论文标题
$（\ Mathbb {p}^2，e）$和nekrasov-shatashvili local $ \ mathbb {p}^2 $的hokrasov-shatashvili限制的holomorphic异常方程
Holomorphic anomaly equation for $(\mathbb{P}^2,E)$ and the Nekrasov-Shatashvili limit of local $\mathbb{P}^2$
论文作者
论文摘要
We prove a higher genus version of the genus $0$ local-relative correspondence of van Garrel-Graber-Ruddat: for $(X,D)$ a pair with $X$ a smooth projective variety and $D$ a nef smooth divisor, maximal contact Gromov-Witten theory of $(X,D)$ with $λ_g$-insertion is related to Gromov-Witten theory of the total space of $ \ MATHCAL {O} _X（-D）$和$ d $的本地Gromov-Witten理论。 专门针对$（x，d）=（s，e）$ for $ s $ a del pezzo表面或有理椭圆形的表面和$ e $ a $ a a a $平滑的反典型分隔线，我们表明，$（s，e）$的最大接触gromov-witten理论是由gromov-witter of gromov-witter of gromov-witter of calabi-yau 3-fold $ $ \ s的calabi-yau-yntercal and cally-n cally-n call and call and cally-n call and call cally-n call and call of）椭圆曲线$ e $的理论。 进一步专注于$ s = \ mathbb {p}^2 $，我们证明，更高的属生成了$（\ mathbb {p}^2，e）$的一系列最大触点gromov-witten不变性，是quasimimodular且满足荷兰型静态态度。该证明结合了准地模性结果和先前以局部$ \ mathbb {p}^2 $和椭圆曲线而闻名的全体形态异常方程。 Furthermore, using the connection between maximal contact Gromov-Witten invariants of $(\mathbb{P}^2,E)$ and Betti numbers of moduli spaces of semistable one-dimensional sheaves on $\mathbb{P}^2$, we obtain a proof of the quasimodularity and holomorphic anomaly equation predicted in the physics literature for the refined topological string nekrasov-shatashvili限制中本地$ \ mathbb {p}^2 $的自由能。
We prove a higher genus version of the genus $0$ local-relative correspondence of van Garrel-Graber-Ruddat: for $(X,D)$ a pair with $X$ a smooth projective variety and $D$ a nef smooth divisor, maximal contact Gromov-Witten theory of $(X,D)$ with $λ_g$-insertion is related to Gromov-Witten theory of the total space of $\mathcal{O}_X(-D)$ and local Gromov-Witten theory of $D$. Specializing to $(X,D)=(S,E)$ for $S$ a del Pezzo surface or a rational elliptic surface and $E$ a smooth anticanonical divisor, we show that maximal contact Gromov-Witten theory of $(S,E)$ is determined by the Gromov-Witten theory of the Calabi-Yau 3-fold $\mathcal{O}_S(-E)$ and the stationary Gromov-Witten theory of the elliptic curve $E$. Specializing further to $S=\mathbb{P}^2$, we prove that higher genus generating series of maximal contact Gromov-Witten invariants of $(\mathbb{P}^2,E)$ are quasimodular and satisfy a holomorphic anomaly equation. The proof combines the quasimodularity results and the holomorphic anomaly equations previously known for local $\mathbb{P}^2$ and the elliptic curve. Furthermore, using the connection between maximal contact Gromov-Witten invariants of $(\mathbb{P}^2,E)$ and Betti numbers of moduli spaces of semistable one-dimensional sheaves on $\mathbb{P}^2$, we obtain a proof of the quasimodularity and holomorphic anomaly equation predicted in the physics literature for the refined topological string free energy of local $\mathbb{P}^2$ in the Nekrasov-Shatashvili limit.
