Symplectic cuts and open/closed strings I

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 $(X,\omega )$ 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.

51 pages + appendix, 8 figures; v3: minor revision, version published in Commun. Math. Phys

References in corpus (34)

• The Topological Vertex
• Knot Invariants and Topological Strings
• Disk Instantons, Mirror Symmetry and the Duality Web
• Open string amplitudes and large order behavior in topological string theory
• An integral structure in quantum cohomology and mirror symmetry for toric orbifolds
• Remodeling the B-model
• Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c=1 Matrix Models
• Nonperturbative effects and nonperturbative definitions in matrix models and topological strings
• Computing Genus-Zero Twisted Gromov-Witten Invariants
• Moduli stacks of stable toric quasimaps
• Exact Results in Supersymmetric Field Theories on Manifolds with Boundaries
• Wall-Crossings in Toric Gromov-Witten Theory I: Crepant Examples
• Riemann-Hilbert problems from Donaldson-Thomas theory
• Vortex partition functions, wall crossing and equivariant Gromov-Witten invariants
• The Crepant Transformation Conjecture for Toric Complete Intersections
• Local Gromov-Witten Invariants are Log Invariants
• Exponential Networks and Representations of Quivers
• Crepant resolutions and open strings
• Disc Instantons in Linear Sigma Models
• Exploring 5d BPS Spectra with Exponential Networks
• On D-branes from Gauged Linear Sigma Models
• Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc
• Mathematical structures of non-perturbative topological string theory: from GW to DT invariants
• Open topological strings and integrable hierarchies: Remodeling the A-model
• Exact multi-instantons in topological string theory
• Quantum Curves, Resurgence and Exact WKB
• Exponential Networks, WKB and Topological String
• Open orbifold Gromov-Witten invariants of [C^3/Z_n]: localization and mirror symmetry
• A-branes, foliations and localization
• On the log-local principle for the toric boundary
• Crepant resolutions and open strings II
• Open/closed Correspondence via Relative/local Correspondence
• From equivariant volumes to equivariant periods
• Symplectic cuts and open/closed strings II

Cited by in corpus (1)

• Symplectic cuts and open/closed strings II