Lattice Vertex Research Articles

Representations of quantum lattice vertex algebras

Let Q be a non-degenerate even lattice, let VQ be the lattice vertex algebra associated to Q, and let VQη be a quantum lattice vertex algebra ([10]). In this paper, we prove that every VQη-module is completely reducible, and the set of simple VQη-modules are in one-to-one correspondence with the set of cosets of Q in its dual lattice.

Completely fixed point free isometry and cyclic orbifold of lattice vertex operator algebras

We continue our study of cyclic orbifolds of lattice vertex operator algebras and their full automorphism groups. We consider some special isometry g∈O(L) such that gi is fixed point free on L for any 1≤i≤|g|−1. We show that when L(2)=∅ and gi is fixed point free on L for any 1≤i≤|g|−1, VLgˆ has extra automorphisms implies either (1) the order of g is a prime or (2) L is isometric to the Leech lattice or some coinvariant sublattices of the Leech lattice.

Semi-automated vertex placement for lattice radiotherapy and dosimetric verification using 3D polymer gel dosimetry.

To evaluate the feasibility of an open-source, semi-automated, and reproducible vertex placement tool to improve the efficiency of lattice radiotherapy (LRT) planning. We used polymer gel dosimetry with a Cone Beam CT (CBCT) readout to commission this LRT technique. We generated a volumetric modulated arc therapy (VMAT)-based LRT plan on a 2 L NIPAM polymer gel dosimeter using our Eclipse Acuros version 15.6 AcurosXB beam model, and also recalculated the plan with a pre-clinical Acuros v18.0 dose calculation algorithm with the enhanced leaf modelling (ELM). With the assistance of the MAAS-SFRThelper software, a lattice vertex diameter of 1.5cm and center-to-center spacing of 3cm were used to place the spheres in a hexagonal, closed packed structure. The verification plan included four gantry arcs with 15°, 345°, 75°, 105° collimator angles. The spheres were prescribed 20Gy to 50% of their combined volume. The 6 MV Flattening Filter Free beam energy was used to deliver the verification plan. The dosimetric accuracy of the LRT delivery was evaluated with 1D dose profiles, 2D isodose maps, and a 3D global gamma analysis. Qualitative comparisons between the 1D dose profiles of the Eclipse plan and measured gel showed good consistency at the prescription dose mark. The average diameter measured 13.3±0.2mm (gel for v15.6), 12.6mm (v15.6 plan), 13.1±0.2mm (gel for v18.0), and 12.3mm (v18.0 plan). 3D gamma analysis showed that all gamma pass percent were>95% except at 1% and 2% at the 1mm distance to agreement criteria. This study presents a novel application of gel dosimetry in verifying the dosimetric accuracy of LRT, achieving excellent 3D gamma results. The treatment planning was facilitated by publicly available software that automatically placed the vertices for consistency and efficiency.

The classification of semi-conformal subalgebras of an even lattice vertex operator algebras with rank two

This work is to understand structures of even lattice vertex operator algebras by using the geometry of the varieties of their semi-conformal vectors. In this paper, we shall describe the varieties of semi-conformal vectors of a class of even lattice vertex operator algebras with rank two to characterize these vertex operator algebras. Moreover, we shall give the G-orbit decomposition of the fixed vertex operator subalgebra VL+ of a rank-two lattice vertex operator algebra VL, where G is the automorphism group preserving the conformal vector of VL+. Based on this decomposition, we classify all maximal semi-conformal subalgebras of VL+.

Properties of Convex Lattice Sets under the Discrete Legendre Transform

The discrete Legendre transform is a powerful tool for analyzing the properties of convex lattice sets. In this paper, for t>0, we study a class of convex lattice sets and establish a relationship between vertices of the polar of convex lattice sets and vertices of the polar of its t−dilation. Subsequently, we show that there exists a class of convex lattice sets such that its polar is itself. In addition, we calculate upper and lower bounds for the discrete Mahler product of a class of convex lattice sets.

Aging properties of the voter model with long-range interactions

We investigate the aging properties of the one-dimensional voter model with long-range interactions in its ordering kinetics. In this system, an agent, Si=±1 , positioned at a lattice vertex i, copies the state of another one located at a distance r, selected randomly with a probability P(r)∝r−α . Employing both analytical and numerical methods, we compute the two-time correlation function G(r;t,s) ( t⩾s ) between the state of a variable Si at time s and that of another one, at distance r, at time t. At time t, the memory of an agent of its former state at time s, expressed by the autocorrelation function A(t,s)=G(r=0;t,s) , decays algebraically for α > 1 as [L(t)/L(s)]−λ , where L is a time-increasing coherence length and λ is the Fisher–Huse exponent. We find λ = 1 for α > 2, and λ=1/(α−1) for 1<α⩽2 . For α⩽1 , instead, there is an exponential decay, as in the mean field. Then, in contrast with what is known for the related Ising model, here we find that λ increases upon decreasing α. The space-dependent correlation G(r;t,s) obeys a scaling symmetry G(r;t,s)=g[r/L(s);L(t)/L(s)] for α > 2. Similarly, for 1<α⩽2 , one has G(r;t,s)=g[r/L(t);L(t)/L(s)] , where the length L regulating two-time correlations now differs from the coherence length as L∝Lδ , with δ=1+2(2−α) .

Modified bosonic integrable hierarchy

In this paper, we construct a new integrable hierarchy by using charged free boson and Friedan–Martinec–Shenker (FMS) bosonization, called modified bosonic integrable hierarchy, which is an anologue of the classical (l−l′)–th modified KP hierarchy. Firstly, the representation of glˆ∞ is given by using charged free bosons. Then the modified bosonic integrable hierarchy is proposed. Next after discussing FMS bosonizations in the language of lattice vertex algebra, bilinear equations of modified bosonic integrable hierarchy are constructed by FMS bosonization, which are rewritten into Hirota bilinear equations by using Hirota derivatives. At last, we consider solutions of this modified bosonic integrable hierarchy.

A rigorous mathematical theory for topological phases and edge modes in spring-mass mechanical systems

In this work, we examine the topological phases of the spring-mass lattices when the spatial inversion symmetry of the system is broken and prove the existence of edge modes when two lattices with different topological phases are glued together. In particular, for the one-dimensional lattice consisting of an infinite array of masses connected by springs, we show that the Zak phase of the lattice is quantized, only taking the value 0 or π . We also prove the existence of an edge mode when two semi-infinite lattices with distinct Zak phases are connected. For the two-dimensional honeycomb lattice, we characterize the valley Chern numbers of the lattice when the masses on the lattice vertices are uneven. The existence of edge modes is proved for a joint honeycomb lattice formed by gluing two semi-infinite lattices with opposite valley Chern numbers together.

On the commutant of the principal subalgebra in the $$A_1$$ lattice vertex algebra

On the commutant of the principal subalgebra in the $$A_1$$ lattice vertex algebra

Extension of Pick’s Theorem to Spherical Geometry using Girard’s Theorem

In this article, Pick’s theorem is extended to three-dimensional bodies with two-dimensional surfaces, namely spherical geometry. The equation for the area of a polygon consisting of equilateral spherical triangles is obtained by combining Girard’s theorem used to find area of any spherical triangle and Pick’s theorem used to find area of a simple polygon with lattice point vertices in Euclidian geometry. Vertices of the polygon are represented by integer points. In this way, an equation to find area of a spherical polygon is presented. This equation could give an idea to be applied on cylindrical surfaces, hyperbolic geometry and more general surfaces. The theorem proposed in this article which is the extension of Pick’s theorem using Girard’s theorem seems to be a special case of a more general theorem.

Vertex Position Optimization for LATTICE Therapy

LATTICE radiation therapy (RT) aims to deliver 3D heterogenous dose of high peak-to-valley dose ratio (PVDR) to the tumor target, with peak dose at lattice vertices inside the target and valley dose for the rest of the target. In current clinical practice the lattice vertex positions are constant during treatment planning. This work proposes a new LATTICE plan optimization method that can optimize lattice vertex positions as plan variables, which is the first lattice vertex position optimization study to the best of our knowledge. The new LATTICE treatment planning method optimizes lattice vertex positions as well as other plan variables (e.g., photon fluences or proton spot weights), with optimization objectives for target PVDR and organs-at-risk (OAR) sparing. To satisfy mathematical differentiability, the lattice vertices are approximated in sigmoid functions. For geometric feasibility, proper geometry constraints are enforced onto the lattice vertex positions. The lattice vertex position optimization problem is solved by iterative convex relaxation method, where lattice vertex positions and photon/proton plan variables are jointly updated via the Quasi-Newton method. Both photon and proton LATTICE RT were considered, and the optimal lattice vertex positions in terms of plan objectives were found by solving all possible combinations on given discrete positions via heuristic searching based on standard IMRT/IMPT, which served as the ground truth for validating the new LATTICE method ("NEW"). That is, the plan with the smallest optimization objective ("BEST"), the plan with the median optimization objective ("MID"), and the plan with the largest optimization objective ("WORST") were selected as the reference plans to be compared with NEW. The table was for an abdomen case with the large bowel as the OAR, where the parameters are total optimization objective f, the mean valley dose of target Dvalley, the mean peak dose of target Dpeak, PVDR = Dpeak/Dvalley, and the mean dose of large bowel Dbowel. The unit of doses is Gy. The results in the table show that the new method indeed provided the optimal lattice vertex positions with the smallest optimization objective, the largest target PVDR, and the best OAR sparing. A new LATTICE treatment planning method is proposed and validated that can optimize lattice vertex positions as well as other photon or proton plan variables for improving target PVDR and OAR sparing.

Study on Lattice Radiotherapy Treatments (LRT) for Head and Neck Bulky Tumors

Lattice radiotherapy (LRT) exploits various effects of radiation, such as the bystander effect and the abscopal effect, and consists on the administration of high dose fraction in small areas with large tumor masses, helping to solve the problem of treating bulky disease, especially if it is located in a critical anatomical area. The optimization of LRT treatment plans is challenging due to the difficulty to generate spots of high dose within the tumor with consequent high gradient. This study compares the plan dosimetry and delivery time of two delivery techniques VMAT and CyberKnife for LRT treatments of bulky head and neck lesions. Six patients with giant head and neck tumors who received LRT at our institution were included in this study. Target and OARs were contoured following international guidelines; to allow easy identification of the desired high gradient zones, an artificial geometrical lattice structure with spherical vertices was arranged inside the target volume (GTV), and the vertices of the lattice representing the high dose boost volumes (GTVboost) were delineated. The GTVboost and GTV were prescribed to receive 12 Gy and 3 Gy, respectively in a single fraction. Separate VMAT and CyberKnife LRT plans were optimized for each patient with lattice vertex of 0.5 diameter and center-to-center distances of 1.5 cm (LRT1.5) and 3 cm (LRT3). The dose heterogeneity was measured as the peak-to-valley dose ratio (PVDR), with the traditional definition being replaced by the D10/D90 ratio, where D10 and D90 represent the doses covering 10% and 90% of the GTV, respectively. For each plan generated, the treatment delivery time, the monitor units (MU), and the PVDR were assessed. Pre-treatment plan verifications were performed with ArcCheck array and Gafchromics film for VMAT and CyberKnife, respectively, using gamma analysis criteria of 3%-3mm. The mean PVDR obtained for VMAT LRT plans were 2.0 and 2.6 for LRT1.5 and LRT3, respectively, and 3.2 and 4.7, respectively for CyberKnife LRT plans. For each pre-treatment plan dose verification, the gamma passing rate (GPR) was higher than 95.0 %; CyberKnife delivery time and MU were more than 10 times higher than that of VMAT, nevertheless, VMAT had a lower PVDR. The detailed results are shown in the table below. CyberKnife LRT has a strong ability to place the peak dose within the target, generating a higher peak-to-valley dose ratio, however its use is partially invalidated by the long beam delivery times and the resulting high MU number; the use of the VMAT LRT technique allows clinically adequate dosimetry with acceptable delivery times.

Dimensional effect and mechanical performance of node-strengthened hybrid lattice structure fabricated by laser powder bed fusion

ABSTRACT Node-strengthened hybrid structures with lower relative density inspired by solid solution strengthening mechanism, namely the edge center interstitial lattice (ECIL) structures and vertex node substitutional lattice (VNSL) structures were designed and fabricated by laser powder bed fusion (LPBF). The geometric feature-dependent defects distribution, the intense microstructure sensitivity as well as the node size effect on the mechanical response were investigated. The microstructure sensitivity induced by geometric feature was found to be related to the different supporting condition and distinctive thermal history. ECIL-1.5 structure possessed the highest plateau stress of 1.79 MPa and the largest crush force efficiency of 59.1%, which increased by 59.8% and 15.2% compared to the initial face centre cubic with z-struts (FCCZ)structure. VNSL-1.5 exhibited the greatest specific energy absorption of 14.6 J/g, demonstrating the highest strengthening efficiency was achieved at the critical sphere diameter to strut thickness (Sph-strut) ratio of 3. This method further improved lightweight efficiency, indicating the inherent strengthening mechanism of crystal materials could guide the design of metamaterials.

Lattice position optimization for LATTICE therapy.

LATTICE radiation therapy delivers 3D heterogenous dose of high peak-to-valley dose ratio (PVDR) to the tumor target, with peak dose at lattice vertices inside the target and valley dose for the rest of the target. Although the lattice vertex positions can impact PVDR inside the target and sparing of organs-at-risk (OAR), they are fixed as constants and not optimized during treatment planning in current clinical practice. This work proposes a new LATTICE plan optimization method that can optimize lattice vertex positions during LATTICE treatment planning, which is the first lattice position optimization study to the best of our knowledge. The new LATTICE treatment planning method optimizes lattice vertex positions as well as other plan variables (e.g., photon fluences or proton spot weights), with optimization objectives for target PVDR and OAR sparing. To satisfy mathematical differentiability, the lattice vertices are approximated in sigmoid functions. For geometric feasibility, proper geometry constraints are enforced onto lattice vertex positions. The lattice position optimization problem is solved by iterative convex relaxation (ICR) method and alternating direction method of multipliers (ADMM), and lattice vertex positions and photon/proton plan variables are jointly updated via the Quasi-Newton method. Both photon and proton LATTICE RT were considered, and the optimal lattice vertex positions in terms of plan objectives were found by solving all possible combinations on given discrete positions via exhaustive searching based on standard IMRT/IMPT, which served as the ground truth for validating the new LATTICE method. The results show that the new method indeed provided the optimal lattice vertex positions with the smallest optimization objective, the largest target PVDR, and the best OAR sparing. A new LATTICE treatment planning method is proposed and validated that can optimize lattice vertex positions as well as other photon or proton plan variables for improving target PVDR and OAR sparing.

Orbifolds of lattice vertex algebras

Orbifolds of lattice vertex algebras

Extra automorphisms of cyclic orbifolds of lattice vertex operator algebras

In this article, we study automorphisms of the cyclic orbifold of a vertex operator algebra associated with a rootless even lattice for a lift of a fixed-point free isometry of odd prime order p. We prove that such a cyclic orbifold contains extra automorphisms, not induced from automorphisms of the lattice vertex operator algebra, if and only if the rootless even lattice can be constructed by Construction B from a code over Zp or is isometric to the coinvariant lattice of the Leech lattice associated with a certain isometry of order p.

Quantum double models coupled to matter fields: A detailed review for a dualization procedure

In this paper, we investigate how it is possible to define a new class of lattice gauge models based on a dualization procedure of a previous generalization of the Kitaev Quantum Double Models. In the case of this previous generalization that will be used as a basis, it was defined by adding new qudits (which were denoted as matter fields in reference to some works) to the lattice vertices with the intention of, for instance, interpreting its models as Kitaev Quantum Double Models coupled with Potts ones. Now, with regard to the generalization that we investigate here, which we want to define as the dual of this previous one, these new qudits were added to the lattice faces. And as the coupling between gauge and matter qudits of the previous generalization was performed by a gauge group action, we show that the dual behavior of these two generalizations was achieved by coupling these same qudits in the second one through a gauge group co-action homomorphism. One of the most striking dual aspects of these two generalizations is that, in both, part of the quasiparticles that were inherited from the Kitaev Quantum Double Models become confined when these action and co-action are nontrivial. But the big news here is that, in addition to the group homomorphism (that defines this gauge group co-action) allows us to classify all the different models of this second generalization, this same group homomorphism also suggests that all these models can be interpreted as two-dimensional restrictions of the 2-lattice gauge theories.

Lattice three-gluon vertex in extended kinematics: Planar degeneracy

We present novel results for the three-gluon vertex, obtained from an extensive quenched lattice simulation in the Landau gauge. The simulation evaluates the transversely projected vertex, spanned on a special tensorial basis, whose form factors are naturally parametrized in terms of individually Bose-symmetric variables. Quite interestingly, when evaluated in these kinematics, the corresponding form factors depend almost exclusively on a single kinematic variable, formed by the sum of the squares of the three incoming four-momenta, q, r, and p. Thus, all configurations lying on a given plane in the coordinate system (q2,r2,p2) share, to a high degree of accuracy, the same form factors, a property that we denominate planar degeneracy. We have confirmed the validity of this property through an exhaustive study of the set of configurations satisfying the condition q2=r2, within the range [0,5GeV]. This drastic simplification allows for a remarkably compact description of the main bulk of the data, which is particularly suitable for future numerical applications. A semi-perturbative analysis reproduces the lattice findings rather accurately, once the inclusion of a gluon mass has cured all spurious divergences.

Dimension formulae and generalised deep holes of the Leech lattice vertex operator algebra

We prove a dimension formula for the weight-1 subspace of a vertex operator algebra $V^{\operatorname{orb}(g)}$ obtained by orbifolding a strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 with a finite-order automorphism $g$. Based on an upper bound derived from this formula we introduce the notion of a generalised deep hole in $\operatorname{Aut}(V)$. Then we show that the orbifold construction defines a bijection between the generalised deep holes of the Leech lattice vertex operator algebra $V_\Lambda$ with non-trivial fixed-point Lie subalgebra and the strongly rational, holomorphic vertex operator algebras of central charge 24 with non-vanishing weight-1 space. This provides the first uniform construction of these vertex operator algebras and naturally generalises the correspondence between the deep holes of the Leech lattice $\Lambda$ and the 23 Niemeier lattices with non-vanishing root system found by Conway, Parker and Sloane.

A lattice theoretical interpretation of generalized deep holes of the Leech lattice vertex operator algebra

Abstract We give a lattice theoretical interpretation of generalized deep holes of the Leech lattice VOA $V_\Lambda $ . We show that a generalized deep hole defines a ‘true’ automorphism invariant deep hole of the Leech lattice. We also show that there is a correspondence between the set of isomorphism classes of holomorphic VOA V of central charge $24$ having non-abelian $V_1$ and the set of equivalence classes of pairs $(\tau , \tilde {\beta })$ satisfying certain conditions, where $\tau \in Co.0$ and $\tilde {\beta }$ is a $\tau $ -invariant deep hole of squared length $2$ . It provides a new combinatorial approach towards the classification of holomorphic VOAs of central charge $24$ . In particular, we give an explanation for an observation of G. Höhn, which relates the weight one Lie algebras of holomorphic VOAs of central charge $24$ to certain codewords associated with the glue codes of Niemeier lattices.