STABILITY CONDITIONS AND THE MIRROR SYMMETRY OF K3 SURFACES IN ATTRACTOR BACKGROUNDS

We study the space of stability conditions on K3 surfaces from the perspective of mirror symmetry. This is done in the attractor backgrounds(moduli). We find certain highly non-generic behaviors of marginal stability walls(a key notion in the study of wall crossings)in the space of stability conditions. These correspond via mirror symmetry to some nongeneric behaviors of special Lagrangians in an attractor background. The main results can be understood as a mirror correspondence in a synthesis of the homological mirror conjecture and SYZ mirror conjecture.

N = 2 SCVA's FROM A GENERALIZED CALABI-YAU MANIFOLD AND MIRROR SYMMETRY

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.

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

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.

Solving the strong CP problem via aˉθ-characterized mirror symmetry

In the standard model QCD Lagrangian,a term of CP violating gluon density is theoretically expected to have a physical coefficientθ¯,which is typically on the order of unity.However,the upper bound on the electric dipole moment of the neutron enforces the value ofθ¯to be extremely small.The significant discrepancy between theoretical expectations and experimental results in this context is widely recognized as the strong CP problem.To solve this puzzle in an appealing context of two Higgs doublets,we propose aθ¯-characterized mirror symmetry between two Higgs singlets with respective discrete symmetries.In our scenario,the parameterθ¯can completely disappear from the full Lagrangian after the standard model fermions take a proper phase rotation as well as the Higgs doublets and singlets.Moreover,all of new physics for solving the strong CP problem can be allowed near the TeV scale.

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

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.

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

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.

Homological Perturbation Theory and Mirror Symmetry

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.

Two-dimensional tetragonal ZnB: A nodalline semimetal with good transport properties

Nodal-line semimetals have become a research hot-spot due to their novel properties and great potential application in spin electronics. It is more challenging to find 2D nodal-line semimetals that can resist the spin–orbit coupling(SOC)effect. Here, we predict that 2D tetragonal Zn B is a nodal-line semimetal with great transport properties. There are two crossing bands centered on the S point at the Fermi surface without SOC, which are mainly composed of the pxy orbitals of Zn and B atoms and the pz orbitals of the B atom. Therefore, the system presents a nodal line centered on the S point in its Brillouin zone(BZ). And the nodal line is protected by the horizontal mirror symmetry M_(z). We further examine the robustness of a nodal line under biaxial strain by applying up to-4% in-plane compressive strain and 5% tensile strain on the Zn B monolayer, respectively. The transmission along the a direction is significantly stronger than that along the b direction in the conductive channel. The current in the a direction is as high as 26.63 μA at 0.8 V, and that in the b direction reaches 8.68 μA at 0.8 V. It is interesting that the transport characteristics of Zn B show the negative differential resistance(NDR) effect after 0.8 V along the a(b) direction. The results provide an ideal platform for research of fundamental physics of 2D nodal-line fermions and nanoscale spintronics, as well as the design of new quantum devices.

Ungraded Matrix Factorizations as Mirrors of Non-orientable Lagrangians

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.

Topological nodal line semimetals

We review the recent,mainly theoretical,progress in the study of topological nodal line semimetals in three dimensions.In these semimetals,the conduction and the valence bands cross each other along a one-dimensional curve in the three-dimensional Brillouin zone,and any perturbation that preserves a certain symmetry group（generated by either spatial symmetries or time-reversal symmetry） cannot remove this crossing line and open a full direct gap between the two bands.The nodal line（s） is hence topologically protected by the symmetry group,and can be associated with a topological invariant.In this review,（ⅰ） we enumerate the symmetry groups that may protect a topological nodal line;（ⅱ） we write down the explicit form of the topological invariant for each of these symmetry groups in terms of the wave functions on the Fermi surface,establishing a topological classification;（ⅲ） for certain classes,we review the proposals for the realization of these semimetals in real materials;（ⅳ） we discuss different scenarios that when the protecting symmetry is broken,how a topological nodal line semimetal becomes Weyl semimetals,Dirac semimetals,and other topological phases;and（ⅴ） we discuss the possible physical effects accessible to experimental probes in these materials.

Electronic properties of SnTe-class topological crystalline insulator materials

The rise of topological insulators in recent years has broken new ground both in the conceptual cognition of condensed matter physics and the promising revolution of the electronic devices.It also stimulates the explorations of more topological states of matter.Topological crystalline insulator is a new topological phase,which combines the electronic topology and crystal symmetry together.In this article,we review the recent progress in the studies of SnTe-class topological crystalline insulator materials.Starting from the topological identifications in the aspects of the bulk topology,surface states calculations,and experimental observations,we present the electronic properties of topological crystalline insulators under various perturbations,including native defect,chemical doping,strain,and thickness-dependent confinement effects,and then discuss their unique quantum transport properties,such as valley-selective filtering and helicity-resolved functionalities for Dirac fermions.The rich properties and high tunability make SnTe-class materials promising candidates for novel quantum devices.

A Tropical View on Landau-Ginzburg Models

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.

Two-dimensional rectangular bismuth bilayer:A novel dual topological insulator

Dual topological insulator(DTI),which simultaneously hosts topological insulator(TI)and topological crystalline insulator(TCI)phases,has attracted extensive attention since it has a better robustness of topological nature and broad application prospects in spintronics.However,the realization of DTI phase in two-dimensional(2D)system is extremely scarce.By first-principles calculations,we predict that the 2D rectangular bismuth(R–Bi)bilayer is a novel DTI,featured by topological invariant=1,mirror Chern number C_(M)=–1,and metallic edge states within the bulk band gap.More interestingly,the TCI phase in bilayer is protected by horizontal glide mirror symmetries,rather than the usual mirror symmetry.The bulk band gap can be effectively tuned by vertical electric field and strain.Besides,the electric field can trigger the transition between TI and metallic phases for the bilayer,accompanied by the annihilation of TCI phase.On this basis,a topological field effect transistor is proposed,which can rapidly manipulate spin and charge carriers via electric field.The KBr(110)surface is demonstrated as an ideal substrate for the deposition of bilayer.These findings provide not only a new strategy for exploiting 2D DTI,but also a promising candidate for spintronic applications.

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

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.

Off-shell superpotentials and Ooguri-Vafa invariants of type Ⅱ/F theory compactification

In this paper,we calculate the off-shell superpotential of two Calabi-Yau manifolds with three parameters by integrating the period of the subsystem.We also obtain the Ooguri-Vafa invariants with open mirror symmetry.

Higher Genus FJRW Theory for Fermat Cubic Singularity

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.

Hypergeometric equations and weighted projective spaces

We compute the Hodge numbers of the polarised(pure) variation of Hodge structure V = grn-1WRn-1f!Z of the Landau-Ginzburg model f:Y → C mirror-dual to a weighted projective space wPn in terms of a variant of Reid's age function of the anticanonical cone over wPn.This implies,for instance,that wPn has canonical singularities if and only if hn-1,0V = 1.We state a conjectural formula for the Hodge numbers of general hypergeometric variations.We show that a general fibre of the Landau-Ginzburg model is birational to a Calabi-Yau variety if and only if a general anticanonical section of wP is Calabi-Yau.We analyse the 104 weighted 3-spaces with canonical singularities,and show that a general anticanonical section is not a K3 surface exactly in those 9 cases where a generic fibre of the Landau-Ginzburg model is an elliptic surface of Kodaira dimension 1.

Quantum McKay Correspondence for Disc Invariants of Toric Calabi–Yau 3-orbifolds

We announce a result on quantum McKay correspondence for disc invariants of outer legs in toric Calabi-Yau 3-orbifolds, and illustrate our method in a special example [C^3/Z5（1, 1, 3）].