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In this paper, we survey some recent results on actions of finite groups on topological manifolds. Given an action of a finite group G on a manifold X , these results provide information on the restriction of the action to a subgroup of G of index bounded above by a number depending only on X. Some of these results refer to the algebraic structure of the group, such as being abelian or nilpotent or admitting a generating subset of controlled size; other results refer to the geometry of the action, e.g. to the existence of fixed points, to the collection of stabilizer subgroups or to the action on cohomology. [ABSTRACT FROM AUTHOR]

We study faithful actions with a dense orbit of abelian unipotent groups on quintic del Pezzo varieties over a field of characteristic zero. Such varieties are forms of linear sections of the Grassmannian of planes in a 5-dimensional vector space. We characterize which smooth forms admit these types of actions and show that in case of existence, the action is unique up to equivalence by automorphisms. We also give a similar classification for mildly singular quintic del Pezzo threefolds and surfaces. [ABSTRACT FROM AUTHOR]

Recently, lattice formulations of Abelian chiral gauge theory in two dimensions have been devised on the basis of the Abelian bosonization. A salient feature of these 2D lattice formulations is that the gauge invariance is exactly preserved for anomaly-free theories and thus is completely free from the question of the gauge mode decoupling. In the present paper, we propose yet another lattice formulation sharing this desired property. A particularly unique point in our formulation is that the vertex operator of the dual scalar field, which carries the vector charge of the fermion and the "magnetic charge" in the bosonization, is represented by a "hole" excised from the lattice; this is the excision method formulated recently by Abe et al. in a somewhat different context. [ABSTRACT FROM AUTHOR]

We solve a case of the Abelian Exponential-Algebraic Closedness Conjecture , a conjecture due to Bays and Kirby, building on work of Zilber, which predicts sufficient conditions for systems of equations involving algebraic operations and the exponential map of an abelian variety to be solvable in the complex numbers. More precisely, we show that the conjecture holds for subvarieties of the tangent bundle of an abelian variety |$A$|⁠ , which split as the product of a linear subspace of the Lie algebra of |$A$| and an algebraic variety. This is motivated by work of Zilber and of Bays–Kirby, which establishes that a positive answer to the conjecture would imply quasiminimality of certain structures on the complex numbers. Our proofs use various techniques from homology (duality between cup product and intersection), differential topology (transversality), and o-minimality (definability of Hausdorff limits), hence we have tried to give a self-contained exposition. [ABSTRACT FROM AUTHOR]

We show that the local von Neumann algebra on convex areas of the frustration-free ground state of abelian quantum double models is of type I I ∞ . [ABSTRACT FROM AUTHOR]

In this paper, we begin the systematic study of group equations with abelian predicates in the main classes of groups where solving equations is possible. We extend the line of work on word equations with length constraints, and more generally, on extensions of the existential theory of semigroups, to the world of groups. We use interpretability by equations to establish model-theoretic and algebraic conditions, which are sufficient to get undecidability. We apply our results to (non-abelian) right-angled Artin groups and show that the problem of solving equations with abelian predicates is undecidable for these. We obtain the same result for hyperbolic groups whose abelianisation has torsion-free rank at least two. By contrast, we prove that in groups with finite abelianisation, the problem can be reduced to solving equations with recognisable constraints, and so this is decidable in right-angled Coxeter groups, or more generally, graph products of finite groups, as well as hyperbolic groups with finite abelianisation. [ABSTRACT FROM AUTHOR]

We show that the dynamics of the scalar Higgs field in the Abelian Higgs–Kibble model supplemented with a dimension-6 derivative operator can be constrained at the quantum level by a certain stability equation. It holds in the Landau gauge and is derived within the recently proposed extended field formalism, where the physical scalar is described by a gauge-invariant field variable. Physical implications of the stability equation are discussed. [ABSTRACT FROM AUTHOR]

Let C be an additive category. The nilpotent category Nil (C) of C , consists of object pairs (X, x) with X ∈ C , x ∈ End C (X) such that xn = 0 for some positive integer n, and a morphism f: (X, x) → (Y, y) is f ∈ Hom C (X , Y) satisfying fx = yf. A general theory of Nil (C) is established and it is abelian in the case that C is abelian. Two abelian categories are equivalent if and only if their nilpotent categories are equivalent, which generalizes a result in [Comm. Algebra, 2018, 46(7): 3062–3070]. As an application, it is proved that all self-adjoint functors are naturally isomorphic to Hom and Tensor functors over the category V of finite-dimensional vector spaces. Both Hom and Tensor can be naturally generalized to HOM and Tensor functor over Nil (V) . They are still self-adjoint, but intrinsically different. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear (1 + n) -dimensional generalized Camassa–Holm Kadomtsev–Petviashvili (g-CH-KP) equation is examined using Lie theory. Lie point symmetries of the equation are computed using MAPLE software and are generalized for the case of any dimension. Moreover, the equation is transformed into a nonlinear ordinary differential equation using the Abelian subalgebra. The nonlinear self-adjoint classification of the equation under consideration is accomplished with the help of which conservation laws for a particular dimension are calculated. Moreover, the new extended algebraic approach is used to compute a wide range of solitonic structures using different set of parameters. Graphic description of some specific applicable solutions for certain physical parameters is portrayed. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a one-dimensional Abelian-Higgs model with Chern-Simons interaction as an effective theory of one-dimensional curves embedded in a three-dimensional space. We demonstrate how this effective model is compatible with the geometry of protein molecules. Using standard field theory techniques, we analyze phenomenologically interesting static configurations of the model and discuss their stability. This simple model predicts some characteristic relations for the geometry of secondary structure motifs of proteins, and we show how this is consistent with the experimental data. After using the data to universally fix basic local geometric parameters, such as the curvature and torsion of the helical motifs, we are left with a single free parameter. We explain how this parameter controls the abundance and shape of the principal motifs (alpha helices, beta strands, and loops connecting them). [ABSTRACT FROM AUTHOR]

An editorial is presented on the theme of noncommutativity and its implications in physics, particularly in the context of noncommutative geometry and its applications in various fields such as the Standard Model, quantum gravity, and condensed matter physics. Topics discussed include the interdisciplinary nature of noncommutative geometry, its theoretical foundations, and practical applications in understanding fundamental aspects of the universe.

We consider the U(1) gauge theory on a four-dimensional torus, where the instanton number is restricted to an integral multiple of p. This theory possesses the nontrivial higher-group structure, which can be regarded as a generalization of the Green–Schwarz mechanism, between Z q 1-form and Z p 3-form symmetries. Here, Z q is a subgroup of the center of U(1). Following the recent study of the lattice construction of the U (1) / Z q principal bundle, we examine how such a structure is realized on the basis of lattice regularization. [ABSTRACT FROM AUTHOR]

The third real de Rham cohomology of compact homogeneous spaces is studied. Given M = G / K with G compact semisimple, we first show that each bi-invariant symmetric bilinear form Q on g such that Q | k × k = 0 naturally defines a G-invariant closed 3-form H Q on M, which plays the role of the so called Cartan 3-form Q ([ · , · ] , ·) on the compact Lie group G. Indeed, every class in H 3 (G / K) has a unique representative H Q . Second, focusing on the class of homogeneous spaces with the richest third cohomology (other than Lie groups), i.e., b 3 (G / K) = s - 1 if G has s simple factors, we give the conditions to be fulfilled by Q and a given G-invariant metric g in order for H Q to be g-harmonic, in terms of algebraic invariants of G/K. As an application, we obtain that any 3-form H Q is harmonic with respect to the standard metric, although for any other normal metric, there is only one H Q up to scaling which is harmonic. Furthermore, among a suitable (2 s - 1) -parameter family of G-invariant metrics, we prove that the same behavior occurs if k is abelian: either every H Q is g-harmonic (this family of metrics depends on s parameters) or there is a unique g-harmonic 3-form H Q (up to scaling). In the case when k is not abelian, the special metrics for which every H Q is g-harmonic depend on 3 parameters. [ABSTRACT FROM AUTHOR]

We demonstrate that the celebrated Stückelberg formalism is modified in the case of a massive four (3 + 1)-dimensional (4D) Abelian 2-form theory due to the presence of a self-duality discrete symmetry in the theory. The latter symmetry entails upon the modified 4D massive Abelian 2-form gauge theory to become a massive model of Hodge theory within the framework of Becchi–Rouet–Stora–Tyutin (BRST) formalism where there is the existence of a set of (anti-)co-BRST transformations corresponding to the usual nilpotent (anti-)BRST transformations. The latter exist in any arbitrary dimension of spacetime for the usual Stückelberg-modified massive Abelian 2-form gauge theory. The modification in the Stückelberg technique is backed by the precise mathematical arguments from the differential geometry where the exterior derivative and Hodge duality operator play the decisive roles. The modified version of the Stückelberg technique remains invariant under the discrete duality transformations which also establish a precise and deep connection between the off-shell nilpotent (anti-)BRST and (anti-)co-BRST transformations. We have clarified a simple trick of using the equations of motion to remove the higher derivative terms in the appropriate Lagrangian densities so that our 4D theory can become consistent. [ABSTRACT FROM AUTHOR]

After fixing the Maximal Abelian gauge in lattice gluodynamics we decompose the nonabelian gauge field into the Abelian field created by Abelian monopoles and the modified nonabelian field with monopoles removed. We then calculate respective static potentials in the fundamental representation and show that the sum of these potentials approximates the nonabelian static potential with good precision at all distances considered. Comparison with other ways of decomposition is made. [ABSTRACT FROM AUTHOR]

Non-Abelian gauge theories with composite fields are examined in the background field method. Generating functionals of Green's functions for a Yang–Mills theory with composite and background fields are introduced, including the generating functional of vertex Green's functions (effective action). The corresponding Ward identities are obtained, and the issue of gauge dependence is investigated. A gauge variation of the effective action is found in terms of a nilpotent operator depending on the composite and background fields. On-shell independence from the choice of gauge fixing for the effective action is established. In the study of the Ward identities and gauge dependence, finite field-dependent BRST transformations with a background field are introduced and employed on a systematic basis. On the one hand, this involves the consideration of (modified) Ward identities with a field-dependent anticommuting parameter, also depending on a non-trivial background. On the other hand, the issue of gauge dependence is studied with reference to a finite variation of the gauge Fermion. The concept of a joint introduction of composite and background fields to non-Abelian gauge theories is exemplified by the Gribov–Zwanziger theory, including the case of a local BRST-invariant horizon, and also by the Volovich–Katanaev model of two-dimensional gravity with dynamical torsion. [ABSTRACT FROM AUTHOR]

In the framework of the Toledano Laredo and Gautam approach, we consider the structures of tensor categories on the analogues of the -category for representations of the super-Yangian of the special linear Lie superalgebra and the quantum loop superalgebra , and study the relation between them. The basic result is the construction of an isomorphism in the category of Hopf superalgebras between completions of the super-Yangian and quantum loop superalgebra equipped with so-called Drinfeld comultiplications. We formulate the theorem on the equivalence of tensor categories of super-Yangian modules and quantum loop superalgebra modules, which enhances the previous result. We also describe the relation between quasitriangular structures and Abelian difference equations, which are defined by the Abelian parts of the universal -matrices. [ABSTRACT FROM AUTHOR]

We investigate a generalization of many functional equations. Namely, we consider the following functional equation ∫S f(x + y + t) dμ(t) + ∫S f(x + φ(y) + t) dν(t) = f(x) + h(y), x, y ∈ S, where (S,+) is an abelian semigroup, φ is a surjective endomorphism of S, E is a linear space over the field K ∈ {R, C} and μ,ν are linear combinations of Dirac measures. Under appropriate conditions on μ and ν and based on Stetkar's result [9], we find and characterize solutions of the previous functional equation. [ABSTRACT FROM AUTHOR]

In this paper, we discuss about finite groups in which, CGH = NGH; for every abelian subgroup H of non prime power order. Also, we classify all such nilpotent and minimal non nilpotent groups. [ABSTRACT FROM AUTHOR]

This document is a correction notice for an article titled "Higher-group structure in lattice Abelian gauge theory under instanton-sum modification." The corrections made in this erratum are related to equations in the continuum theory discussed in Section 2. The original discussions falsely attributed the periodicity of the compact scalar to the 2π periodicity, and the definition of the 4-form gauge field was incorrect. However, these modifications do not affect any conclusions in the lattice theory. The erratum also introduces new equations and conditions to maintain the periodicity of certain variables and to admit nontrivial configurations. The authors express gratitude to Ryo Yokokura for discussions and apologize for an error in the text. [Extracted from the article]

This paper discusses the existence of solitary waves and periodic waves for a generalized (2+1)-dimensional Kadomtsev-Petviashvili modified equal width-Burgers (KP-MEW-Burgers) equation with small damping and a weak local delay convolution kernel by using the dynamical systems approach, specifically based on geometric singular perturbation theory and invariant manifold theory. Moreover, the monotonicity of the wave speed is proved by analyzing the ratio of Abelian integrals. The upper and lower bounds of the limit wave speed are given. In addition, the upper and lower bounds and monotonicity of the period T T of traveling wave when the small positive parameter τ → 0 τ → 0 are also obtained. Perhaps this paper is the first discussion on the solitary waves and periodic waves for the delayed KP-MEW-Burgers equations and the Abelian integral theory may be the first application to the study of the (2+1)-dimensional equation. [ABSTRACT FROM AUTHOR]

This paper explores the nature of the solution sets of systems of equations in virtually abelian groups. We view this question from two angles. From a formal language perspective, we prove that the set of solutions to a system of equations forms an EDT0L language, with respect to a natural normal form. Looking at growth, we show that the growth series of the language of solutions is rational. Furthermore, considering the set of solutions as a set of tuples of group elements, we show that it has rational relative growth series with respect to any finite generating set. [ABSTRACT FROM AUTHOR]

Pseudo-Hermiticity ramifies novel symmetries and enriches unique topological phases distinct from existing non-Hermitian frameworks. Here, we introduce new types of pseudo-Hermitian models across various spatial dimensions, constructed based on the q-deformed Clifford algebra. Each model preserves specific symmetries, and we characterize their band topology via (Abelian and non-Abelian) quantum metric and tensor Berry connections. Concrete relations between quantum metric, Berry curvature, and simple connections between topological invariants and quantum volume are established. Interestingly, these pseudo-Hermitian systems have different geometries and symmetries compared to their Hermitian counterparts, while maintaining the same band topology. We show that the emergence of the non-Hermitian skin effect in these models is a consequence of pseudo-Hermiticity breakdown. Finally, we modify their phase diagram using non-Bloch band theory when various open boundary conditions are imposed. Moreover, we propose a practical scheme for implementing these pseudo-Hermitian models in the dilated Hermitian Hamiltonian. Our work provides a systematic theoretical framework for a comprehensive and essential understanding of the concept of topology and symmetry in such systems. Some of the work carried out in this paper includes. • Constructing new types of pseudo-Hermitian models in various spatial dimensions based on q-deformed Clifford algebra. • Determining the symmetries preserved by each of these pseudo-Hermitian systems. • Characterizing the band topology of these systems and demonstrating that they exhibit different geometries and symmetries compared to their Hermitian counterparts, despite sharing the same topology. • Establishing concrete relations between quantum metric, Berry curvature, and simple relations between topological invariants and quantum volume. • Demonstrating the occurrence of the non-Hermitian skin effect in these models as a result of pseudo-Hermitian breakdown, and modifying their phase diagrams using non-Bloch band theory under various open boundary conditions. • Proposing a practical scheme for implementing these pseudo-Hermitian models in the dilated Hermitian Hamiltonian. [ABSTRACT FROM AUTHOR]

In this work, we investigate the thermodynamic properties of black holes (BHs) that have non-trivial topological features in their phase diagrams. We consider three different models of BHs: (1) a class of BHs in dRGT massive gravity, which adds a mass term to general relativity; (2) a class of BHs in 5D Yang–Mills massive gravity, which combines dRGT massive gravity with a non-Abelian gauge field; and (3) a D-dimensional RN-AdS BH surrounded by Quintessence and a cloud of strings, which are strange forms of matter that change the thermodynamics of the BH. Our goal is to find the critical points of these BHs, which provide the location of first-order phase transitions, and figure out their corresponding topological charges. Topological charges are numbers that show how complicated the BH topology is. Then, we look at these BHs as topological defects in the thermodynamic domain, which is the space of thermodynamic variables like pressure and temperature. We calculate winding numbers to analyze topology on a global and local scale at these defects, which are integers that indicate how many times a curve encircling the defect wraps around the origin. Our analysis reveals that the total topological charge is either equal to 0 or 1 for all models, meaning that the BHs have either a trivial or simple topology. In some cases, we see that the BHs' topology belongs to a different thermodynamic topological class. This means that the BHs can go through topological phase transitions. • We investigate the thermodynamic properties of black holes (BHs). • We consider dRGT, Yang–Mills and D-dimensional RN-AdS BH massive gravity models. • We find the critical points of these BHs. • Then, we look at these BHs as topological defects in the thermodynamic domain. • We calculate winding numbers to analyze topology on a global and local scale at these defects. [ABSTRACT FROM AUTHOR]

The off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetries are obtained by using the (anti-)chiral superfield approach (ACSA) to Becchi–Rouet–Stora–Tyutin (BRST) formalism for the four (3 + 1) -dimensional (4D) Stückelberg-modified massive Abelian 2-form gauge theory. We perform exactly similar kind of exercise for the derivation of the off-shell nilpotent (anti-)co-BRST symmetry transformations, too. In the above derivations, the symmetry invariant restrictions on the superfields play very important and decisive roles. To prove the sanctity of the above nilpotent symmetries, we generalize our 4D ordinary theory (defined on the 4D flat Minkowskian space–time manifold) to its counterparts (4 , 1) -dimensional (anti-)chiral super-submanifolds of the (4 , 2) -dimensional supermanifold which is parametrized by the superspace coordinates Z M = (x μ , , ̄) where x μ (μ = 0 , 1 , 2 , 3) are the bosonic coordinates and a pair of Grassmannian variables (, ̄) are fermionic: 2 = 2 ̄ = 0 , ̄ + ̄ = 0 in nature. One of the novel observations of our present endeavor is the derivation of the Curci–Ferrari (CF)-type restrictions from the requirement of the symmetry invariance of the coupled (but equivalent) Lagrangian densities of our theory within the framework of ACSA to BRST formalism. We also exploit the standard techniques of ACSA to capture the off-shell nilpotency and absolute anticommutativity of the conserved (anti-)BRST as well as the conserved (anti-)co-BRST charges. In a subtle manner, the proof of the absolute anticommutativity of the above conserved charges also implies the existence of the appropriate CF-type restrictions on our theory. This proof is also a novel result. [ABSTRACT FROM AUTHOR]

Generalizing the Moret-Bailly pencil of supersingular abelian surfaces to higher dimensions, we construct for each field of characteristic p > 0 a smooth projective variety with trivial dualizing sheaf that does not lift to characteristic zero. Our approach heavily relies on local unipotent group schemes, the Beauville-Bogomolov decomposition for Kähler manifolds with c1 = 0, and equivariant deformation theory in mixed characteristics. [ABSTRACT FROM AUTHOR]

In this paper, we study the monotonicity and convexity of the period function associated with centers of a specific class of symmetric Newtonian systems of degree 8. In this regard, we prove that if the period annulus surrounds only one elementary center, then the corresponding period function is monotone; but, for the other cases, the period function has exactly one critical point. We also prove that in all cases, the period function is convex. [ABSTRACT FROM AUTHOR]

The static isotropic gravitational field equation, governing the geometry and dynamics of stellar structure, is considered in Einstein–Gauss–Bonnet (EGB) gravity. This is a nonlinear Abelian differential equation which generalizes the simpler general relativistic pressure isotropy condition. A gravitational potential decomposition is postulated in order to generate new exact solutions from known solutions. The conditions for a successful integration are examined. Remarkably we generate a new exact solution to the Abelian equation from the well known Schwarzschild interior seed metric. The metric potentials are given in terms of elementary functions. A physical analysis of the model is performed in five and six spacetime dimensions. It is shown that the six-dimensional case is physically more reasonable and is consistent with the conditions restricting the physics of realistic stars. [ABSTRACT FROM AUTHOR]

We discuss the nilpotent Becchi–Rouet–Stora–Tyutin (BRST), anti-BRST and (anti-) co-BRST symmetry transformations and derive their corresponding conserved charges in the case of a two (1 + 1) -dimensional (2D) self-interacting non-Abelian gauge theory (without any interaction with matter fields). We point out a set of novel features that emerge out in the BRST and co-BRST analysis of the above 2D gauge theory. The algebraic structures of the symmetry operators (and corresponding conserved charges) and their relationship with the cohomological operators of differential geometry are established too. To be more precise, we demonstrate the existence of a single Lagrangian density that respects the continuous symmetries which obey proper algebraic structure of the cohomological operators of differential geometry. In the literature, such observations have been made for the coupled (but equivalent) Lagrangian densities of the 4D non-Abelian gauge theory. We lay emphasis on the existence and properties of the Curci–Ferrari (CF)-type restrictions in the context of (anti-)BRST and (anti-)co-BRST symmetry transformations and pinpoint their key differences and similarities. All the observations, connected with the (anti-)co-BRST symmetries, are completely novel. [ABSTRACT FROM AUTHOR]

We describe how to efficiently construct the quantum chemical Hamiltonian operator in matrix product form. We present its implementation as a density matrix renormalization group (DMRG) algorithm for quantum chemical applications. Existing implementations of DMRG for quantum chemistry are based on the traditional formulation of the method, which was developed from the point of view of Hilbert space decimation and attained higher performance compared to straightforward implementations of matrix product based DMRG. The latter variationally optimizes a class of ansatz states known as matrix product states, where operators are correspondingly represented as matrix product operators (MPOs). The MPO construction scheme presented here eliminates the previous performance disadvantages while retaining the additional flexibility provided by a matrix product approach, for example, the specification of expectation values becomes an input parameter. In this way, MPOs for different symmetries -- abelian and non-abelian -- and different relativistic and non-relativistic models may be solved by an otherwise unmodified program. [ABSTRACT FROM AUTHOR]

We consider the Kac–Bernstein functional equation f (x + y) g (x - y) = f (x) f (y) g (x) g (- y) , x , y ∈ X , on an arbitrary Abelian group X. We give a complete description of the solutions of this equation in the class of positive functions. This result is a generalization of a theorem by Pl. Kannappan. We also study the solutions of this equation in the class of complex-valued Hermitian functions. [ABSTRACT FROM AUTHOR]

Let G be a group and AutΦ(G) denote the group of all automorphisms of G centralizing G/Φ(G) elementwise. In this paper, we give a necessary and sufficient condition on a finite p-group G for the group AutΦ(G) to be elementary abelian. [ABSTRACT FROM AUTHOR]

Given a finite group G and an abelian variety A acted on by G , to any subgroup H of G , we associate an abelian subvariety A H on which the associated Hecke algebra H H for H in G acts. Any irreducible rational representation W ˜ of H H induces an abelian subvariety of A H in a natural way. In this paper we give equations for this abelian subvariety. In a special case these equations become much easier. We work out some examples. [ABSTRACT FROM AUTHOR]

In this paper we show that the Diophantine problem in solvable Baumslag-Solitar groups BS(1,k) and in wreath products A ≀ Z, where A is a finitely generated abelian group and Z is an infinite cyclic group, is decidable, i.e., there is an algorithm that, given a finite system of equations with constants in such a group, decides whether or not the system has a solution in the group. [ABSTRACT FROM AUTHOR]

This paper advances the relations by A. Ja. Aleksandrov between axisymmetric and plane strain states to the case of mixed boundary-value problems. We revise two models of a strip-shaped and a circular stamp indented into a half-plane and a half-space, respectively, when the normal and tangential traction components are unknown a priori. Also, two models of a strip-shaped and a penny-shaped interfacial crack are considered. By using the theory of Abelian integral operators and the Riemann-Hilbert problem on a segment we derive the solutions to the model problems and establish the relations between the governing systems of integral equations associated with the four models and their solutions. These relations can be interpreted as mappings between (i) plane and axisymmetric contact problems, (ii) plane and axisymmetric fracture models, (iii) plane contact and fracture problems, and (iv) axisymmetric contact and fracture problems. The mappings enable us to write down the governing systems of integral equations and the solutions to any three models by making use of the governing system and the solution to the fourth problem. The transformations are specified in the scalar cases when there is no friction in the contact zone and when a crack is in a homogeneous elastic medium. By considering the contact frictionless problem of an annulus stamp it is shown that, although an exact solution to the plane strain frictionless contact problem of two stamps is available, a transformation of the plane to the axisymmetric solution in this case is not possible to obtain, and derivation of a closed-form solution to the annulus stamp model is still an open mathematical problem. [ABSTRACT FROM AUTHOR]

We consider the Abelian-Higgs model with two complex scalar fields and arbitrary positive integer charges with the addition of a higher-order generalization of the Josephson term. The theory possesses vortices of both local and global variants. The only finite-energy configurations are shown to be the local vortices for which a certain combination of vortex numbers and electric charges — called the global vortex number — vanishes. The local vortices have rational fractional magnetic flux, as opposed to the global counterparts that can have an arbitrary fractional flux. The global vortices have angular domain walls, which we find good analytic approximate solutions for. Finally, we find a full classification of the minimal local vortices as well as a few nonminimal networks of vortices, using numerical methods. [ABSTRACT FROM AUTHOR]

Kaluza‐Klein Theory states that a metric on the total space of a principal bundle P→M, if it is invariant under the principal action of P, naturally reduces to a metric together with a gauge field on the base manifold M. We propose a generalization of this Kaluza‐Klein principle to higher principal bundles and higher gauge fields. For the particular case of the abelian gerbe of Kalb‐Ramond field, this Higher Kaluza‐Klein geometry provides a natural global formulation for Double Field Theory (DFT). In this framework the doubled space is the total space of a higher principal bundle and the invariance under its higher principal action is exactly a global formulation of the familiar strong constraint. The patching problem of DFT is naturally solved by gluing the doubled space with a higher group of symmetries in a higher category. Locally we recover the familiar picture of an ordinary para‐Hermitian manifold equipped with Born geometry. Infinitesimally we recover the familiar picture of a higher Courant algebroid twisted by a gerbe (also known as Extended Riemannian Geometry). As first application we show that on a torus‐compactified spacetime the Higher Kaluza‐Klein reduction gives automatically rise to abelian T‐duality, while on a general principal bundle it gives rise to non‐abelian T‐duality. As final application we define a natural notion of Higher Kaluza‐Klein monopole by directly generalizing the ordinary Gross‐Perry one. Then we show that under Higher Kaluza‐Klein reduction, this monopole is exactly the NS5‐brane on a 10d spacetime. If, instead, we smear it along a compactified direction we recover the usual DFT monopole on a 9d spacetime. Kaluza‐Klein Theory states that a metric on the total space of a principal bundle P ‐> M, if it is invariant under the principal action of P, naturally r'educes to a metric together with a gauge field on the base manifold M. Here a generalization of this Kaluza‐Klein principle to higher principal bundles and higher gauge fields is proposed. For the particular case of the abelian gerbe of Kalb‐Ramond field, this Higher Kaluza‐Klein geometry provides a natural global formulation for Double Field Theory (DFT). In this framework the doubled space is the total space of a higher principal bundle and the invariance under its higher principal action is exactly a global formulation of the familiar strong constraint. The patching problem of DFT is naturally solved by gluing the doubled space with a higher group of symmetries in a higher category. Locally one recovers the familiar picture of an ordinary para‐Hermitian manifold equipped with Born geometry. Infinitesimally one sees the familiar picture of a higher Courant algebroid twisted by a gerbe (also known as Extended Riemannian Geometry). As first application it is shown that on a torus‐compactified spacetime the Higher Kaluza‐Klein reduction gives automatically rise to abelian T‐duality, while on a general principal bundle it gives rise to non‐abelian T‐duality. As final application the author defines a natural notion of Higher Kaluza‐Klein monopole by directly generalizing the ordinary Gross‐Perry one. Then it is shown that under Higher Kaluza‐Klein reduction, this monopole is exactly the NS5‐brane on a 10d spacetime. If, instead, it is smeared along a compactified direction one recovers the usual DFT monopole on a 9d spacetime. [ABSTRACT FROM AUTHOR]

We introduce non-Abelian Kuramoto model on S3 in the most general form. Following an analogy with the classical Kuramoto model (on the circle S1), we study some interesting variations of the model on S3 that are obtained for particular coupling functions. As a partial case, by choosing "standard" coupling function one obtains a previously known model, that is referred to as Kuramoto-Lohe model on S3.We briefly address two particular models: Kuramoto models on S3 with frustration and with external forcing. These models on higher dimensional manifolds have not been studied so far. By choosing suitable values of parameters we observe different nontrivial dynamical regimes even in the simplest setup of globally coupled homogeneous population.Although non-Abelian Kuramoto models can be introduced on various symmetric spaces, we restrict our analysis to the case when underlying manifold is the 3-sphere. Due to geometric and algebraic properties of this specific manifold, variations of this model are meaningful and geometrically well justified. [ABSTRACT FROM AUTHOR]

We investigate topological T-duality in the framework of non-abelian gerbes and higher gauge groups. We show that this framework admits the gluing of locally defined T-duals, in situations where no globally defined ("geometric") T-duals exist. The gluing results into new, higher-geometrical objects that can be used to study non-geometric T-duals, alternatively to other approaches like non-commutative geometry. [ABSTRACT FROM AUTHOR]

We prove that the wired uniform spanning forest exhibits mean-field behaviour on a very large class of graphs, including every transitive graph of at least quintic volume growth and every bounded degree nonamenable graph. Several of our results are new even in the case of Z d , d ≥ 5 . In particular, we prove that every tree in the forest has spectral dimension 4/3 and walk dimension 3 almost surely, and that the critical exponents governing the intrinsic diameter and volume of the past of a vertex in the forest are 1 and 1/2 respectively. (The past of a vertex in the uniform spanning forest is the union of the vertex and the finite components that are disconnected from infinity when that vertex is deleted from the forest.) We obtain as a corollary that the critical exponent governing the extrinsic diameter of the past is 2 on any transitive graph of at least five dimensional polynomial growth, and is 1 on any bounded degree nonamenable graph. We deduce that the critical exponents describing the diameter and total number of topplings in an avalanche in the Abelian sandpile model are 2 and 1/2 respectively for any transitive graph with polynomial growth of dimension at least five, and are 1 and 1/2 respectively for any bounded degree nonamenable graph. In the case of Z d , d ≥ 5 , some of our results regarding critical exponents recover earlier results of Bhupatiraju et al. (Electron J Probab 22(85):51, 2017). In this case, we improve upon their results by showing that the tail probabilities in question are described by the appropriate power laws to within constant-order multiplicative errors, rather than the polylogarithmic-order multiplicative errors present in that work. [ABSTRACT FROM AUTHOR]

We introduce a natural stochastic extension, called SSP, of the abelian sandpile model (ASM), which shares many mathematical properties with ASM, yet radically differs in its physical behavior, for example in terms of the shape of the steady state and of the avalanche size distribution. We establish a basic theory of SSP analogous to that of ASM, and present a brief numerical study of its behavior. Our original motivation for studying SSP stems from its connection to the LLL algorithm established in another work by Ding et al. (LLL and stochastic sandpile models, https://sites.google.com/view/seungki/). The importance of understanding how LLL works cannot be stressed more, especially from the point of view of lattice-based cryptography. We believe SSP serves as a tractable toy model of LLL that would help further our understanding of it. [ABSTRACT FROM AUTHOR]

Within the framework of Becchi-Rouet-Stora-Tyutin (BRST) approach, we discuss mainly the fermionic (i.e., off-shell nilpotent) (anti-)BRST, (anti-)co-BRST, and some discrete dual symmetries of the appropriate Lagrangian densities for a two (1 + 1)-dimensional (2D) modified Proca (i.e., a massive Abelian 1-form) theory without any interaction with matter fields. One of the novel observations of our present investigation is the existence of some kinds of restrictions in the case of our present Stückelberg-modified version of the 2D Proca theory which is not like the standard Curci-Ferrari (CF) condition of a non-Abelian 1-form gauge theory. Some kinds of similarities and a few differences between them have been pointed out in our present investigation. To establish the sanctity of the above off-shell nilpotent (anti-)BRST and (anti-)co-BRST symmetries, we derive them by using our newly proposed (anti-)chiral superfield formalism where a few specific and appropriate sets of invariant quantities play a decisive role. We express the (anti-)BRST and (anti-)co-BRST conserved charges in terms of the superfields that are obtained after the applications of (anti-)BRST and (anti-)co-BRST invariant restrictions and prove their off-shell nilpotency and absolute anticommutativity properties, too. Finally, we make some comments on (i) the novelty of our restrictions/obstructions and (ii) the physics behind the negative kinetic term associated with the pseudoscalar field of our present theory. [ABSTRACT FROM AUTHOR]

Following arXiv:1907.04737, we continue our investigation of the relation between the renormalizability (with finitely many couplings) and integrability in 2d σ- models. We focus on the "λ-model," an integrable model associated to a group or symmetric space and containing as special limits a (gauged) WZW model and an "interpolating model" for non-abelian duality. The parameters are the WZ level k and the coupling λ, and the fields are g, valued in a group G, and a 2d vector A± in the corresponding algebra. We formulate the λ-model as a σ-model on an extended G × G × G configuration space (g, h, h ¯ ), defining h and h ¯ by A+ = h∂+h−1, A_ = h ¯ ∂− h ¯ −1. Our central observation is that the model on this extended configuration space is renormalizable without any deformation, with only λ running. This is in contrast to the standard σ-model found by integrating out A±, whose 2-loop renormalizability is only obtained after the addition of specific finite local counterterms, resulting in a quantum deformation of the target space geometry. We compute the 2-loop β-function of the λ-model for general group and symmetric spaces, and illustrate our results on the examples of SU(2)/U(1) and SU(2). Similar conclusions apply in the non-abelian dual limit implying that non-abelian duality commutes with the RG flow. We also find the 2-loop β-function of a "squashed" principal chiral model. [ABSTRACT FROM AUTHOR]

The image registration problem on a group G asks, given two functions f , g : G → R that are related by a translation f (x) = g (s − 1 ⋅ x) by an element s ∈ G , to find s. For abelian groups, the Fourier transform provides an elegant and fast solution to this problem. For nonabelian groups, the problem is much more involved. This paper shows how this applied problem can shed light on the constructions of noncommutative harmonic analysis, in particular the theory of Gelfand pairs. The abstract theory then suggests a novel two-step approach to solving such problems. The Gelfand pair (SO (3) , SO (2)) then provides us with an intuitive solution of the registration problem for images on S 2. [ABSTRACT FROM AUTHOR]

We will analyze through a first order perturbative formulation the local loss of symmetry when a source of non-Abelian Yang–Mills and gravitational fields interacts with an external agent that perturbes the original geometry associated to the source. Then, the symmetry in Abelian and non-Abelian field structures in four-dimensional Lorentzian spacetimes is displayed through the existence of local planes of symmetry that we previously called blades one and two. These orthogonal local planes diagonalize the stress-energy tensor and every vector in these planes is an eigenvector of the stress-energy tensor. The loss of symmetry will be manifested by the tilting of these planes under the influence of the external agent. It was also found already that there is an algorithm to block-diagonalize the Yang–Mills field strength in a local gauge invariant way. The loss of symmetry will also be manifested by the tilting of these planes that block-diagonalize the Yang–Mills field strength under the influence of the external agent. Using perturbative analysis from a previous manuscript dealing with the Abelian case, we will demonstrate how to develop an algorithm for constructing local conserved currents inside both local orthogonal planes of stress-energy diagonalization. As the interaction proceeds, the planes will tilt perturbatively, and in this strict sense the original local symmetry will be lost. But we will prove that the new orthogonal planes or blades at the same point will correspond after the tilting generated by perturbation to a new symmetry, with associated new local currents, both on each new local planes of symmetry. Old symmetries will be broken, new symmetries will arise. There will be a local symmetry evolution in the non-Abelian case as well. This result will produce a new theorem on dynamic symmetry evolution. This new classical model will be useful in order to better understand anomalies in quantum field theories. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Camassa-Holm-KP equation, which is a model for shallow water waves. By using the analysis of the phase space, we obtain some qualitative properties of equilibrium points and existence results of solitary wave solutions for the Camassa-Holm-KP equation without delay. Furthermore we show the existence of solitary wave solutions for the equation with a special local delay convolution kernel by combining the geometric singular perturbation theory and invariant manifold theory. In addition, we discuss the existence of solitary wave solutions for the Camassa-Holm-KP equation with strength 1 of nonlinearity, and prove the monotonicity of the wave speed by analyzing the ratio of the Abelian integral. [ABSTRACT FROM AUTHOR]

We investigate Néron models of Jacobians of singular curves over strictly Henselian discretely valued fields and their behavior under tame base change. For a semiabelian variety, this behavior is governed by a finite sequence of (a priori) real numbers between 0 and 1, called jumps. The jumps are conjectured to be rational, which is known in some cases. The purpose of this paper is to prove this conjecture in the case where the semiabelian variety is the Jacobian of a geometrically integral curve with a push-out singularity. Along the way, we prove the conjecture for algebraic tori which are induced along finite separable extensions and generalize Raynaud's description of the identity component of the Néron model of the Jacobian of a smooth curve (in terms of the Picard functor of a proper, flat, and regular model) to our situation. The main technical result of this paper is that the exact sequence that decomposes the Jacobian of one of our singular curves into its toric and Abelian parts extends to an exact sequence of Néron models. Previously, only split semiabelian varieties were known to have this property. [ABSTRACT FROM AUTHOR]

We classify all recurrent configurations of the Abelian sandpile model (ASM) on Ferrers graphs. The classification is in terms of decorations of EW-tableaux, which undecorated are in bijection with the minimal recurrent configurations. We introduce decorated permutations, extending to decorated EW-tableaux a bijection between such tableaux and permutations, giving a direct bijection between the decorated permutations and all recurrent configurations of the ASM. We also describe a bijection between the decorated permutations and the intransitive trees of Postnikov, the breadth-first search of which corresponds to a canonical toppling of the corresponding configurations. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of multiple solutions for the non-Abelian Chern–Simons–Higgs (N × N) -system: Δ u i = λ (∑ j = 1 N ∑ k = 1 N K k j K j i e u j e u k − ∑ j = 1 N K j i e u j ) + 4 π ∑ j = 1 n i δ p i j , i = 1 , ... , N ; over a doubly periodic domain Ω, with coupling matrix K given by the Cartan matrix of S U (N + 1) , (see (1.2) below). Here, λ > 0 is the coupling parameter, δ p is the Dirac measure with pole at p and n i ∈ N , for i = 1 , ... , N. When N = 1 , 2 many results are now available for the periodic solvability of such system and provide the existence of different classes of solutions known as: topological, non-topological, mixed and blow-up type. On the contrary for N ≥ 3 , only recently in [27] the authors managed to obtain the existence of one doubly periodic solution via a minimization procedure, in the spirit of [46]. Our main contribution in this paper is to show (as in [46]) that actually the given system admits a second doubly periodic solutions of "Mountain-pass" type, provided that 3 ≤ N ≤ 5. Note that the existence of multiple solutions is relevant from the physical point of view. Indeed, it implies the co-existence of different non-Abelian Chern–Simons condensates sharing the same set (assigned component-wise) of vortex points, energy and fluxes. The main difficulty to overcome is to attain a "compactness" property encompassed by the so-called Palais–Smale condition for the corresponding "action" functional, whose validity remains still open for N ≥ 6. [ABSTRACT FROM AUTHOR]

The class of Abelian p-groups are an example of some very interesting phenomena in computable structure theory. We will give an elementary first-order theory T p whose models are each bi-interpretable with the disjoint union of an Abelian p-group and a pure set (and so that every Abelian p-group is bi-interpretable with a model of T p ) using computable infinitary formulas. This answers a question of Knight by giving an example of an elementary first-order theory of "Ulm type": Any two models, low for ω 1 C K , and with the same computable infinitary theory, are isomorphic. It also gives a new example of an elementary first-order theory whose isomorphism problem is Σ 1 1 -complete but not Borel complete. [ABSTRACT FROM AUTHOR]