Elliptic Fibrations Research Articles

Modular curves and Mordell-Weil torsion in F-theory

In this work we prove a bound for the torsion in Mordell-Weil groups of smooth elliptically fibered Calabi-Yau 3- and 4-folds. In particular, we show that the set of torsion groups which can occur on a smooth elliptic Calabi-Yau n-fold is contained in the set of subgroups which appear on a rational elliptic surface if n ≥ 3 and is slightly larger for n = 2. The key idea in our proof is showing that any elliptic fibration with sufficiently large torsion1 has singularities in codimension 2 which do not admit a crepant resolution. We prove this by explicitly constructing and studying maps to a modular curve whose existence is predicted by a universal property. We use the geometry of modular curves to explain the minimal singularities that appear on an elliptic fibration with prescribed torsion, and to determine the degree of the fundamental line bundle (hence the Kodaira dimension) of the universal elliptic surface which we show to be consistent with explicit Weierstrass models. The constraints from the modular curves are used to bound the fundamental group of any gauge group G in a supergravity theory obtained from F-theory. We comment on the isolated 8-dimensional theories, obtained from extremal K3’s, that are able to circumvent lower dimensional bounds. These theories neither have a heterotic dual, nor can they be compactified to lower dimensional minimal SUGRA theories. We also comment on the maximal, discrete gauged symmetries obtained from certain Calabi-Yau threefold quotients.

L‐equivalence for degree five elliptic curves, elliptic fibrations and K3 surfaces

We construct non-trivial L-equivalence between curves of genus one and degree five, and between elliptic surfaces of multisection index five. These results give the first examples of L-equivalence for curves (necessarily over non-algebraically closed fields) and provide a new bit of evidence for the conjectural relationship between L-equivalence and derived equivalence. The proof of the L-equivalence for curves is based on Kuznetsov's Homological Projective Duality for Gr ( 2 , 5 ) , and L-equivalence is extended from genus one curves to elliptic surfaces using the Ogg–Shafarevich theory of twisting for elliptic surfaces. Finally, we apply our results to K3 surfaces and investigate when the two elliptic L-equivalent K3 surfaces we construct are isomorphic, using Neron–Severi lattices, moduli spaces of sheaves and derived equivalence. The most interesting case is that of elliptic K3 surfaces of polarization degree ten and multisection index five, where the resulting L-equivalence is new.

Elliptic fibrations on K3 surfaces with a non-symplectic involution fixing rational curves and a curve of positive genus

In this paper we complete the classification of the elliptic fibrations on K3 surfaces which admit a non-symplectic involution acting trivially on the Néron–Severi group. We use the geometric method introduced by Oguiso and moreover we provide a geometric construction of the fibrations classified. If the non-symplectic involution fixes at least one curve of genus 1, we relate all the elliptic fibrations on the K3 surface with either elliptic fibrations or generalized conic bundles on rational elliptic surfaces. This description allows us to write the Weierstrass equations of the elliptic fibrations on the K3 surfaces explicitly and to study their specializations.

Temperature compensated three-dimension fiber optic vector magnetic field sensor based on an elliptical core micro fiber Bragg grating.

In this paper, a temperature-compensated three-dimension vector fiber optic magnetic field sensor based on an elliptical core micro fiber Bragg grating (FBG) has been proposed and experimentally demonstrated. The elliptical core fiber was tapered to form a microfiber, in which a FBG was inscribed. Due to the magnetism-manipulation of the anisotropic aggregation of ferromagnetism nanoparticles around the fiber surface, the effective refractive index of the evanescent field for two orthogonal polarization modes was modulated, and the magnetic field orientation can be detected by interrogating the wavelength interval between two reflection peaks. However, two reflection peaks show the identical response to ambient temperature. Hence the proposed sensor can achieve the measurements of the magnetic field intensity and the orientation simultaneously without the temperature cross-sensitivity. The experimental results show that the magnetic field orientation sensitivity of 15 pm/deg and intensity sensitivity of 81 pm/mT can be achieved, and the maximum standard variation of the temperature cross-sensitivity is only 0.02 nm. The proposed elliptical core micro FBG appears to have potential applications in navigation, vehicle detection, and current sensing.

More on a dessin on the base: Kodaira exceptional fibers and mutually (non-)local branes

A “dessin d'enfant” is a graph embedded on a two-dimensional oriented surface named by Grothendieck. Recently we have developed a new way to keep track of non-localness among 7-branes in F-theory on an elliptic fibration over P1 by drawing a triangulated “dessin” on the base. To further demonstrate the usefulness of this method, we provide three examples of its use. We first consider a deformation of the I0⁎ Kodaira fiber. With a dessin, we can immediately find out which pairs of 7-branes are (non-)local and compute their monodromies. We next identify the paths of string(-junction)s on the dessin by solving the mass geodesic equation. By numerically computing their total masses, we find that the Hanany-Witten effect has not occurred in this example. Finally, we consider the orientifold limit in the spectral cover/Higgs bundle approach. We observe the characteristic configuration presenting the cluster sub-structure of an O-plane found previously.

Elliptical fibre dielectric waveguides: a transverse transmission line analysis

The authors present a new method for determining the optical properties of arbitrary refractive index elliptical core fibres. The method applies the elliptic cylindrical coordinate system to Maxwell's equations and appropriately, using Fourier Transforms, operated on an elliptical core optical fibre waveguide, derives a flexible algorithm, which can be used to determine the optical properties of elliptical fibre waveguides. By dividing the fibre into a series of uniform consecutive thin dielectric layers transversely across the elliptical cross section with suitable transforming functions representing voltage and current themselves based on the electromagnetic field components, leads to transverse transmission line equations. The resonance frequencies of the cascaded matrices determine the mode propagation constants. The analysis leads to a numerical algorithm for calculating the exact modes of propagation constants. The method is analytically exact and computationally accurate, leading to algorithmic implementation, without making use of Mathieu functions which are solutions for step index waveguide. The advantage of this work is also the fact that it can be applied to arbitrary refractive index profiles. They present results of modal b ¯ − V diagrams for various fibre ellipticities and also apply this method to determine the birefringence of graded index profile fibres and mode cut-off frequencies.

From the signature theorem to anomaly cancellation

We survey the Hirzebruch signature theorem as a special case of the Atiyah–Singer index theorem. The family version of the Atiyah–Singer index theorem in the form of the Riemann–Roch–Grothendieck–Quillen (RRGQ) formula is then applied to the complexified signature operators varying along the universal family of elliptic curves. The RRGQ formula allows us to determine a generalized cohomology class on the base of the elliptic fibration that is known in physics as (a measure of) the local and global anomaly. Combining several anomalous operators allows us to cancel the local anomaly on a Jacobian elliptic surface, a construction that is based on the construction of the Poincaré line bundle over an elliptic surface.

Bott vanishing for algebraic surfaces

Bott proved a strong vanishing theorem for sheaf cohomology on projective space. It holds for toric varieties, but not for most other varieties. We prove Bott vanishing for the quintic del Pezzo surface, also known as the moduli space M_{0,5}^bar of 5-pointed stable curves of genus zero. This is the first non-toric Fano variety for which Bott vanishing has been shown, answering a question by Achinger, Witaszek, and Zdanowicz. In another direction, we prove sharp results on which K3 surfaces satisfy Bott vanishing. For example, a K3 surface with Picard number 1 satisfies Bott vanishing if and only if the degree is 20 or at least 24. For K3 surfaces of any Picard number, we have complete results when the degree is big enough. We build on Beauville, Mori, and Mukai's work on moduli spaces of K3 surfaces, as well as recent advances by Arbarello-Bruno-Sernesi, Ciliberto-Dedieu-Sernesi, and Feyzbakhsh. The most novel aspect of the paper is our analysis of what happens when there is an elliptic curve of low degree. (In other terminology, this concerns K3 surfaces that are unigonal, hyperelliptic, trigonal, or tetragonal.) It turns out that the crucial issue is whether an elliptic fibration has a certain special type of singular fiber.

Mode Selective Conversion Enabled by the Long-Period Gratings Inscribed in Elliptical Core Few-Mode Fiber

We demonstrate the fabrication of long-period fiber gratings (LPFGs) in an elliptical core few-mode fiber (e-FMF) by CO2 laser. The mode conversion from fundamental mode to high-order modes (LP11a, LP11b, and LP21a mode) can be realized by the LPFGs with high efficiency of more than 95%. The effect of bending, torsion, temperature, and strain on the resonance wavelength of the e-FMF based LPFG mode converters are experimentally investigated. The temperature and strain sensitivities of the LPFGs are characterized to be −36 pm/°C and −4.9 pm/μϵ, and the LPFGs are found to be insensitive to the bending and torsion. Due to the existence of trench in the e-FMF, the higher-order LP02 and LP31a mode can also be stimulated by the e-FMF based LPFGs with a grating period of 515 μm and 400 μm, respectively. The proposed e-FMF based LPFGs could have a promising prospect in the application of fiber sensing and mode division multiplexing optical communication system.

Study on filtration performance of elliptical fiber with different arrangements

The fibrous media composed of elliptical fibers is widely used owing to the high filtration efficiency. However, there are few studies on the arrangement of non-circular fibers, although the single non-circular fiber has been clearly investigated. In this article, two-dimensional numerical geometries of fibrous media with different elliptical fiber arrangements, namely, random distribution structure, dense–sparse structure, and bimodal structure, are developed for studying filtration performance. The results show that the large aspect ratio and solid volume fraction represent low particle penetration. When the particle diameter ( Dp) is small, the quality factor of bimodal structure is higher than the dense–sparse structure, especially at Dp = 50 nm. For the large Dp, the opposite is true. Meanwhile, reducing fiber diameter ( Df) is more significant than increasing solid volume fraction in terms of improving penetration. As for dense–sparse structure, replacing the elliptical fibers in sparse layers with circular fibers can comprehensively improve the quality factor of fibrous media. However, if the replacement between elliptical fiber and circular fiber occurs in dense layer, it will result in high quality factor at Dp ⩽ 500 nm, while low quality factor at Dp > 500 nm.

Stringy Hirzebruch classes of Weierstrass fibrations

A Weierstrass fibration is an elliptic fibration $Y\to B$ whose total space $Y$ may be given by a global Weierstrass equation in a $\mathbb{P}^2$-bundle over $B$. In this note, we compute stringy Hirzebruch classes of singular Weierstrass fibrations associated with constructing non-Abelian gauge theories in $F$-theory. For each Weierstrass fibration $Y\to B$ we then derive a generating function $\chi^{\text{str}}_y(Y;t)$, whose degree-$d$ coefficient encodes the stringy $\chi_y$-genus of $Y\to B$ over an unspecified base of dimension $d$, solely in terms of invariants of the base. To facilitate our computations, we prove a formula for general characteristic classes of blowups along (possibly singular) complete intersections.

Arithmetic and geometry of a K3 surface emerging from virtual corrections to Drell–Yan scattering

We study a K3 surface, which appears in the two-loop mixed electroweak-quantum chromodynamic virtual corrections to Drell--Yan scattering. A detailed analysis of the geometric Picard lattice is presented, computing its rank and discriminant in two independent ways: first using explicit divisors on the surface and then using an explicit elliptic fibration. We also study in detail the elliptic fibrations of the surface and use them to provide an explicit Shioda--Inose structure. Moreover, we point out the physical relevance of our results.

Air-Core Ring Fiber With >1000 Radially Fundamental OAM Modes Across O, E, S, C, and L Bands

In this work, an air-core ring fiber is designed with a record high 1322 orbital angular momentum (OAM) modes at 1550 nm while maintaining radially single-mode condition. Moreover, it can support over 1004 OAM modes across all O, E, S, C, and L bands, exploiting to our knowledge the highest number of OAM modes ever supported in the optical fiber within a wide wavelength range. Simulations show that, across the C and L bands, the fiber with 55-μm air-core radius and 0.45-μm ring width can preserve 3.3 × 10 -3 effective refraction index difference between the two highest-order OAM modes HE 340,1 and EH 271,1 . This enables efficient mode separation, and thus achieving stable OAM modes transmission. The effective refractive index differences between the even and odd fiber eigenmodes are also analyzed in the elliptical and bent fibers. We note that higher-order OAM modes are more tolerant to the fiber ellipticity and bending. This ring fiber design has the potential to increase the spectral efficiency and the overall capacity in fiber-based communications system.

Analysis of Elliptical Waveguides With Anisotropic Dielectric Layers

Our main objective is to extend the discrete mode matching (DMM) method to analyze multilayered elliptical fibers or transmission lines with elliptical dielectric layers. We generate the formulation for the full-wave equivalent circuit (FWEC) and derive the hybrid matrix elements for the dielectric layers in an elliptical coordinate system and then obtain the system equation of the structure. To demonstrate the technique, we analyze elliptical waveguides and compare the results with the data obtained by other authors and by commercial software.

Dessin d’enfant and a description of mutually nonlocal 7-branes without using branch cuts

We consider the special roles of the zero loci of the Weierstrass invariants $g_2(\tau(z))$, $g_3(\tau(z))$ in F-theory on an elliptic fibration over $P^1$ or a further fibration thereof. They are defined as the zero loci of the coefficient functions $f(z)$ and $g(z)$ of a Weierstrass equation. They are thought of as complex co-dimension one objects and correspond to the two kinds of critical points of a dessin d'enfant of Grothendieck. The $P^1$ base is divided into several cell regions bounded by some domain walls extending from these planes and D-branes, on which the imaginary part of the $J$-function vanishes. This amounts to drawing a dessin with a canonical triangulation. We show that the dessin provides a new way of keeping track of mutual non-localness among 7-branes without employing unphysical branch cuts or their base point. With the dessin we can see that weak- and strong-coupling regions coexist and are located across an $S$-wall from each other. We also present a simple method for computing a monodromy matrix for an arbitrary path by tracing the walls it goes through.

Selective transverse mode operation of a fiber laser based on few-mode FBG for rotation sensing.

A fiber laser with selective transverse mode operation based on few-mode FBG was designed, and rotation sensing via hybrid mode operation was successfully demonstrated. The mode selection was achieved by a few-mode FBG inscribed in homemade elliptical multilayer-core fiber (EMCF). The particular designed EMCF only supports LP01 and LP11 even mode groups, and the resonance of a few-mode FBG could be adjusted through mode excitation. Therefore, selective transverse mode operation was realized by switching the resonance wavelengths corresponding to the self-coupling mode or cross-coupling mode. Besides, rotation sensing was achieved in the hybrid mode operation due to the asymmetric multilayer-core design. A sensitivity of 0.074 mW/° was preliminarily demonstrated. The measured angle of the rotation sensing system is within ± 2° in the temperature range of 10-90 °C, showing that this system was inherently insensitive to temperature, eliminating the requirement for temperature compensation.

Design of an elliptical air-hole dual-core photonic crystal fiber for over two octaves spanning supercontinuum generation

A highly birefringent polarization-maintaining dual-core photonic crystal fiber (DC-PCF) structure with elliptical air holes around the cores has been designed for possible supercontinuum generation over two octaves spanning a spectral bandwidth in the 400- to 1800-nm wavelength range. The supercontinuum (SC) spectrum covers the near-UV, visual, and near-infrared regions of electromagnetic spectra when a hyperbolic secant pulse of 50 fs duration, 322 mW average power, and 80 MHz repetition rate corresponding to 4.03 nJ energy is launched into the DC-PCF around a 1060-nm pumping wavelength. The generated SC spectrum satisfies the necessary requirements for applications in biophotonics, telecommunications, and frequency metrology.

Heterotic-F-theory duality with Wilson line symmetry-breaking

We begin with an E8 x E8 Heterotic model broken to an SU(5)gauge and a mirror SU(5)gauge, where one SU(5) and its spectrum is identified as the visible sector while the other can be identified as a hidden mirror world. In both cases we obtain the minimal supersymmetric standard model spectrum after Wilson-line symmetry-breaking enhanced by a low energy R-parity enforced by a local (or global) U(1)x-symmetry. Using Heterotic/F-theory duality, we show how to eliminate the vector-like exotics which were obtained in previous constructions. In these constructions, the Calabi-Yau [CY] four-fold was defined by an elliptic fibration with section over a base B3 and a GUT surface given by K3/ℤ2 = Enriques surface. In the present paper we construct a quotient CY four-fold fibered by tori with two elliptic structures given by a pair of sections fibered over the Enriques surface. Using Heterotic/F-theory duality we are able to define the cohomologies used to derive the massless spectrum. Our model for the 'correct' F-theory dual of a Heterotic model with Wilson-line symmetry-breaking builds on prior literature but employs the stack-theoretic version of the dictionary between the Heterotic semi-stable Es-bundles with Yang-Mills connection and the dP9-fibrations used to construct the F-theory dual.

Symplectic surgeries along certain singularities and new Lefschetz fibrations

We define a new 4-dimensional symplectic cut and paste operations arising from the generalized star relations(ta0ta1ta2⋯ta2g+1)2g+1=tb1tb2gtb3, also known as the trident relations, in the mapping class group Γg,3 of an orientable surface of genus g≥1 with 3 boundary components. We also construct new families of Lefschetz fibrations by applying the (generalized) star relations and the chain relations to the families of words (tc1tc2⋯tc2g−1tc2gtc2g+12tc2gtc2g−1⋯tc2tc1)2n=1, (tc1tc2⋯tc2gtc2g+1)(2g+2)n=1 and (tc1tc2⋯tc2g−1tc2g)2(2g+1)n=1 in the mapping class group Γg of the closed orientable surface of genus g≥1 and n≥1. Furthermore, we show that the total spaces of some of these Lefschetz fibrations are irreducible exotic symplectic 4-manifolds. Using the degenerate cases of the generalized star relations, we also realize all elliptic Lefschetz fibrations and genus two Lefschetz fibrations over S2 with non-separating vanishing cycles.

Finite groups acting on elliptic surfaces

We show that automorphism groups of Hopf and Kodaira surfaces have unbounded finite subgroups. For elliptic fibrations on Hopf, Kodaira, bielliptic, and K3 surfaces, we make some observations on finite groups acting along the fibers and on the base of such a fibration.