Crossing-Line-Node Semimetals: General Theory and Application to Rare-Earth Trihydrides

Multiple line nodes in energy-band gaps are found in semimetals preserving mirror-reflection symmetry. We classify possible configurations of multiple line nodes with crossing points (crossing line nodes) under point-group symmetry. Taking the spin-orbit interaction (SOI) into account, we also classify topological phase transitions from crossing-line-node Dirac semimetals to other topological phases, e.g., topological insulators and point-node semimetals. This study enables one to find crossing-line-node semimetal materials and their behavior in the presence of SOI from the band structure in the absence of SOI without detailed calculations. As an example, the theory applies to hexagonal rare-earth trihydrides with the HoD3 structure and clarifies that it is a crossing-line-node Dirac semimetal hosting three line nodes.Comment: 16 pages, 9 figure

Real nodal sextics without real nodes

We present a rigid isotopy classification of irreducible sextic curves in $\mathbb{RP}^2$ which have non-real ordinary double points as their only singularities. Our approach uses periods of K3 surfaces and V. Nikulin's classification of involutions with condition on unimodular lattices. The classification obtained generalizes Nikulin's rigid isotopy classification of non-singular sextics in $\mathbb{RP}^2$

Relating virtual knot invariants to links in S3\mathbb{S}^{3}

Geometric interpretations of some virtual knot invariants are given in terms of invariants of links in $\mathbb{S}^3$. Alexander polynomials of almost classical knots are shown to be specializations of the multi-variable Alexander polynomial of certain two-component boundary links of the form $J \sqcup K$ with $J$ a fibered knot. The index of a crossing, a common ingredient in the construction of virtual knot invariants, is related to the Milnor triple linking number of certain three-component links $J \sqcup K_1 \sqcup K_2$ with $J$ a connected sum of trefoils or figure-eights. Our main technical tool is virtual covers. This technique, due to Manturov and the first author, associates a virtual knot $\upsilon$ to a link $J \sqcup K$, where $J$ is fibered and $\text{lk}(J,K)=0$. Here we extend virtual covers to all multicomponent links $L=J \sqcup K$, with $K$ a knot. It is shown that an unknotted component $J_0$ can be added to $L$ so that $J_0 \sqcup J$ is fibered and $K$ has algebraic intersection number zero with a fiber of $J_0 \sqcup J$. This is called fiber stabilization. It provides an avenue for studying all links with virtual knots.Comment: Improved main theorem and standardized orientation conventions. This version to appear in New York Journal of Mathematic

Geometry and arithmetic of certain log K3 surfaces

Let $k$ be a field of characteristic $0$. In this paper we describe a classification of smooth log K3 surfaces $X$ over $k$ whose geometric Picard group is trivial and which can be compactified into del Pezzo surfaces. We show that such an $X$ can always be compactified into a del Pezzo surface of degree $5$, with a compactifying divisor $D$ being a cycle of five $(-1)$-curves, and that $X$ is completely determined by the action of the absolute Galois group of $k$ on the dual graph of $D$. When $k=\mathbb{Q}$ and the Galois action is trivial, we prove that for any integral model $\mathcal{X}/\mathbb{Z}$ of $X$, the set of integral points $\mathcal{X}(\mathbb{Z})$ is not Zariski dense. We also show that the Brauer-Manin obstruction is not the only obstruction for the integral Hasse principle on such log K3 surfaces, even when their compactification is "split"

Mirror pairs of Calabi-Yau threefolds from mirror pairs of quasi-Fano threefolds

We present a new construction of mirror pairs of Calabi-Yau manifolds by smoothing normal crossing varieties, consisting of two quasi-Fano manifolds. We introduce a notion of mirror pairs of quasi-Fano manifolds with anticanonical Calabi-Yau fibrations using recent conjectures about Landau-Ginzburg models. Utilizing this notion, we give pairs of normal crossing varieties and show that the pairs of smoothed Calabi-Yau manifolds satisfy the Hodge number relations of mirror symmetry. We consider quasi-Fano threefolds that are some blow-ups of Gorenstein toric Fano threefolds and build 6518 mirror pairs of Calabi-Yau threefolds, including 79 self-mirrors.Comment: This version will appear in J. Math. Pures App

K3 metrics from little string theory

Certain six-dimensional (1,0) supersymmetric little string theories, when compactified on $T^3$, have moduli spaces of vacua given by smooth K3 surfaces. Using ideas of Gaiotto-Moore-Neitzke, we show that this provides a systematic procedure for determining the Ricci-flat metric on a smooth K3 surface in terms of BPS degeneracies of (compactified) little string theories.Comment: 34 page

A Five Dimensional Generalization of the Topological Weyl Semimetal

We generalize the concept of three-dimensional topological Weyl semimetal to a class of five dimensional (5D) gapless solids, where Weyl points are generalized to Weyl surfaces which are two-dimensional closed manifolds in the momentum space. Each Weyl surface is characterized by a U(1) second Chern number $C_2$ defined on a four-dimensional manifold enclosing the Weyl surface, which is equal to its topological linking number with other Weyl surfaces in 5D. In analogy to the Weyl semimetals, the surface states of the 5D metal take the form of topologically protected Weyl fermion arcs, which connect the projections of the bulk Weyl surfaces. The further generalization of topological metal in $2n+1$ dimensions carrying the $n$-th Chern number $C_n$ is also discussed.Comment: 5 pages, 1 figur

Wall-Crossing implies Brill-Noether. Applications of stability conditions on surfaces

Over the last few years, wall-crossing for Bridgeland stability conditions has led to a large number of results in algebraic geometry, particular on birational geometry of moduli spaces. We illustrate some of the methods behind these result by reproving Lazarsfeld's Brill-Noether theorem for curves on K3 surfaces via wall-crossing. We conclude with a survey of recent applications of stability conditions on surfaces. The intended reader is an algebraic geometer with a limited working knowledge of derived categories. This article is based on the author's talk at the AMS Summer Institute on Algebraic Geometry in Utah, July 2015.Comment: 25 pages, 3.5 figures. v2: expanded comparison to Lazarsfeld's methods and results; addressed referee comment

Emergent Momentum-Space Skyrmion Texture on the Surface of Topological Insulators

The quantum anomalous Hall effect has been theoretically predicted and experimentally verified in magnetic topological insulators. In addition, the surface states of these materials exhibit a hedgehog-like "spin" texture in momentum space. Here, we apply the previously formulated low-energy model for Bi$_2$Se$_3$, a parent compound for magnetic topological insulators, to a slab geometry in which an exchange field acts only within one of the surface layers. In this sample set up, the hedgehog transforms into a skyrmion texture beyond a critical exchange field. This critical field marks a transition between two topologically distinct phases. The topological phase transition takes place without energy gap closing at the Fermi level and leaves the transverse Hall conductance unchanged and quantized to $e^2/2h$. The momentum-space skyrmion texture persists in a finite field range. It may find its realization in hybrid heterostructures with an interface between a three-dimensional topological insulator and a ferromagnetic insulator.Comment: Revised manuscript, modified title, supplemental materials adde

Algebraic Surfaces Holomorphically Dominable by C2

An n-dimensional complex manifold M is said to be (holomorphically) dominable by \CC^n if there is a map F:\CC^n \ra M which is holomorphic such that the Jacobian determinant $\det(DF)$ is not identically zero. Such a map F is called a dominating map. In this paper, we attempt to classify algebraic surfaces X which are dominable by \CC^2 using a combination of techniques from algebraic topology, complex geometry and analysis. One of the key tools in the study of algebraic surfaces is the notion of Kodaira dimension (defined in section 2). By Kodaira's pioneering work and its extensions, an algebraic surface which is dominable by \CC^2 must have Kodaira dimension less than two. Using the Kodaira dimension and the fundamental group of X, we succeed in classifying algebraic surfaces which are dominable by \CC^2 except for certain cases in which X is an algebraic surface of Kodaira dimension zero and the case when X is rational without any logarithmic 1-form. More specifically, in the case when X is compact (namely projective), we need to exclude only the case when X is birationally equivalent to a K3 surface (a simply connected compact complex surface which admits a globally non-vanishing holomorphic 2-form) that is neither elliptic nor Kummer (see sections 3 and 4 for the definition of these types of surfaces).Comment: 39 page