Toric Surfaces Research Articles

Equivariant Elliptic Cohomology, Toric Varieties, and Derived Equivalences

Abstract In this article we study the equivariant elliptic cohomology of complex toric varieties. We prove a reconstruction theorem showing that equivariant elliptic cohomology encodes considerable non-trivial information on the equivariant 1-skeleton of a toric variety $X$ (although it is not a complete invariant of its GKM graphs). We obtain a complete characterization of smooth and proper toric surfaces with isomorphic equivariant elliptic cohomology. Contrary to ordinary cohomology and K-theory, elliptic cohomology is expected not to be a derived invariant of algebraic varieties. We verify this prediction by showing that elliptic cohomology distinguishes derived equivalent varieties. More precisely, we show that there exist pairs of equivariantly derived equivalent toric varieties with non-isomorphic equivariant elliptic cohomology.

Non-planar corrections in ABJM theory from quantum M2 branes

The quantization of semiclassical strings in AdS spacetimes yields predictions for the strong-coupling behaviour of the scaling dimensions of the corresponding operators in the planar limit of the dual gauge theory. Finding non-planar corrections requires computing string loops (corresponding to torus and higher genus surfaces), which is a challenging task. It turns out that in the case of the Uk(N) × U−k(N) ABJM theory there is an alternative approach: one may semiclassically quantize M2 branes in AdS4 × S7/ℤk which are wrapped around the 11d circle of radius 1/k = λ/N . Such M2 branes are the M-theory generalization of the strings in AdS × CP3. In this work, we show that by expanding in large M2 brane tension T2 ~ kN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{kN} $$\\end{document} for fixed k, followed by an expansion in large k, we can predict the large λ asymptotics of the non-planar corrections to the dimensions of the dual ABJM operators. As a specific example, we consider the M2 brane configuration that generalizes the long folded string with large spin in AdS4, and compute the 1-loop correction to its energy. This calculation allows us to determine non-planar corrections to the universal scaling function or cusp anomalous dimension. We extend our analysis to the semiclassical M2 branes that generalize the “short” and “long” circular strings with two equal angular momenta in CP3. The “short” M2 brane corresponds to a dual operator whose dimension at strong coupling scales as ∆ ∼ λ1/4 + …, and we derive the leading non-planar correction to it.

Logarithmic Pandharipande–Thomas spaces and the secondary polytope

Maulik and Ranganathan have recently introduced moduli spaces of logarithmic stable pairs. In the case of toric surfaces we recast this theory using three ingredients: Gelfand, Kapranov and Zelevinsky secondary polytopes, Hilbert schemes of points, and tautological vector bundles. In particular, logarithmic stable pairs spaces are expressed as the zero set of an explicit section of a vector bundle on a logarithmically smooth space, thus providing an explicit description of their virtual fundamental class. A key feature of our construction is that moduli spaces are completely canonical, unlike the existing construction, which is only well-defined up to logarithmic modifications. We calculate the Euler–Satake characteristics of our moduli spaces in a number of basic examples. These computations indicate the complexity of the spaces we construct.

Arithmetic counts of tropical plane curves and their properties

Abstract Recently, the first and third author proved a correspondence theorem which recovers the Levine- Welschinger invariants of toric del Pezzo surfaces as a count of tropical curves weighted with arithmetic multiplicities. In this paper, we study properties of the arithmetic count ofplane tropical curves satisfying point conditions. We prove that this count is independent of the configuration of point conditions. Moreover, a Caporaso- Harris formula for the arithmetic count of plane tropical curves is obtained by moving one point to the very left. Repeating this process until all point conditions are stretched, we obtain an enriched count of floor diagrams which coincides with the tropical count. Finally, we prove polynomiality properties for the arithmetic counts using floor diagrams.

Bio-inspired powder bed fusion 3D printed surfaces for enhanced critical heat flux and boiling heat transfer

The thermal performance and sizing of two-phase heat exchangers employed in various industries can be significantly improved by modifying the surface topology of tubes. The advancement in metal 3D printing technology enables the modification of surfaces to generate distinct patterns that enhance the heat transfer coefficient. This study aligns with these advances by investigating the boiling heat transfer performance of 3D printed surfaces using methanol as the working fluid. The enhanced surfaces studied include four bio-inspired 3D printed surfaces (Mushroom, Cactus, Petal, and Torus), each strategically designed to improve the heat transfer coefficient (HTC) and delay the onset of critical heat flux (CHF). The primary objective of this research is to discuss and analyze the heat transfer mechanism in terms of bubble dynamics, particularly focusing on the interaction between surface morphology and bubble behavior. Experimental results reveal significant enhancements in CHF and HTC for 3D printed surfaces compared to plain surfaces. Among the 3D printed surfaces, the Mushroom surface exhibits the highest enhancement in CHF of 88 % compared to the plain surface. Similarly, the maximum HTC enhancements are 138 %, 86 %, 65 %, and 46 % for the Mushroom, Cactus, Petal, and Torus surfaces, respectively, compared to the plain surface. The methodologies and findings of this study are poised to have a lasting impact on the field, potentially opening new avenues for improving the efficiency and effectiveness of various two-phase heat exchangers.

The volume polynomial of lattice polygons

We prove that every indefinite quadratic form with non-negative integer coefficients is the volume polynomial of a pair of lattice polygons. This solves the discrete version of the Heine–Shephard problem for two bodies in the plane. As an application, we show how to construct a pair of planar tropical curves (or a pair of divisors on a toric surface) with given intersection number and self-intersection numbers.

Tropical Refined Curve Counting with Descendants

We prove a q-refined tropical correspondence theorem for higher genus descendant logarithmic Gromov–Witten invariants with a λg\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda _g$$\\end{document} class in toric surfaces. Specifically, a generating series of such logarithmic Gromov–Witten invariants agrees with a q-refined count of rational tropical curves satisfying higher valency conditions. As a corollary, we obtain a geometric proof of the deformation invariance of this tropical count. In particular, our results give an algebro-geometric meaning to the tropical count defined by Blechman and Shustin. Our strategy is to use the logarithmic degeneration formula, and the key new technique is to reduce to computing integrals against double ramification cycles and connect these integrals to the non-commutative KdV hierarchy.

Investigating Secant Varieties

A variety is a geometric object that locally looks like the set of points that are the solutions of some polynomial equations. Any given variety gives rise to a sequence of secant varieties indexed by the nonnegative integers. The kth secant variety is the union of all planes spanned by k+1 points on the original variety. Like vector spaces in linear algebra, varieties have an intrinsic dimension. A naive parameter count produces an upper bound on the dimension of the kth secant variety called its expected dimension. A secant variety is called defective if its actual dimension is less than its expected dimension. Despite being well understood in a few interesting cases, the classification of varieties with defective secant varieties in general remains far from complete. For a specific class of varieties called smooth toric varieties, our goal is to determine which ones admit defective secant varieties. Using the computer algebra system Macaulay2, we conducted experiments on this class of varieties and collected data on defectiveness, focusing on varieties of low dimension and Picard rank. Toric varieties have strong connections to combinatorics and convex polytopes; for any toric variety, there is a corresponding polytope that contains data about the variety. From the polytope, we are able to construct a particular kind of matrix that encodes information about the variety and its secants, which allowed us to prove defectivity for various Picard rank 2 toric varieties. Conversely, applying results from a paper by Laface–Massarenti–Rischter (2022) to the polytopes, we were able to prove non-defectivity of all secant varieties of certain toric surfaces with Picard rank 2. These statements together yield a classification of defective secant varieties for a subset of 2-dimensional smooth toric varieties with Picard rank 2.

Hypergeometric Function and its Application on Heat Kernel in a Torus Surface with a Rectangular Lattice

Hypergeometric functions are transcendental functions that are applicable in various branches of mathematics, physics, and engineering. They are solutions to a class of differential equations called hypergeometric differential equations. In this paper we will be using theta functions, null zeta function and Ramanujan’s identities to study the behavior of heat kernel in a torus surface with rectangular Lattice.

Extended legality of curved boundary integral method.

The angular spectrum method is an efficient approach for synthesizing electromagnetic beams from planar electric field distributions. The electric field definition is restricted to a plane, which can introduce inaccuracy when applying the synthesized beam to curved surface features. The angular spectrum method can also be interpreted as a pure source method defining the field symmetrically with respect to the creation plane. Recently, we generalized that symmetric field method to arbitrary source distributions, which are valid at any point on compact, regular surface Ω in R 3. We call this approach the Curved Boundary Integral method. The electromagnetic fields synthesized with this method satisfy the Helmholtz equation and are adjusted via amplitude and phase at the desired surface. The fields are obtained as a relatively simple integral. However, restrictions on where in space the synthesized field is valid were included in the mathematical proof length to avoid obscuring the main points. These restrictions can be significant depending on the shape and degree of curvature of surface Ω. In this article, we remove these restrictions so that the integral representation of the electromagnetic beam becomes valid at all points r∈R 3∖Ω, with a minor restriction. Its modification can work even on Ω. We demonstrate the importance of this extended legality with a source field parametrized into the torus surface. The electromagnetic radiation of this structure would not be valid at any point in space without this extension. Finally, we show that by changing the order of integration, the field singularity at each source point is eliminated.

Thresholds for the distributed surface code in the presence of memory decoherence

In the search for scalable, fault-tolerant quantum computing, distributed quantum computers are promising candidates. These systems can be realized in large-scale quantum networks or condensed onto a single chip with closely situated nodes. We present a framework for numerical simulations of a memory channel using the distributed toric surface code, where each data qubit of the code is part of a separate node, and the error-detection performance depends on the quality of four-qubit Greenberger–Horne–Zeilinger (GHZ) states generated between the nodes. We quantitatively investigate the effect of memory decoherence and evaluate the advantage of GHZ creation protocols tailored to the level of decoherence. We do this by applying our framework for the particular case of color centers in diamond, employing models developed from experimental characterization of nitrogen-vacancy centers. For diamond color centers, coherence times during entanglement generation are orders of magnitude lower than coherence times of idling qubits. These coherence times represent a limiting factor for applications, but previous surface code simulations did not treat them as such. Introducing limiting coherence times as a prominent noise factor makes it imperative to integrate realistic operation times into simulations and incorporate strategies for operation scheduling. Our model predicts error probability thresholds for gate and measurement reduced by at least a factor of three compared to prior work with more idealized noise models. We also find a threshold of 4×102 in the ratio between the entanglement generation and the decoherence rates, setting a benchmark for experimental progress.

Cohomology Chambers on Complex Surfaces and Elliptically Fibered Calabi–Yau Three-Folds

We determine several classes of smooth complex projective surfaces on which Zariski decomposition can be combined with vanishing theorems to yield cohomology formulae for all line bundles. The obtained formulae express cohomologies in terms of divisor class intersections, and are adapted to the decomposition of the effective cone into Zariski chambers. In particular, we show this occurs on generalised del Pezzo surfaces, toric surfaces, and K3 surfaces. In the second part we use these surface results to derive formulae for all line bundle cohomology on a simple class of elliptically fibered Calabi–Yau three-folds. Computing such quantities is a crucial step in deriving the massless spectrum in string compactifications.

SYZ mirror of Hirzebruch surfaces [formula omitted] and Morse homotopy

We study homological mirror symmetry for Hirzebruch surfaces Fk as complex manifolds by using the Strominger-Yau-Zaslow construction of mirror pair and Morse homotopy. For toric Fano surfaces, Futaki-Kajiura and the author proved homological mirror symmetry by using Morse homotopy in [9,10,16]. In this paper, we extend Futaki-Kajiura's result of the Hirzebruch surface F1 to Fk. We discuss Morse homotopy and show that homological mirror symmetry in the sense above holds true.

Cross-Cap Defects and Fault-Tolerant Logical Gates in the Surface Code and the Honeycomb Floquet Code

We consider the Z2 toric code, surface code, and Floquet code defined on a nonorientable surface, which can be considered as families of codes extending Shor’s nine-qubit code. We investigate the fault-tolerant logical gates of the Z2 toric code in this setup, which corresponds to e↔m exchanging symmetry of the underlying Z2 gauge theory. We find that nonorientable geometry provides a new way for the emergent symmetry to act on the code space, and discover the new realization of the fault-tolerant Hadamard gate of the two-dimensional surface code with a single cross cap connecting the vertices nonlocally along a slit, dubbed a nonorientable surface code. This Hadamard gate can be realized by a constant-depth local unitary circuit modulo nonlocality caused by a cross cap. Via folding, the nonorientable surface code can be turned into a bilayer local quantum code, where the folded cross cap is equivalent to a bilayer twist terminated on a gapped boundary and the logical Hadamard only contains local gates with intralayer couplings when being away from the cross cap, as opposed to the interlayer couplings on each site needed in the case of the folded surface code. We further obtain the complete logical Clifford gate set for a stack of nonorientable surface codes and similarly for codes defined on Klein-bottle geometries. We then construct the honeycomb Floquet code in the presence of a single cross cap, and find that the period of the sequential Pauli measurements acts as a HZ logical gate on the single logical qubit, where the cross cap enriches the dynamics compared with the orientable case. We find that the dynamics of the honeycomb Floquet code is precisely described by a condensation operator of the Z2 gauge theory, and illustrate the exotic dynamics of our code in terms of a condensation operator supported at a nonorientable surface. Published by the American Physical Society 2024

MS-GIFT: Multi-Sided Geometry-Independent Field ApproximaTion Approach for Isogeometric Analysis

The Geometry-Independent Field approximaTion (GIFT) technique, an extension of isogeometric analysis (IGA), allows for separate spaces to parameterize the computational domain and approximate solution field. Based on the GIFT approach, this paper proposes a novel IGA methodology that incorporates toric surface patches for multi-sided geometry representation, while utilizing B-spline or truncated hierarchical B-spline (THB-spline) basis for analysis. By creating an appropriate bijection between the parametric domains of distinct bases for modeling and approximation, our method ensures smoothness within the computational domain and combines the compact support of B-splines or the local refinement potential of THB-splines, resulting in more efficient and precise solutions. To enhance the quality of parameterization and consequently boost the accuracy of downstream analysis, we suggest optimizing the composite toric parameterization. Numerical examples validate the effectiveness and superiority of our suggested approach.

On the finite generation of valuation semigroups on toric surfaces

We provide a combinatorial criterion for the finite generation of a valuation semigroup associated with an ample divisor on a smooth toric surface and a non-toric valuation of maximal rank. As an application, we construct a lattice polytope such that none of the valuation semigroups of the associated polarized toric variety coming from one-parameter subgroups and centered at a non-toric point are finitely generated.

Homological mirror symmetry of toric Fano surfaces via Morse homotopy

Strominger–Yau–Zaslow (SYZ) proposed a way of constructing mirror pairs as pairs of torus fibrations. We apply this SYZ construction to toric Fano surfaces as complex manifolds, and discuss the homological mirror symmetry, where we consider Morse homotopy of the moment polytope instead of the Fukaya category.

Equilibrium analysis of surface-constrained elastic rods: Unveiling contact and internal forces through local geometry

Confinement scenarios of thin elastic rods are prevalent in both natural and engineered systems. The accurate quantification of the mechanical interplay between confined rods and their confining surfaces remains a formidable challenge, primarily due to the intricate nonlinear nature of thin rods and their contact with surfaces. Here, we present a theoretical framework designed to characterize equilibrium states of thin elastic rods constrained to general surfaces, accounting for the influence of both conservative force and rod inhomogeneity. The framework proposes a general scheme to directly unveil contact and internal forces of confined rods through the consideration of local geometry, based on balance equations for local forces and moments. Characteristics of the contact force are contingent upon whether the deformed rod exhibits vanishing geodesic curvature. Case studies involving closed rods constrained to spherical, cylindrical, and torus surfaces are conducted to elucidate geometric effects on various rod behavior, encompassing equilibrium configurations, internal forces and moments, and contact forces. The outcomes of our study offer fundamental insights into how geometric and material properties lead to the intricate mechanical interactions between one- and two-dimensional materials.

Characterization of droplet impact dynamics onto a stationary solid torus

The impingement mechanism of a liquid droplet on a solid torus surface is demonstrated using numerical simulations and an analytical approach. A computational model employing the volume of fluid method is developed to conduct simulations for the present investigation. Several influencing parameters, namely, diameter ratio (Dt/Do), contact angle (θ), initial droplet velocity (described by Weber number, We), surface tension (specified by Bond number, Bo), and viscosity of liquid drop (described by Ohnesorge number, Oh) are employed to characterize the impacting dynamics of a water drop onto a stationary toroidal substrate. The pattern of temporal and maximum deformation factors is elaborated by considering various relevant influencing factors to describe the fluidic behavior of the drop impingement mechanism. The key findings indicate that the developed central film gets ruptured at the early stage when the value of Dt/Do is lower because a relatively thin film is developed. Concomitantly, the very tiny drops get pinched off at Dt/Do= 0.83, whereas the detached drops are relatively large-sized in the case of lower Dt/Do= 0.16 due to the higher drainage rate of liquid mass through the hole at lower Dt/Do. It is also revealed that the first pinch-off is found to be faster with the continual upsurge of We for a specific value of Dt/Do and θ. Aside from that, efforts are made to show a scattered regime map in order to differentiate the pattern of droplet configuration during impingement. We have also attempted to establish a correlation that effectively characterizes the maximum deformation factor, which closely matches with the numerical findings. The developed correlation exhibits a firm agreement with the numerical data within deviations of 8.5%. Finally, an analytical framework is formulated to predict the deformations factor, which closely agrees with the computational findings.

Computation of symmetries of rational surfaces

<p>In this paper, we provided, first, a general symbolic algorithm for computing the symmetries of a given rational surface, based on the classical differential invariants of surfaces, i.e., <italic>Gauss curvature</italic> and <italic>mean curvature</italic>. In practice, the algorithm works well for sparse parametrizations (e.g., toric surfaces) and PN surfaces. Additionally, we provided a specific, and symbolic, algorithm for computing the symmetries of ruled surfaces. This algorithm works extremely well in practice, since the problem is reduced to that of rational space curves, which can be efficiently solved by using existing methods. The algorithm for ruled surfaces is based on the fact, proven in the paper, that every symmetry of a rational surface must also be a symmetry of its <italic>line of striction</italic>, which is a rational space curve. The algorithms have been implemented in the computer algebra system Maple, and the implementations have been made public. Evidence of their performance is given in the paper.</p>