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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., Comment: 45 pages + appendix, 8 figures

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 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 K\"ahler 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., Comment: 51 pages + appendix, 8 figures; v3: minor revision, version published in Commun. Math. Phys

We consider generalizations of equivariant volumes of abelian GIT quotients obtained as partition functions of 1d, 2d, and 3d supersymmetric GLSM on $S^1$, $D^2$ and $D^2 \times S^1$, respectively. We define these objects and study their dependence on equivariant parameters for non-compact toric K\"ahler quotients. We generalize the finite-difference equations (shift equations) obeyed by equivariant volumes to these partition functions. The partition functions are annihilated by differential/difference operators that represent equivariant quantum cohomology/K-theory relations of the target and the appearance of compact divisors in these relations plays a crucial role in the analysis of the non-equivariant limit. We show that the expansion in equivariant parameters contains information about genus-zero Gromov-Witten invariants of the target., Comment: v2: typos fixed and references added, version to appear in Adv.Theor.Math.Phys