Twisted Group Research Articles

Nanorings Resembling Beehives for Ultrasensitive Fluorescence Detection.

In this study, we showcase the fabrication of two nanorings resembling beehives using intricately designed donor-acceptor (D-A) fluorophores. The D-A fluorophores, featuring three twisted fluorene groups on each side of the acceptor group, adopt a bent conformation that promotes the creation of a nanoring morphology upon aggregation. With porosity for maximum binding sites, high emission efficiency, and well-organized arrangements, the nanoring-based hives offer exceptional sensitivity and selectivity in the detection of organic sulfides. Particularly, nanorings formed from benzselenodiazole-containing molecules exhibit heightened sensitivity, achieving a limit of detection (LOD) of 0.2 ppb for dimethyl sulfide and 17 ppb for dimethyl disulfide. Due to its unparalleled sensitivity and selectivity, which was not achievable with previous optical sensors, this technology enables the continuous monitoring of meat spoilage in its early stages on an hourly basis. This provides crucial insights into the exact moments when freshness begins to deteriorate and how long the meat can be stored for.

One-loop double copy relation from twisted (co)homology

We propose a geometric relation between closed and open string amplitudes at one-loop. After imposing a homological splitting on the world-sheet torus, twisted intersection theory is used to establish a one-loop double copy relation. The latter expresses a closed string amplitude by a pair of open string amplitudes and twisted intersection numbers. These inner products on the vector space of twisted differential forms are related to the twisted homology and cohomology groups associated with the Riemann-Wirtinger integral.

Strange duality at level one for alternating vector bundles

In this paper, we show a strange duality isomorphism at level one for the space of generalized theta functions on the moduli spaces of alternating anti-invariant vector bundles in the ramified case. These anti-invariant vector bundles constitute one of the non-trivial examples of parahoric G-torsors, where G is a twisted (not generically split) parahoric group scheme.

Intersection numbers, polynomial division and relative cohomology

We present a simplification of the recursive algorithm for the evaluation of intersection numbers for differential n-forms, by combining the advantages emerging from the choice of delta-forms as generators of relative twisted cohomology groups and the polynomial division technique, recently proposed in the literature. We show that delta-forms capture the leading behaviour of the intersection numbers in presence of evanescent analytic regulators, whose use is, therefore, bypassed. This simplified algorithm is applied to derive the complete decomposition of two-loop planar and non-planar Feynman integrals in terms of a master integral basis. More generally, it can be applied to derive relations among twisted period integrals, relevant for physics and mathematical studies.

On unstable twisted rational cohomology groups of the automorphism groups of free groups

In this paper, we consider low-dimensional twisted rational cohomology groups of the automorphism groups of free groups. In particular, we study behaviors of Kawazumi’s cocycles by observing their restrictions to the IA-automorphism groups. We show the nontrivialities of certain unstable twisted cocycles constructed from Kawazumi’s cocycles, and compute the second and the third cohomology groups of the automorphism group of the free group of rank three with coefficients in the exterior powers of the abelianization of the free group.

The compact exceptional Lie algebra 𝔤^{𝔠}₂ as a twisted ring group

A new highly symmetrical model of the compact Lie algebra g 2 c \mathfrak {g}^c_2 is provided as a twisted ring group for the group Z 2 3 \mathbb {Z}_2^3 and the ring R ⊕ R \mathbb {R}\oplus \mathbb {R} . The model is self-contained and can be used without previous knowledge on roots, derivations on octonions or cross products. In particular, it provides an orthogonal basis with integer structure constants, consisting entirely of semisimple elements, which is a generalization of the Pauli matrices in s u ( 2 ) \mathfrak {su}(2) and of the Gell-Mann matrices in s u ( 3 ) \mathfrak {su}(3) . As a bonus, the split Lie algebra g 2 \mathfrak {g}_2 is also seen as a twisted ring group.

The Twisted Partial Group Algebra and (Co)homology of Partial Crossed Products

The Twisted Partial Group Algebra and (Co)homology of Partial Crossed Products

The inverse of a quantum bilinear form of the oriented braid arrangement

We follow here the results of Varchenko, who assigned to each weighted arrangement \(\mathcal{A}\) of hyperplanes in the \(n\)-dimensional real space a bilinear form, which he called the quantum bilinear form of the arrangement \(\mathcal{A}\). We briefly explain the quantum bilinear form of the oriented braid arrangement in the \(n\)-dimensional real space. The main concern of this paper is to compute the inverse of the matrix of the quantum bilinear form of the oriented braid arrangement in \(\mathbb{R}^n\), \({n\ge 2}\). To solve this problem, in [3] the authors used some special matrices and their factorizations in terms of simpler matrices. So, to simplify some matrix calculations, we first introduce a twisted group algebra \({\mathcal{A}(S_{n})}\) of the symmetric group \(S_{n}\) with coefficients in the polynomial ring in \(n^2\) commutative variables and then use a natural representation of some elements of the algebra \({\mathcal{A}(S_{n})}\) on the generic weight subspaces of the multiparametric quon algebra \({\mathcal{B}}\), which immediately gives the corresponding matrices of the quantum bilinear form.

Feynman integrals in two dimensions and single-valued hypergeometric functions

We show that all Feynman integrals in two Euclidean dimensions with massless propagators and arbitrary non-integer propagator powers can be expressed in terms of single-valued analogues of Aomoto-Gelfand hypergeometric functions. The latter can themselves be written as bilinears of hypergeometric functions, with coefficients that are intersection numbers in a twisted homology group. As an application, we show that all one-loop integrals in two dimensions with massless propagators can be written in terms of Lauricella FDr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {F}_D^{(r)} $$\\end{document} functions, while the L-loop ladder integrals are related to the generalised hypergeometric L+1FL functions.

On twisted group ring isomorphism problem for p-groups

Abstract In this article, we explore the problem of determining isomorphisms between the twisted complex group algebras of finite $p$ -groups. This problem bears similarity to the classical group algebra isomorphism problem and has been recently examined by Margolis-Schnabel. Our focus lies on a specific invariant, referred to as the generalized corank, which relates to the twisted complex group algebra isomorphism problem. We provide a solution for non-abelian $p$ -groups with generalized corank at most three.

Global anomalies & bordism of non-supersymmetric strings

The three tachyon-free non-supersymmetric string theories in ten dimensions provide a handle on quantum gravity away from the supersymmetric lamppost. However, they have not been shown to be fully consistent; although local anomalies cancel due to versions of the Green-Schwarz mechanism, there could be global anomalies, not cancelled by the Green-Schwarz mechanism, that could become fatal pathologies. We compute the twisted string bordism groups that control these anomalies via the Adams spectral sequence, showing that they vanish completely in two out of three cases (Sugimoto and SO(16)2) and showing a partial vanishing also in the third (Sagnotti 0’B model). We also compute lower-dimensional bordism groups of the non-supersymmetric string theories, which are of interest to the classification of branes in these theories via the Cobordism Conjecture. We propose a worldvolume content based on anomaly inflow for the SO(16)2 NS5-brane, and discuss subtleties related to the torsion part of the Bianchi identity. As a byproduct of our techniques and analysis, we also reprove that the outer ℤ2 automorphism swapping the two E8 factors in the supersymmetric heterotic string is also non-anomalous.

Trace maps in motivic homotopy and local terms

We define a trace map for every cohomological correspondence in the motivic stable homotopy category over a general base scheme, which takes values in the twisted bivariant groups. Local contributions to the trace map give rise to quadratic refinements of the classical local terms, and some A 1 \mathbb {A}^1 -enumerative invariants, such as the local A 1 \mathbb {A}^1 -Brouwer degree and the Euler class with support, can be interpreted as local terms. We prove an analogue of a theorem of Varshavsky, which states that for a contracting correspondence, the local terms agree with the naive local terms.

A computational investigation of twelve phenylurea herbicides including photoexcitation and structural relaxation

Phenylurea herbicides are widely used to control unwanted plant growth, but can linger in the environment. Photodegradation is a key mechanism by which these compounds can be transformed. This paper investigates the direct photoabsorption and relaxation of twelve phenylurea compounds (chlorbromuron, chlortoluron, diuron, fenuron, fluometuron, isoproturon, linuron, methiuron, metobromuron, metoxuron, monolinuron, monuron) using density functional theory. Ground state structures are optimized at the B3LYP-D3/def2-SVP level of theory and excited states are computed using CAM-B3LYP-D3/def2-SVPD. These molecules are found to be mostly planar and structurally similar in the ground state, with the exception of methiuron which has a significant distortion. The vertical excitation energies are affected by the presence of halogen substituents, and trends in the relaxation of these states are observed. Specifically, halogenated compounds generally have a local excited state minimum corresponding to a stretched carbon-halogen bond, phenylureas containing a methoxy group on the nitrogen have a local minimum corresponding to a twisted urea group and/or a minimum corresponding to the breaking of the nitrogen–oxygen bond between the urea and the methoxy group. The results are discussed in the context of known photodegradation mechanisms for this class of compounds.

Quotient gradings and the intrinsic fundamental group

Quotient grading classes are essential participants in the computation of the intrinsic fundamental group π1(A) of an algebra A. In order to study quotient gradings of a finite-dimensional semisimple complex algebra A it is sufficient to understand the quotient gradings of twisted group algebra gradings. We establish the graded structure of such quotients using Mackey's obstruction class. Then, for matrix algebras A=Mn(C) we tie up the concepts of braces, group-theoretic Lagrangians and elementary crossed products. We also manage to compute the intrinsic fundamental group of the diagonal algebras A=C4 and A=C5.

Bulk-to-boundary anyon fusion from microscopic models

Topological quantum error correction based on the manipulation of the anyonic defects constitutes one of the most promising frameworks towards realizing fault-tolerant quantum devices. Hence, it is crucial to understand how these defects interact with external defects such as boundaries or domain walls. Motivated by this line of thought, in this work, we study the fusion events between anyons in the bulk and at the boundary in fixed-point models of 2 + 1-dimensional non-chiral topological order defined by arbitrary fusion categories. Our construction uses generalized tube algebra techniques to construct a bi-representation of bulk and boundary defects. We explicitly derive a formula to calculate the fusion multiplicities of a bulk-to-boundary fusion event for twisted quantum double models and calculate some exemplary fusion events for Abelian models and the (twisted) quantum double model of S3, the simplest non-Abelian group-theoretical model. Moreover, we use the folding trick to study the anyonic behavior at non-trivial domain walls between twisted S3 and twisted Z2 as well as Z3 models. A recurring theme in our construction is an isomorphism relating twisted cohomology groups to untwisted ones. The results of this work can directly be applied to study logical operators in two-dimensional topological error correcting codes with boundaries described by a twisted gauge theory of a finite group.

On the semisimplicity of twisted group rings and generalized Maschke’s theorem

Classical Maschke’s theorem gives a condition for the semisimplicity of group algebras over a field. As group algebras are a particular case of twisted group rings, we extend Maschke’s theorem to twisted group rings over division rings by proving that if the characteristic of a division ring [Formula: see text] does not divide the order of a finite group [Formula: see text], twisted group ring [Formula: see text] is semisimple. Then we prove the converse of this theorem for twisted group rings arising from split extensions of [Formula: see text] by [Formula: see text]. As there is a one-to-one correspondence between twisted group rings and group lattices over division rings, we conclude with the consequence of the semisimplicity of twisted group rings on group lattices.

Vertex-source pairs and the Clifford theory of finite group crossed product algebras

Let G be a finite group and let A be a G-crossed product -algebra where is a commutative neotherian local complete ring such that the field is of characteristic p. We study the Clifford Theory of A over the 1-component A 1 of A with emphasis on the vertex-source pairs of indecomposable A-modules. We present the following application of our results (Section 5): Let be a field of characteristic p that is “big enough”, let so that , let that is covered by b and let T be the G-stabilizer of e. Thus e is a central idempotent of AT . As is known, there is a twisted group algebra C of T over k such that the categories C-mod and Ab-mod are equivalent. Let X be an indecomposable (resp. simple) C-module. Then from a vertex-source pair of X, we give a formula for a vertex-source pair of a corresponding indecomposable (resp. simple) Ab-module .

Crossed product codes

Bernal et al. provided a necessary and sufficient condition for a linear code to be realized as an ideal in a finite group algebra and De La Cruz and Willems proved a similar result for ideals in twisted group algebras. In this paper, we extend this characterization to crossed products. Furthermore, we determine conditions for some crossed product codes to be self-dual.

The Explicit Twisted Group Algebra Structure of the Cayley–Dickson Algebra

The Explicit Twisted Group Algebra Structure of the Cayley–Dickson Algebra

Twisted Weyl group multiple Dirichlet series over the rational function field

Weyl group multiple Dirichlet series are Dirichlet series in r complex variables, with analytic continuation to Cr and a group of functional equations isomorphic to the Weyl group of a reduced root system of rank r. Such series may be defined for any global field K, but in the case when K is an algebraic function field they are expected to be, up to a variable change, rational functions in several variables. We verify the rationality of these functions in the case when K=Fq(T), and describe the denominators and support of the numerators.