The Crepant Transformation Conjecture for toric complete intersections

Coates, Tom; Iritani, Hiroshi; Jiang, Yunfeng

doi:10.1016/j.aim.2017.11.017

Cited by 49 publications

(84 citation statements)

References 70 publications

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“…A good reference on genus-zero Givental formalism could be [18]. Besides, a lot of other works contain good introductions to the Lagrangian cones including [4,6,27,28], among others. In fact, [27] and [28,Section 3] adopt an axiomatic approach which also applies to our situation.…”

Section: Givental Formalism In Genus Zeromentioning

confidence: 99%

Structures in genus‐zero relative Gromov–Witten theory

Fan

You

2020

Journal of Topology

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Section: Givental Formalism In Genus Zeromentioning

confidence: 99%

Structures in genus‐zero relative Gromov–Witten theory

Fan

You

2020

Journal of Topology

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“…Since Ruan's influential conjecture [69], an intensely studied problem in Gromov-Witten theory has been to determine the relation between GW invariants of target spaces related by a crepant birational transformation (CRC). The most general formulation of the CRC is framed in terms of Givental formalism ( [29]; see also [30] for an expository account); the conjecture has been proved in a number of examples [24,26,29] and has by now gained folklore status, with a general proof in the toric setting announced for some time [25,28]. A natural question one can ask is whether similar relations exist in the context of open Gromov-Witten theory.…”

Section: 2mentioning

confidence: 99%

Crepant resolutions and open strings

Brini

Cavalieri

Ross

2017

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. We formulate a Crepant Resolution Correspondence for open Gromov-Witten invariants (OCRC) of toric Lagrangian branes inside Calabi-Yau 3-orbifolds by encoding the open theories into sections of Givental's symplectic vector space. The correspondence can be phrased as the identification of these sections via a linear morphism of Givental spaces. We deduce from this a Bryan-Graber-type statement for disk invariants, which we extend to arbitrary topologies in the Hard Lefschetz case. Motivated by ideas of Iritani, Coates-CortiIritani-Tseng and Ruan, we furthermore propose 1) a general form of the morphism entering the OCRC, which arises from a geometric correspondence between equivariant K-groups, and 2) an all-genus version of the OCRC for Hard Lefschetz targets. We provide a complete proof of both statements in the case of minimal resolutions of threefold An-singularities; as a necessary step of the proof we establish the all-genus closed Crepant Resolution Conjecture with descendents in its strongest form for this class of examples. Our methods rely on a new description of the quantum D-modules underlying the equivariant Gromov-Witten theory of this family of targets.

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“…For this reason, we need to generalize shift operators to big quantum cohomology. We also observe that shift operators are closely related to the Γ-integral structure [Iri09,KKP08,CIJ14]. We show that a flat section of the quantum connection associated to an equivariant vector bundle in the formalism of Γ-integral structure is invariant under shift operators (Proposition 3.18).…”

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confidence: 85%

“…As is discussed in [CIJ14,§3], the divisor equation shows that the specialization Q = 1 of the Novikov variable is well-defined for s(E). In view of the intertwining property in Theorem 3.14, it suffices to show that…”

Section: 5mentioning

confidence: 99%

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Shift operators and toric mirror theorem

Iritani

2017

Geom. Topol.

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Abstract. We give a new proof of Givental's mirror theorem for toric manifolds using shift operators of equivariant parameters. The proof is almost tautological: it gives an A-model construction of the I-function and the mirror map. It also works for non-compact or non-semipositive toric manifolds. IntroductionIn 1995, Seidel [Sei97] introduced an invertible element of quantum cohomology associated to a Hamiltonian circle action. This has had many applications in symplectic topology. Seidel himself used it to construct non-trivial elements of π 1 of the group of Hamiltonian diffeomorphisms. calculated Seidel's elements in a more general setting and obtained Batyrev's ring presentation of quantum cohomology of toric manifolds. Their method, however, does not yield explicit structure constants of quantum cohomology, i.e. genus-zero Gromov-Witten invariants.Recently, Braverman, Maulik, Okounkov and Pandharipande [OP10, BMO11, MO12] introduced a shift operator of equivariant parameters for equivariant quantum cohomology. Their shift operators reduce to Seidel's invertible elements under the nonequivariant limit. In this paper, we show that equivariant genus-zero Gromov-Witten invariants of toric manifolds are reconstructed only from formal properties of shift operators. This means that the equivariant quantum topology of toric manifolds is determined by its classical counterpart.More specifically, we give a new proof of Givental's mirror theorem for toric manifolds, which is stated as follows: Giv98b,LLY99,Iri08,Bro09], see §4.2 for more details). Let X Σ be a semi-projective toric manifold having a torus fixed point. Let I(y, z) be the cohomologyvalued hypergeometric series defined bywhere u i , i = 1, . . . , m is the class of a prime toric divisor. Then I(y, −z) lies in Givental's Lagrangian cone L X Σ associated to X Σ .We prove this theorem in the following way. Recall that equivariant genus-zero Gromov-Witten invariants of a T -variety X can be encoded by an infinite-dimensional Lagrangian submanifold L X of the symplectic vector space H X [Giv04]:The space H X is called the Givental space and L X is called the Givental cone. By the general theory, each C × -subgroup k : C × → T defines a shift operator S k acting on the Givental space H X and induces a vector field on L X :The operator S k is determined by T -fixed loci in X and their normal bundles (see Definition 3.13). For toric manifolds, we have a shift operator S i for each torus-invariant prime divisor. Then we identify the I-function I(y, z) with an integral curve of the commuting vector fields f → z −1 S i f. Theorem 1.2. Givental's I-function I(y, z) is a unique integral curve which satisfies the differential equation:and is of the form I(y, z) = ze, where we setThe I-function defines a mirror map y → τ (y) ∈ H * T (X) via Birkhoff factorization [CG07, Iri08]. As a corollary to our proof, we obtain the following relationship between the equivariant Seidel elements S i (τ ) and the mirror map. This generalizes a previous result [GI12] in the semipo...

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The Crepant Transformation Conjecture for toric complete intersections

Cited by 49 publications

References 70 publications

Structures in genus‐zero relative Gromov–Witten theory

Structures in genus‐zero relative Gromov–Witten theory

Crepant resolutions and open strings

Shift operators and toric mirror theorem

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