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Mirror symmetry

About: Mirror symmetry is a research topic. Over the lifetime, 2422 publications have been published within this topic receiving 90786 citations.


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TL;DR: In this article, the mirror map q-series of certain families of Calabi-Yau manifolds were shown to be automorphic functions, and a hierarchy of algebraic instanton corrections correlated with the differential Galois group of the Picard-Fuchs equation was proposed.
Abstract: Arithmetic properties of mirror symmetry (type IIA-IIB string duality) are studied. We give criteria for the mirror map q-series of certain families of Calabi–Yau manifolds to be automorphic functions. For families of elliptic curves and lattice polarized K3 surfaces with surjective period mappings, global Torelli theorems allow one to present these criteria in terms of the ramification behavior of natural algebraic invariants – the functional and generalized functional invariants respectively. In particular, when applied to one parameter families of rank 19 lattice polarized K3 surfaces, our criterion demystifies the Mirror-Moonshine phenomenon of Lian and Yau and highlights its non-monstrous nature. The lack of global Torelli theorems and presence of instanton corrections makes Calabi–Yau threefold families more complicated. Via the constraints of special geometry, the Picard–Fuchs equations for one parameter families of Calabi–Yau threefolds imply a differential equation criterion for automorphicity of the mirror map in terms of the Yukawa coupling. In the absence of instanton corrections, the projective periods map to a twisted cubic space curve. A hierarchy of “algebraic” instanton corrections correlated with the differential Galois group of the Picard–Fuchs equation is proposed.

39 citations

Posted Content
TL;DR: In this paper, the BKMP Remodeling Conjecture for all genus open-closed orbifolds Gromov-Witten invariants of an arbitrary semi-projective toric Calabi-Yau 3-orbifold relative to an outer framed Aganagic-Vafa Lagrangian brane was proved.
Abstract: The Remodeling Conjecture proposed by Bouchard-Klemm-Marino-Pasquetti (BKMP) [arXiv:0709.1453, arXiv:0807.0597] relates the A-model open and closed topological string amplitudes (the all genus open and closed Gromov-Witten invariants) of a semi-projective toric Calabi-Yau 3-manifold/3-orbifold to the Eynard-Orantin invariants of its mirror curve. It is an all genus open-closed mirror symmetry for toric Calabi-Yau 3-manifolds/3-orbifolds. In this paper, we present a proof of the BKMP Remodeling Conjecture for all genus open-closed orbifold Gromov-Witten invariants of an arbitrary semi-projective toric Calabi-Yau 3-orbifold relative to an outer framed Aganagic-Vafa Lagrangian brane. We also prove the conjecture in the closed string sector at all genera.

39 citations

Journal ArticleDOI
TL;DR: In this article, a non-toric Lagrangian torus fibration on a toric Calabi-Yau (CY) orbifold, called the Gross fibration, was constructed using the Strominger and Yau-Zaslow recipe.
Abstract: For a toric Calabi–Yau (CY) orbifold $\mathcal{X}$ whose underlying toric variety is semi-projective, we construct and study a non-toric Lagrangian torus fibration on $\mathcal{X}$, which we call the Gross fibration. We apply the Strominger–Yau–Zaslow (SYZ) recipe to the Gross fibration of $\mathcal{X}$ to construct its mirror with the instanton corrections coming from genus $0$ open orbifold Gromov–Witten (GW) invariants, which are virtual counts of holomorphic orbi-disks in $\mathcal{X}$ bounded by fibers of the Gross fibration. We explicitly evaluate all these invariants by first proving an open/closed equality and then employing the toric mirror theorem for suitable toric (parital) compactifications of $\mathcal{X}$. Our calculations are then applied to (1) prove a conjecture of Gross-Siebert on a relation between genus $0$ open orbifold GW invariants and mirror maps of $\mathcal{X}$—this is called the open mirror theorem, which leads to an enumerative meaning of mirror maps, and (2) demonstrate how open (orbifold) GW invariants for toric CY orbifolds change under toric crepant resolutions—an open analogue of Ruan’s crepant resolution conjecture.

39 citations

Journal ArticleDOI
TL;DR: In this article, the topological aspect of the spin-triplet $f$-wave superconductor UPt$_3$ through microscopic calculations of edge and vortex-bound states based on the quasiclassical Eilenberger and Bogoliubov-de Gennes theories is investigated.
Abstract: We investigate the topological aspect of the spin-triplet $f$-wave superconductor UPt$_3$ through microscopic calculations of edge- and vortex-bound states based on the quasiclassical Eilenberger and Bogoliubov-de Gennes theories. It is shown that a gapless and linear dispersion exists at the edge of the $ab$-plane. This forms a Majorana valley, protected by the mirror chiral symmetry. We also demonstrate that, with increasing magnetic field, vortex-bound quasiparticles undergo a topological phase transition from topologically trivial states in the double-core vortex to zero-energy states in the normal-core vortex. As long as the $d$-vector is locked into the $ab$-plane, the mirror symmetry holds the Majorana property of the zero-energy states, and thus UPt$_3$ preserves topological crystalline superconductivity that is robust against the crystal field and spin-orbit interaction.

38 citations

Posted Content
TL;DR: In this article, it was shown that certain hypergeometric series used to formulate mirror symmetry for Calabi-Yau hypersurfaces, in string theory and algebraic geometry, satisfy a number of interesting properties.
Abstract: We show that certain hypergeometric series used to formulate mirror symmetry for Calabi-Yau hypersurfaces, in string theory and algebraic geometry, satisfy a number of interesting properties. Many of these properties are used in separate papers to verify the BCOV prediction for the genus one Gromov-Witten invariants of a quintic threefold and more generally to compute the genus one Gromov-Witten invariants of any Calabi-Yau projective hypersurface.

38 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202351
2022116
2021138
2020130
2019139
2018125