Symplectic cuts and open/closed strings II

TL;DR

This work generalizes open-string mirror symmetry for toric Calabi–Yau threefolds by embedding symplectic cuts into higher-dimensional Calabi–Yau geometries. The authors show that the equivariant disk potential $W(t,c,\epsilon)$ is naturally uplifted to an equivariant period of a Calabi–Yau fourfold $X_4$, with extended Picard–Fuchs equations capturing joint open and closed moduli, and they extend the framework to two branes via Calabi–Yau fivefolds $X_5$, giving rise to doubled PF systems and quarter-volumes. A key technical achievement is the explicit expression of the quantum Lebesgue measure $\mathcal{H}^D$ in terms of Lauricella’s $F_D^{(n)}$ hypergeometric functions, and the precise monodromy relations that connect $\mathcal{H}^D$ to the toric brane disk potentials. The results unify open/closed string data across CY3, CY4, and CY5 in a single hierarchy, enabling systematic computation of equivariant disk potentials and their extended PF equations, and laying groundwork for exploring quantum cohomology of cuts and potential annulus invariants.

Abstract

In arXiv:2306.07329 we established a connection between symplectic cuts of Calabi-Yau threefolds and open topological strings, and used this to introduce an equivariant deformation of the disk potential of toric branes. In this paper we establish a connection to higher-dimensional Calabi-Yau geometries by showing that the equivariant disk potential arises as an equivariant period of certain Calabi-Yau fourfolds and fivefolds, which encode moduli spaces of one and two symplectic cuts (the maximal case) by a construction of Braverman arXiv:alg-geom/9712024. Extended Picard-Fuchs equations for toric branes, capturing dependence on both open and closed string moduli, are derived from a suitable limit of the equivariant quantum cohomology rings of the higher Calabi-Yau geometries.

Figures (8)

• Figure 1: Hyperplanes for a double symplectic cut of $\mathbb{C}^3$.
• Figure 2: The total space of Braverman's construction, which realizes $X_4$ as a fibration over the $w$-plane, with generic fiber $X_3^{}$ and a degenerate fiber $X_3^{<}\cup_{X_2}X_3^{>}$ over the origin.
• Figure 3: Picture of the monodromy of $\rho(c,\epsilon_\pm)$ for $c>0$ and $c<0$, respectively.
• Figure 4: Hyperplane for a symplectic cut of $\mathbb{C}^3$ in two different phases. The axes here are labeled by the variables $p_i:=|z_i|^2$.
• Figure 5: Hyperplane for a symplectic cut of local $\mathbb{P}^2$ in three different phases with $t>0$. For $t<0$, there are instead two phases corresponding to $c>-\frac{1}{3} t$ and $c<-\frac{1}{3} t$, respectively.

Theorems & Definitions (4)

• Remark 2.1: Conventions
• Remark 2.2
• Remark 3.1
• Remark 3.2: Non-equivariant limit