Mirror Symmetry Constructions

Clader, Emily; Ruan, Yongbin

doi:10.1007/978-3-319-94220-9_1

Cited by 2 publications

(2 citation statements)

References 32 publications

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“…To avoid lengthy reviews of these techniques we refer to Refs. [4,45,6,50] and references therein for further background.…”

Section: Lie Algebra Descriptionmentioning

confidence: 99%

Algebraic Structure of tt * Equations for Calabi-Yau Sigma Models

Alim

2017

Commun. Math. Phys.

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The tt * equations define a flat connection on the moduli spaces of 2d, N = 2 quantum field theories. For conformal theories with c = 3d, which can be realized as nonlinear sigma models into Calabi-Yau d-folds, this flat connection is equivalent to special geometry for threefolds and to its analogs in other dimensions. We show that the non-holomorphic content of the tt * equations in the cases d = 1, 2, 3 is captured in terms of finitely many generators of special functions, which close under derivatives. The generators are understood as coordinates on a larger moduli space. This space parameterizes a freedom in choosing representatives of the chiral ring while preserving a constant topological metric. Geometrically, the freedom corresponds to a choice of forms on the target space respecting the Hodge filtration and having a constant pairing. Linear combinations of vector fields on that space are identified with generators of a Lie algebra. This Lie algebra replaces the non-holomorphic derivatives of tt * and provides these with a finer and algebraic meaning. For sigma models into lattice polarized K3 manifolds, the differential ring of special functions on the moduli space is constructed, extending known structures for d = 1 and 3. The generators of the differential rings of special functions are given by quasi-modular forms for d = 1 and their generalizations in d = 2, 3. Some explicit examples are worked out including the case of the mirror of the quartic in P 3 , where due to further algebraic constraints, the differential ring coincides with quasi modular forms.

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“…To avoid lengthy reviews of these techniques we refer to Refs. [4,45,6,50] and references therein for further background.…”

Section: Lie Algebra Descriptionmentioning

confidence: 99%

Algebraic Structure of tt * Equations for Calabi-Yau Sigma Models

Alim

2017

Commun. Math. Phys.

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“…Additionally, Θ X should be reflexive. Both of these conditions are satisfied only if w is divisible by w i , i.e., when there is a Fermat polynomial W 0 [26]. As we wish to consider more general cases, we need some generalization of this construction.…”

Section: Mirror Glsmmentioning

confidence: 99%

On Calabi-Yau manifolds in weighted projective spaces and their mirror GLSMs

Kochergin

2021

Preprint

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The goal of the present paper is to calculate the complex structure moduli space Kähler potentials for hypersurfaces in weighted projective spaces and compare with the partition functions of their mirror GLSMs. We explicitly perform the Kähler potential computation and show that the corresponding formula is well-defined in case of quasismooth hypersurfaces. We then construct the mirror GLSM with an appropriate number of Kähler parameters and discuss the interpretation of its partition function in terms of mirror symmetry. Namely, it is shown that different contributions to the partition function are related to various charts of the complex structure moduli space.

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Mirror Symmetry Constructions

Cited by 2 publications

References 32 publications

Algebraic Structure of tt * Equations for Calabi-Yau Sigma Models

Algebraic Structure of tt * Equations for Calabi-Yau Sigma Models

On Calabi-Yau manifolds in weighted projective spaces and their mirror GLSMs

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