Open Gromov–Witten Theory of $K_{{\mathbb {P}}^2}, K_{{{\mathbb {P}}^1}\times {{\mathbb {P}}^1}}, K_{W{\mathbb {P}}\left[ 1,1,2\right] }, K_{{{\mathbb {F}}}_1}$ and Jacobi Forms

Abstract

It was known through the efforts of many works that the generating functions in the closed Gromov–Witten theory of $K_{{\mathbb {P}}^2}$ are meromorphic quasi-modular forms (Coates and Iritani in Kyoto J Math 58(4):695–864, 2018; Lho and Pandharipande in Adv Math 332:349–402, 2018; Coates and Iritani in Gromov–Witten invariants of local ${\mathbb {P}}^{2}$ and modular forms, arXiv:1804.03292 [math.AG], 2018) basing on the B-model predictions (Bershadsky et al. in Commun Math Phys 165:311–428, 1994; Aganagic et al. in Commun Math Phys 277:771–819, 2008; Alim et al. in Adv Theor Math Phys 18(2):401–467, 2014). In this article, we extend the modularity phenomenon to $K_{{{\mathbb {P}}^1}\times {{\mathbb {P}}^1}}, K_{W{\mathbb {P}}[1,1,2]}, K_{{\mathbb {F}}_1}$. More importantly, we generalize it to the generating functions in the open Gromov–Witten theory using the theory of Jacobi forms where the open Gromov–Witten parameters are transformed into elliptic variables.

Some Explicit Formulae

Some explicit formulae for the disk potential, annulus potential, $\omega _{0,3}$, and $\omega _{1,1}$ for certain special one-parameter families of our four examples are collected in this appendix. The general expressions are displayed below.

• Disk potential

  $$\begin{aligned} \partial _x{W}=\log y\cdot {1\over x}. \end{aligned}$$

  (A.1)

• Annulus potential

  $$\begin{aligned} \omega _{0,2}(u_{1},u_{2})=B(u_{1},u_{2})=(\wp (u_{1}-u_{2})+ {\eta }_{1})du_1\boxtimes du_2. \end{aligned}$$

  (A.2)

• Recursion kernel $K=d^{-1}S/\Lambda$,

  $$\begin{aligned} S(u_{1},u_{2})= & {} (\wp (u_{1}-u_{2})+\widehat{\eta }_{1}) du_1\boxtimes du_2,\nonumber \\ \Lambda= & {} 2 \sum _{k=0}^{\infty } {1\over 2k+1} \left( {y-y^{*}\over y+y^{*}}\right) ^{2k+1}\partial _{u}x {1\over x} du. \end{aligned}$$

  (A.3)

  Here $d^{-1}S$ is as defined in (4.15), and the expression $y^{*}=-y-2h(x)$ in (3.3) is determined from the mirror curve equation as in (3.1) and (3.3).

• $\omega _{0,3}$

  $$\begin{aligned} \omega _{0,3}(u_{1},u_{2},u_{3})= & {} \sum _{r\in R^{\circ }} \left( 2[{1\over \Lambda } ]_{-2}\cdot \prod _{k=1}^{3} (\wp (u_{k}-u_{r})+\eta _{1}) \right) \nonumber \\&\quad du_1\boxtimes du_2\boxtimes du_{3}, \end{aligned}$$

  (A.4)

• $\omega _{1,1}$

  $$\begin{aligned} \omega _{1,1}(u_1)= & {} \sum _{r\in R^{\circ }}\left( {1\over 24}\left[ {1 \over \Lambda }\right] _{-2}\wp ^{(2)}(u_1-u_r)+{{\eta _1}}\left[ {1\over \Lambda }\right] _{-2}\wp (u_1-u_r)\right. \nonumber \\&\quad \left. +{1\over 4}\left[ {1\over \Lambda }\right] _{0}\wp (u_1-u_r) \right) du_1. \end{aligned}$$

  (A.5)

In the above we have used the notation $[-]_{n}$ to denote the degree n Laurent coefficient at the corresponding point in consideration. Direct computations show that

$$\begin{aligned} \left[ {1\over \Lambda }\right] _{-2}= {1\over [ \Lambda ]_{2}}={1\over a_{0}} , \quad \left[ {1\over \Lambda }\right] _{0}= -{a_{2}\over a_{0}^{2}}+{a_{1}^{2}\over a_{0}^{3}}, \end{aligned}$$

(A.6)

where

$$\begin{aligned} a_{0}= & {} {2 [x']_{1} [y-y^{*}]_{1}\over [x]_{0} [y+y^{*}]_{0} },\end{aligned}$$

(A.7)

$$\begin{aligned} a_{1}= & {} {2 [x']_{1} [y-y^{*}]_{1}} \cdot - { [x]_{0} [y+y^{*}]_{1}+ [x]_{1} [y+y^{*}]_{0} \over [x]_{0}^2 [y+y^{*}]^2_{0} }\end{aligned}$$

(A.8)

$$\begin{aligned}&+\,2{ [x']_{1} [y-y^{*}]_{2}+ [x']_{2} [y-y^{*}]_{1} \over [x]_{0} [y+y^{*}]_{0} } ,\end{aligned}$$

(A.9)

$$\begin{aligned} a_{2}= & {} {2 [x']_{1} [y-y^{*}]^{3}_{1}\over 3 [x]_{0} [(y+y^{*})^3]_{0} } \end{aligned}$$

(A.10)

$$\begin{aligned}&+\,{2 [x']_{1} [y-y^{*}]_{3}+2 [x']_{2} [y-y^{*}]_{2}+2 [x']_{3} [y-y^{*}]_{1}\over [x]_{0} [(y+y^{*})]_{0}}\end{aligned}$$

(A.11)

$$\begin{aligned}&- { 2 ( [x']_{1} [y-y^{*}]_{2} + [x']_{2} [y-y^{*}]_{1}) ( [x]_{0} [y+y^{*}]_{1} +[x]_{1} [y+y^{*}]_{0}) \over [x]_{0}^2 [(y+y^{*})]_{0}^2 } \qquad \end{aligned}$$

(A.12)

$$\begin{aligned}&\quad + {2 ( [x']_{1} [y-y^{*}]_{1}) ( [x]_{2} [y+y^{*}]_{0} +[x]_{1} [y+y^{*}]_{1}+[x]_{0} [y+y^{*}]_{2}) \over [x]_{0}^2 [(y+y^{*})]_{0}^2 } \qquad \end{aligned}$$

(A.13)

$$\begin{aligned}&\quad - {2 ( [x']_{1} [y-y^{*}]_{1}) ( [x]_{1} [y+y^{*}]_{0} +[x]_{1} [y+y^{*}]_{0})^2 \over [x]_{0}^3 [(y+y^{*})]_{0}^3 }. \end{aligned}$$

(A.14)

1.1 $K_{{\mathbb {P}}^{2}}$

The affine part of the mirror curve given in Example 2.1 is equivalent to

$$\begin{aligned} y^{2}+(x+1)y+q_{1}x^{3}=0, \quad q_{1}=(-3\phi )^{-3}. \end{aligned}$$

(A.15)

The set of finite ramification points is $R^{\circ }=\{{1\over 2}, {\tau \over 2}, {1+\tau \over 2}\}$. Uniformization gives

$$\begin{aligned} x=-3(-4)^{1\over 3}\kappa ^{2} \phi \wp (u)-{9\over 4}\phi ^{3}, \quad y=\kappa ^{3}\wp '(u)-{1+x\over 2} \end{aligned}$$

(A.16)

with

$$\begin{aligned} \phi (\tau )=\Theta _{A_{2}}(2\tau ) {\eta (3\tau ) \over \eta (\tau )^{3} }, \quad \kappa =\zeta _{6}\,2^{-{4\over 3}} 3^{1\over 2} \pi ^{-1}{\eta (3\tau )\over \eta (\tau )^{3}}. \end{aligned}$$

(A.17)

1.2 $K_{{\mathbb {F}}_1}$

The affine part of the mirror curve given in Example 2.4 is

$$\begin{aligned} y^2+y+xy+q_1x+q_2x^2y=0. \end{aligned}$$

(A.18)

The set of finite ramification points is $R^{\circ }=\{0, {1\over 2}, {\tau \over 2}, {1+\tau \over 2}\}$. The uniformization is given by the iteration of the following changes of coordinates (for some $\epsilon$ and $\kappa$)

$$\begin{aligned} \alpha= & {} 4^{1\over 3}\kappa ^{2} \wp (u+\epsilon )-{1\over 3}\left( {1\over 4}+{1\over 2}q_2\right) ,\nonumber \\ \beta= & {} \kappa ^{3}\wp '(u+\epsilon )-\left( \left( {1\over 2}-q_1\right) \alpha +{1\over 4}{q_2})\right) , \end{aligned}$$

(A.19)

$$\begin{aligned} x= & {} \beta ^{-1}{\left( \alpha +{q_2\over 2}-q_1^2+q_1\right) },\nonumber \\ y= & {} -{1\over 2}(1+x+q_2x^2)-{1\over 2}+x\left( x\alpha -\left( {1\over 2}-q_1 \right) \right) . \end{aligned}$$

(A.20)

Taking the special one-parameter family $q_{1}=1,q_{2}=s$, we have

$$\begin{aligned} s=2^{-8} {\eta ^{8}(\tau )\over \eta ^{8}(4\tau )}, \quad \kappa =2^{-{13\over 3}}\pi ^{-1} {\eta (2\tau )^2\over \eta (4\tau )^4} . \end{aligned}$$

(A.21)

1.3 $K_{{\mathbb {P}}^{1}\times {\mathbb {P}}^{1}}$

Then affine part of the mirror curve given in Example 2.2 is equivalent to

$$\begin{aligned} y^{2}+(1+x+q_{1}x^2) y+q_{2}x^{2}=0. \end{aligned}$$

(A.22)

The set of finite ramification points is $R^{\circ }=\{0, {1\over 2}, {\tau \over 2}, {1+\tau \over 2}\}$. The uniformization is given by the iteration of the following changes of coordinates (for some $\epsilon$ and $\kappa$)

$$\begin{aligned} \alpha= & {} 2^{2\over 3}\kappa ^{2} \wp (u+\epsilon )+{1\over 12} (-1-2q_{1}+4q_{2}),\nonumber \\ \beta= & {} \kappa ^{3} \wp '(u+\epsilon )- {1\over 2} \left( \alpha +{1\over 2}q_{1}\right) , \end{aligned}$$

(A.23)

$$\begin{aligned} x= & {} \beta ^{-1}\left( 2^{2\over 3}\kappa ^{2} \wp (u+\epsilon ) +{1\over 6} (1+2q_{1}-4q_{2})\right) ,\nonumber \\ y= & {} -{1\over 2}+ x \left( \alpha x -{1\over 2}\right) - {1\over 2} (1+x +q_{1} x^2). \end{aligned}$$

(A.24)

Taking the special one-parameter subfamily $q_{1}=q_{2}=s$, we have

$$\begin{aligned} s=-2^{-8} {\eta ^{8}(\tau )\over \eta ^{8}(4\tau )}, \quad \kappa =2^{-{7\over 3}}\pi ^{-1}\theta _{2}^{-2}(2\tau ). \end{aligned}$$

(A.25)

1.4 $K_{W{\mathbb {P}}[1,1,2]}$

The affine part of the mirror curve given in Example 2.3 is equivalent to

$$\begin{aligned} y^2+{x^{4}}+y+ b_{4} x^{2}y+ b_{0} xy=0, \quad q_{1}=b_{4}b_{0}^{-4}, q_{2}=b_{0}^{-2}. \end{aligned}$$

(A.26)

The set of finite ramification points is $R^{\circ }=\{0, {1\over 2}, {\tau \over 2}, {1+\tau \over 2}\}$. The following combination is independent of the specialization to an one-parameter subfamily

$$\begin{aligned} (b_{0}^2-4b_{4})^2 = 64 { (\theta _{2}^{4}(2\tau ) +\theta _{3}^{4} (2\tau ))^{2} \over \theta _{4}^{8}(2\tau )}, \end{aligned}$$

(A.27)

up to an $SL_{2}({\mathbb {Z}})$-transform on $\tau$.

The uniformization is given by the iteration of the following changes of coordinates (for some $\epsilon$ and $\kappa$)

$$\begin{aligned} \alpha= & {} 2^{3\over 2}\kappa ^{2} \wp (u+\epsilon ) -{1\over 12} (b_{0}^{2}+2 b_{4}),\nonumber \\ \beta= & {} \kappa ^{3} \wp '(u+\epsilon )-{1\over 2}b_{0} \left( \alpha +{1\over 2}b_{4}\right) , \end{aligned}$$

(A.28)

$$\begin{aligned} x= & {} \beta ^{-1}\left( 2^{3\over 2}\kappa ^{2} \wp (u+\epsilon ) +{1\over 3} (b_{4}-{1\over 4}b_{0}^{2}) \right) ,\nonumber \\ y= & {} -{1\over 2} +x \left( \alpha x-{1\over 2}b_{0}\right) -{1\over 2} (1+b_{0}x+b_{4}x^2). \end{aligned}$$

(A.29)

Taking the special one-parameter subfamily $(q_{1},q_{2})=(0,s)$ that is $b_{4}=0$, we have

$$\begin{aligned} s=64^{-1} { \theta _{4}^{8}(2\tau ) \over (\theta _{2}^{4}(2\tau ) +\theta _{3}^{4} (2\tau ))^{2} }, \quad \kappa =2^{-{1\over 3}} \pi ^{-1} \theta _{4}^{-2} (2\tau ). \end{aligned}$$

(A.30)