Flat Space Research Articles

Dirac gauge theory for topological spinors in 3+1 dimensional networks

Gauge theories on graphs and networks are attracting increasing attention not only as approaches to quantum gravity but also as models for performing quantum computation. Here we propose a Dirac gauge theory for topological spinors in dimensional networks associated to an arbitrary metric. Topological spinors are the direct sum of 0-cochains and 1-cochains defined on a network and describe a matter field defined on both nodes and links of a network. Recently in Bianconi (2021 J. Phys. Complex. 2 035022) it has been shown that topological spinors obey the topological Dirac equation driven by the discrete Dirac operator. In this work we extend these results by formulating the Dirac equation on weighted and directed dimensional networks which allow for the treatment of a local theory. The commutators and anti-commutators of the Dirac operators are non vanishing an they define the curvature tensor and magnetic field of our theory respectively. This interpretation is confirmed by the non-relativistic limit of the proposed Dirac equation. In the non-relativistic limit of the proposed Dirac equation the sector of the spinor defined on links follows the Schrödinger equation with the correct giromagnetic moment, while the sector of the spinor defined on nodes follows the Klein–Gordon equation and is not negligible. The action associated to the proposed field theory comprises of a Dirac action and a metric action. We describe the gauge invariance of the action under both Abelian and non-Abelian transformations and we propose the equation of motion of the field theory of both Dirac and metric fields. This theory can be interpreted as a limiting case of a more general gauge theory valid on any arbitrary network in the limit of almost flat spaces.

Fuzzy style flat-based clustering

The recently developed fuzzy style k-plane clustering (S-KPC) algorithm displays promising clustering quality by leveraging both similarities and distinguishable styles between samples on stylistic data. However, S-KPC becomes vulnerable to similar styles that are not easily distinguishable. In this study, a novel fuzzy style flat-based clustering (FSFC) algorithm is proposed to overcome this vulnerability. In FSFC, a style flat matrix (SFM) is designed to project samples onto appropriate flats while maintaining the styles of different clusters in a reasonable manner. Based on SFM, the core of FSFC is to learn the potentially intersecting manifold structures of clusters in the projected flat space to make samples with the same style close to the cluster center and simultaneously far away from the other cluster centers. Furthermore, the objective function of FSFC can provide scale flexibility for each flat in the projected flat space. In particular, the optimization problem of FSFC can be decomposed into a series of sub-problems about the flat parameters, which can be locally optimized using the concave-convex procedure (CCCP). Extensive experiments on both synthetic and real-world datasets demonstrate the competitive clustering performance of FSFC. Moreover, FSFC outperforms some state-of-the-art manifold clustering algorithms on six case studies about stylistic data.

Classical Lie bialgebras for AdS/CFT integrability by contraction and reduction

Integrability of the one-dimensional Hubbard model and of the factorised scattering problem encountered on the worldsheet of AdS strings can be expressed in terms of a peculiar quantum algebra. In this article, we derive the classical limit of these algebraic integrable structures based on established results for the exceptional simple Lie superalgebra \mathfrak{d}(2,1;\epsilon)𝔡(2,1;ϵ) along with standard \mathfrak{sl}(2)𝔰𝔩(2) which form supersymmetric isometries on 3D AdS space. The two major steps in this construction consist in the contraction to a 3D Poincaré superalgebra and a certain reduction to a deformation of the \mathfrak{u}(2|2)𝔲(2|2) superalgebra. We apply these steps to the integrable structure and obtain the desired Lie bialgebras with suitable classical r-matrices of rational and trigonometric kind. We illustrate our findings in terms of representations for on-shell fields on AdS and flat space.

A 10d construction of Euclidean axion wormholes in flat and AdS space

Euclidean wormhole geometries sourced by axions and dilatons are puzzling objects in quantum gravity. From one side of the wormhole to the other, the scalar fields traverse a few Planck lengths in field space and so corrections from the UV might potentially affect the consistency of the solution, even when the wormholes are large. Motivated by this, we carry out the first explicit 10d lifts of regular Euclidean axion wormholes. We start off with the lift of Giddings-Strominger wormholes in mathcal{N} = 8 Euclidean supergravity over a 6-torus to 10d type IIA supergravity and find the solution can be everywhere tuned into the parametrically controlled supergravity regime. Secondly, we construct explicit wormholes in AdS spaces and find them again to be under parametric control. We find the first wormhole solutions in massive type IIA on S3 × S3 and in type IIB on T1,1. The latter has an explicit holographic dual, and similar to the earlier constructions in AdS5 × S5/ℤk, the wormholes violate operator positivity since Tr(F ± ⋆ F)2< 0. This puzzle might arise from subtleties related to computing holographic n-point functions in the presence of multiple boundaries.

Satisfaction Evaluation for Underpass Green Spaces in Mountainous Cities under the Perspective of Environmental Perception

The overpasses and the terrain under them in Chongqing, a mountainous city in China, are complex and diverse, and some spaces under the overpasses are integrated and reconstructed into the underpass green space for citizens to stroll about or have a rest. From the perspective of visitor perception, this paper constructs a perception evaluation system of the environmental characteristics of underpass green space in mountainous cities from the following five environmental perception dimensions: path organization, security, aesthetic value, physical environment, activities and cultural. The IPA-Kano model is used to quantify environmental perception, and the main environmental factors affecting the improvement of recreation satisfaction of underpass green space in three types of terrain are explored, with a view to improving the environment and service functions of underpass green spaces in high-density interchange networks in mountainous cities, and enhancing the attractiveness of underpass green spaces. It can be found from the study that: (1) Among the five environmental perception dimensions, visitors pay more attention to the physical environment quality of the underpass green space and their physical and psychological activity experience, while their demands for visual senses are relatively low. Due to the deficiency or lack of leisure facilities, sports facilities, children’s playgrounds and amusement equipment, the dimension of “activities and cultural perception” of the underpass green space has the lowest scores of all. (2) The existing sites, facilities and landscape resources of the underpass green space, different terrain types and underpass environment are the important reasons that affect the performance of environmental perception factors and their priority ranking results. (3) The improvement of security of the arrival path or sports facilities is beneficial to improve visitor satisfaction of underpass green space of three types of terrain. The number of environmental factors to be optimized of the three types of terrain are ranked as: mountainous green space &gt; flat green space &gt; concave green space. Among them, four environmental factors have a high priority in two kinds of underpass green space, which are the distribution and quantity of leisure facilities, the effect of noise reduction and sound insulation, the adequacy of activity venues and the distribution and quantity of sports facilities. Finally, according to the particularity of the underpass environment and the characteristics of three types of terrain, this paper puts forward some suggestions for optimizing the service function of underpass green space from five perceptual dimensions.

Holomorphic isometries into homogeneous bounded domains

We prove two rigidity theorems on holomorphic isometries into homogeneous bounded domains. The first shows that a Kähler-Ricci soliton induced by the homogeneous metric of a homogeneous bounded domain is trivial, i.e. Kähler-Einstein. In the second one we prove that a homogeneous bounded domain and the flat (definite or indefinite) complex Euclidean space are not relatives, i.e. they do not share a common Kähler submanifold (of positive dimension). Our theorems extend the results proved by us earlier [Proc. Amer. Math. Soc. 149 (2021), pp. 4931–4941] and by Xiaoliang Cheng and Yihong Hao [Ann. Global Anal. Geom. 60 (2021), pp. 167–180].

Quantum scattering of spinless particles in Riemannian manifolds

Quantum mechanics is sensitive to the geometry of the underlying space. Here we present a framework for quantum scattering of a nonrelativistic particle confined to a two-dimensional space. When the motion manifold hosts localized curvature modulations, scattering occurs from an emergent geometric potential and the metric tensor field. Analytical and full numerical simulations identify the geometric potential as the primary source for low-energy scattering, while the metric tensor field of the curved space governs high-energy diffraction. Compared to flat spaces, important differences in the validity range of perturbation approaches are found and demonstrated by full numerical simulations using combined finite element and boundary element methods. As an illustration, we consider a Gaussian-shaped dent leading to effects known as gravitational lensing. Experimentally, the considered setup is realizable based on geometrically engineered two-dimensional materials.

Casimir energy for spinor fields with δ-shell potentials

This work analyzes the Casimir energy of a massive spinor field propagating in flat space endowed with a spherically symmetric δ-function potential. By utilizing the spectral zeta function regularization method, the Casimir energy is evaluated after performing a suitable analytic continuation. Explicit numerical results are provided for specific cases in which the Casimir energy is unambiguously defined. The results described in this work represent a generalization of the MIT bag model for spinor fields.

Phases of theories with fermions in AdS

We study the phases of Yukawa theories at weak coupling and the Gross-Neveu models in AdS spaces at zero and finite temperature. Following the method used in [15], we first compute the one-loop partition functions, using the generalized eigenfunctions of the Dirac and Laplace operators on Euclidean AdS in the Poincaré coordinates. These functions satisfy desired periodicities under thermal identification. The method replicates results for partition functions known in the literature. We then study the phases of these field theories with fermions as regions in the corresponding parameter spaces at zero temperature. The phases and the corresponding phase boundaries are further identified as a function of the mass-squared of the scalar field and temperature for the Yukawa theories. While for the Gross-Neveu models, the changes in the phases as a function of the fermionic mass and the coupling constant at finite temperature are discussed. The Gross-Neveu-Yukawa model is studied for AdS4. We also note certain deviations from phases of these theories in flat space.

The Maslov index and some applications to dispersion relations in curved space times

The aim of the present work is to generalize the results given in Osorio Morales and Santillán [Eur. Phys. J. C 82, 353 (2022)] to a generic situation for causal geodesics. It is argued that these results may be of interest for causality issues. Recall that the presence of superluminal signals in a generic space time (M, gμν) does not necessarily imply violations of the principle of causality {[G. M. Shore, Nucl. Phys. B 778, 219 (2007)] and [T. J. Hollowood and G. M. Shore, Phys. Lett. B 655, 67 (2007)]}. In flat spaces, global Lorenz invariance leads to the conclusion that closed time-like curves appear if these signals are present. In a curved space instead, there is only local Poincare invariance, and the presence of closed causal curves may be avoided even in the presence of a superluminal mode, especially when terms violating the strong equivalence principle appear in the action. This implies that the standard analytic properties of the spectral components of these functions are therefore modified, and in particular, the refraction index n(ω) is not analytic in the upper complex ω plane. The emergence of these singularities may also take place for non-superluminal signals due to the breaking of global Lorenz invariance in a generic space time. In the present work, it is argued that the homotopy properties of the Maslov index are useful for studying how the singularities of n(ω) vary when moving along a geodesic congruence. In addition, several conclusions obtained in Shore [Nucl. Phys. B 778, 219 (2007)] and Hollowood and Shore [Phys. Lett. B 655, 67 (2007)] are based on the Penrose limit along a null geodesic, and they are restricted to GR with matter satisfying strong energy conditions. The use of the Maslov index may allow a more intrinsic description of singularities, not relying on that limit, and a generalization of these results about non-analyticity to generic gravity models with general matter content.

Advanced Lanchester Combat Model for Inhomogeneous Armies in RTS Games

This paper introduces an approach describing the battle dynamics between two inhomogeneous (consisting of units of different types) armies in real-time strategy games like Starcraft 2. This approach is based on representing the armies as distributions of densities of damage per second and attributing the strengths to the armies, which in homogeneous case match the well-known Lanchester strengths. The proposed approach can be used to optimize the army composition in various conditions: in an open flat space, in narrow passages or in ground versus air battles.

BMS field theories with u(1) symmetry

We investigate quantum field theories in two dimensions (2D) with an underlying Bondi--van der Burgh--Metzner-Sachs symmetry augmented by $\mathfrak{u}(1)$ currents. These field theories are expected to holographically capture features of charged versions of cosmological solutions in asymptotically flat 3D spacetimes called flat space cosmologies. We conduct a study of the modular properties of these field theories. The characters for the highest weight representations of the symmetry algebra are constructed, and the partition function of the theory is obtained from them. We derive the density of (primary) states and find the entropy and asymptotic values of the structure constants exploiting the modular properties of the partition function and the torus one-point function. The expression for the asymptotic structure constants shows shifts in the weights and one of the central terms and an extra phase compared to the earlier results in the literature for Bondi--van der Burgh--Metzner-Sachs invariant theories without $\mathfrak{u}(1)$ currents present. We reproduce our field results for the structure constants by a bulk computation involving a scalar probe in the background of a charged flat space cosmology.

Four-point functions of gravitons and conserved currents of CFT in momentum space: testing the nonlocal action with the TTJJ

We present an analysis of the perturbative realization of the TTJJ correlator, with two stress energy tensors and two conserved currents, using free field theory, integrating out conformal sectors in the quantum corrections. This allows to define, around flat space, an exact perturbative expansion of the complete anomaly effective action – up to 4-point functions – whose predictions can be compared against those of the anomaly induced action. The latter is a variational solution of the conformal anomaly constraint at d=4 in the form of a nonlocal Wess–Zumino action. The renormalization procedure and the degeneracies of the tensor structures of this correlator are discussed, valid for a generic conformal field theory, deriving its anomalous trace Ward identities (WIs). In this application, we also illustrate a general procedure that identifies the minimal number of tensorial structures and corresponding form factors for the TTJJ and any 4-point function. The approach is implemented for three, four and five dimensions, addressing the tensor degeneracies of the expansion in momentum space. We show that the renormalized TTJJ can be split into two contributions, a non anomalous and an anomalous part, each separately conserved. The first satisfies ordinary trace WIs, while the second satisfies anomalous trace WIs. The result of the direct computation is compared against the expression of the same 4-point function derived from the nonlocal anomaly induced action. We show that the prediction for the anomalous part of the TTJJ derived from such action, evaluated in two different conformal decompositions, the Riegert and Fradkin–Vilkovisky (FV) choices, differ from the anomaly part identified in the perturbative TTJJ, in the flat spacetime limit. The anomaly part of the correlator computed with the Riegert choice is affected by double poles, while the one computed with the FV choice does not satisfy the conservation WIs. We present the correct form of the expansion of the anomaly induced action at the second order in the metric perturbations around flat space that reproduces the perturbative result.

Chiral current induced by torsional Weyl anomaly

Torsion can be realized as dislocation in the crystal lattice of material. It is particularly interesting if the material has fermions in the spectrum, such as graphene, topological insulators, and Dirac and Weyl semimetals, as it's transport properties can be affected by the torsion. In this paper, we find that, due to Weyl anomaly, torsion in Dirac and Weyl semimetals can induce novel chiral currents, either near a boundary or in a ``conformally flat space.'' We briefly discuss how to measure this interesting effect in experiment. It is remarkable that these experiments can help to clarify the theoretical controversy of whether an imaginary Pontryagin density could appear in the Weyl anomaly.

Gravitational wave memory in conformally flat spacetimes

We study the gravitational wave memory effect in spacetimes related to flat space by a conformal transformation. The discussion is general but the gravitational wave length scale is assumed to be small compared with the background curvature radius. The general formulas are applied to Friedmann-Robertson-Walker metrics of all spatial curvatures. The effect of new terms stemming from spatial curvature is potentially detectable in future gravitational wave measurements.

PSO-Based Sparse Source Location in Large-Scale Environments With a UAV Swarm

Locating multiple sources in an unknown environment based on their signal strength is called a multi-source location problem. In recent years, there has been great interest in deploying autonomous devices to solve it. A particle swarm optimizer (PSO) is a widely employed source location method. Yet most work in this field focuses on a flat search space while ignoring height information. An unmanned aerial vehicle (UAV) has a coarser but wider view as it flies higher. Inspired by such facts, this paper focuses on improving the efficiency of locating sources by utilizing height information through UAVs. A novel source location model is designed where their sensing range gradually increases as their flying height rises, but their obtained signal strength fades away. It can be directly deployed to existing PSO-based multi-source location methods and improve their performance, especially in a large-scale environment with sparse sources. UAVs can spontaneously switch their search schemes between a rough search at a higher height and a fine one at a lower height. Experimental results of three PSO-based methods show their significant improvement after deploying our model. Given the same computation resources, its deployment leads to over 30% hike in both location accuracy and speed. This represents a great advance to the field of source location.

Particles of Negative and Zero Energy in Black Holes and Cosmological Models

Particles with negative energies are considered for three different cases: inside the horizon of a Schwarzschild black hole, Milne’s coordinates in flat Minkowski space–time (Milne’s universe using nonsynchronous coordinates) and in the cosmological Gödel model of the rotating universe. It is shown that, differently from the Gödel model with a nondiagonal term, where it occurs that negative energies are impossible, they are present in all other cases considered in the paper. Particles with zero energy are also possible in the first two cases.

A spherically symmetric gravitational solution of nearly conformally flat metric measure space

In this manuscript, we study the nearly flat approximation of a conformally invariant gravitational theory in metric measure space (MMS). In addition, we investigate the transformation of the energy–momentum tensor in this context and obtain the vacuum solution of MMS and its weak field limit in the spherically symmetric coordinates. We show that while it is already a vacuum solution, it can simulate dark matter when restricted to the framework of general relativity, i.e., a symmetry-broken conformal frame. This is done by employing a density function which is an essential part of MMS. We derive an equation for the density function for a general profile of a rotation curve obtained from observations. Specifically, the density function corresponding to two well-known profiles PSS and NFW are provided.

Topological effects on non-relativistic eigenvalue solutions under AB-flux field with pseudoharmonic- and Mie-type potentials

In this paper, we investigate the quantum dynamics of a non-relativistic particle confined by the Aharonov–Bohm quantum flux field with pseudoharmonic-type potential in the background of topological defect produced by a point-like global monopole. We solve the radial Schrödinger equation analytically and determine the exact eigenvalue solution of the quantum system. Afterwards, we consider a Mie-type potential in the quantum system and solve the radial equation analytically and obtain the eigenvalue solution. We analyze the effects of the topological defect and the quantum flux with these potentials on the energy eigenvalue and wave function of the non-relativistic particles. In fact, it is shown that the energy levels and wave functions are influenced by the topological defect shifted the result compared to the flat space results. In addition, the quantum flux field also shifted the eigenvalue solutions and an analogue of the Aharonov–Bohm effect for bound-states is observed. Finally, we utilize these eigenvalue solutions to some known diatomic molecular potential models and presented the energy eigenvalue and wave function.

CERTAIN COSMOLOGICAL MODELS WITH VARIATION OF HUBBLE PARAMETER

The present paper deals with the FRW-Cosmological Model of universe for W2 flat perfect fluid space time. Einstein field equations with variable cosmological constant (Λ) has been obtained for such spacetime and in order to get the complete cosmological solution the law of variation for Hubble’s parameter is considered. A new class of solution have been discussed for the Einstein field equations with variable cosmological constant in which the pressure, energy density, and cosmological constant Λ are found to be decreasing function of cosmic time. The physical and kinematical properties of models are also discussed.