  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
$(X,\omega)$ with a Hamiltonian $U(1)$ action into two pieces glued along the
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 GW invariants. Conversely, the monodromies of the
quantum measure around cycles in K\"ahler moduli space encode open GW
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.
