Holomorphic anomaly equation for and the Nekrasov-Shatashvili limit of local

Abstract

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 $\lambda _g$ -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.

On the log–local principle for the toric boundary

“…The first is to investigate extensions to correspondences with other invariants. At the level of BPS invariants [12-14, 19, 29], this is proven for the pair of ℙ 2 and smooth cubic in [5,6] and in higher genus in [9]. In [7,8], we extend the correspondences to the non-toric and higher genus/refined setting and include open Gromov-Witten invariants, their underlying open BPS counts, as well as quiver Donaldson-Thomas invariants to the set of correspondences.…”

Curves on K3 surfaces in divisibility two

“…We use a similar argument as above. This time, however, we consider the [a 5 1 ]-coefficient of ( 7)…”

“…This term does not contribute to the [a 5 1 ]-coefficient because the weight from the edge is a multiple of a 2 . Therefore the [a 5 1 ]-coefficient of ( 7) gives…”

“…In the physics literature, the (conjectural) holomorphic anomaly equation [3,4] predicts hidden structures of the Gromov-Witten partition function associated to Calabi-Yau varieties. For the past few years, there has been an extensive work to prove the holomorphic anomaly equation in many cases: local P 2 [21], the quintic threefold [10,13], K3 surface with primitive curve classes [27], elliptic fibration [28] and P 2 relative to a smooth cubic [5].…”

“…Section 6 contains all relevant definitions 5. The genus 0 and genus 1 cases are proved by Lee and Leung in[19,20].…”