{"title":"Symplectic Cuts and Open/Closed Strings I","authors":"Luca Cassia, Pietro Longhi, Maxim Zabzine","doi":"10.1007/s00220-024-05190-5","DOIUrl":null,"url":null,"abstract":"<div><p>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 <i>A</i>-brane in the same threefold. Symplectic cutting is a natural operation that decomposes a symplectic manifold <span>\\((X,\\omega )\\)</span> with a Hamiltonian <i>U</i>(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 <i>A</i>-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.\n</p></div>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":"406 1","pages":""},"PeriodicalIF":2.2000,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00220-024-05190-5.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s00220-024-05190-5","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
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.
期刊介绍:
The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards.