modularity in gromov-witten theorypeople.brandeis.edu/~lian/frg_workshop_2015/zhou.pdfjie zhou...

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

based on joint works

M. Alim, E. Scheidegger, S.-T. Yau, J. Z arxiv: 1306.0002

Y. Shen, J. Z arxiv:1411.2078

S.-C. Lau, J. Z arxiv: 1412.1499

Y. Ruan, Y. Shen, J. Z work in progress


Outline

1 Introduction and motivationGromov-Witten theory and mirror symmetryExploring modularity

2 Why is modularity expected?Quick introduction to modular formsElliptic curvesLessons learnt from studying elliptic curves

3 ExamplesSome non-compact Calabi-Yau 3-foldsOpen, orbifold Gromov-Witten theory of elliptic orbifold curvesLattice polarized K3 surfaces

4 Conclusions and discussions


Outline






IntroductionGromov-Witten theory

Gromov-Witten theory (topological string A-model) is a theory of countingof holomorphic curves. Given a CY 3-fold Y ,

NGWd ,β ” = ” number of holomorphic curves of genus g degree β ∈ H2(Y ,Z)

Define the generating series to be

Fg (Y , t) =∑

β∈H2(Y ,Z)

NGWg ,β qβ, qβ = e2πi

∫β ω(t)

Here ω(t) =∑h1,1(Y )

i=1 tiωi , ωi , i = 1, 2 · · · h1,1(Y ) are the generators forthe Kahler cone of Y .



Mathematically, the above generating series is defined using theintersection theory of the moduli space of stable maps

Fg (Y , t) =∑

β∈H2(Y ,Z)

〈eω(t)〉g ,β

〈ωi1 · · ·ωik 〉g ,β =

∫[Mg,k (Y ,β)]vir

k∏j=1

ev∗j ωij



The q parameter is a formal parameter in the formal generating series ofGW invariants. Right now we can not really say the generating series is afunction since it might be the case that the series is divergent for anynonzero value of q.

It is not easy to see whether and how the formal generating seriesconverge. To do that we would have to know the information of theinfinite sequence of GW invariants.

For some special CY 3-folds, the generating functions Fg (Y , t) could becomputed by using the localization technique Kontsevich (1994), topologicalvertex Aganagic, Klemm, Marino & Vafa (2003), etc. For general CY 3-folds, it’sextremely difficult to compute them.


IntroductionGromov-Witten theory defined on CY families

One can think of putting the CY 3-fold Y in a family p : Y → K. Thenthe quantities Fg (Y , t) are formal series defined near the point t = i∞corresponding to q = 0, called the large volume limit, inside thecomplexified Kahler cone of Y which is contained in the complexifiedKahler moduli space K.

The complexified Kahler moduli space K is in general strictly larger thanthe complexfied Kahler cone. For toric varieties the former is nicelydescribed by a fan (secondary fan) constructed using the toric data andthe latter is some cone sitting inside the fan.


IntroductionMirror symmetry

The mirror symmetry conjecture says that these generating series areactually global objects defined on the moduli space. More precisely, it saysfor the family p : Y → K, there is another family of CY 3-foldsπ : X →M such that

The moduli spaces are isomorphic. The map giving the isomorphismis called the mirror map. Throughout my talk, moduli space meanscoarse moduli space or the base of the CY family.

The quantities Fg (Y , t) are essentially identical to their counter partsF (g)(X , t) which are called topological string free energies, where t isnow thought of as some coordinate system on the moduli space Mvia the mirror map.



For genus g = 0, mirror symmetry is initiated by the celebrated workCandelas, 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) sectionof L2−2g , where L is the Hodge line bundle of the familyπ : X →M.Bershadsky, Cecotti, Ooguri & Vafa (1993) (This puts the storyin the language of complex geometry).

Furthermore, {F (g)}g≥1 are related recursively via the holomorphicanomaly equations. Bershadsky, Cecotti, Ooguri & Vafa (1993) (This allowsone to translate the story using only the language of differentialequations).



Thanks to mirror symmetry, one can try to extract Gromov-Witteninvariants of Y by studying properties of (the moduli space of) X and bysolving F (g)(X , t) from the holomorphic anomaly equations.

Techniques are developed by Bershadsky, Cecotti, Ooguri & Vafa (1993),

Yamaguchi& Yau (1995), Alim & Lanege (2007) to solve the holomorphicanomaly equations (To a large extent, the problem of fixing the integrationconstant called holomorphic ambiguity in solving the differential equationsis still open, though). For example, the F (g)s for the mirror quintic familycan be solved up to g = 51 Huang, Klemm& Quackbush (2006).


IntroductionImportance of topological string free energies

So far we have seen that the topological string partition functions F (g) areinteresting from the perspective of enumerative geometry.

These topological quantities also compute physics quantities. For example,according to Bershadsky, Cecotti, Ooguri & Vafa (1993), they compute certaincoupling constants in the lower energy effective field theory action.

Hence knowing how they behave as functions, rather than formal series, isboth interesting and important.


Outline






MotivationChallenges

In the A-model the topological string free energies are defined as formalseries in the first place. The Kahler moduli space K is usually theextension of the Kahler chambers glued together. Some Kahler chambersrepresents Kahler cones of some geometries, but not all of them. That is,not every chamber is in the geometric phase. This makes it more difficultto study the global properties of the generating series.

In the mirror B-model, the moduli space is a global object and thetopological string functions are global non-holomorphic objects (sections ofline bundles over a variety). But the (real) analytic properties (e.g,analytic continuation, which is potentially related to dualitytransformations) of these non-holomorphic objects are not clear.


MotivationChallenges

The reason that we care about global properties of these quantities is thatin practice we are often interested in the local series expansions of them atdifferent points on the moduli space. For example, at the mirror point ofthe large volume limit, called large complex structure limit, the Fourierseries are related to GW invariants. We usually start from the(perturbative) series expansion at a distinguished point, and want to knowthe series expansion at any other point that we are interested in on themoduli space. This is essentially analytic continuation.


MotivationModularity 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 arithmeticlocally symmetric variety of the form Γ\G/K called modular variety).

In these cases, the quantities F (g)(X , t) (and hence Fg (Y , t)) happen tobe the most natural objects defined on the modular curve, that is, modularforms. These are objects enjoying very nice symmetries under the action ofSL(2,Z) or its subgroup.


MotivationWhy is modularity useful

Modularity (the fact that M is a modular variety) gives a naturalcompletion of the Kahler cone into the Kahler moduli space K.

Modularity tells how F (g) and thus Fg behave globally.

Modularity is often related to the arithmeticity, and in particular,integrality of the Fourier development of Fg , which in our case givesthe Gromov-Witten invariants.

Modular transformations is sometimes related to dualities of thephysics theory. Also potentially, the re-summation phenomenonarising in the context of modularity (e.g., the shadow in Mockmodular forms Dabholkar, Murthy, & Zagier (2012) is related towall-crossing.

· · ·


Goal of this talkExploring 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 theseexamples.

Discuss modularity of GW theory of some non-compact CY 3-foldfamilies by solving the holomorphic anomaly equations. Alim,

Scheidegger, Yau & Zhou (2013), Zhou (2014 thesis)

Discuss modularity of genus zero open GW and all genera orbifoldGW theory for elliptic orbifold P1’s. Satake & Takahashi (2011), Shen &

Zhou (2014), Lau & Zhou (2014)

Discuss modularity of orbifold GW theory for some quotients of K3surfaces Ruan, Shen & Zhou (work in progress), if time permits.


Relation to previous resultsKnown examples: motivated by physics and proved mathematically

Y = elliptic curve Douglas (1993), Rudd (1994), Dijkgraaf (1995), Kaneko &

Zagier (1995), Okunkov & Pandharipande (2002)...

F1(t) = − log η(q), q = exp 2πit

F2(t) =1

103680(10E2(q)3 − 6E2(q)E4(q)− 4E6(q))

Fg (t) is a quasi-modular form of weight 6g − 6...

Y = elliptic orbifold P1 Milanov & Ruan (2011), Satake & Takahashi (2011)...

Y = a special K3 fibration (STU model), Y = K3× T 2/Z2 (FHSVmodel). IIA− HE duality tells that Fg (Y ) have nice modularproperties. This is generalized gradually and is now known as theKKV conjecture for K3 surfaces Kachru & Vafa (1995), Marino & Moore

(1998), Katz, Klemm & Vafa (1999), Klemm & Marino (2005), Maulik &

Pandharipande (2006), Pandharipande & Thomas (2014) ...


Outline






Quick introduction to modular formsDefinition

A function f : H → C is modular form of weight k for a congruencesubgroup 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 themodular group.


Quick introduction to modular formsDefinition

Alternatively, we can define the slash operator so that the above conditiontakes the form of a symmetry condition.

Define the slash operator |γ , γ ∈ SL(2,R) to be the following

|γ : f 7→ f |γ(: τ 7→ (cτ + d)−k f (γτ))

Then f is a modular form for Γ iff

f |γ = f , ∀γ ∈ Γ


Quick review of modular formsEquivalent 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 Kk2 → 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, yetimportant, descriptions of modular forms will appear later in my talk.


Quick review of modular formsQuasi-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∑i=1

c i (cτ + d)k−i fi (τ), ∀(a, b; c , d) ∈ Γ

for some fi , 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.


Quick review of modular formsExample of modular forms

Example: Take Γ = SL(2,Z), and defineE2k(τ) =

∑(m,n)∈Z2−{(0,0)}

1(mτ+n)2k .

Then one can check

E2(aτ + b

cτ + d) = (cτ + d)2E2(τ) +

12

2πic(cτ + d)

E2k(aτ + b

cτ + d) = (cτ + d)2kE2k(τ), k ≥ 2

1

Im aτ+bcτ+d

= (cτ + d)2 1

Imτ− 2ic(cτ + d)

Hence one can see for example thatE4,E6 are modular formsE2 is a quasi-modular form (of course a modular form is also aquasi-modular form)E2 := E2 − 3

π1

Imτ is an almost-holomorphic modular formJie Zhou (Perimeter Institute) Modularity in Gromov-Witten Theory FRG Workshop, Brandeis 25 / 92

Outline






Elliptic curvesWhy 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 latticeZ⊕ Zτ∗ gives rise to the complex structure of the Riemann surface C .

Take ω0 = i2 Imτ∗dz∗ ∧ dz∗, dz∗ = dx + τ∗dy . Then the Kahler moduli

space is this case is R+ω0, the complexified Kahler moduli space is thusK = {t = t1 + it2)ω0|t2 > 0} ∼= H. While it is trivial to realized theT (: t 7→ t + 1) ∈ SL(2,Z) transform on a generating series like∑

n ane2πint , it is not easy to interpret the S-transform, which is the other

generator of the group SL(2,Z), on the generating series.


Elliptic curvesMirror of elliptic curves

Now let’s go to the mirror side. It is well known that the mirror of anelliptic curve is an elliptic curve. The moduli space complex structures ofthe mirror curve (with marking=a choice of symplectic basis of H1(E ,Z))is parametrized by another copy of H, the mirror map is then the identitymap K → H, t 7→ τ which sends the area of the elliptic curve (C , tω0) tothe ”shape” (i.e., the τ -modulus) of the mirror curve.

The true moduli space M is the quotient of the moduli space of mirrorelliptic curves with markings by SL(2,Z) (the mapping class group) byforgetting about the marking. Hence on the mirror side the mapH →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-modelshould mirror to something in the A-model. And the quotient by theS-transform is, correspondingly, induced by forgetting the extra structure.


Elliptic curvesOrbifold construction for mirror of elliptic curves

Let’s formally apply the orbifold construction for mirror families to theelliptic curve case.

For the A-model, take an elliptic curve, say, a Fermat cubic. Thenaccording to the procedure of orbifold construction, the mirror familyshould be the desingularization of the Z3 quotient of the Hesse pencil

3∑i=1

x3i − z−

13

3∏i=1

xi = 0

Here the Z3 action is given by

ρ = exp2πi

3: [x1, x2, x3] 7→ [x1, ρx2, ρ

2x3]

It is easy to see that the quotient has no singular point, hence nodesingularization is actually needed.


Elliptic curvesmirror map and isogeny

Hence the Z3 quotient of the Hesse pencil π : X →M is the mirror familyof the family of Fermat cubic curve with varying Kahler structures. (Alittle work shows that the Z3 action on a fiber is generated by thetranslation 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 underlyingspace is the P1 parametrized by z) of the mirror family is a modular curve

M∼= Γ0(3)\H∗

where Γ0(3) is a nice congruence subgroup of the full modular groupSL(2,Z).


Elliptic curvesmoduli space as a modular curve

Hence we have argued that the ”correct” moduli space K of complexfiedkahler structure for the Fermat cubic (as the A-model) should be themodular curve Γ0(3)\H∗ according to mirror symmetry.

In particular, the modular group that enters the picture is not the fullmodular group SL(2,Z) but the subgroup Γ0(3).


Elliptic curvesExtra 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, butsome subfamily. The term Fermat cubic means that the extra structure weput on the genus one Riemann surface is not just the complexfied Kahlerstructure, but also some complex structure of the Riemann surface suchthat it is the Fermat cubic in P2. Forgetting this extra structure in theA-model, the moduli space would change from Γ0(3)\H∗ to SL(2,Z)\H∗.



If we shave chosen the elliptic curve to be not the Fermat cubic, butx4

1 + x42 + x2

3 = 0. Then the naıve application of the orbifold constructionwould tell that the moduli space for the mirror family (and hence themoduli space in the A-side) is Γ0(2)\H∗.

If we have chosen x61 + x3

2 + x23 = 0, we get the modular curve Γ0(1∗)\H∗

as the moduli space.

· · ·



In retrospect, the reason why the moduli space M = SL(2,Z)\H∗ is asdescribed at the beginning of this section is probably that

in the A-model of the elliptic curve, we put no extra structure at allbesides the complexified Kahler structure

A natural conclusion is that the mirror of the marking (symplectic basis ofhomology) on the B-side, which makes the moduli space H∗, should bemirror to some ”largest structure” in the A-side. If we start from themoduli space of complexified Kahler structures with ”the largest structure”and forget about it ”gradually”, the modular curve would change from H∗to SL(2,Z)\H∗ ”gradually”.


Outline






Lessons learnt from studying elliptic curvesSummary of elliptic curves

The moduli space involved in the A-model is visible to the extrastructure one puts, which is mirror to the corresponding extrastructure on the B-model and determines the modular group.

In the B-model, one can have some nice subfamilies of elliptic curvesso that the bases M are parameterized by modular curves Γ\H∗. Inthe examples discussed above, these bases are moduli spaces ofcomplex structures with level structures (which make the modulargroups Γ0(N) for some N). One can equally consider some othersubfamilies corresponding to some other extra structures, e.g.,polarization, endormorphism, etc. Then the modular curves would bereplaced by the so-called Shimura varieties.


Lessons learnt from studying elliptic curvesK3s 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(Γ3,19)+\D

Here D is called a Hermitian symmetric domain, M is called an arithmeticlocally symmetric variety. On M the generalization of modular forms,called automorphic forms, can be defined.



One can add extra structures in the B-model for K3s, which not onlychanges the discrete group O(Γ3,19)+, but also D. For example, one canconsider the moduli space of CY metrics on K3s, the extra structure wouldbe a Kahler structure compatible with the complex structure.

In this talk, we will be mainly interested in moduli space of latticepolarized algebraic K3 surfaces by The extra structure is a latticeM < H2(K3,Z). We require the Picard lattice of the K3 contain thislattice. Then one has the decomposition H2(K3,Z) = M ⊕ T . It can beshown that (see e.g., Aspinwall (1996)) the moduli space is

MM ∼ O(T )\O(2, 20− rankM)/(O(2)× O(20− rankM))

Here M is called the Neron-Severi lattice while T is the transcendentallattice.



Example: the orbifold construction applied to the Fermat quartic familygives the mirror family for the Fermat quartic. It is a lattice-polarized by

M = E8 ⊕ E8 ⊕ U ⊕ 〈−4〉 ⊂ H2(K3,Z) = E8 ⊕ E8 ⊕ U ⊕ U ⊕ U

Hence T = U ⊕ 〈4〉, this is a special case studied by Dolgachev (1995).

It turns out that

MM∼= Γ0(2)+\H∗

where Γ0(2)+ = Γ0(2) ∪ 〈τ 7→ − 12τ 〉.



For CY 3-folds and higher dimensional ones, the natural analogue of D is acomplex homogenous space but not a Hermitian symmetry domain, forwhich the existing theory of automorphic forms do not apply, at least in astraightforward way Griffiths & Schmid (1969).

What is worse, in general, it is not easy to establish the isomorphismbetween 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 aremodular curves (or nice modular varieties), so there is some hope thatmodularity can be established for those special families. I will give anexample later in this talk.


Lessons learnt from studying elliptic curvesInvariant can be obtained by looking at special subfamilies

Since we are often interested in extracting invariants of the CY variety, sotaking a special family, instead of the universal family, does not hurt andcan often help recover the most important information that we want.

To illustrate this, a familiar example is as follows. To extract GWinvariants of the quintic CY3, one can look at the Dwork pencil of quinticsinstead of the universal family of quintics. There are some subtitlesthough, 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 subfamiliesinstead of the universal family.


ExamplesStrategy in exploring modularity in Gromov-Witten theory

Let’s get back to the problem of studying modularity for theGromov-Witten theory of a CY variety Y .

Step 1: Identify the mirror family (mirror B-model) π : X →M ofthe CY family (A-model) p : Y → K.

Step 2: Establish the isomorphism M∼= a modular variety. This isthe 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 andsymmetries can be obtained for free.

Step 3: Prove/check that the generating functions have themodularity predicted in Step 2 (especially the modular group whichreflects the ”extra structure” carried by the CY family) by usingvarious ways of identifying modular forms.


Outline






Some non-compact Calabi-Yau 3-foldsKP2 : geometry of A-model

Consider the example Y = KP2 . The geometry is given byY = (C4 − Z )/C∗ with the (Cox ring) coordinates are X1,X2,X3,P andthe weights of the action of C∗ are 1, 1, 1,−3. One can alternatively usethe action by λ ∈ U(1) : (X1,X2,X3,P) 7→ (λX1, λX2, λX3, λ

−3P) withthe moment map given by µ = |X1|2 + |X2|2 + |X3|2 − 3|P|2.

Physically Witten (1993), the coordinates parametrize the vacuum fieldconfiguration of a 2d N = (2, 2) GLSM. They are the vacuum expectationvalues of some chiral fields transforming under U(1). The moment mapµ = r corresponds to the D-term constraint.


Some non-compact Calabi-Yau 3-foldsKP2 : Kahler moduli space

The r parameter in the D-term constraint µ− r = 0 is the Kahlerparameter r =

∫C ω ∈ R+, where ω is the Kahler form and C is the curve

spanning the Mori cone which is dual to the Kahler cone. The complexfiedKahler 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) gaugegroup action.

The complexfied Kahler cone is one-dimensional and is parametrized byα = exp 2πi(θ + ir). Since r ≥ 0, this Kahler cone is topologically ahemisphere giving the ”Calabi-Yau phase”. The other hemisphere is the”orbifold phase”. The Kahler moduli space K is then a copy of P1

parametrized by α. This is given by the secondary fan and is larger thanthe complexfied Kahler cone. It can be interpreted as the moduli space of(GIT) stability conditions in this particular example Witten (1993).


Some non-compact Calabi-Yau 3-foldsKP2 : Kahler moduli space

As reasoned by Witten (1993) using physics, the singularities on the modulispace should be α = 0, 1,∞.

However, it is not clear from geometry what the orbifold phase means forthe geometry KP2 . In particular, the meaning of singularities is obscure byonly looking at the A-model geometry.


Some non-compact Calabi-Yau 3-foldsKP2 : geometry of B-model

We now use the mirror picture to look at the moduli space K. Inparticular, this helps us understand the (more) mathematical meaning ofthe singularities more clearly.

The Hori-Vafa Hori & Vafa (2000) mirror family π : X →M of the geometryp : Y → K is the following family

uv − H(x , y ;α) = 0, ((u, v), (x , y)) ∈ C2 × (C∗)2

where

H(x , y ;α) = y2 − (x + 1)y − αx3


Some non-compact Calabi-Yau 3-foldsKP2 : geometry of mirror curve family

The information (e.g, periods, Weil-Petersson metric on the base M) ofthe mirror family π : X →M is fully encoded in the family of ellipticcurves πelliptic : E →M given by

H(x , y ;α) = y2 − (x + 1)y − αx3

For example, the Picard-Fuchs equations which are satisfied by the periodsare related as follows:

LCY 3 = Lelliptic ◦ θ, θ = α∂

∂α

Lelliptic = θ2 − α(θ +1

3)(θ +

2

3)

The singularities of the Picard-Fuchs equation are given by α = 0, 1,∞.


Some non-compact Calabi-Yau 3-foldsKP2 : arithmetic aspects of mirror curve family

Now we study modularity of the mirror elliptic curve familyπelliptic : E →M. It is a standard fact that this family is 3-isogenous tothe Hesse pencil πHesse : EHesse →M

x31 + x3

2 + x33 − (

α

27)−

13 x1x2x3 = 0

That is, the mirror family for the CY 3-fold is the same as the mirror ofthe Fermat cubic which is discussed earlier.

The base is parametrized by the modular curve

M∼= Γ0(3)\H∗


Some non-compact Calabi-Yau 3-foldsKP2 : periods are modular forms

The above identification M∼= Γ0(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 thefollowing expression of the parameter α (called Hauptmodul)

α(τ) =

33η(3τ)9

η(τ)3

33η(3τ)9

η(τ)3 + η(τ)9

η(3τ)3

, η(τ) = q−1

24

∏n

(1− qn), q = e2πiτ

The fundamental period (the analytic one at α = 0) annihilated by Lellipticis a Gauss hypergeometric series (and transcendental) in α and is also amodular form

2F1(1

3,

2

3; 1;α) =

∑n

(3n)!

n!(2n)!(α

27)n

2F1(1

3,

2

3; 1;α(τ)) =

(33η(3τ)12 + η(τ)12)13

η(τ)η(3τ)= ΘA2(τ) =

∑qm

2+mn+n2


Some non-compact Calabi-Yau 3-foldsKP2 : singularities on modular curve and Fricke involution

More importantly, the modular curve Γ0(3)\H∗ tells the information of thesingularities easily.

The points α = 0, 1 are the two cusps [τ ] = [i∞], [0], respectively. Thepoint α = 1 is the unique elliptic fixed point [τ ] = [ST−1(exp 2πi

3 )].

Moreover, the two cusps are related by the Fricke involutionWN=3 : τ 7→ − 1

3τ . It is more natural than the S-transformation in thesense that Γ0(3)\H∗ is the moduli space of complex structures of ellipticcurves with certain extra structure (in this case the level structure,meaning a cyclic subgroup of order 3 of 3-torsion points on the ellipticcurve) and it is the Fricke involution rather than the S-transformationwhich acts on the extra structure.


Some non-compact Calabi-Yau 3-foldsKP2 : singularities on moduli spaces

Geometrically, from the perspective of elliptic curve families, the pointsα = 0, 1 corresponds to singular elliptic curves (with j =∞) and shouldcorrespond to punctures on the moduli space. The point α =∞ gives anelliptic curve with extra automorphism and hence is an orbifold point onthe moduli space.

From the perspective of CY 3-fold families, the points α = 0, 1,∞ are thelarge complex structure limit, conifold point, orbifold point on the modulispace, respectively.


Some non-compact Calabi-Yau 3-foldsKP2 : implications of modularity

Now let’s look at some implications of the fact that the moduli space is amodular curve.

According to Bershadsky, Cecotti, Ooguri & Vafa (1993), Yamaguchi& Yau (1995),

Alim & Lange (2007), solving the holomorphic anomaly equations is reducedto calculation of Kahler potentials of the Weil-Petersson metric on M. Tobe a little more precise, F (g) are polynomials of

propagators S ij ,S i ,S , Ki and Yukawa couplingsor equivalentlyderivatives of the Kahler potential Ki ,Kij ,Kijk and Yukawa couplings

Up to holomorphic ambiguities which presumably can be fixed by boundaryocnditions, the explicit polynomials can be determined recursively in acombinatorial way.


Some non-compact Calabi-Yau 3-foldsKP2 : topological string free energies are modular forms

Modulo the problem of fixing the holomorphic ambiguities, computingderivatives of the Kahler potential is reduced to calculation of periods.

As explained above, periods are modular objects. Straightforwardcomputation Alim, Scheidegger, Yau& Zhou (2013) shows that everythinginvolved in the computation are modular forms. More precisely,almost-holomorphic modular forms (things like E2,E4,E6), which form aring that is closed upon taking derivatives.

So: topological string free energies are modular forms.



Modular forms for modular groups of small level often have integral Fourierexpansions. Assuming mirror symmetry, an easy consequence of modularityis that the GW invariants have certain integrality Zhou (2014 thesis)

The Fricke involution acting on the modular curve induces an action onthe modular forms. It turns out that the series expansions of F (g) at thetwo cusps [τ = i∞], [τ = 0] or equivalently at the large complex structurelimit and at the conifold point, are related by the Fricke involution.



Physically, at the large complex structure limit, F (g) is mirror to GWtheory, at the conifold point, F (g) is related to the so-called c = 1 stringGhoshal & Vafa (1995).

At the large complex structure limit, we choose a distinguished localcoordinate tGW (called flat coordinate), and expand the topological stringfree energy Fg in tGW , then we get FGW

g (tGW ).

That is, near the large complex structure limit, FGWg (tGW ) is mirror to the

generating series of the GW invariants

FGWg (tGW ) =

∑β

NGWg ,β qβ



At the conifold point, we choose another distinguished local coordinatebased at this point, denoted by tcon, and expand the same function F (g)

in terms of tcon. Then we get a new function F cong (tcon).

That is, near the conifold point, we have (called gap condition)

F cong (tcon) =

c

t2g−2con

+ regular terms



Of course the two functions FGWg ,F con

g 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 waythat

F cong = FGW

g |W3

Here |γ is the previously defined slash operator on modular forms and W3

is the Fricke involution τ 7→ − 13τ .


Some non-compact Calabi-Yau 3-foldsOther examples

In retrospect, that the geometry KP2 has something to do with modularityat all is due to the combinatorics of the toric data (e.g, the charge vectoris the same as that for the cubic surface in P2).

That is, the toric diagram used to define the geometry KP2 encodes theequivalent information that is used to define the anti-canonical divisor ofP2, which is the Hesse cubic family. This is why the mirror curve family ofKP2 coincides with the mirror elliptic curve family of the Hesse pencil.


Some non-compact Calabi-Yau 3-foldsOther examples

It is easy to imagine that if one starts with KWP2[1,1,2], then the toric data

would be the same as the anticanonial divisor of WP2[1, 1, 2] which gives

rise to the pencil x41 + x2

2 + x23 − z−

12 x1x2x3 = 0. The mirror curve or the

CY 3-fold would be the mirror of the above pencil obtained using theorbifold construction. In any case, the moduli space M, that is, the baseP1 of the family parametrized by z , is a modular curve.

The same approach of exploring modularity works for some othernon-compact CY 3-folds like some local del Pezzo surfaces, as discussed inAlim, Scheidegger, Yau & Zhou (2013).


Some non-compact Calabi-Yau 3-foldsSummary

In the non-compact CY 3-fold examples discussed above, modularity isused to:

Clarify the meaning of the Kahler moduli space in the A-model

Study the sequence of Gromov-Witten invariants

Explore the duality of the physics theories


Some non-compact Calabi-Yau 3-foldsAnther example in which modularity gives duality

All of the properties, especially integrality and duality, are obtained almostfor free once the identification M∼= Γ\H∗ is established.

This phenomenon appears in some other context as well. For example, inthe well-studied 4d N = 2 pure supersymmetric gauge theory withG = SU(2) Seiberg & Witten (1994).



If the Seiberg-Witten curve family is taken to be

z +Λ4

z= 2(x2 − u)

or equivalentlyy2 = (x − u)(x − Λ2)(x + Λ2)

Then the base parametrized by uΛ2 is the modular curve Γ0(4)\H∗. The

singular points u =∞,Λ2,−Λ2 corresponds to the cusps τ = i∞, 0, 1/2.

The transformation exchanging the two cusps τ = i∞, 0 is the Frickeinvolution WN=4 : τ 7→ − 1

4τ . This is the electro-magnetic duality inSeiberg-Witten theory.



If the Seiberg-Witten curve family is taken to be

y2 = (x − u)(x − Λ2)(x + Λ2)

Then the base parametrized by uΛ2 is the modular curve Γ(2)\H∗. The

singular points u =∞,Λ2,−Λ2 corresponds to the cusps τ = i∞, 0, 1.The transformation exchanging the two cusps τ = i∞, 0 is theS-transformation τ 7→ − 1

τ .

This is consistent with the previous model since the two modular groupsare isomorphic: Γ0(4)→ Γ(2), τ 7→ 2τ . Therefore, the Fricke involution onΓ0(4)\H∗ is translated to the S-transform on Γ(2)\H∗.


Outline






Open, orbifold Gromov-Witten theory of elliptic orbifoldcurvesProving modularity from the A-model directly

In the above examples, we used mirror symmetry to predicts modularity ofGW theory. We can also do this directly in the A-model in some cases,basing on the expectations from mirror symmetry.


Open, orbifold Gromov-Witten theory of elliptic orbifoldcurvesGenus zero open GW of elliptic orbifold P1s

Consider the quotient of an elliptic curve E by its automorphic group (thecurve 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 P1, we get the followingclassification for the quotient

P12,2,2,2,P1

3,3,3,P12,2,4,P1

2,3,6

Here the subscripts indicate the orders for the stalibizers groups of theorbifold points on the P1. These are conjectured to be mirror to theelliptic singularities. For example, the geometry part of the mirror of P1

3,3,3

is conjectured to be the Hesse pencil∑

x3i − ψ

∏xi = 0.



One can put Lagrangians on the elliptic orbifold P1s and count theholomorphic disks bounded by them. This procedure can be made rigorousfor some particularly chosen Lagrangians.

In the following, we shall consider the example P13,3,3 and explain why the

genus zero potential is modular, following Lau & Zhou (2014).



The mirror of (X = P13,3,3, L = Seidel Langrangian) is a LG model (V ,W )

where V = Def L and W is the superpotential. In the A-model, W is theobstruction that appears in the MC equation in the A∞-algebra thatappears in Langrangian Floer theory

m2(b) = W (b) · IdL,m(b) =∑k

mk(b, · · · b), ∀b ∈ Def L

In the A-model, coefficients of W are obtained by counting holomorphicdisks bounded by the Seidel Lagrangian L.

Further the LG/CY correspondence tells that the LG model (V ,W ) is dualto the CY model W = 0, hence they share the same moduli space.



So the mirror is conjectured to be described by the Hesse pencil W = 0

x31 + x3

2 + x33 − ψx1x2x3 = 0

It is a standard fact that the base of the Hesse pencil is the modular curveM = P1

ψ∼= Γ(3)\H∗.

Now we can conjecture that the genus zero potential is a modular form forΓ(3). The next step is to prove this rigorously, if possible.



Since the elliptic orbifold is covered by an elliptic curve E which is furthercovered by the complex plane C, everything lifts in the complex planeincluding the Seidel Langrangian.

The holomorphic disks are now presented by polygons in the complex planeand the configuration of holomorphic disks can be put into a few groupsaccording to their shapes. All polygons in the same group are similar.

This means that the number of holomorphic disks within each group isindependent of its degree (=area). This fact turns the problem of countinginto 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 workedout explicitly.



For the P13,3,3, the genus zero GW generating function, that is, the

potential is given by

W = φ(q)(x3 + y3 + z3)− ψ(q)xyz

with

φ(q) =∞∑k=0

(−1)3k+1(2k + 1)q3(12k2+12k+3)

ψ(q) = −q +∞∑k=1

((−1)3k+1(6k + 1))q(6k+1)2+ (−1)3k(6k − 1)q(6k−1)2

)



We have conjectured φ, ψ to be modular forms for Γ(3). The set ofmodular forms for the modular group forms a finitely generated ring (dueto the interpretation that they are holomorphic sections for line bundlesover the modular curve).

Simple algebra turns φ, ψ into products of η or θ-functions. We find themto be certain monomials of the generators of the ring of modular forms forΓ(3). This then proves the expected modularity.



Similar ideas work for the other examples P14,4,2, P1

2,3,6, except that for thelatter the combinatorics is much more difficult. We expect it to work forP1

2,2,2,2 as well.



The examples that I just described are very very special in the sense thatall of the invariants can be computed explicitly and we obtain a niceclosed-form expression for the generating series.

Now let’s study an example in which the invariants can not be directlycomputed in an easy way, yet in which the modularity can be proved.

Before that we need to look at another way of describing quasi-modularforms, that is, through differential equations. We believe that this way oflooking at quasi-modular forms applies to more general situations than theexamples that I shall discuss below.



The set of quasi-modular forms for a nice congruence subgroup Γ ofSL(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 ringgenerated by E2,E4,E6 and they satisfy the following differential equationscalled Ramanujan identities (q = exp 2πiτ)

1

2πi

∂

∂τE2 =

1

12(E 2

2 − E4),

1

2πi

∂

∂τE4 =

1

3(E2E4 − E6),

1

2πi

∂

∂τE6 =

1

2(E2E6 − E 2

4 )

E2(τ) = 1− 24q + · · · ,E4(τ) = 1 + 240q + · · · ,E6(τ) = 1− 504q + · · · ,



We can look at the Ramanujan identities in the opposite way. Suppose wehave some formal q-series (not even convergent, as what we get in GWtheory) satisfying the Ramanujan identities and the same boundaryconditions are E2,E4,E6, then by the existence and uniqueness of solutionsto ODEs we can conclude that these formal series must be THE Eisensteinseries.

In particular, they are convergent series and are analytic everywhere on theupper half plane. The modular transformations also tell the symmetriesthey enjoy.



For some other nice groups, we have similar Ramanujan identities. Anotherexamples is Γ = Γ0(4). The ring of quasi-modular forms is generated byany two of the three functions θ2

2(2τ), θ23(2τ), θ2

4(2τ) and E2(τ). Theysatisfy slightly more complicated but very explicit differential equations:

∂τθ22(2τ) = P1(θ2

2(2τ), θ23(2τ), θ2

4(2τ),E2(τ))

∂τθ23(2τ) = P2(θ2

2(2τ), θ23(2τ), θ2

4(2τ),E2(τ))

∂τθ24(2τ) = P3(θ2

2(2τ), θ23(2τ), θ2

4(2τ),E2(τ))

∂τE2(τ) = P4(θ22(2τ), θ2

3(2τ), θ24(2τ),E2(τ)) (∗)

where P1,P2,P3,P4 are explicit polynomials. The details are notimportant in this talk so I shall not display them.


Open, orbifold Gromov-Witten theory of elliptic orbifoldcurvesOrbifold GW of elliptic orbifold P1s

Now we consider the orbifold GW theory of the elliptic orbifold P12,2,2,2 as

an example.

The cohomology, as a graded vector space, has the basis

1 ∈ H0, x , y , z ,w ∈ H1,H ∈ H2

with the relations

x2 = y2 = z2 = w2 =1

2H

These will give the insertions in the correlation functions.



The correlation functions of genus g ≥ 2 are proved to be polynomials ofgenus zero and genus ones using the tautological relations Ionel (2002),

Faber, & Pandharipande (2005), while genus one calculation is further reducedto genus zero case Getzler (1997). Hence we only need to discuss genus zeroGW theory. Axioms in GW theory (string equation, dilation equation,divisor equation, WDVV equation, etc) tells that genus zero correlationfunctions are polynomials and derivatives of only THREE basic correlationfunctions.

That is, these three genus zero correlation function are the build blocks ofall genera correlation functions.



The three generating functions are

A = 〈x , y , z ,w〉 =∑β

qβ(

∫[M0,β(X ,β)]vir

ev∗x ∪ ev∗y ∪ ev∗z ∪ ev∗w)

B = 〈x , x , x , x〉 =∑β

qβ(


· · · )

C = 〈x , x , y , y〉 =∑β

qβ(


· · · )


Open, orbifold Gromov-Witten theory of elliptic orbifoldcurvesOrbifold GW of elliptic orbifold P1s: WDVV is equivalent to Ramanujan Shen & Zhou(2014).

These three build blocks are not independent. In fact, they are constraintby the axioms in GW theory (mostly WDVV equations which reflects theassociatively of the quantum product or the operator product in theunderlying TFT).



Mirror symmetry predicts that these three build blocks are modular formsfor Γ0(2) and also their weights.

In practice, we can compare a generating series with the Fourier series ofthe modular forms of expect weight, then using the fact that the vectorspace of a fixed weight is finite dimensional, we can guess a formula forthe generating series in terms of modular forms by looking at only the firstfew terms in the series expansion (the fact the first few terms are the sameimplies the rest are so is called the q-expansion principle).



To prove the modularity rigorously we compare the WDVV equations withthe Ramanujan identities

∂τθ2(2τ) = P1(θ2(2τ), θ3(2τ), θ4(2τ),E2(τ))

∂τθ3(2τ) = P2(θ2(2τ), θ3(2τ), θ4(2τ),E2(τ))

∂τθ4(2τ) = P3(θ2(2τ), θ3(2τ), θ4(2τ),E2(τ))

∂τE2(τ) = P4(θ2(2τ), θ3(2τ), θ4(2τ),E2(τ)) (∗)

It turns out that up to linear combination, these two sets of equations areidentical, and the corresponding boundary conditions are the same.

Therefore, by the existence and uniqueness of the solutions to ODEs, weconclude that the build blocks are quasi-modular forms.



Since derivatives of quasi-modular forms are quasi-modular forms, the allgenera correlation functions, as polynomials of the derivatives of the buildblocks, are automatically quasi-modular forms.

Similar to the CY 3-fold case, one can obtain symmetries of the correlationfunctions and integrality of orbifold GW invariants for free.



Moreover, at the orbifold point, we are supposed to get the LG model andthe correlation functions in the LG model are generating functions ofsolutions to the Witten equation Witten (1991). The theory of countingthese solutions is established rigorously by Fan, Jarvis & Ruan (2007). Thecorrelation functions can be computed in a way similar to the GW theory.

Since modularity is particularly useful in relating series expansions atdifferent points on the moduli space, this then makes it very easy tocompare the correlation functions of the two theories and hence prove theso-called LG/CY correspondence.


Outline






Lattice polarization K3 surfacesComparing WDVV to Ramanujan Ruan, Shen & Zhou (in progress)

The same idea works for the other elliptic orbifold P1s.

It should also work for the qorbifold GW of Fermat quartic K3. For theFermat quartic, the mirror family is the lattice polarized K3 family byM = E8 ⊕ E8 ⊕ U ⊕ 〈−4〉. The base M, as discussed earlier, is themodular curve Γ0(2)+\H∗.

Hence we know what to expect: the correlation functions of orbifold GWfor the quotient of Fermat quartic are quasi-modular forms for Γ0(2)+.

What is left is to find the ”building blocks” and compare the WDVVequations satisfied by them with the Ramanujan identities forquasi-modular forms for Γ0(2)+. This is work in progress Ruan, Shen & Zhou

(in progress).


ConclusionsModularity can be obtained from both A- and B-models in examples and is useful

Modularity, if exists, is particularly useful in studying the global propertiesand symmetries of the moduli spaces and the generating series that comefrom enumerative geometry.

For certain special families, one can prove that moduli spaces themselvesare modular varieties. This tells what to expect. The modular groupsuggested by using the mirror picture tells what to prove or check in theA-model.

To actually prove modularity of the generating series, one needs to usedifferent ways of describing modular forms.


DiscussionsPossible modularity from summations

For those cases where the series can be worked out explicitly, one uses thedescription that modular objects can often be described as summationsover lattices. These summation are not necessarily elliptic modular forms,but could be more general modular objects, like Jacobi forms,Hilbert/Siegel/Mock modular forms... See for example Kachru & Vafa

(1995), Marino & Moore (1998), Klemm & Marino (2005), Maulik & Pandharipande

(2006), Dabholkar, Murthy, & Zagier (2012), Pandharipande & Thomas (2014) ...

On the other hand, summations and partition functions from physics alsoproduce objects whose mathematical properties are not known yet, e.g.,the total free energy of GW invariants (Hurwitz numbers, more precisely,

Z (λ, τ) = q−1

24

∫dz

2πiz

∏p∈Z≥0+ 1

2(1 + zqpeλ

p2

2 )(1 + z−1qpe−λp2

2 ) ) of

elliptic curves Dijkgraff (1993), Kaneko & Zagier (1995) , Nekrasov partitionfunction Nekrasov (2003).


DiscussionsPossible modularity from differential equations

Generating series from enumerative geometry often satisfy nice differentialequations (from constraints in the underlying physics theory). Solutions tothese equations are usually special functions related to geometricquantities (e.g. periods, Kahler potentials) and form a differential ring.

It might be helpful to study modularity from the perspective of differentialequations. For example, in Alim, Scheidegger, Yau & Zhou (2013), differentialrings are constructed on the moduli spaces of one-parameter CY 3-foldfamilies, in the context of topological string B-model. The ring reduces tothe ring of modular objects in case that the moduli space can indeed beidentified to be a modular curve (this is not surprising now since thegenerators of the rings are eventually related to periods which are modularforms). These differential rings give generalizations of modular formsotherwise (along the lines studied in Shimura (1986) in which similar butsimpler rings are studied. The elements in the rings are shown to becertain automorphic forms.).


Thank you!


modularity in gromov-witten theorypeople.brandeis.edu/~lian/frg_workshop_2015/zhou.pdfjie zhou...

Documents