A mirror construction for the big equivariant quantum cohomology of toric manifolds

Iritani, Hiroshi

doi:10.1007/s00208-016-1437-7

Cited by 11 publications

(3 citation statements)

References 61 publications

(112 reference statements)

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“…The methods used here, such as our use of quantum sheaf cohomology to determine the vacua of the correct dual theory, reminiscent of methods in e.g. [32], should be straightforward to extend to more general Fano toric varieties.…”

Section: Discussionmentioning

confidence: 99%

Toda-like (0,2) mirrors to products of projective spaces

Chen¹,

Sharpe²,

Wu³

2016

J. High Energ. Phys.

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Abstract:One of the open problems in understanding (0,2) mirror symmetry concerns the construction of Toda-like Landau-Ginzburg mirrors to (0,2) theories on Fano spaces. In this paper, we begin to fill this gap by making an ansatz for (0,2) Toda-like theories mirror to (0,2) supersymmetric nonlinear sigma models on products of projective spaces, with deformations of the tangent bundle, generalizing a special case previously worked out for P 1 × P 1 . We check this ansatz by matching correlation functions of the B/2-twisted Todalike theories to correlation functions of corresponding A/2-twisted nonlinear sigma models, computed primarily using localization techniques. These (0,2) Landau-Ginzburg models admit redundancies, which can lend themselves to multiple distinct-looking representatives of the same physics, which we discuss.

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Section: Discussionmentioning

confidence: 99%

Toda-like (0,2) mirrors to products of projective spaces

Chen¹,

Sharpe²,

Wu³

2016

J. High Energ. Phys.

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“…Many similar results have been proved in this setting, but at present their relationship with the analogous symplectic results is unclear. For instance, Iritani [24] computed the big equivariant quantum cohomology of (possibly non-compact) toric varieties using the algebro-geometric version of the Seidel representation (see [25]), and his mirror map should be related to the superpotential of Fukaya-Oh-Ohta-Ono which appears in our results. In algebraic geometry the Seidel representation exists for formal reasons, using torus localisation on moduli spaces, and does not require the delicate analysis carried out by Ritter in the symplectic case.…”

Section: 4mentioning

confidence: 84%

Quantum Cohomology and Closed-String Mirror Symmetry for Toric Varieties

Smith¹

2020

The Quarterly Journal of Mathematics

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We give a short new computation of the quantum cohomology of an arbitrary smooth toric variety X, by showing directly that the Kodaira-Spencer map of Fukaya-Oh-Ohta-Ono defines an isomorphism onto a suitable Jacobian ring. The proof is based on the purely algebraic fact that a class of generalised Jacobian rings associated to X are free as modules over the Novikov ring. In contrast to previous results of this kind, X need not be compact. When X is monotone the presentation we obtain is completely explicit, using only well-known computations with the standard complex structure.

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“…. , λ d are the equivariant parameters for T -(S 1 ) d acting on X. Iritani [Iri17b,Iri17a] has made an in depth study of the big equivariant quantum cohomology for toric varieties, which computes the highly non-trivial extension of W λ in the bulk directions.…”

Section: Introductionmentioning

confidence: 99%

$T$-equivariant disc potential and SYZ mirror construction

Kim,

Lau,

Zheng

2019

Preprint

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We develop a G-equivariant Lagrangian Floer theory by counting pearly trees in the Borel construction L G . We apply the construction to smooth moment-map fibers of toric semi-Fano varieties and obtain the T-equivariant Landau-Ginzburg mirrors. We also apply this to the typical S 1 -invariant SYZ singular fiber, which is the single-pinched torus, and compute its S 1 -equivariant disc potential. Contents 1. Introduction 1 2. A Morse model for equivariant Lagrangian Floer theory 4 2.1. The non-equivariant singular chain model 5 2.2. The non-equivariant Morse model with a strict unit 8 2.3. The equivariant Morse model 15 2.4. Homotopy partial units and the equivariant disc potentials 20 2.5. Equivariant Floer theory for Lagrangian immersions 25 3. The T-equivariant disc potentials of toric manifolds 26 3.1. Morse theory on the approximation spaces 27 3.2. Computing the T-equivariant disc potentials 31 4. S 1 -equivariant disc potential for the immersed 2-sphere 36 References 40

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A mirror construction for the big equivariant quantum cohomology of toric manifolds

Cited by 11 publications

References 61 publications

Toda-like (0,2) mirrors to products of projective spaces

Toda-like (0,2) mirrors to products of projective spaces

Quantum Cohomology and Closed-String Mirror Symmetry for Toric Varieties

$T$-equivariant disc potential and SYZ mirror construction

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