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Topological superconductors are an essential component for topologically protected quantum computation and information processing. Although signatures of topological superconductivity have been reported in heterostructures, material realizations of intrinsic topological superconductors are rather rare. Here we use scanning tunnelling spectroscopy to study the transition metal dichalcogenide 4Hb-TaS2 that interleaves superconducting 1H-TaS2 layers with strongly correlated 1T-TaS2 layers, and find spectroscopic evidence for the existence of topological surface superconductivity. These include edge modes running along the 1H-layer terminations as well as under the 1T-layer terminations, where they separate between superconducting regions of distinct topological nature. We also observe signatures of zero-bias states in vortex cores. All the boundary modes exhibit crystallographic anisotropy, which—together with a finite in-gap density of states throughout the 1H layers—allude to the presence of a topological nodal-point superconducting state. Our theoretical modelling attributes this phenomenology to an inter-orbital pairing channel that necessitates the combination of surface mirror symmetry breaking and strong interactions. It, thus, suggests a topological superconducting state realized in a natural compound. Scanning tunnelling microscopy provides evidence for topological superconductivity in a naturally occurring transition metal dichalcogenide material.

Among the most actively studied issues in the cuprates are the natures of the pseudogap and strange metal states and their relationship to superconductivity1. There is general agreement that the low-energy physics of the Mott-insulating parent state is well captured by a two-dimensional spin S = 1/2 antiferromagnetic Heisenberg model2. However, recent observations of a large thermal Hall conductivity in several parent cuprates appear to defy this simple model and suggest proximity to a magneto-chiral state that breaks all mirror planes that are perpendicular to the CuO2 layers3–6. Here we use optical second harmonic generation to directly resolve the point group symmetries of the model parent cuprate Sr2CuO2Cl2. We report evidence of an order parameter that breaks all perpendicular mirror planes and is consistent with a magneto-chiral state in zero magnetic field. Although this order is clearly coupled to the antiferromagnetism, we are unable to realize its time-reversed partner by thermal cycling through the antiferromagnetic transition temperature or by sampling different spatial locations. This suggests that the order onsets above the Neel temperature and may be relevant to the mechanism of pseudogap formation. Several cuprate superconductors were recently shown to have chiral phonons. Here second harmonic generation measurements show that antiferromagnetism breaks all mirror symmetries in a related compound, consistent with a chiral state.

We review some classical and modern aspects of hypergeometric differential equations, including A-hypergeometric systems of Gel$$'$$ ′ fand, Graev, Kapranov and Zelevinsky. Some recent advances in this theory, such as Euler–Koszul homology, rank jump phenomena, irregularity questions and Hodge theoretic aspects are discussed with more details. We also give some applications of the theory of hypergeometric systems to toric mirror symmetry.

The recently conjectured knots-quivers correspondence (Kucharski et al. in Phys Rev D 96(12):121902, 2017. arXiv:1707.02991 , Adv Theor Math Phys 23(7):1849–1902, 2019. arXiv:1707.04017 ) relates gauge theoretic invariants of a knot K in the 3-sphere to the representation theory of a quiver $$Q_{K}$$ associated to the knot. In this paper we provide geometric and physical contexts for this conjecture within the framework of Ooguri-Vafa large N duality (Ooguri and Vafa in Nucl Phys B 577:419–438, 2000), that relates knot invariants to counts of holomorphic curves with boundary on $$L_{K}$$ , the conormal Lagrangian of the knot in the resolved conifold, and corresponding M-theory considerations. From the physics side, we show that the quiver encodes a 3d $${\mathcal {N}}=2$$ theory $$T[Q_{K}]$$ whose low energy dynamics arises on the worldvolume of an M5 brane wrapping the knot conormal and we match the (K-theoretic) vortex partition function of this theory with the motivic generating series of $$Q_{K}$$ . From the geometry side, we argue that the spectrum of (generalized) holomorphic curves on $$L_{K}$$ is generated by a finite set of basic disks. These disks correspond to the nodes of the quiver $$Q_{K}$$ and the linking of their boundaries to the quiver arrows. We extend this basic dictionary further and propose a detailed map between quiver data and topological and geometric properties of the basic disks that again leads to matching partition functions. We also study generalizations of A-polynomials associated to $$Q_{K}$$ and (doubly) refined version of LMOV invariants (Ooguri and Vafa 2000; Labastida and Marino in Commun Math Phys 217(2):423–449, 2001. arXiv:hep-th/0004196 ; Labastida et al. in JHEP 11:007, 2000. arXiv:hep-th/0010102 ; Aganagic and Vafa in Large N duality, mirror symmetry, and a Q-deformed A-polynomial for knots. arXiv:1204.4709 ; Fuji et al. in Nucl Phys B 867:506–546, 2013. arXiv:1205.1515 ).

We present experimental data on water repetitively brought in contact with cellophane. Although this hydrophilic polymer is insoluble in water, repetitively immersing it in water changes the liquid’s properties. We compare the physicochemical properties of the water left over after removing the cellophane to those of previously published data on water repetitively brought in contact with other in water insoluble polymers (cotton wool or Nafion). Some of the properties are similar. All the properties considerably differ from those of the Milli-Q® water used. On lyophilizing these waters, solid residues remain. The residues are soluble in water. The chemical nature of the residues differs from that of the perturbing polymers.

Van der Waals heterostructures form a massive interdisciplinary research field, fueled by the rich material science opportunities presented by layer assembly of artificial solids with controlled composition, order and relative rotation of adjacent atomic planes. Here we use atomic resolution transmission electron microscopy and multiscale modeling to show that the lattice of MoS$_2$ and WS$_2$ bilayers twisted to a small angle, $\theta

By mirror symmetry, the quantum connection of a weighted projective line is closely related to the localized Fourier–Laplace transform of some Gauß–Manin system. Following an article of D’Agnolo, Hien, Morando, and Sabbah, we compute the Stokes matrices for the latter at $$\infty $$ ∞ for the cases $${\mathbb {P}}(1,3)$$ P ( 1 , 3 ) and $${\mathbb {P}}(2,2)$$ P ( 2 , 2 ) by purely topological methods. We compare them to the Gram matrix of the Euler–Poincaré pairing on $$D^b(\mathrm{Coh}({\mathbb {P}}(1,3)))$$ D b ( Coh ( P ( 1 , 3 ) ) ) and $$D^b(\mathrm{Coh}({\mathbb {P}}(2,2)))$$ D b ( Coh ( P ( 2 , 2 ) ) ) , respectively. This article is based on the doctoral thesis of the author.

Conservation laws are deeply related to any symmetry present in a physical system1,2. Analogously to electrons in atoms exhibiting spin symmetries3, it is possible to consider neutrons and protons in the atomic nucleus as projections of a single fermion with an isobaric spin (isospin) of t = 1/2 (ref. 4). Every nuclear state is thus characterized by a total isobaric spin T and a projection Tz—two quantities that are largely conserved in nuclear reactions and decays5,6. A mirror symmetry emerges from this isobaric-spin formalism: nuclei with exchanged numbers of neutrons and protons, known as mirror nuclei, should have an identical set of states7, including their ground state, labelled by their total angular momentum J and parity π. Here we report evidence of mirror-symmetry violation in bound nuclear ground states within the mirror partners strontium-73 and bromine-73. We find that a J π = 5/2− spin assignment is needed to explain the proton-emission pattern observed from the T = 3/2 isobaric-analogue state in rubidium-73, which is identical to the ground state of strontium-73. Therefore the ground state of strontium-73 must differ from its J π = 1/2− mirror bromine-73. This observation offers insights into charge-symmetry-breaking forces acting in atomic nuclei. Observations of the decay of 73Sr, when compared to its mirror nucleus 73Br, indicate that the spin assignment of their ground states differ, demonstrating mirror-symmetry violation.

A compact complex manifold is Kobayashi non-hyperbolic if there exists an entire curve on it. Using mirror symmetry we establish that there are (possibly singular) elliptic or rational curves on any Calabi-Yau manifold $X$, whose mirror dual $\check X$ exists and is not "Hodge degenerate", therefore proving that $X$ is Kobayashi non-hyperbolic. We are not aware of any higher dimensional simply connected Calabi-Yau manifolds that satisfy the "Hodge degenerate" condition., Comment: 8 pages, comments are welcome

We consider a pair consisting of an invertible polynomial and a finite abelian group of its symmetries. Berglund, Hubsch, and Henningson proposed a duality between such pairs giving rise to mirror symmetry. We define an orbifoldized signature for such a pair using the orbifoldized elliptic genus. In the case of three variables and based on the homological mirror symmetry picture, we introduce two integral lattices, a transcendental and an algebraic one. We show that these lattices have the same rank and that the signature of the transcendental one is the orbifoldized signature. Finally, we give some evidence that these lattices are interchanged under the duality of pairs.

We address the symmetry-breaking bifurcation of optical solitons in cubic–quintic media with an imprinted partially parity-time-symmetric potential. An increase in soliton power leads to symmetry breaking, which occurs through a pitchfork bifurcation. In contrast to all previously reported symmetry breaking, restoring bifurcation occurs when soliton power exceeds another critical value. Symmetry-breaking region between symmetry-breaking point and restoration point shrinks with the growth of the strength of quintic nonlinearity. While the base branch of solitons loses their stabilities in the symmetry-breaking region, the stabilities of two branches of bifurcated asymmetric solitons respecting mirror symmetry are opposite. Nonreciprocal propagation of unstable asymmetric solitons is also observed. We thus put forward the first example of restoring bifurcation of symmetric solitons in two dimensions.

We study the behavior of a family of Hermitian–Yang–Mills connections on complex two-dimensional Kahler tori, when the metrics degenerate. We prove a convergence theorem of connections for this family. From this result and an idea coming from mirror symmetry, we construct a (special) Lagrangian submanifold on the mirror torus. Conjecturally this gives the inverse of the homological mirror symmetry construction of Fukaya (J Algebraic Geom 11(3):393–512, 2002).

Inserting of any sort of irregularity in a photonic crystal (PC) with a perfect periodicity can produce defect modes. Great interest has been directed towards defect modes due to many potential applications. In this work, a binary PC consisting of silica (SiO2) and silver (Ag) is assumed. The PC is supposed to have a mirror symmetry with the structure (SiO2/Ag)N(Ag/SiO2) N, where 2 N is the number of periods of the PC. The transmission spectra of the PC are investigated in the visible and IR wavelengths. It is found that when the thickness of Ag layer increases, band gaps are broadened, more band gaps appear starting at low wavelengths and all these band gaps are shifted toward loner wavelengths. Defect modes appear at certain thicknesses of Ag, diminish gradually and then entirely disappear. When the incidence angle increases, band gaps are blue shifted in contrary to their behavior with the increase of the thickness of the metal and SiO2 layers. Inserting of any sort of irregularity in a photonic crystal (PC) with a perfect periodicity can produce defect modes. Great interest has been directed towards defect modes due to many potential applications. In this work, a binary PC consisting of silica (SiO2) and silver (Ag) is assumed. The PC is supposed to have a mirror symmetry with the structure (SiO2/Ag)N(Ag/SiO2) N, where 2 N is the number of periods of the PC. The transmission spectra of the PC are investigated in the visible and IR wavelengths. It is found that when the thickness of Ag layer increases, band gaps are broadened, more band gaps appear starting at low wavelengths and all these band gaps are shifted toward loner wavelengths. Defect modes appear at certain thicknesses of Ag, diminish gradually and then entirely disappear. When the incidence angle increases, band gaps are blue shifted in contrary to their behavior with the increase of the thickness of the metal and SiO2 layers.

We present the holographic dual for the strongly coupled, low energy dynamics of balanced ${\cal N}=4$ field theories in $(2+1)$ dimensions. The infinite family of Type IIB backgrounds with AdS$_4\times S^2\times S^2$ factors is described in terms of a Laplace problem with suitable boundary conditions. The system describes an array of D3, NS5 and D5 branes. We study various aspects of these Hanany--Witten set-ups (number of branes, linking numbers, dimension of the Higgs and Coulomb branches) and encode them in holographic calculations. A generic expression for the Free Energy/Holographic Central Charge is derived. These quantities are then calculated explicitly in various general examples. We also discuss how Mirror Symmetry is encoded in our Type IIB backgrounds. The connection with previous results in the bibliography is made., 25 pages plus appendices

Topological crystalline insulators (TCI) represent a new state of quantum matter which is topologically protected by crystalline symmetry. We construct a minimal tight-binding model for 2D thin TCI films with C3-symmetry honeycomb lattice. Four Dirac cones are observed at the high-symmetry points K, K′ and Γ points, considering the basic term spin-orbit interaction. By introducing the mass term, Dirac gaps can be opened to trigger the topological phase transition. We then numerically investigate the band structure, conductance and local density of state of the TCI films consisting of zigzag and armchair nanoribbons. The results indicate that there is a one-to-one correspondence between gapless topological edge states and nonzero mirror-Chern number. Finally, we apply an external electric field perpendicular to the TCI films, which breaks mirror symmetry. As a result, the nontrivial edge modes are destroyed, and the conductance is turned off by the electric field.

In the context of 4d effective gravity theories with 8 supersymmetries, we propose to unify, strenghten, and refine the several swampland conjectures into a single statement: the structural criterion, modelled on the structure theorem in Hodge theory. In its most abstract form the new swampland criterion applies to all 4d $\mathcal{N}=2$ effective theories (having a quantum-consistent UV completion) whether supersymmetry is \emph{local} or rigid: indeed it may be regarded as the more general version of Seiberg-Witten geometry which holds both in the rigid and local cases. As a first application of the new swampland criterion we show that a quantum-consistent $\mathcal{N}=2$ supergravity with a cubic pre-potential is necessarily a truncation of a higher-$\mathcal{N}$ \textsc{sugra}. More precisely: its moduli space is a Shimura variety of `magic' type. In all other cases a quantum-consistent special K\"ahler geometry is either an arithmetic quotient of the complex hyperbolic space $SU(1,m)/U(m)$ or has no \emph{local} Killing vector. Applied to Calabi-Yau 3-folds this result implies (assuming mirror symmetry) the validity of the Oguiso-Sakurai conjecture in Algebraic Geometry: all Calabi-Yau 3-folds $X$ without rational curves have Picard number $\rho=2,3$; in facts they are finite quotients of Abelian varieties. More generally: the K\"ahler moduli of $X$ do not receive quantum corrections if and only if $X$ has infinite fundamental group. In all other cases the K\"ahler moduli have instanton corrections in (essentially) all possible degrees., Comment: 94 pages, 2 figures

Spectral flow, spacetime supersymmetry, topological twists, chiral primaries related to marginal deformations, mirror symmetry: these are important consequences of the worldsheet $$ \mathcal{N} $$ N = 2 superconformal symmetry of strings on Calabi-Yau manifolds. To various degrees of certainty, these features were also established when the target is either 7d or 8d with exceptional holonomy G2 or Spin(7) respectively. We show that these are more than mere analogies. We exhibit an underlying symmetry $$ \mathcal{SW}\left(\frac{3}{2},2\right) $$ SW 3 2 2 making a bridge between the latter cases and K3 target spaces. Reviewing unitary representations of $$ \mathcal{SW}\left(\frac{3}{2},2\right) $$ SW 3 2 2 leads us to speculate on further roles of this algebra in string theory compactifications and on the existence of topologically twisted versions of $$ \mathcal{SW}\left(\frac{3}{2},2\right) $$ SW 3 2 2 theories.

We have obtained and analyzed the S1 ← S0 fluorescence excitation spectrum of jet-cooled 2-methylpropanal ((CH3)2CHCHO). In addition, the ab initio calculations of the 2-methylpropanal structure in the ground (S0) and lowest excited singlet (S1) electronic states have been carried out by the CASPT2/def2-TZVPP method. The calculations have shown that the S1 ← S0 electronic excitation of the 2-methylpropanal conformers (cis and gauche) causes large conformational changes: the isopropyl group rotates around the central C–C bond and the carbonyl fragment CCHO becomes non-planar. The potential energy surface of 2-methylpropanal in the S1 state has the six minima, corresponding to three pairs of mirror symmetry conformers: 1ab, 2ab, and 3ab. In the S1 ← S0 jet-cooled spectra of (CH3)2CHCHO and (CH3)2CHCDO three systems of vibronic transitions from the gauche (S0) conformer to vibrational levels of the 1, 2, and 3 conformers (S1) with the origins ( $$ {0}_0^0 $$ ) at 29,887; 30,198; and 29,963 cm−1 for (CH3)2CHCHO and 29,926; 30,221; and 29,990 cm−1 for (CH3)2CHCDO have been identified. The high activity of torsional and inversional vibrations of the 1, 2, and 3 conformers has been found for both molecules. Intensities of bands in torsional sequences pass through a maximum; it confirms the theoretical prediction about the minima shifts between the gauche and 1, 2, and 3 conformers along the torsional coordinate. Using the experimental inversional energy levels, the 1↔3 and 2a↔2b potential functions of inversion have been determined for both molecules. The corresponding calculated and experimental data are in satisfactory agreement.

Highly enantioselective production of chiral compounds by chiral catalysis is one of the most challenging forms of catalytic selectivity. In this perspective, we argue by examples that the key to achieving high enantioselectivity lies in processes with non-linear kinetics or equilibria that effectively amplify small differences in enantiospecific energetics. Examples of such processes have been uncovered over the past decade and include autocatalysis, surface explosion reactions, stirring or grinding of crystallites, and cooperative self-assembly.

From string theory, the notion of deformed Hermitian Yang–Mills connections has been introduced by Marino et al. (J High Energy Phys Paper 5, 2000). After that, Leung et al. (Adv Theor Math Phys 4(6):1319–1341, 2000) proved that it naturally appears as mirror objects of special Lagrangian submanifolds via Fourier–Mukai transform between dual torus fibrations. In their paper, some conditions are imposed for simplicity. In this paper, data to glue their construction on tropical manifolds are proposed and a generalization of the correspondence is proved without the assumption that the Lagrangian submanifold is a section of the torus fibration.

Three-dimensional (3D) topological nodal points, such as Weyl and Dirac nodes have attracted wide-spread interest across multiple disciplines and diverse material systems. Unlike nodal points that contain little structural variations, nodal lines can have numerous topological configurations in the momentum space, forming nodal rings, nodal chains and potentially nodal links and nodal knots. However, nodal lines have much less development for the lack of ideal material platforms. In condensed matter for example, nodal lines are often fragile to spin-orbit-coupling, locating off the Fermi level, coexisting with energy-degenerate trivial bands and dispersing strongly in energy of the line degeneracy. Here, overcoming all above difficulties, we theoretically predict and experimentally observe nodal chains in a metallic-mesh photonic crystal having frequency-isolated linear bandtouching rings chained across the entire Brillouin zone (BZ). These nodal chains are protected by mirror symmetries and have a frequency variation less than 1%. We used angle-resolved transmission (ART) to probe the projected bulk dispersions and performed Fourier-transformed field scan (FTFS) to map out the surface dispersions, which is a quadratic touching between two drumhead surface bands. Our results established an ideal nodal-line material for further studies of topological line-degeneracies with nontrivial connectivities, as well as the consequent wave dynamics richer than 2D Dirac and 3D Weyl materials., Comment: 5 pages, 4 figures

Chirality as a physical phenomenon originating from lacking mirror symmetry in many natural substances exposes fascinating optical activities, including circular dichroism (CD) and optical circular conversion dichroism (O-CCD). While naturally occurring chiroptical properties are negligible in real substances, boosting these properties to significant levels is one of the main challenges for designing functional photonic and optoelectronic devices. Here, we introduce a new paradigm for the realization of planar metasurfaces with substantial O-CCD, reporting that considerable chiral effects of three-dimensional chiral structure are practically provided by feasible metasurfaces by picking out nanooscillators with strong near-field interaction, different optical properties including loss and radiation, and high overlap between their spectral positions. We apply this concept to heterogeneous nanoplasmonic structures, composed of L-pair of Au–Ag nanorods, as archetypical configurations for substantially raising dichroism compared to materially homogenous ones, i.e., Au–Au and Ag–Ag pairs with different length, respectively. The essential prerequisite for phase lag associated with the third dimension along lightwave propagation is compensated by proposed heterogeneity in metasurface design. Theoretical findings based on the simple hybridization model are adequately supported by the computational results.

We explain the proof, obtained in collaboration with Chenyang Xu, of a 1999 conjecture of Veys about poles of maximal order of Igusa zeta functions. The proof technique is based on the Minimal Model Program in birational geometry, but the proof was heavily inspired by ideas coming from non-archimedean geometry and mirror symmetry; we will outline these relations at the end of the paper. This text is intended to be a low-tech introduction to these topics; we only assume that the reader has a basic knowledge of algebraic geometry.

We explore the distribution of topological numbers in Calabi-Yau manifolds, using the Kreuzer-Skarke dataset of hypersurfaces in toric varieties as a testing ground. While the Hodge numbers are well-known to exhibit mirror symmetry, patterns in frequencies of combination thereof exhibit striking new patterns. We find pseudo-Voigt and Planckian distributions with high confidence and exact fit for many substructures. The patterns indicate typicality within the landscape of Calabi-Yau manifolds of various dimension., Comment: 62 pages, 22 figures in main text, LaTeX, v.3: section 2.5 added; minor edits; matches version in CMP

The principle of mirror symmetry, which states that nuclear structure remains the same when protons are swapped for neutrons and vice versa, has been found to be broken in the lowest-energy forms of a mirror pair of nuclei. Bromine-73 and strontium-73 have different ground-state configurations.

We explore the geometric interpretation of the twisted index of 3d ${\mathcal N} =4$ gauge theories on $S^1\times \Sigma$ where $\Sigma$ is a closed Riemann surface. We focus on a rich class of supersymmetric quiver gauge theories that have isolated vacua under generic mass and FI parameter deformations. We show that the path integral localises to a moduli space of generalised vortex equations on $\Sigma$, which can be understood algebraically as quasi-maps to the Higgs branch. We show that the twisted index reproduces the virtual Euler characteristic of the moduli spaces of twisted quasi-maps and demonstrate that this agrees with the contour integral representation introduced in previous work. Finally, we investigate 3d ${\mathcal N} = 4$ mirror symmetry in this context, which implies an equality of enumerative invariants associated to mirror pairs of Higgs branches under the exchange of equivariant and degree counting parameters., Comment: 65 pages, 2 figures, minor corrections, version to appear in JHEP

This paper presents a new class of anomalous objects that partly disappear when they are reflected in a mirror. They can be constructed by combination of mathematical properties of mirror symmetry and psychological properties of optical illusion. A general method for designing these objects is proposed together with examples.

The aim of this paper is to apply ideas from the study of Legendrian singularities to a specific example of interest within mirror symmetry. We calculate the Landau–Ginzburg A-model with \(M={\mathbb {C}}^{3},W=z_1 z_2 z_3\) in its guise as microlocal sheaves along the natural singular Lagrangian thimble \(L={\hbox {Cone}}(T^2)\subset M\). The description we obtain is immediately equivalent to the B-model of the pair-of-pants \(\mathbb {P}^{1}{\setminus }\{0,1,\infty \}\) as predicted by mirror symmetry.

We classify non-smooth del Pezzo surfaces with \(\frac{1}{3}(1,1)\) points in 29 qG-deformation families grouped into six unprojection cascades (this overlaps with work of Fujita and Yasutake in Classification of log del Pezzo surfaces of index three, arXiv:1401.1283 [math.AG]), we tabulate their biregular invariants, we give good model constructions for surfaces in all families as degeneracy loci in rep quotient varieties, and we prove that precisely 26 families admit qG-degenerations to toric surfaces. This work is part of a program to study mirror symmetry for orbifold del Pezzo surfaces (Akhtar et al. in Proc Am Math Soc 144(2):513–527, 2016).

We are considering the class of heterotic $\mathcal{N}=(2,2)$ Landau-Ginzburg orbifolds with 9 fields corresponding to $A_1^9$ Gepner models. We classify all of its Abelian discrete quotients and obtain 152 inequivalent models closed under mirror symmetry with $\mathcal{N}=1,2$ and $4$ supersymmetry in 4D. We compute the full massless matter spectrum at the Fermat locus and find a universal relation satisfied by all models. In addition we give prescriptions of how to compute all quantum numbers of the 4D states including their discrete R-symmetries. Using mirror symmetry of rigid geometries we describe orbifold and smooth Calabi-Yau phases as deformations away from the Landau-Ginzburg Fermat locus in two explicit examples. We match the non-Fermat deformations to the 4D Higgs mechanism and study the conservation of R-symmetries. The first example is a $\mathbb{Z}_3$ orbifold on an E$_6$ lattice where the R-symmetry is preserved. Due to a permutation symmetry of blow-up and torus K\"{a}hler parameters the R-symmetry stays conserved also smooth Calabi-Yau phase. In the second example the R-symmetry gets broken once we deform to the geometric $\mathbb{Z}_3 \times \mathbb{Z}_{3,\text{free}}$ orbifold regime., Comment: 35 pages, 2 figures

We consider mirror symmetry for (essentially arbitrary) hypersurfaces in (possibly noncompact) toric varieties from the perspective of the Strominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface $H$ in a toric variety $V$ we construct a Landau-Ginzburg model which is SYZ mirror to the blowup of $V\times\mathbb{C}$ along $H\times 0$, under a positivity assumption. This construction also yields SYZ mirrors to affine conic bundles, as well as a Landau-Ginzburg model which can be naturally viewed as a mirror to $H$. The main applications concern affine hypersurfaces of general type, for which our results provide a geometric basis for various mirror symmetry statements that appear in the recent literature. We also obtain analogous results for complete intersections., Comment: 83 pages; v2: added appendix discussing the analytic structure on moduli of objects in the Fukaya category; v3: further clarifications in response to referee report; v4: further clarifications throughout, especially sections 4 and 7 and appendix A; added appendix B on the geometry of reduced spaces

We generalize the standard combinatorial techniques of toric geometry to the study of log Calabi-Yau surfaces. The character and cocharacter lattices are replaced by certain integral linear manifolds described by Gross, Hacking, and Keel, and monomials on toric varieties are replaced with the canonical theta functions which GHK defined using ideas from mirror symmetry. We describe the tropicalizations of theta functions and use them to generalize the dual pairing between the character and cocharacter lattices. We use this to describe generalizations of dual cones, Newton and polar polytopes, Minkowski sums, and finite Fourier series expansions. We hope that these techniques will generalize to higher-rank cluster varieties., Comment: 40 pages, 2 figures. The final publication is available at Springer via http://dx.doi.org/10.1007/s00029-015-0221-y, Selecta Math. (2016)

Non-symmorphic crystals are generating great interest as they are commonly found in quantum materials, like iron-based superconductors, heavy-fermion compounds, and topological semimetals. A new type of surface state, a floating band, was recently discovered in the nodal-line semimetal ZrSiSe, but also exists in many non-symmorphic crystals. Little is known about its physical properties. Here, we employ scanning tunneling microscopy to measure the quasiparticle interference of the floating band state on ZrSiSe (001) surface and discover rotational symmetry breaking interference, healing effect and half-missing-type anomalous Umklapp scattering. Using simulation and theoretical analysis we establish that the phenomena are characteristic properties of a floating band surface state. Moreover, we uncover that the half-missing Umklapp process is derived from the glide mirror symmetry, thus identify a non-symmorphic effect on quasiparticle interferences. Our results may pave a way towards potential new applications of nanoelectronics., A new type of surface state has recently been reported in ZrSiSe but its physical properties remain unclear. Here, Zhu, Chang, Huang, et al. report rotational symmetry breaking, healing effect and anomalous Umklapp scattering arising from the nonsymmorphic effect of the floating band state on ZrSiSe (001) surface.

We prove that if two Calabi–Yau invertible pencils have the same dual weights, then they share a common factor in their zeta functions. By using Dwork cohomology, we demonstrate that this common factor is related to a hypergeometric Picard–Fuchs differential equation. The factor in the zeta function is defined over the rationals and has degree at least the order of the Picard–Fuchs equation. As an application, we relate several pencils of K3 surfaces to the Dwork pencil, obtaining new cases of arithmetic mirror symmetry.

$R$-coloured knot polynomials for $m$-strand torus knots $Torus_{[m,n]}$ are described by the Rosso-Jones formula, which is an example of evolution in $n$ with Lyapunov exponents, labelled by Young diagrams from $R^{\otimes m}$. This means that they satisfy a finite-difference equation (recursion) of finite degree. For the gauge group $SL(N)$ only diagrams with no more than $N$ lines can contribute and the recursion degree is reduced. We claim that these properties (evolution/recursion and reduction) persist for Khovanov-Rozansky (KR) polynomials, obtained by additional factorization modulo $1+{\bf t}$, which is not yet adequately described in quantum field theory. Also preserved is some weakened version of differential expansion, which is responsible at least for a simple relation between {\it reduced} and {\it unreduced} Khovanov polynomials. However, in the KR case evolution is incompatible with the mirror symmetry under the change $n\longrightarrow -n$, what can signal about an ambiguity in the KR factorization even for torus knots. }, Comment: 23 pp

6d superconformal field theories (SCFTs) are the SCFTs in the highest possible dimension. They can be geometrically engineered in F-theory by compactifying on non-compact elliptic Calabi-Yau manifolds. In this paper we focus on the class of SCFTs whose base geometry is determined by $-2$ curves intersecting according to ADE Dynkin diagrams and derive the corresponding mirror Calabi-Yau manifold. The mirror geometry is uniquely determined in terms of the mirror curve which has also an interpretation in terms of the Seiberg-Witten curve of the four-dimensional theory arising from torus compactification. Adding the affine node of the ADE quiver to the base geometry, we connect to recent results on SYZ mirror symmetry for the $A$ case and provide a physical interpretation in terms of little string theory. Our results, however, go beyond this case as our construction naturally covers the $D$ and $E$ cases as well., version 2: typos corrected, 30 pages, 8 figures

The Milnor fibre of any isolated hypersurface singularity contains many exact Lagrangian spheres: the vanishing cycles associated to a Morsification of the singularity. Moreover, for simple singularities, it is known that the only possible exact Lagrangians are spheres. We construct exact Lagrangian tori in the Milnor fibres of all non-simple singularities of real dimension four. This gives examples of Milnor fibres whose Fukaya categories are not generated by vanishing cycles. Also, this allows progress towards mirror symmetry for unimodal singularities, which are one level of complexity up from the simple ones., Comment: v2: 66 pages, 37 figures; minor changes

The BKMP conjecture (2006–2008) proposed a new method to compute closed and open Gromov–Witten invariants for every toric Calabi–Yau 3-folds, through a topological recursion based on mirror symmetry. So far, this conjecture has been verified to low genus for several toric CY3folds, and proved to all genus only for $${\mathbb{C}^3}$$ . In this article we prove the general case. Our proof is based on the fact that both sides of the conjecture can be naturally written in terms of combinatorial sums of weighted graphs: on the A-model side this is the localization formula, and on the B-model side the graphs encode the recursive algorithm of the topological recursion. One can slightly reorganize the set of graphs obtained in the B-side, so that it coincides with the one obtained by localization in the A-model. Then it suffices to compare the weights of vertices and edges of graphs on each side, which is done in two steps: the weights coincide in the large radius limit, due to the fact that the toric graph is the tropical limit of the mirror curve. Then the derivatives with respect to Kahler radius coincide due to the special geometry property implied by the topological recursion.

The observation of symmetry breaking in a coupled nanolaser system could yield new types of switchable devices.

We consider examples of extremal transitions between families of Calabi-Yau complete intersection threefolds in toric varieties, which are induced by toric embeddings of one toric variety into the other. We show that the toric map induced by the linear dual of the embedding induces a birational morphism between the mirror Calabi-Yau families, and in one case show that it can be extended to the full mirror transition between mirror families. Note: this version uses a different approach to give a shorter proof of the same result. Also, Lemma 5.8 in the previous versions is definitely wrong., Comment: 21 pages. Major revision, a different method is used to give an improved proof and correct a mistake in previous versions

The temperature dependence of the defect mode in a 1D ternary superconducting photonic crystal (PC) with mirror symmetry was theoretically investigated by simultaneously considering thermal expansion effect and thermal-optical effect. Three different materials H (high-refractive index dielectric), L (low-refractive index dielectric), and S (superconducting material) are used to construct the ternary superconducting PC structure. For comparison, six structures are discussed. It is found that the structure (HLS) N (SLH) N has the highest temperature sensitivity. Here, N is the number of periods. The average change in the central wavelength of the defect mode is 31.18 nm/K as the thickness of superconducting layer is 50 nm. Thus, based on the structure (HLS) N (SLH) N , a low temperature sensor with higher sensitivity can be designed to realize the low temperature sensor.

Thermal properties of Zn2(C8H4O4)2•C6H12N2 metal-organic framework compound at 8–300 K suggest the possibility of subbarrier tunnelling transitions between left-twisted (S) and right-twisted (R) forms of C6H12N2 dabco molecules with D3 point symmetry. The data agree with those obtained for the temperature behavior of nuclear spin-lattice relaxation times. It is shown that there is a temperature range where the transitions are stopped. Therefore, Zn2(C8H4O4)2•C6H12N2 and related compounds are interesting objects to study the effect of spontaneous mirror-symmetry breaking and stabilization of chiral isomeric molecules in solids at low temperatures.

We investigate the cosmology of the minimal model of neutral naturalness, the mirror Twin Higgs. The softly-broken mirror symmetry relating the Standard Model to its twin counterpart leads to significant dark radiation in tension with BBN and CMB observations. We quantify this tension and illustrate how it can be mitigated in several simple scenarios that alter the relative energy densities of the two sectors while respecting the softly-broken mirror symmetry. In particular, we consider both the out-of-equilibrium decay of a new scalar as well as reheating in a toy model of twinned inflation, Twinflation. In both cases the dilution of energy density in the twin sector does not merely reconcile the existence of a mirror Twin Higgs with cosmological constraints, but predicts contributions to cosmological observables that may be probed in current and future CMB experiments. This raises the prospect of discovering evidence of neutral naturalness through cosmology rather than colliders., 44 pages, 12 figures, v2 updated to match published version

Brane brick models are Type IIA brane configurations that encode the $2d$ $\mathcal{N}=(0,2)$ gauge theories on the worldvolume of D1-branes probing toric Calabi-Yau 4-folds. We use mirror symmetry to improve our understanding of this correspondence and to provide a systematic approach for constructing brane brick models starting from geometry. The mirror configuration consists of D5-branes wrapping 4-spheres and the gauge theory is determined by how they intersect. We also explain how $2d$ $(0,2)$ triality is realized in terms of geometric transitions in the mirror geometry. Mirror symmetry leads to a geometric unification of dualities in different dimensions, where the order of duality is $n-1$ for a Calabi-Yau $n$-fold. This makes us conjecture the existence of a quadrality symmetry in $0d$. Finally, we comment on how the M-theory lift of brane brick models connects to the classification of $2d$ $(0,2)$ theories in terms of 4-manifolds., Comment: 74 pages, 56 figures

In this paper, we find the extended charge vectors for disc invariants of outer branes in a toric Calabi–Yau 3-orbifold, and use these extended charge vectors to establish a quantum McKay correspondence for disc invariants of branes intersecting effective outer legs, and give an enumerative explanation to the integrality of open–closed mirror map in terms of Ooguri–Vafa invariants.

According to our present knowledge, the spontaneous resolution of racemic mixtures of chiral molecules or chiral conformers of achiral molecules into macroscopic chiral superstructures requires the confinement of these molecules in a crystal lattice, on surfaces or in other well-ordered assemblies. Herein we provide the first experimental evidence that mirror-symmetry breaking can also take place at a liquid-liquid phase transition in isotropic liquids of achiral molecules, even at relatively high temperatures around 200 °C. It is proposed that cooperative segregation of enantiomorphic molecular conformations gives rise to a conglomerate of two chiral and immiscible liquids. In these liquid conglomerates a strong chiral amplification was observed, which led to degeneracy from a stochastic distribution and eventually provided uniform chirality. We anticipate that this work will contribute to the understanding of symmetry breaking in soft matter and provide a new tool for the identification of chirality traces, and possibly affect the discussion of the emergence of chirality in prebiotic systems.

We discuss the relation between transposition mirror symmetry of Berglund and Hubsch for bimodal singularities and polar duality of Batyrev for associated toric K3 hypersurfaces. We also show that homological mirror symmetry for singularities implies the geometric construction of Coxeter–Dynkin diagrams of bimodal singularities by Ebeling and Ploog.

We hope to understand the Hodge theoretic aspect of mirror symmetry in the framework of the fundamental diagram of log mixed Hodge theory. We give a formulation of mirror conjecture for Calabi–Yau threefolds as the coincidence of log period maps with specified sections under the mirror map. Since a variation of Hodge structure with unipotent monodromies on a product of punctured discs uniquely extends over the puncture to a log Hodge structure, we can work on the boundary point over which the log Riemann–Hilbert correspondence exists, and we can observe clearly in high resolution the behavior of Z-structure over the boundary point (cf. notes in Introduction below). This is an advantage of log Hodge theory.

We consider mirror symmetry (A-side vs B-side, namely singularity side) in the framework of quantum differential systems. We focuse on the logarithmic non-resonant case, which describes the geometric situation and we show that such systems provide a good framework in order to generalize the construction of the rational structure given by Katzarkov, Kontsevich and Pantev for the complex projective space. As an application, we give a closed formula for the rational structure defined by the Lefschetz thimbles on the flat sections of the Gauss-Manin connection associated with the Landau–Ginzburg models of weighted projective spaces (a class of Laurent polynomials). As a by-product, using a mirror theorem, we get a rational structure on the orbifold cohomology of weighted projective spaces. The formula on the B-side is more complicated than the one on the A-side (the latter agrees with one of Iritani’s results), depending on numerous combinatorial data which are rearranged after the mirror transformation.

A Lagrangian submanifold in an almost Calabi-Yau manifold is called positive if the real part of the holomorphic volume form restricted to it is positive. An exact isotopy class of positive Lagrangian submanifolds admits a natural Riemannian metric. We compute the Riemann curvature of this metric and show all sectional curvatures are non-positive. The motivation for our calculation comes from mirror symmetry. Roughly speaking, an exact isotopy class of positive Lagrangians corresponds under mirror symmetry to the space of Hermitian metrics on a holomorphic vector bundle. The latter space is an infinite-dimensional analog of the non-compact symmetric space dual to the unitary group, and thus has non-positive curvature., 21 pages, added reference, corrected minor errors