Quasi-modularity and holomorphic anomaly equation for the twisted Gromov-Witten theory: $\mathcal{O}(3)$ over $\mathbb{P}^2$

TL;DR

This paper establishes the quasi-modularity and derives the holomorphic anomaly equation for the twisted Gromov-Witten theory of the line bundle 3 over the projective plane, advancing understanding in enumerative geometry.

Contribution

It proves the quasi-modularity property and formulates the holomorphic anomaly equation for this specific twisted Gromov-Witten theory.

Findings

Proved quasi-modularity of the theory.

Derived the holomorphic anomaly equation.

Enhances understanding of enumerative invariants.

Abstract

In this paper, we prove quasi-modularity property for the twisted Gromov-Witten theory of $\mathcal{O}(3)$ over $\mathbb{P}^2$. Meanwhile, we derive its holomorphic anomaly equation.

Equations

$$N_{g,d}:={\int }_{[{\overline{M}}_{g,n}(X,d){]}^{\text{vir}}}1$$

$$F_g:=\sum _{d=0}^{\infty }{N}_{g,d}{q}^d$$

$$\mathcal{R}\in \mathbb{Q}[[q]],$$

$$F_g\in \mathcal{R}.$$

$$\frac{\partial }{\partial \bar{q}}{F}_g(q,\bar{q})=\text{exact formula of}{F}_{h<g}.$$

$$\frac{\partial }{\partial E_2}{{\Omega }^{\tau (q)}}_{g,n}(H,...,H)$$

$$=$$

$$-\frac{1}{36}{i}_{\ast }\left({{\Omega }^{\tau (q)}}_{g-1,n+2}(H,...,H,1,1)\right)$$

$$-\frac{1}{36}{j}_{\ast }\sum _{\begin{array}{c}{g}_1+g_2=g\\ S_1\bigsqcup S_2=\{1,...,n\}\end{array}}{{\Omega }^{\tau (q)}}_{g_1,|S_1|+1}(H^{S_1},1)\otimes {{\Omega }^{\tau (q)}}_{g_2,|S_2|+1}(H^{S_2},1)$$

$$i:{\overline{\mathcal{M}}}_{g-1,n+2}\to {\overline{\mathcal{M}}}_{g,n},j:{\overline{\mathcal{M}}}_{g_1,|S_1|+1}\times {\overline{\mathcal{M}}}_{g_2,|S_2|+1}\to {\overline{\mathcal{M}}}_{g,n}.$$

$${\overline{M}}_{g+1,n}(X,\beta )\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$E_{g,n,\beta }=R^0{\pi }_{\ast }{f}^{\ast }E-R^1{\pi }_{\ast }{f}^{\ast }E\in K^0({\overline{M}}_{g,n}(X,\beta ))$$

$$(v_1,v_2{)}^{tw}:={\int }_X{v}_1{v}_2\mathbf{c}(E)$$

$${\Omega }_{g,n}(v_1,...,v_n):=\sum _{\beta =0}^{\infty }{q}^{\beta }{p}_{\ast }\left([{\overline{M}}_{g,n}(X,\beta ){]}^{vir}e{v}_1^{\ast }{v}_1...e{v}_n^{\ast }{v}_n\mathbf{c}(E_{g,n,\beta })\right)\in H^{\ast }({\overline{M}}_{g,n})[[q]]$$

$$⟨{{\psi }_1}^{k_1}{v}_1,...,{{\psi }_n}^{k_n}{v}_n{⟩}_{g,n,\beta }={\int }_{{\overline{M}}_{g,n}}{{\psi }_1}^{k_1}e{v}_1^{\ast }{v}_1...{{\psi }_n}^{k_n}e{v}_n^{\ast }{v}_n\mathbf{c}(E_{g,n,\beta }).$$

$$(t_0,t_1,t_2)\cdot [x_0,x_1,x_2]=[{t_0}^{-1}{x}_0,{t_1}^{-1}{x}_1,{t_2}^{-1}{x}_2]$$

$$(t_0,t_1,t_2)\cdot [x_0,x_1,x_2,v]=[{t_0}^{-1}{x}_0,{t_1}^{-1}{x}_1,{t_2}^{-1}{x}_2,v]$$

$$J^{tw}(t(z),-z)=-1z+t(z)+\sum _{n,d}\frac{q^d}{n!}⟨t(\psi ),...,t(\psi ),\frac{{\varphi }^{\alpha }}{-z-{\varphi }_{n+1}}{⟩}_{0,n+1,d}^{tw}{\varphi }_{\alpha }$$

$${\mathcal{L}}^{tw}=\left\{z\sum _{i=1}^n{c}_i(t,z)S^{\text{tw}}(t,-z{)}^{\ast }{\varphi }_i\right\}$$

$$S^{\text{tw}}(t,z{)}^{\ast }(v)=v+\sum _{n,d,\alpha }\frac{q^d}{n!}⟨v,t,...,t,\frac{{\varphi }_{\alpha }}{z-\psi }{⟩}_{0,n+2,d}^{\text{tw}}{\varphi }^{\alpha }$$

$$zdS(t,q)=dt{\ast }_{t}S(t,q).$$

$$\Omega =R_{\cdot }\omega$$

$$(t_0,t_1,t_2)\cdot [z_0,z_1,z_2]:=[{t_0}^{-1}{z}_0,{t_1}^{-1}{z}_1,{t_2}^{-1}{z}_2]$$

$$H_{({\mathbb{C}}^{\ast }{)}^3}^{\ast }({\mathbb{P}}^2)\cong \mathbb{C}[H]/⟨\prod _{i=0}^2(H-{\lambda }_i)=0⟩$$

$$(v_1,v_2):={\int }_{{\mathbb{P}}^2}{v}_1{v}_2{e}_{({\mathbb{C}}^{\ast }{)}^3}(\mathcal{O}(3))={\int }_{{\mathbb{P}}^2}{v}_1{v}_2(3H)$$

$$\left(\begin{array}{ccc}0& 3& 0\\ 3& 0& 0\\ 0& 0& 3{\lambda }^3\end{array}\right).$$

$${\mathfrak{e}}_{\alpha }=\frac{{\prod }_{\beta \ne \alpha }(H-{\xi }^{\beta }\lambda )}{{\prod }_{\beta \ne \alpha }({\xi }^{\alpha }\lambda -{\xi }^{\beta }\lambda )}=\frac{{\prod }_{\beta \ne \alpha }(H-{\xi }^{\beta }\lambda )}{3{\xi }^{2\alpha }{\lambda }^2}$$

$$\sum _{\alpha }{\mathfrak{e}}_{\alpha }=1,{\mathfrak{e}}_{\alpha }\cdot {\mathfrak{e}}_{\beta }={\delta }_{\alpha \beta }{\mathfrak{e}}_{\alpha }.$$

$$H\cdot {\mathfrak{e}}_{\alpha }=({\xi }^{\alpha }\lambda ){\mathfrak{e}}_{\alpha }=:{\mathfrak{h}}_{\alpha }{\mathfrak{e}}_{\alpha }$$

$$\Vert {\mathfrak{e}}_{\alpha }\Vert =(1,{\bar{\mathfrak{e}}}_{\alpha }{)}^{\lambda }=\sqrt{\frac{1}{{\xi }^{\alpha }\lambda }}.$$

$${\Omega }_{g,n}(v_1,...,v_n)=\sum _{d=0}^{\infty }{q}^d{p}_{\ast }\left(e{v}_1^{\ast }{v}_1...e{v}_n^{\ast }{v}_n{e}_{({\mathbb{C}}^{\ast }{)}^3}(R{\pi }_{\ast }{f}^{\ast }\mathcal{O}(3))\cap {\left[{\overline{M}}_{g,n}({\mathbb{P}}^2,d)\right]}^{vir}\right)$$

$$({\Omega }^{\tau (q)}{)}_{g,n}(v_1,...,v_n)=\sum _{k=0}^{\infty }\frac{1}{k!}({\pi }_k{)}_{\ast }\left({\Omega }_{g,n+k}(v_1,...,v_n,\tau (q{)}^{\otimes k})\right)$$

Full text

1. 1 Introduction

2. 2 Preliminary on twisted Gromov-Witten theory

3. 2.1 Genus-0 twisted Gromov-Witten theory

4. 2.1.1 Twisted Lagarian cone

5. 2.1.2 Quantum differential equation

6. 2.2 Higher genus twisted Gromov-Witten invariants.

7. 3 Twisted theory $\mathcal{O}(3)$ over $\mathbb{P}^{2}$

8. 3.1 Classical $(\mathbb{C}^{*})^{3}$ equivariant theory for $\mathbb{P}^{2}$

9. 3.2 $I$ function and mirror theorem

10. 3.3 Computation of $R(z)^{*}1$

11. 3.4 Oscillatory integral and asymptotic expansion

12. 3.5 Quantum differential equation

13. 4 Finite generation property

14. 4.1 Basic generators

15. 4.2 Total $R$ matrix

16. 4.3 Finite generation property

17. 5 Quasi-modularity property

18. 5.1 Review of quasi-modular forms of $\Gamma_{0}(3)$

19. 5.2 Relate generators to quasi-modular form

20. 5.3 Proof of quasi-modularity property of the twisted theory $\mathcal{O}(3)$ over $\mathbb{P}^{2}$

21. 6 Holomorphic anomaly equation

1. Introduction

Let $X$ be a smooth compact Calabi-Yau threefold, denote

[TABLE]

to be the genus-$g$, degree $d\in H_{2}(X;\mathbb{Z})$ Gromov-Witten invariants, where $[\overline{M}_{g,n}(X,d)]^{\text{vir}}$ is the virtual fundamental class of the moduli space of stable map $\overline{M}_{g,n}(X,d)$ (c.f. [12] and [16]). Define

[TABLE]

to be the genus-$g$ generating funtions. Then there are two predictions from mirror symmetry:

• •

There is a finitely generated subring of “quasi-modular objects”

[TABLE]

such that for all $g$

[TABLE]

• •

$F_{g}$ satisfies a recursion formula, which is called holomorphic anomaly equation. i.e.

[TABLE]

where $F_{g}(q,\bar{q})$ is the modular completion of $F_{g}(q)$.

In general, it is usually very difficult to compute the Gromov-Witten invariants for compact Calabi-Yau threefold. Recently, it is proved that these two properties are satisfied for Quintic 3-fold (Ref. [9]). Instead, the twisted Gromov-Witten theory is much easier and also expected to satisfy the two properties in the above predictions. Many examples such as local $\mathbb{P}^{2}$ has been studied (Ref. [10], [3] and [4]). In this paper, inspired by the calculations in [9], we discuss one simple example $\mathcal{O}(3)$ over $\mathbb{P}^{2}$ and prove the quasi-modularity property and holomorphic anomaly equation.

The reason why we are interested in this twisted theory $\mathcal{O}(3)$ over $\mathbb{P}^{2}$ is: it has close relation with geometry of cubic elliptic curve, since the zeros of the generic section defines a cubic elliptic curve.

Our main theorem is

Theorem 1.1**.**

The formal twisted theory $\mathcal{O}(3)$ over $\mathbb{P}^{2}$ satisfies

• •

Quasi-modularity property: the complex degree $g-1+n$ part $[{\Omega^{\tau(q)}}_{g,n}(H,...,H)]_{g-1+n}$ is a quasi modular form of modular group $\Gamma_{0}(3)$ with weight $2g-2+2n$ with valued in $A^{g-1+n}(\overline{\mathcal{M}}_{g,n};\mathbb{Q})$.

• •

The holomorphic anomaly equation:

[TABLE]

where $\tau(q)$ is the mirror map (see section 3). $E_{2}$ is the second Eisenstein series. $i$ and $j$ are the gluing maps between moduli space of curves.

[TABLE]

Remark 1.2**.**

Our theorem is somehow parallel to the quasi modularity and holomorphic anomaly equation of elliptic curve proved in [14]. The main difference of the quasi modularity of twisted theory $\mathcal{O}(3)$ over $\mathbb{P}^{2}$ and elliptic curve is about the modular ring: one is $\Gamma_{0}(3)$, the other is $SL(2;\mathbb{Z})$. The associated holomorphic anomaly equation is the same shape.

This paper is organised as follows: In section 2, we review some basic knowledge in twisted Gromov-Witten theory. In section 3, we focus on the genus-0 twisted Gromov-Witten theory of the example $\mathcal{O}(3)$ over $\mathbb{P}^{2}$ and the computation of $R$ matrix. In section 4 and section 5, we prove the finite generation and quasi modularity property of twisted Gromov-Witten theory of the example $\mathcal{O}(3)$ over $\mathbb{P}^{2}$. In section 6, we prove the associated holomorphic anomaly equation.

Acknowledgements. The authors would like to special thank Shuai Guo and Felix Janda for discussing Givental theory and Calabi-Yau geometry. The results are obtained during the visit of the author in University of Michigan. The author would like thank professor Melissa Liu and Yongbin Ruan for their help during the visit in Columbia University and University of Michigan. The author is partially supported by NSFC grant 11601279.

2. Preliminary on twisted Gromov-Witten theory

Let $X$ be a smooth projective variety, $E$ is a holomorphic vector bundle over $X$. $\overline{M}_{g,n}(X,\beta)$ is the moduli space of stable maps to $X$ of genus-$g$, with $n$ markings, degree $\beta$. We consider the universal stable maps over $\overline{M}_{g,n}(X,\beta)$

[TABLE]

Set the K-theoretical push forward

[TABLE]

Let $\mathbf{c}$ be an invertible multiplicative character class. For any $v_{1},v_{2}\in H^{*}(X;\mathbb{Q})$, the twisted paring is defined by

[TABLE]

and the correlator is defined by

[TABLE]

Then the Cohomological field theory $(H^{*}(X,\mathbb{Q}),(,)^{tw},\Omega,1)$ with unit $1\in H^{*}(X,\mathbb{Q})$ is our twisted theory. Similarly, we can define the (decedant) twisted genus-$g$ GW invariants by

[TABLE]

In general, there are many different type of twisted theory associated to a holomorphic vector bundle $E$ over a smooth projective variety $X$. In this paper, we study a certain type of twisted theory, called formal theory (c.f. [10]). As an example, we give the formal theory associated to $\mathcal{O}(3)$ over $\mathbb{P}^{2}$.

Example 2.1**.**

Consider the $(\mathbb{C}^{*})^{3}$ action on the base $\mathbb{P}^{2}$, with action

[TABLE]

then this torus action has a natural lift to the total space $\mathcal{O}_{\mathbb{P}^{2}}(3)$ by

[TABLE]

denote the associated equivariant parameter of the torus action $\lambda_{0},\lambda_{1},\lambda_{2}$, then taking the specialization $\lambda_{i}=\xi^{i}\lambda$, where $\xi=e^{\frac{2\pi i}{3}}$. The invertible multiplier character class is the equivariant Euler class $\mathbf{e}_{(\mathbb{C}^{*})^{3}}(\mathcal{O}_{\mathbb{P}^{2}}(3))$ (taking specialization). Then we obtain the associated formal twisted theory.

2.1. Genus-0 twisted Gromov-Witten theory

The genus-0 twisted GW invariants was introduced and studied in [2] and [1]. There are two important ingredients in genus-0 twisted theory: one is the twisted Lagrange cone, the other one is the quantum differential equation.

2.1.1. Twisted Lagarian cone

Assume $\{\phi_{\alpha}\}_{\alpha=1}^{N}$ is a linear basis of $H^{*}(X,\mathbb{Q})$. A generic element in the Lagrangian cone $\mathcal{L}^{tw}$ has the form

[TABLE]

where $t(z)\in H^{*}(X,\mathbb{Q})[[z]]$. By string equation, dilaton equation and genus-0 topological recursion relation, Givental proved the Lagrangian cone has very special geometry (c.f. [8]):

• •

$\mathcal{L}^{tw}$ is a cone at the origin

• •

Each tangent space $T$ is tangent to the cone $\mathcal{L}^{tw}$ exactly along $zT$

This implies that

• •

The cone $\mathcal{L}^{tw}$ is determined by big $J$ function: $J(t,-z)$

• •

Each tangent space $T$ is a $\Lambda[[z]]$ module with basis $S^{\text{tw}}(t,-z)^{*}\phi_{1},...,S^{\text{tw}}(t,-z)^{*}\phi_{n}$.

• •

[TABLE]

where $S$ operator is defined by 2-pt function

[TABLE]

2.1.2. Quantum differential equation

Quantum differential equation at point $t\in H^{*}(X;\mathbb{Q})$ has the form

[TABLE]

By genus-0 topological recursion relation, the two point function $S(t,q)$ is always a fundamental solution of the QDE (1). Near the semisimple point $t=\tau$, there is a formal solution of the form $\Psi Re^{\frac{U}{z}}$, where $R$ is unique up to some constant matrix of power series of $z$: $e^{\sum_{k=0}^{\infty}z^{2k+1}diag(a^{(k)}_{1},...,a^{(k)}_{n})}$, where all $a_{j}^{(i)}\in\mathbb{C}$ (c.f. [7]).

Remark 2.2**.**

In general, there is no relation between these two solutions. In some good cases, for example, the full equi-variant theory of a smooth toric variety. These 2 solution can be related by localization and quantum Riemann-Roch. The ambiguity of the $R$ matrix can be uniquely determined by quantum Riemann-Roch (c.f. [7].).

2.2. Higher genus twisted Gromov-Witten invariants.

Most examples of twisted theory we are interested are semi-simple. The semisimplicity of our examples can be seen from the proof. So we can directly apply Givental-Teleman classification theorem ([17]) to compute higher genus twisted GW invariants from genus-0 twisted GW invariants. Formally, the formula is (for precise formula, see[15])

[TABLE]

where $\omega_{g,n}=\Omega_{g,n}\cap H^{0}(\overline{M}_{g,n})$ is the topological part of $\Omega$. It is a graph sum formula. All the contributions in the graph sum are expressed in terms of $R$ matrix.

So if we want to compute higher genus twisted GW invariants, the key point is to compute the $R$ matrix here. Before compute it, we should first know where $R$ matrix comes from and its geometric meaning. The original proof in [17] relies heavily on the topological structure of the moduli space of curves. For equivariant Gromov-Witten theory, there is a more geometric way to get the $R$ matrix. For example, for smooth toric variety, first taking localization, then contract the unstable rational components to get stable graphs, in this process, we get a matrix $R_{loc}$, which comes from the contribution of all the unstable rational components. Then using quantum Riemann-Roch theorem, string and dilaton equations, we get the $R$ marix.

3. Twisted theory $\mathcal{O}(3)$ over $\mathbb{P}^{2}$

In this section, we consider the formal twisted theory associated to $\mathcal{O}(3)$ over $\mathbb{P}^{2}$.

3.1. Classical $(\mathbb{C}^{*})^{3}$ equivariant theory for $\mathbb{P}^{2}$

In this subsection, we compute $(\mathbb{C}^{*})^{3}$ equivariant theory for $\mathbb{P}^{2}$, where the action is given by

[TABLE]

then the classical equivariant cohomology ring is

[TABLE]

where $H=c_{1}^{\mathbb{C}^{*}}(\mathcal{O}(1))$ under the lift of the torus action to $\mathcal{O}(1)$, which satisfy $H|_{p_{\alpha}}=\lambda_{\alpha}$. Choose a flat basis $\{1,H,H^{2}\}$ of $H^{*}_{(\mathbb{C}^{*})^{3}}(\mathbb{P}^{2})$. The twisted paring by twisted bundle $\mathcal{O}(3)$ is: any $v_{1},v_{2}$,

[TABLE]

From now on, for simplicity, we specialize the equivariant parameter by $\lambda_{\alpha}:=\xi^{\alpha}\lambda$. Then the twisted paring matrix under basis $\{1,H,H^{2}\}$ is

[TABLE]

The algebra $H^{*}_{\mathbb{C}^{*}}(\mathbb{P}^{2})$ is semisimple. It has a canonical basis

[TABLE]

satisfying the properties of idempotent of the semisimple algebra:

[TABLE]

It is easy to compute

[TABLE]

and its norm

[TABLE]

3.2. $I$ function and mirror theorem

First, the correlator of the twisted theory associated to $\mathcal{O}(3)$ over $\mathbb{P}^{2}$ now is defined to be

[TABLE]

In the following, we consider the associated shifted CohFT

[TABLE]

with initial valuation

[TABLE]

To compute $R$ matrix, we should use another family of elements $I(q,-z)$ in the twisted Lagrangian cone $\mathcal{L}^{\mathcal{O}(3)}_{\mathbb{P}^{2}}$, which is a hypergeometric series.

[TABLE]

with $I_{0}(q)=1+\sum_{d=1}^{\infty}q^{d}\frac{(3d)!}{(d!)^{3}}$, $I_{1}(q)=\sum_{d=1}^{\infty}q^{d}\frac{(3d)!}{(d!)^{3}}\left(\sum_{m=d+1}^{3d}\frac{3}{m}\right)$. A very important property of the $I$ function is: it is the solution of Picard-Fuchs equation

[TABLE]

where $D_{H}=H+zq\frac{d}{dq}$. Take paring with normalized fixed point basis $\{\bar{\mathfrak{e}}_{\alpha}\}_{\alpha=1}^{3}$, we can replace $H$ with $\{\mathfrak{h}_{\alpha}\}$, then the Picard-Fuchs equation becomes

[TABLE]

3.3. Computation of $R(z)^{*}1$

Via genus-0 mirror theorem (c.f. [6] and [13]),

[TABLE]

where $\tau(q)=\frac{I_{1}(q)}{I_{0}(q)}$ is the mirror map. By localization and quantum Riemann-Roch theorem (c.f. [2]), we have the relation between $S$ operator and $R$ operator,

[TABLE]

where

[TABLE]

which comes from contribution of quantum Riemann-Roch theorem. Thus $I_{0}R_{0\bar{\alpha}}(\tau(q),q,z)$ satisfies Picard-Fuchs equation

[TABLE]

with initial condition i.e.

[TABLE]

where $L_{\alpha}(q)=\mathfrak{h}_{\alpha}+q\frac{d}{dq}u^{\alpha}(\tau(q),q)$. By looking at coefficient of $z^{0}$ in the equation (3).

[TABLE]

where $L:=(1-27q)^{-\frac{1}{3}}$. Then by looking at the coefficients of $z^{k+1},k\geq 0$, we can get all the ${(R_{k})_{0}}^{\alpha}$ recursively by solving the Picard-Fuchs equation and initial conditions. For example, here we list ${(R_{k})_{0}}^{\alpha}$ for $k=1,2,3$.

[TABLE]

In general, we have the following property of $R(z)^{*}1$

Proposition 3.1**.**

[TABLE]

Proof.

For simplicity, we write

[TABLE]

then the Picard-Fuchs equation becomes

[TABLE]

with initial condition

[TABLE]

where

[TABLE]

Notice

[TABLE]

Observing that the coefficient of $z^{k+1}$ in equation (5) gives us

[TABLE]

for any $k\geq 3$.

Then we solve the equation (5) by induction, assume any $i<k$,

[TABLE]

then

[TABLE]

After integration, we have

[TABLE]

Then the proposition 3.1 follows from the following initial condition which will be proved by proposition 3.2 and lemma 3.3 via oscillatory integral and its asymptotic expansion.

[TABLE]

∎

3.4. Oscillatory integral and asymptotic expansion

111The argument in this section is similar to the Appendix A in [11].

Now we consider the Landau-Ginzburg potential $W:(\mathbb{C}^{*})^{4}\rightarrow\mathbb{C}$ for the $(\mathbb{C}^{*})^{3}$ equivariant twisted theory $\mathcal{O}(3)$ over $\mathbb{P}^{2}$,

[TABLE]

Now we consider the oscillatory integral

[TABLE]

where $\gamma$ is the Lefschetz thimble in a sub-torus in $(\mathbb{C}^{*})^{4}$

[TABLE]

We consider the following 3 local charts of the sub-torus

[TABLE]

[TABLE]

[TABLE]

then we have the following proposition, which gives the relation between the $R(z)^{*}1$ and oscillatory integral.

Proposition 3.2**.**

For $\alpha=0,1,2$, the derivative of $\mathcal{I}(q,z,\lambda)|_{U_{\alpha}}$ has the asymptotic expansion

[TABLE]

with

[TABLE]

Proof.

For simplicity, we only prove for $\alpha=0$. In the local chart $U_{0}$, the oscillatory integral $\mathcal{I}(q,z)$ becomes:

[TABLE]

In the above, we use integration by parts repeatedly and the vanishing of the integration equals to zero at the ends of the Lefchetz thimble.

Notice that

[TABLE]

Then we have

[TABLE]

Here we use the standard asymptotic expansion of Gamma function:

[TABLE]

in complex domain as $x\rightarrow\infty$ uniformly on $|argx|\leq\pi-\epsilon$, $\epsilon$ is given in advance.

∎

Lemma 3.3**.**

[TABLE]

Proof.

Now we compute the oscillatory integral using coordinates $x_{0},x_{1},x_{2}$.

[TABLE]

Notice that $\left(-q{L_{\alpha}}^{3}\right)|_{L=0}=\frac{\lambda^{3}}{27},$ so at $L=0$, we have

[TABLE]

In the second step in the above, we use the replacement $\frac{x_{i}}{-z}\rightarrow x_{i}$. The last step follows from the asymptotic expansion of Gamma function (6). ∎

3.5. Quantum differential equation

The quantum differential equation has the form $dS(t,q)=dt*_{t}S(t,q)$, where $d$ only action on $t$, not on $q$. For our use, we write it in terms of $S(t,z)^{*}$. At point $\tau=\tau(q)$, the quantum differential equation becomes

[TABLE]

where $D_{H}=zq\frac{d}{dq}+H$ and $\dot{\tau}=H+q\frac{d\tau(q)}{dq}=I_{1,1}H$. Assume

[TABLE]

Via Birkhoff factorization, we obtain

Lemma 3.4**.**

The quantum differential matrix

[TABLE]

where for any $n\geq m\geq 1$

[TABLE]

Proof.

Via Genus-0 mirror theorem (c.f.[6] and [13]), we have

[TABLE]

Taking derivative along $D_{H}$

[TABLE]

So we get

[TABLE]

then taking derivatives along $D_{H}$ recursively, we obtain the

[TABLE]

∎

From symmetry of the quantum produc $\dot{\tau}*_{\tau}$ and character polynomial of matrix $A$, we get following relations among the entries of matrix $A$.

Lemma 3.5**.**

Let $L(q):=(1-27q)^{-\frac{1}{3}}$, then

[TABLE]

Proof.

The equations $(i)$ are exactly the Zinger-Zagier relation (c.f. [18]). Now we give the proof of equation $(ii)$. Let

[TABLE]

then

[TABLE]

The $\widetilde{I}$ function satisfies PF equation

[TABLE]

It is equivalent to

[TABLE]

It is easy to compute the left hand side of the above equation is

[TABLE]

Thus $\widetilde{I}(q,z)$ lies in the tangent space $T\mathcal{L}^{\text{tw}}$ of the twisted Lagrangian cone. By the expansion

[TABLE]

we have

[TABLE]

then comparing the coefficient of $S^{*}1$ in the equation (7), we get equation $(ii)$. ∎

4. Finite generation property

4.1. Basic generators

Define the following degree $k$ generator

[TABLE]

Lemma 4.1**.**

For generators $\{X_{k}\}$ and $\{Y_{k}\}$, we have

[TABLE]

Proof.

The first equation just follows from $(ii)$ in Lemma3.5. The second one follows easily by induction from the following identities

[TABLE]

∎

4.2. Total $R$ matrix

Together the quantum differential equation (1) and the relation between $S$ operator and $R$ operator, we obtain the recursion relation of total $R$ matrix.

[TABLE]

where $R_{i\bar{\alpha}}:=\langle H^{i},R(\tau(q),z)(\bar{e}_{\alpha}(q))\rangle$. From the recursion formula of $R$ matrix, we obtain the important property of $R$ matrix

Proposition 4.2**.**

For any $k\geq 0$,

[TABLE]

Proof.

The first one is just a restatement of proposition 3.1. The first two columns of (8) give us

[TABLE]

[TABLE]

Then the recursion relation for $\{{R(z)_{i}}^{\alpha}\}$ is

[TABLE]

[TABLE]

By using equation (4) and equations $(i)$ in lemma 3.5, we get

[TABLE]

Notice that

[TABLE]

Combining with proposition 3.1, we prove the proposition. ∎

4.3. Finite generation property

Theorem 4.3**.**

[TABLE]

where $[\quad]_{d}$ means the complex cohomological degree $d$ part.

Proof.

For simplicity, we first compute the contribution of the trivial stable graph to ${\Omega^{\tau(q)}}_{g,n}(H^{i_{1}},...,H^{i_{n}})$:

[TABLE]

Recall

[TABLE]

so

[TABLE]

then the degree $d$ term equals to

[TABLE]

For general genus-$g$, $n$ marking stable graph $\Gamma$, the associated contribution of $\Gamma$ in Givental-Teleman graph sum formula is

• •

On each vertex, the contribution is

[TABLE]

• •

On the legs, the contribution is

[TABLE]

• •

On the kappa tails, the contribution is

[TABLE]

• •

On the edges, the contribution is

[TABLE]

where $(\eta^{ij})$ is the inverse matrix of the paring matrix 2. Notice that

[TABLE]

Assume the kappa tails contributes degree $t$ algebraic cohomological classes. Then we can analysis each factor in the contribution of $\Gamma$ to $\left[{\Omega^{\tau(q)}}_{g,n}\left(R(z)^{-1}(H^{i_{1}}),..,R(z)^{-1}(H^{i_{n}})\right)\right]_{\deg=d}$: The power of $\lambda$ is

[TABLE]

The power of $\frac{I_{00}}{L}$ is

[TABLE]

The rest coefficients lies in the the ring

[TABLE]

Lastly, since the graph sum formula is symmetric to any index $\alpha$, so after taking summation, the factors $\xi^{\alpha},\xi^{\beta}...$ become some rational numbers. ∎

5. Quasi-modularity property

5.1. Review of quasi-modular forms of $\Gamma_{0}(3)$

Let

[TABLE]

be the cubic AGM theta functions (c.f. [5]). The ring of modular form and quasi-modular form of $\Gamma_{0}(3)$ is

[TABLE]

[TABLE]

where $E_{2},E_{4},E_{6}$ are the Eisenstein series of weight 2,4,6 respectively.

Remark 5.1**.**

By computing finitely many Fourier coefficients, we can check that $a^{2}(Q)$, $a(Q)b^{3}(Q)$ and $b^{6}(Q)$ is another set of generators of the ring of modular form of $\Gamma_{0}(3)$ of weight 2,4 and 6 respectively.

Since the ring of quasi-modular form is closed under the derivative operator $Q\frac{d}{dQ}$, then by comparing finitely many Fourier coefficient, we get the following identities.

Lemma 5.2**.**

[TABLE]

Proof.

If $f(Q)$ is a modular form of weight $k$ and level $n$ for some character $\chi$, then the Serre derivative

[TABLE]

is a modular form of weight $k+2$ and level $n$ for the same character. Computing finitely many Fourier coefficient, we find the identities. ∎

5.2. Relate generators to quasi-modular form

Recall that

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and

[TABLE]

On the other hand, the cubic AGM theta function

[TABLE]

Naturally we take the identification

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then

[TABLE]

Moreover, we have the following lemma

Lemma 5.3**.**

Under the identification (10), the variable $Q$ is exactly the mirror map

[TABLE]

Proof.

By definition, $x=qe^{\frac{I_{1}}{I_{0}}}$ satisfies the differential equation

[TABLE]

Using lemma 5.2, the differential equation (12) simplifies to

[TABLE]

Since $Q$ has no constant term when written in terms of $x$, this implies that $x=Q$. ∎

Now we can relate the basic generators to the quasi-modular from of $\Gamma_{0}(3)$.

Lemma 5.4**.**

[TABLE]

Proof.

By lemma 5.2

[TABLE]

The second and third ones follow from lemma5.2 and equations (11). ∎

Remark 5.5**.**

From lemma 5.4, we see that

[TABLE]

is another set of generators of the ring of modular form of $\Gamma_{0}(3)$. And

[TABLE]

is another set of generators of the ring of quasi-modular form of $\Gamma_{0}(3)$.

5.3. Proof of quasi-modularity property of the twisted theory $\mathcal{O}(3)$ over $\mathbb{P}^{2}$

First, we study the relations between basic generators and modular forms. Taking $d=g-1+n$ in the theorem 4.3, combining the lemma 5.4, we obtain the following quasi-modularity property for twisted theory $\mathcal{O}(3)$ over $\mathbb{P}^{2}$.

Corollary 5.6**.**

For $\lambda$-twisted theory $\mathcal{O}(3)$ over $\mathbb{P}^{2}$, the degree $g-1+n$ part of ${\Omega^{\tau(q)}}_{g,n}(H,...,H)$ is a cycle-valued quasi modular form of $\Gamma_{0}(3)$ with weight $2g-2+2n$.

[TABLE]

Remark 5.7**.**

For other degree parts, we can also express the coefficients as the rational function of the generators of quasi-modular form of ring $\Gamma_{0}(3)$.

6. Holomorphic anomaly equation

In this section, we prove the holomorphic anomaly equation. This part is similar as the proof in [10] for the case of local $\mathbb{P}^{2}$. The key point is the following proposition:

Proposition 6.1**.**

For $R$ matrix,

[TABLE]

For $V$ matrix,

[TABLE]

Proof.

Equation (13) follows from equation (9) and lemma 5.4. Equation (14) just follows from equation (13). ∎

Using proposition 6.1 and Givental-Teleman classification theorem, we obtain

Theorem 6.2**.**

For the formal twisted theory $\mathcal{O}(3)$ over $\mathbb{P}^{2}$, the holomorphic anomaly equation holds

[TABLE]

Proof.

By Teleman-Givental theorem, ${\Omega^{\tau(q)}}_{g,n}(H^{i_{1}},...,H^{i_{n}})$ can be expressed as a graph sum formula. Taking derivative along $\frac{\partial}{\partial E_{2}(Q)}$, using Leibniz’s rule, together with proposition 6.1, we prove the theorem. ∎