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 $$(X,\omega )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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.

Luca Cassia

Pietro Longhi

Maxim Zabzine

Communications in Mathematical Physics

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The mission of "Communications in Mathematical Physics" is to provide a high-level forum for works inspired by the foresight and challenges of modern physics, while adhering to the highest mathematical standards.

《Communications in Mathematical Physics》的使命是为受到现代物理学的前瞻性思维和挑战所启发的研究提供一个高水平的交流平台，同时确保满足最高的数学标准。

Category	Quartile
物理与天体物理	1
物理与天体物理, 物理数学物理	1

Symplectic Cuts and Open/Closed Strings I

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