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mirror symmetry,D-brane superpotentials and Ooguri-Vafa invariants of Calabi-Yau manifolds

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Chinese Physics C 2015年第12期39卷 7-12页

作者：张姗姗杨富中 School of Physical Sciences University of Chinese Academy of Sciences

The D-brane superpotential is very important in the low energy effective theory. As the generating function of all disk instantons from the worldsheet point of view, it plays a crucial role in deriving some important properties of the compact Calabi Yau manifolds. By using the generalized GKZ hypergeometric system, we will calculate the D-brahe superpotentials of two non-Fermat type compact CMabi Yau hypersurfaces in toric varieties, respectively. Then according to the mirror symmetry, we obtain the A-model superpotentials and the Ooguri Vafa invariants for the mirror Calabi-Yau manifolds.

关键词： D-brahe superpotential mirror symmetry Ooguri-Vafa invariant

A Tropical View on Landau-Ginzburg Models

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Acta Mathematica Sinica,English Series 2024年第1期40卷 329-382页

作者： Michael CARL Max PUMPERLA Bernd SIEBERT Alfred-Delp-Str.2 73430 AalenGermany IU Internationale Hochschule Juri-Gagarin-Ring 15299084 ErfurtGermany AnyscaleInc.2080 Addison St Ste 7BerkeleyCA 94704USA Department of Mathematics The Univ.of Texas at Austin2515 SpeedwayAustinTX 78712USA

This paper,largely written in 2009/2010,fits Landau-Ginzburg models into the mirror symmetry program pursued by the last author jointly with Mark Gross since 2001.This point of view transparently brings in tropical disks of Maslov index 2 via the notion of broken lines,previously introduced in two dimensions by Mark Gross in his study of mirror symmetry for P2.A ma jor insight is the equivalence of properness of the Landau-Ginzburg potential with smoothness of the anticanonical divisor on the mirror side.We obtain proper superpotentials which agree on an open part with those classically known for toric varieties.Examples include mirror LG models for non-singular and singular del Pezzo surfaces,Hirzebruch surfaces and some Fano threefolds.

关键词： Landau-Ginzburg model mirror symmetry tropical disk broken line wall structure scattering diagram Gross-Siebert program del Pezzo surface toric degeneration

Ungraded Matrix Factorizations as mirrors of Non-orientable Lagrangians

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Acta Mathematica Sinica,English Series 2024年第1期40卷 26-42页

作者： Lino AMORIM Cheol-Hyun CHO Department of Mathematics Kansas State University138 Cardwell Hall1228 N.17th StreetManhattanKS 66506USA Department of Mathematical Sciences Research Institute of MathematicsSeoul National UniversitySan 56-1ShinrimdongGwanakguSeoul 47907South Korea

We introduce the notion of ungraded matrix factorization as a mirror of non-orientable Lagrangian submanifolds.An ungraded matrix factorization of a polynomial W,with coefficients in a field of characteristic 2,is a square matrix Q of polynomial entries satisfying Q^(2)=W·Id.We then show that non-orientable Lagrangians correspond to ungraded matrix factorizations under the localized mirror functor and illustrate this construction in a few instances.Our main example is the Lagrangian submanifold RP^(2)⊂CP^(2)and its mirror ungraded matrix factorization,which we construct and study.In particular,we prove a version of Homological mirror symmetry in this setting.

关键词： mirror symmetry non-orientable Lagrangian submanifold matrix factorization

Lattice relaxation,mirror symmetry and magnetic field effects on ultraflat bands in twisted trilayer graphene

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Science China(Physics,Mechanics & Astronomy) 2021年第6期64卷 114-120页

作者： Zewen Wu Zhen Zhan Shengjun Yuan Key Laboratory of Artificial Mic ro-and Nano-strucures of the Minisry of Education School of Physics and TechnologyWuhan UniversityWuhan 430072China

Twisted graphene multilayers exhibit strongly correlated insulating states and superconductivity due to the presence of ultraflat bands near the charge neutral point.In this paper,the response of ultraflat bands to lattice relaxation and a magnetic field in twisted trilayer graphene(tTLG)with different stacking arrangements is investigated by using a full tight-binding model.We show that lattice relaxations are indispensable for understanding the electronic properties of tTLG,in particular,of tTLG in the presence of mirror symmetry.Lattice relaxations renormalize the quasiparticle spectrum near the Fermi energy and change the localization of higher energy flat bands.Furthermore,different from the twisted bilayer graphene,the Hofstadter butterfly spectrum can be realized at laboratory accessible strengths of magnetic field.Our work verifies tTLG as a more tunable platform than the twisted bilayer graphene in strongly correlated phenomena.

关键词： flat bands twisted trilayer graphene lattice relaxation mirror symmetry magnetic field tight-binding

N = 2 SCVA's FROM A GENERALIZED CALABI-YAU MANIFOLD AND mirror symmetry

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Acta Mathematica Scientia 2009年第4期29卷 961-972页

作者：胡森马文晔邱敬佩 Department of Mathematics University of Science and Technology of China USTC Shanghai Institute for Advanced Studies

We construct an N = 2 superconformal vertex algebra（SCVA） from a generalized Calabi-Yau manifold and compute the BRST cohomology of its associated topological vertex algebras. We show that the BRST cohomology coincides with the generalized Dobeault cohomology. We show that the two topological vertex algebras constructed from the N = 2 SCVA by A and B twist respectively are mirror pairs.

关键词： superconformal vertex algebra generalized Calabi-Yau mirror symmetry

Homological Perturbation Theory and mirror symmetry

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Acta Mathematica Sinica,English Series 2003年第4期19卷 695-714页

作者： JianZHOU DepartmentofMathematicalSciences TsinghuaUniversityBeijing100084P.R.China

We explain how deformation theories of geometric objects such as complexstructures, Poisson structures and holomorphic bundle structures lead to differential Gerstenhaberor Poisson algebras. We use homological perturbation theory to construct A_∞ algebra structures onthe cohomology, and their canonically defined deformation. Such constructions are used to formulatea version of A_∞ algebraic mirror symmetry.

关键词： homological perturbation theory mirror symmetry

Hubble Scale Dark Energy Meets Nano Scale Casimir Energy and the Rational of Their T-Duality and mirror symmetry Equivalence

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World Journal of Nano Science and Engineering 2015年第3期5卷 57-67页

作者： M.S.El Naschie Department of Physics University of AlexandriaAlexandriaEgypt

We establish that ordinary energy, Casimir energy and dark energy are not only interlinked but are basically the same thing separated merely by scale and topology. Casimir energy is essentially a nano scale spacetime phenomenon produced by the boundary condition of the two Casimir plates constituting the Casimir experimental set up for measuring the Casimir force. By contrast dark energy is the result of the cosmic boundary condition, i.e. the boundary of the universe. This one sided M?bius-like boundary located at vast cosmic distance and was comparable only to the Hubble radius scales of the universe. All the Casimir energy spreads out until the majority of it reaches the vicinity of the edge of the cosmos. According to a famous theorem due to the Ukrainian-Israeli scientist I. Dvoretzky, almost 96% of the total energy will be concentrated at the boundary of the universe, too far away to be measured directly. The rest of the accumulated Casimir energy density is consequently the nearly 4% to 4.5%, the existence of which is confirmed by various sophisticated cosmic measurements and observations. When all is said and done, the work is essentially yet another confirmation of Witten’s T-duality and mirror symmetry bringing nano scale and Hubble scale together in an unexpected magical yet mathematically rigorous way.

关键词： mirror symmetry Casimir Energy Dark Energy Zero Point Vacuum Energy T-Duality Nano Scale-Hubble Scale Mobius Holographic Boundary Dvoretzky’s Theorem Banach-Tarski Theorem

Higher Genus FJRW Theory for Fermat Cubic Singularity

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Acta Mathematica Sinica,English Series 2021年第8期37卷 1179-1204页

作者： Xin WANG School of Mathematics Shandong UniversityJinan 250100P.R.China

In this paper,we study the higher genus FJRW theory of Fermat cubic singularity with maximal group of diagonal symmetries using Giventai formalism.As results,we prove the finite generation property and holomorphic anomaly equation for the associated FJRW theory.Via general LG-LG mirror theorem,our results also hold for the Saito-Givental theory of the Fermat cubic singularity.

关键词： FJRW theory Saito-Givental theroy mirror symmetry finite generation holomorphic anomaly equation

Symm Sketch: Creating symmetric 3D free-form shapes from 2D sketches

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Computational Visual Media 2015年第1期1卷 3-16页

作者： Yongwei Miao Feixia Hu Xudong Zhang Jiazhou Chen Renato Pajarola College of Computer Science and Technology Zhejiang University of Technology Department of Informatics University of Zürich

This paper presents Symm Sketch—a system for creating symmetric 3D free-form shapes from 2D sketches. The reconstruction task usually separates a 3D symmetric shape into two types of shape components, that is, the self-symmetric shape component and the mutual-symmetric shape components. Each type can be created in an intuitive manner. Using a uniform symmetry plane, the user first draws 2D sketch lines for each shape component on a sketching plane. The z-depth information of the hand-drawn input sketches can be calculated using their property of mirror symmetry to generate 3D construction curves. In order to provide more freedom for controlling the local geometric features of the reconstructed free-form shapes(e.g., non-circular crosssections), our modeling system creates each shape component from four construction curves. Using one pair of symmetric curves and one pair of general curves, an improved cross-sectional surface blending scheme is applied to generate a parametric surface for each component. The final symmetric free-form shape is progressively created, and is represented by 3D triangular mesh. Experimental results illustrate that our system can generate complex symmetric free-form shapes effectively and conveniently.

关键词： 3D reconstruction construction curves free-form shapes mirror symmetry sketch-based modeling