Mirror Construction Research Articles

Comparison of nonarchimedean and logarithmic mirror constructions via the Frobenius structure theorem

AbstractFor a log Calabi Yau pair () with smooth affine, satisfying either a maximal degeneracy assumption or contains a Zariski dense torus, we prove under the condition that D is the support of a nef divisor that the structure constants defining a trace form on the mirror algebra constructed by Gross–Siebert are given by the naive curve counts defined by Keel–Yu. As a corollary, we deduce that the equality of the mirror algebras constructed by Gross–Siebert and Keel–Yu in the case contains a Zariski dense torus. In addition, we use this result to prove a mirror conjecture proposed by Mandel for Fano pairs satisfying the maximal degeneracy assumption.

A Derived Equivalence of the Libgober–Teitelbaum and the Batyrev–Borisov Mirror Constructions

Abstract In this paper, we study a particular mirror construction to the complete intersection of two cubics in $\operatorname{{\mathbb{P}}}^{5}$, due to Libgober and Teitelbaum. Using variations of geometric invariant theory and methods of Favero and Kelly, we prove a derived equivalence of this mirror to the Batyrev–Borisov mirror of the complete intersection.

A Gromov–Witten Theory for Simple Normal-Crossing Pairs Without Log Geometry

We define a new Gromov–Witten theory relative to simple normal crossing divisors as a limit of Gromov–Witten theory of multi-root stacks. Several structural properties are proved including relative quantum cohomology, Givental formalism, Virasoro constraints (genus zero) and a partial cohomological field theory. Furthermore, we use the degree zero part of the relative quantum cohomology to provide an alternative mirror construction of Gross and Siebert (Intrinsic mirror symmetry, arXiv:1909.07649) and to prove the Frobenius structure conjecture of Gross et al. (Publ Math Inst Hautes Études Sci 122:65–168, 2015) in our setting.

The Coherence of Surah Al-Dzariyat in Perspective of Semitic Rhetoric

The order of the verses and their coherence has been debated among orientalists in qur’anic structure studies. This article aims to reveal the coherence of surah Al-Dzariyat questioned by Richard Bell, regarding the correlation between the verses in the surah. The method of analysing this surah is Semitic Rhetorical Analysis (SRA), utilising the verses’ symmetry explanations from Muslim scholars’ interpretations. This library research adopts descriptive and content analysis methods for analysing data. As the result of the study, surah adz-Dzariyat consists of two passages and seven parts in mirror construction based on Semitic Rhetoric principles. The relation between the verses shows their coherence and excellent verses order without jumpy arrangement. Several groups of verses which at first appeared to be irregular became to appear orderly and have a specific connection which can be understood through Semitic Rhetoric principles. This analysis strengthens the miracles of the Qur’an and refutes various opinions that doubt how good the order of the verses of the Qur’an is.

Non-intrusive, imaging-based method for shock wave characterization in bubbly gas–liquid fluids

A novel experimental imaging-based method is presented for the non-intrusive determination of shock wave characteristics (i.e. shock wave speed and magnitude, and shock-induced liquid velocity) in a bubbly flow solely from gas bubble velocities. Shock wave speeds are estimated by the relative motion between gas bubbles at two locations by splitting the camera field-of-view using a mirror construction, increasing the dynamic spatial range of the measurement system. Although gas bubbles have in general poor tracing properties of the local fluid velocity, capturing the relative dynamics provides accurate estimates for the shock wave properties. This proposed imaging-based method does not require pressure transducers, the addition of tracer particles, or volumetric reconstruction of the gas bubbles. The shock wave magnitude and shock-induced liquid velocity are computed with a hydrodynamic model, which only requires non-intrusively measured variables as input. Two reference measurements, based on pressure transducers and the liquid velocity field by particle image velocimetry, show that the proposed method provides reliable estimates for the shock wave front speed and the shock-induced liquid velocity within the experimental range of 70 < U_textrm{s}< 400 m/s.Graphical abstract

Disk counting and wall-crossing phenomenon via family Floer theory

We use the wall-crossing formula in the non-archimedean SYZ mirror construction (arXiv: 2003.06106) to compute the Landau-Ginzburg superpotential and the one-pointed open Gromov-Witten invariants for a Chekanov-type Lagrangian torus in any smooth toric Fano compactification of C^n. It agrees with the works of Auroux, Chekanov-Schlenk, and Pascaleff-Tonkonog.

Hyperelliptic integrals and mirrors of the Johnson–Kollár del Pezzo surfaces

For all integers k &gt; 0 k&gt;0 , we prove that the hypergeometric function \[ I ^ k ( α ) = ∑ j = 0 ∞ ( ( 8 k + 4 ) j ) ! j ! ( 2 j ) ! ( ( 2 k + 1 ) j ) ! 2 ( ( 4 k + 1 ) j ) ! α j \widehat {I}_k(\alpha )=\sum _{j=0}^\infty \frac {\bigl ((8k+4)j\bigr )!j!}{(2j)!\bigl ((2k+1)j\bigr )!^2 \bigl ((4k+1)j\bigr )!} \ \alpha ^j \] is a period of a pencil of curves of genus 3 k + 1 3k+1 . We prove that the function I ^ k \widehat {I}_k is a generating function of Gromov–Witten invariants of the family of anticanonical del Pezzo hypersurfaces X = X 8 k + 4 ⊂ P ( 2 , 2 k + 1 , 2 k + 1 , 4 k + 1 ) X=X_{8k+4} \subset \mathbb {P}(2,2k+1,2k+1,4k+1) . Thus, the pencil is a Landau–Ginzburg mirror of the family. The surfaces X X were first constructed by Johnson and Kollár. The feature of these surfaces that makes our mirror construction especially interesting is that | − K X | = | O X ( 1 ) | = ∅ |-K_X|=|\mathcal {O}_X (1)|=\varnothing . This means that there is no way to form a Calabi–Yau pair ( X , D ) (X,D) out of X X and hence there is no known mirror construction for X X other than the one given here. We also discuss the connection between our construction and work of Beukers, Cohen and Mellit on hypergeometric functions.

Notes on two-dimensional pure supersymmetric gauge theories

In this note we study IR limits of pure two-dimensional supersymmetric gauge theories with semisimple non-simply-connected gauge groups including SU(k)/ℤk, SO(2k)/ℤ2, Sp(2k)/ℤ2, E6/ℤ3, and E7/ℤ2 for various discrete theta angles, both directly in the gauge theory and also in nonabelian mirrors, extending a classification begun in previous work. We find in each case that there are supersymmetric vacua for precisely one value of the discrete theta angle, and no supersymmetric vacua for other values, hence supersymmetry is broken in the IR for most discrete theta angles. Furthermore, for the one distinguished value of the discrete theta angle for which supersymmetry is unbroken, the theory has as many twisted chiral multiplet degrees of freedom in the IR as the rank. We take this opportunity to further develop the technology of nonabelian mirrors to discuss how the mirror to a G gauge theory differs from the mirror to a G/K gauge theory for K a subgroup of the center of G. In particular, the discrete theta angles in these cases are considerably more intricate than those of the pure gauge theories studied in previous papers, so we discuss the realization of these more complex discrete theta angles in the mirror construction. We find that discrete theta angles, both in the original gauge theory and their mirrors, are intimately related to the description of centers of universal covering groups as quotients of weight lattices by root sublattices. We perform numerous consistency checks, comparing results against basic group-theoretic relations as well as with decomposition, which describes how two-dimensional theories with one-form symmetries (such as pure gauge theories with nontrivial centers) decompose into disjoint unions, in this case of pure gauge theories with quotiented gauge groups and discrete theta angles.

A proposal for nonabelian $(0,2)$ mirrors

In this paper we give a proposal for mirrors to (0,2) supersymmetric gauged linear sigma models (GLSMs), for those (0,2) GLSMs which are deformations of (2,2) GLSMs. Specifically, we propose a construction of (0,2) mirrors for (0,2) GLSMs with E terms that are linear and diagonal, reducing to both the Hori-Vafa prescription as well as a recent (2,2) nonabelian mirrors proposal on the (2,2) locus. For the special case of abelian (0,2) GLSMs, two of the authors have previously proposed a systematic construction, which is both simplified and generalized by the proposal here.

Construction of freeform mirrors for an off-axis telecentric scanning system through multiple surfaces expansion and mixing

In this paper, a direct design method for an off-axis two-mirror telecentric scanning system with a linear field of view (FOV) is proposed. A single freeform mirror structure is firstly considered, in which the aberration free geometry of the off-axis parabolic (OAP) surface is leveraged to provide the focusing function and build surface contour of the sub-region on the mirror for each FOV. Multiple OAP surfaces for construction of the freeform mirror are located at an OAP base to satisfy the telecentric condition. The imaging distortion of this single freeform mirror structure is analyzed and found unavoidable due to the unsymmetrical geometry of the OAP base. A freeform reflective corrector is supplemented, and it is constructed from multiple plane surfaces located at a curved base to fulfill the f-theta scanning geometry. Thus, a two-mirror structure composed of one freeform primary mirror and one freeform reflective corrector is established. Each plane-OAP surfaces pair corresponds to a specific FOV. These multiple OAP surfaces and multiple plane surfaces are then expanded and mixed respectively, to construct the freeform primary mirror and freeform reflective corrector. An f-theta two-mirror freeform scanning system with ±10.4° linear FOV is designed using the proposed construction method. The design result is diffraction-limited, and a scanning error less than 5 μm and telecentricity angle less than 0.2° are achieved.

BHK mirror symmetry for K3 surfaces with non-symplectic automorphism

In this paper we consider the class of K3 surfaces defined as hypersurfaces in weighted projective space, that admit a non-symplectic automorphism of non-prime order, excluding the orders 4, 8, and 12. We show that on these surfaces the Berglund–Hübsch–Krawitz mirror construction and mirror symmetry for lattice polarized K3 surfaces constructed by Dolgachev agree; that is, both versions of mirror symmetry define the same mirror K3 surface.

On mirror maps for manifolds of exceptional holonomy

We study mirror symmetry of type II strings on manifolds with the exceptional holonomy groups G2 and Spin(7). Our central result is a construction of mirrors of Spin(7) manifolds realized as generalized connected sums. In parallel to twisted connected sum G2 manifolds, mirrors of such Spin(7) manifolds can be found by applying mirror symmetry to the pair of non-compact manifolds they are glued from. To provide non-trivial checks for such geometric mirror constructions, we give a CFT analysis of mirror maps for Joyce orbifolds in several new instances for both the Spin(7) and the G2 case. For all of these models we find possible assignments of discrete torsion phases, work out the action of mirror symmetry, and confirm the consistency with the geometrical construction. A novel feature appearing in the examples we analyse is the possibility of frozen singularities.

Finishing of metal optics by ion beam technologies

Ultraprecise mirror devices show considerable potential with view to applications in the visible and the ultraviolet spectral ranges. Aluminum alloys gather good mechanical and excellent optical properties and thus they emerge as important mirror construction materials. However, ultraprecision machining and polishing of optical aluminum surfaces are challenging, which originates from the high chemical reactivity and the heterogeneous matrix structure. Recently, several ion beam-based techniques have been developed to qualify aluminum mirrors for short-wavelength applications. We give an overview of the state-of-the-art ion beam-processing techniques for figure error correction and planarization, either by direct aluminum machining or with the aid of polymer or inorganic, amorphous surface films.

Vertex Operator Algebras and 3d mathcal{N} = 4 gauge theories

We introduce two mirror constructions of Vertex Operator Algebras associated to special boundary conditions in 3d mathcal{N} = 4 gauge theories. We conjecture various relations between these boundary VOA’s and properties of the (topologically twisted) bulk theories. We discuss applications to the Symplectic Duality and Geometric Langlands programs.

JKLMR conjecture and Batyrev construction

We study a mirror interpretation of the relation between the exact partition functions of gauged linear sigma-models (GLSM) on S2 and Kähler potentials on the moduli spaces of the CY manifolds proposed by Jockers et al. We use the Batyrev mirror construction for establishing the explicit relation between GLSM and the corresponding mirror family of the Calabi–Yau manifolds, defined as hypersurfaces in weighted projective spaces. We demonstrate how to do this by the explicit calculation in the case of the quintic threefold and its mirror.

Improvement of the polar magneto-optical Kerr effect in the spatial tunable one-dimensional magnetophotonic crystals

We have studied the magnetophotonic crystals (MPCs) with excellent capabilities to enhance reflection and Kerr rotation simultaneously. In order to simultaneously investigate the symmetric and nonsymmetric spatial structures, we have considered the different repetition numbers for dielectric photonic crystals around the magnetic defect layer. To construction of dielectric Bragg mirrors, we have used of Al2O3 due to low refractive index and its unique optical features. The Ce:YIG with high magneto-optical features is utilized as magnetic material to increasing the magneto-optical responses of structures. By adjusting spatial features such as repetition numbers of PCs and thickness of magnetic defect layers, we have achieved the magneto-optical Kerr rotation more than 70 degrees and reflection close to 1. We have introduced the structures with single and several cavities. For investigation of light localization at defect layer, the electric field distribution along the MPCs is simulated using finite element method (FEM). With light localization inside the cavity and multiple reflections in it, the magneto-optical (MO) responses of these MPCs are significantly increased. Our proposed structures having high MO responses, are suitable cases to design MO elements incorporated in integrated photonic devices and optical telecommunication tools.

Leonardo da Vinci’s design for the construction of burning mirrors

The use of burning mirrors has been an issue for Leonardo since his apprenticeship under Andrea Verrocchio (1468–70). From about 1508 he dealt more closely with the theoretical prerequisites of reflection. In his Codex Arundel and on numerous sheets of the Codex Atlanticus we find more than 200 designs belonging to the subject. Afterwards (1513–15), in the service of Giuliano de’ Medici, he planned the process-steps for the production of large concave mirrors. He set great expectations on this technology (ms. G). In this stage he complains about lack of support by two German craftsmen and encrypts his notes in cipher. Several times he remembers the problems of soldering, which his teacher Verocchio had met while soldering the spherical segments on the lantern of Florence cathedral. Numerous are the drawings of precise grinding and polishing machines for the production of mirrors. Leonardo finds a solution to avoid shading if the receptor of the solar rays obstructed the front of a mirror. Burning mirrors should heat fishponds or dyeing tubs, and they would save fire-wood during lime-burning. In the service of Giuliano, commander-in-chief of the papal army, Leonardo may well have trusted in the legend of Archimedes destructing Roman ships by burning mirrors.

Equivalences of families of stacky toric Calabi-Yau hypersurfaces

Given the same anti-canonical linear system on two distinct toric varieties, we provide a derived equivalence between partial crepant resolutions of the corresponding stacky hypersurfaces. The applications include: a derived unification of toric mirror constructions, calculations of Picard lattices for linear systems of quartics in $\mathbf {P}^3$, and a birational reduction of Reid’s list to 81 families.

On Clifford double mirrors of toric complete intersections

We present a construction of noncommutative double mirrors to complete intersections in toric varieties. This construction unifies existing sporadic examples and explains the underlying combinatorial and physical reasons for their existence.

A proposal for (0,2) mirrors of toric varieties

In this paper we propose (0,2) mirrors for general Fano toric varieties with special tangent bundle deformations, corresponding to subsets of toric deformations. Our mirrors are of the form of (B/2-twisted) (0,2) Landau-Ginzburg models, matching Hori-Vafa mirrors on the (2,2) locus. We compare our predictions to (0,2) mirrors obtained by Chen et al. for certain examples of toric varieties, and find that they match. We also briefly outline conjectures for analogous results for hypersurfaces in Fano toric varieties. Our methods utilize results from supersymmetric localization, which allows us to incidentally gain occasional further insights into GLSM-based (2,2) mirror constructions. For example, we explicitly verify that closed string correlation functions of the original A-twisted GLSM match those of the mirror B-twisted Landau-Ginzburg model, as well as (0,2) deformations thereof.