learnt from studying elliptic curves 3 Examples Some non-compact Calabi-Yau 3-folds Open, orbifold Gromov-Witten theory of elliptic orbifold curves Lattice polarized K3 surfaces 4 Conclusions and discussions Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 65 / 92

66 Open, orbifold Gromov-Witten theory of elliptic orbifold curves Proving modularity from the A-model directly In the above examples, we used mirror symmetry to predicts modularity of GW theory. We can also do this directly in the A-model in some cases, basing on the expectations from mirror symmetry. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 66 / 92

67 Open, orbifold Gromov-Witten theory of elliptic orbifold curves Genus zero open GW of elliptic orbifold P 1 s Consider the quotient of an elliptic curve E by its automorphic group (the curve E has to be special to allow for a nontrivial automorphic group. That is, τ is SL(2, Z)-equivalent to i or exp 2πi 3 ). Requiring the quotient to be a copy of P 1, we get the following classification for the quotient P 1 2,2,2,2, P 1 3,3,3, P 1 2,2,4, P 1 2,3,6 Here the subscripts indicate the orders for the stalibizers groups of the orbifold points on the P 1. These are conjectured to be mirror to the elliptic singularities. For example, the geometry part of the mirror of P 1 3,3,3 is conjectured to be the Hesse pencil x 3 i ψ x i = 0. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 67 / 92

68 Open, orbifold Gromov-Witten theory of elliptic orbifold curves Genus zero open GW of elliptic orbifold P 1 s One can put Lagrangians on the elliptic orbifold P 1 s and count the holomorphic disks bounded by them. This procedure can be made rigorous for some particularly chosen Lagrangians. In the following, we shall consider the example P 1 3,3,3 and explain why the genus zero potential is modular, following Lau & Zhou (2014). Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 68 / 92

69 Open, orbifold Gromov-Witten theory of elliptic orbifold curves Genus zero open GW of elliptic orbifold P 1 s The mirror of (X = P 1 3,3,3, L = Seidel Langrangian) is a LG model (V, W ) where V = Def L and W is the superpotential. In the A-model, W is the obstruction that appears in the MC equation in the A -algebra that appears in Langrangian Floer theory m 2 (b) = W (b) Id L, m(b) = k m k (b, b), b Def L In the A-model, coefficients of W are obtained by counting holomorphic disks bounded by the Seidel Lagrangian L. Further the LG/CY correspondence tells that the LG model (V, W ) is dual to the CY model W = 0, hence they share the same moduli space. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 69 / 92

70 Open, orbifold Gromov-Witten theory of elliptic orbifold curves Genus zero open GW of elliptic orbifold P 1 s So the mirror is conjectured to be described by the Hesse pencil W = 0 x x x 3 3 ψx 1 x 2 x 3 = 0 It is a standard fact that the base of the Hesse pencil is the modular curve M = P 1 ψ = Γ(3)\H. Now we can conjecture that the genus zero potential is a modular form for Γ(3). The next step is to prove this rigorously, if possible. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 70 / 92

71 Open, orbifold Gromov-Witten theory of elliptic orbifold curves Genus zero open GW of elliptic orbifold P 1 s Since the elliptic orbifold is covered by an elliptic curve E which is further covered by the complex plane C, everything lifts in the complex plane including the Seidel Langrangian. The holomorphic disks are now presented by polygons in the complex plane and the configuration of holomorphic disks can be put into a few groups according to their shapes. All polygons in the same group are similar. This means that the number of holomorphic disks within each group is independent of its degree (=area). This fact turns the problem of counting into an easy combinatoric problem. The whole process is carried out carefully in Cho, Hong, & Lau (2012), Cho, Hong, Kim & Lau (2014) within the framework of Lagrangian Floer theory. The summations for the potential and the matrix factorizations are worked out explicitly. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 71 / 92

72 Open, orbifold Gromov-Witten theory of elliptic orbifold curves Genus zero open GW of elliptic orbifold P 1 s For the P 1 3,3,3, the genus zero GW generating function, that is, the potential is given by with ψ(q) = q + W = φ(q)(x 3 + y 3 + z 3 ) ψ(q)xyz φ(q) = ( 1) 3k+1 (2k + 1)q 3(12k2 +12k+3) k=0 (( 1) 3k+1 (6k + 1))q (6k+1)2 + ( 1) 3k (6k 1)q (6k 1)2 ) k=1 Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 72 / 92

73 Open, orbifold Gromov-Witten theory of elliptic orbifold curves Genus zero open GW of elliptic orbifold P 1 s We have conjectured φ, ψ to be modular forms for Γ(3). The set of modular forms for the modular group forms a finitely generated ring (due to the interpretation that they are holomorphic sections for line bundles over the modular curve). Simple algebra turns φ, ψ into products of η or θ-functions. We find them to be certain monomials of the generators of the ring of modular forms for Γ(3). This then proves the expected modularity. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 73 / 92

74 Open, orbifold Gromov-Witten theory of elliptic orbifold curves Genus zero open GW of elliptic orbifold P 1 s Similar ideas work for the other examples P 1 4,4,2, P1 2,3,6, except that for the latter the combinatorics is much more difficult. We expect it to work for P 1 2,2,2,2 as well. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 74 / 92

75 Open, orbifold Gromov-Witten theory of elliptic orbifold curves Genus zero open GW of elliptic orbifold P 1 s The examples that I just described are very very special in the sense that all of the invariants can be computed explicitly and we obtain a nice closed-form expression for the generating series. Now let s study an example in which the invariants can not be directly computed in an easy way, yet in which the modularity can be proved. Before that we need to look at another way of describing quasi-modular forms, that is, through differential equations. We believe that this way of looking at quasi-modular forms applies to more general situations than the examples that I shall discuss below. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 75 / 92

76 Open, orbifold Gromov-Witten theory of elliptic orbifold curves Genus zero open GW of elliptic orbifold P 1 s The set of quasi-modular forms for a nice congruence subgroup Γ of SL(2, Z) form a ring that is closed under derivative Kaneko, & Zagier (1995). For example, take Γ = SL(2, Z). Then this ring is the polynomial ring generated by E 2, E 4, E 6 and they satisfy the following differential equations called Ramanujan identities (q = exp 2πiτ) 1 2πi τ E 2 = 1 12 (E 2 2 E 4 ), 1 2πi τ E 4 = 1 3 (E 2E 4 E 6 ), 1 2πi τ E 6 = 1 2 (E 2E 6 E4 2 ) E 2 (τ) = 1 24q +, E 4 (τ) = q +, E 6 (τ) = 1 504q +, Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 76 / 92

77 Open, orbifold Gromov-Witten theory of elliptic orbifold curves Genus zero open GW of elliptic orbifold P 1 s We can look at the Ramanujan identities in the opposite way. Suppose we have some formal q-series (not even convergent, as what we get in GW theory) satisfying the Ramanujan identities and the same boundary conditions are E 2, E 4, E 6, then by the existence and uniqueness of solutions to ODEs we can conclude that these formal series must be THE Eisenstein series. In particular, they are convergent series and are analytic everywhere on the upper half plane. The modular transformations also tell the symmetries they enjoy. Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 77 / 92

78 Open, orbifold Gromov-Witten theory of elliptic orbifold curves Genus zero open GW of elliptic orbifold P 1 s For some other nice groups, we have similar Ramanujan identities. Another examples is Γ = Γ 0 (4). The ring of quasi-modular forms is generated by any two of the three functions θ 2 2 (2τ), θ2 3 (2τ), θ2 4 (2τ) and E 2(τ). They satisfy slightly more complicated but very explicit differential equations: τ θ 2 2(2τ) = P 1 (θ 2 2(2τ), θ 2 3(2τ), θ 2 4(2τ), E 2 (τ)) τ θ 2 3(2τ) = P 2 (θ 2 2(2τ), θ 2 3(2τ), θ 2 4(2τ), E 2 (τ)) τ θ 2 4(2τ) = P 3 (θ 2 2(2τ), θ 2 3(2τ), θ 2 4(2τ), E 2 (τ)) τ E 2 (τ) = P 4 (θ 2 2(2τ), θ 2 3(2τ), θ 2 4(2τ), E 2 (τ)) where P 1, P 2, P 3, P 4 are explicit polynomials. The details are not important in this talk so I shall not display them. ( ) Jie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 78 / 92