Symplectic cuts and open/closed strings I

TL;DR

The paper develops a unified framework to relate genus-zero closed and open Gromov-Witten invariants of toric Calabi-Yau threefolds through a quantum uplift of symplectic cuts implemented via equivariant GLSMs. The central object, the quantum Lebesgue measure $\mathcal{H}^D$, encodes both closed data (via $\mathcal{F}^D=\int \mathcal{H}^D dc$) and open data (via the monodromy of $\mathcal{H}^D$ yielding the equivariant open superpotential $W$), providing globally defined, phase-independent expressions across the Kähler moduli space. The authors demonstrate the framework in explicit examples—$\mathbb{C}^3$, the resolved conifold, and local $\mathbb{P}^2$—where they compute equivariant volumes, quantum cuts, and holomorphic disk potentials, showing that regular parts reproduce known open-string superpotentials while monodromies reproduce disk counts in various chambers. This construction unifies open- and closed-string data through analytic continuation and instanton resummation, offering a new perspective on the open-closed string relationship with potential extensions to higher genus and broader string-theoretic interpretations.

Abstract

This paper introduces a concrete relation between genus zero closed Gromov-Witten invariants of Calabi-Yau threefolds and genus zero open Gromov-Witten invariants of a Lagrangian $A$-brane in the same threefold. Symplectic cutting is a natural operation that decomposes a symplectic manifold $(X,ω)$ with a Hamiltonian $U(1)$ action into two pieces glued along an invariant divisor. In this paper we study a quantum uplift of the cut construction defined in terms of equivariant gauged linear sigma models. The nexus between closed and open Gromov-Witten invariants is a quantum Lebesgue measure associated to a choice of cut, that we introduce and study. Integration of this measure recovers the equivariant quantum volume of the whole CY3, thereby encoding closed Gromov-Witten invariants. Conversely, the monodromies of the quantum measure around cycles in Kähler moduli space encode open Gromov-Witten invariants of a Lagrangian $A$-brane associated to the cut. Both in the closed and the open string sector we find a remarkable interplay between worldsheet instantons and semiclassical volumes regularized by equivariance. This leads to equivariant generating functions of GW invariants that extend smoothly across the entire moduli space, and which provide a unifying description of standard GW potentials. The latter are recovered in the non-equivariant limit in each of the different phases of the geometry.

Figures (8)

• Figure 1: Moment polytope of $O_{ \mathbb{P}^2}(-3)$, with a hyperplane defining a symplectic cut.
• Figure 2: The quotient $\mathbb{C}^2\sslash U(1)$ with moment map $\mu(z)=|z_1|^2+|z_2|^2$. $M_0\cong S^3$ is the locus where $\mu(z)=t>0$, which in this case, is described by a $U(1)^2$ fibration over an interval such that on each of the endpoints one of the circles shrinks to a point.
• Figure 3: Moment polytopes of $\overline M_-$ and $\overline M_+$.
• Figure 4: Symplectic cut of $\mathbb{C}^2$ along $\mathbb{P}^1$.
• Figure 5: Hyperplane for a symplectic cut of $\mathbb{C}^3$ in two different phases.

Theorems & Definitions (5)

• Example 2.1: Symplectic quotient
• Definition 2.2: Symplectic cut
• Example 2.3: Symplectic cut of $\mathbb{C}^2$
• Remark 2.4: Conventions
• Remark 2.5: Note added