Gromov–Witten Theory of Toric Birational Transformations

Acosta, Pedro; Shoemaker, Mark

doi:10.1093/imrn/rnz001

Cited by 2 publications

(9 citation statements)

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“…We choose a sufficiently big η 0 > 0 such that Γ i 's are contained in A q * (η 0 ) and that ∂A q * (η 0 ) F −1 q * (u 0 ). Arguing as in parts (1), (4) in the proof of Theorem 7.22, we see that Γ i can be continuously deformed to a relative cycle Γ i (q) of (A q (η 0 ), A q (η 0 ) ∩ F −1 q (u 0 )) as q varies in a small neighbourhood of q * in V ss − . This shows that the vanishing cycles ∂Γ 1 (q) and ∂Γ i (q) (with 2 ≤ i ≤ k) have zero intersection number.…”

Section: 42mentioning

confidence: 72%

“…Part (3) has been already achieved. We show part (4). Recall that the decomposition (7.28) corresponds to the basis {V •± i } of K(X ± ) under the Γ-integral structure and it gives rise to the asymptotic basis (see §6.2) associated with τ • ± and φ.…”

Section: 42mentioning

confidence: 97%

“…It compares the equivariant quantum D-modules over the open set U 0 that contains the non-semisimple loci. It is also important that the two quantum D-modules are connected through the explicit mirror LG model 4. More precisely, the analytic lift is defined over functions in z which admit asymptotic expansions along an angular sector (with vertex at z = 0); such functions form a sheaf A over the real oriented blowup of C at the origin [94].…”

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confidence: 99%

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Global Mirrors and Discrepant Transformations for Toric Deligne-Mumford Stacks

Iritani

2020

SIGMA

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We introduce a global Landau-Ginzburg model which is mirror to several toric Deligne-Mumford stacks and describe the change of the Gromov-Witten theories under discrepant transformations. We prove a formal decomposition of the quantum cohomology D-modules (and of the all-genus Gromov-Witten potentials) under a discrepant toric wallcrossing. In the case of weighted blowups of weak-Fano compact toric stacks along toric centres, we show that an analytic lift of the formal decomposition corresponds, via the Γintegral structure, to an Orlov-type semiorthogonal decomposition of topological K-groups. We state a conjectural functoriality of Gromov-Witten theories under discrepant transformations in terms of a Riemann-Hilbert problem.

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Section: 42mentioning

confidence: 72%

Section: 42mentioning

confidence: 97%

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confidence: 99%

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Global Mirrors and Discrepant Transformations for Toric Deligne-Mumford Stacks

Iritani

2020

SIGMA

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“…Proof. First we prove (2). Using Definition 6.6 and Proposition 6.9, we find that P (Ω 1 , Ω 2 ) equals (6.7)…”

Section: And Make a Linear Change Of Variables Smentioning

confidence: 82%

“…Therefore Λ contains all homology classes of stable maps. We will prove in Lemma 4.8 (2) below that Λ is the dual lattice of the Picard group of the coarse moduli space. On the other hand, O should correspond to the group generated by classes in H 2 (X, L; Q) ∼ = Q m of orbi-discs with boundaries in a Lagrangian torus orbit L ⊂ X.…”

Section: 3mentioning

confidence: 99%

Hodge-theoretic mirror symmetry for toric stacks

Coates

Corti

Iritani

et al. 2020

J. Differential Geom.

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Using the mirror theorem [15], we give a Landau-Ginzburg mirror description for the big equivariant quantum cohomology of toric Deligne-Mumford stacks. More precisely, we prove that the big equivariant quantum D-module of a toric Deligne-Mumford stack is isomorphic to the Saito structure associated to the mirror Landau-Ginzburg potential. We give a GKZ-style presentation of the quantum D-module, and a combinatorial description of quantum cohomology as a quantum Stanley-Reisner ring. We establish the convergence of the mirror isomorphism and of quantum cohomology in the big and equivariant setting. TOM COATES, ALESSIO CORTI, HIROSHI IRITANI, AND HSIAN-HUA TSENG 7.1. Result 47 7.2. Estimates for the Gauss-Manin connection 49 7.3. Gauge fixing 51 7.4. Proof of Theorem 7.2 and Corollary 7.3 55 References 55 IntroductionThis paper is the last in a series of papers [14, 15] that study the genus-zero Gromov-Witten theory of toric Deligne-Mumford stacks. Let X be a toric Deligne-Mumford stack, or toric stack for short, that satisfies a mild semi-projectivity hypothesis (spelled out below). In [15] we proved a mirror theorem that says that a certain hypergeometric function, called the I-function, lies on the Givental cone for X. This determines all genus-zero Gromov-Witten invariants of X. The present paper builds on this mirror theorem to establish Hodgetheoretic mirror symmetry for toric stacks in a very general setting -without assuming that X is compact, or imposing any positivity condition on c 1 (X). We prove that the big and equivariant quantum cohomology D-module of X can be described as the Saito structure of the Landau-Ginzburg model mirror to X.It has been proposed by Givental [32] (see also [46]) that the mirror of a toric manifold X is a Landau-Ginzburg model, or more precisely, a Laurent polynomial function F = F (x 1 , . . . , x n ) with Newton polytope equal to the fan polytope of X. In particular, Givental [37] showed that, for weak Fano toric manifolds X, oscillatory integrals e F/z dx 1 ···dxnx 1 ···xn give solutions of the small quantum cohomology D-module of X. His result also implies that the quantum cohomology ring of X is isomorphic to the Jacobian ring of F , via an isomorphism which matches the Poincaré pairing with the residue pairing. Givental-style mirror symmetry has been extended to big quantum cohomology by Barannikov, and Mann [3,27,60]; this compares the Frobenius manifold structure [28] defined by the big quantum cohomology of X with K. Saito's flat structure [71, 72] associated to a miniversal unfolding of F .Let us briefly review our main construction. Let X be a toric Deligne-Mumford stack with semi-projective coarse moduli space. (This means that the coarse moduli space is projective over affine and contains a torus-fixed point.) We introduce an unfolding F (x; y) of Givental's Landau-Ginzburg potential by choosing a finite subset G in the fan lattice N:where b 1 , . . . , b m are generators of one-dimensional cones of the stacky fan of X, Q is the Novikov variable, λ(b i ) and λ...

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Gromov–Witten Theory of Toric Birational Transformations

Cited by 2 publications

References 42 publications

Global Mirrors and Discrepant Transformations for Toric Deligne-Mumford Stacks

Global Mirrors and Discrepant Transformations for Toric Deligne-Mumford Stacks

Hodge-theoretic mirror symmetry for toric stacks

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