Flat Space Research Articles

Supersymmetric black hole hair and AdS3× S3

The 4D-5D connection allows us to view the same near horizon geometry as part of a 4D black hole or a 5D black hole. A much studied example of this phenomenon is the BMPV black hole uplifted to 6D with flat base space versus Taub-NUT base space. These black holes have identical near horizon AdS3 × S3 geometry. In this paper, we study modes in AdS3 × S3 and identify those that correspond to supersymmetric hair modes in the full black hole spacetimes. We show that these modes satisfy non-normalisable boundary conditions in AdS3. The non-normalisable boundary conditions are different for different hair modes and for different asymptotic completion. We also discuss how the supersymmetric hair modes on BMPV black holes fit into the classification of supersymmetric solutions of 6D supergravity.

Piecewise Time Polynomials-Based Control Methods for Obstacle Avoidance and Precision Positioning of Tower Crane Systems with Varying Cable Lengths

During the hoisting and lowering operations of a tower crane, dynamic variations in cable lengths significantly influence the oscillation frequency and amplitude of the load. These variations complicate the oscillation characteristics, heightening the challenge of balancing obstacle avoidance with precise positioning. To tackle this issue, we propose a trajectory planning and tracking control method that integrates hoisting control to reduce the impact of varying cable lengths on load swinging and achieve accurate positioning during obstacle navigation. A novel definition of swing angle is introduced to model the crane’s rigid and swinging components separately, enhancing model accuracy while simplifying complexity. A piecewise polynomial constructs the load trajectory in a low-dimensional flat space, which is then mapped to a high-dimensional generalized state space through a homeomorphic transformation, ensuring trajectory smoothness and traceability. A fractional-order sliding mode controller is employed to facilitate rapid and precise tracking of the actuated degrees of freedom, suppressing load oscillation while maintaining positioning accuracy. Experimental validation on a tower crane platform shows that the proposed strategy enables smooth obstacle avoidance and precise target point reaching, even with varying cable lengths.

Towards bootstrapping F-theory

We consider type IIB string theory with N D3 branes and various configurations of sevenbranes, such that the string coupling gs is fixed to a constant finite value. These are the simplest realizations of F-theory, and are holographically dual to rank N Argyres-Douglas conformal field theories (CFTs) with SU(2) and SU(3) flavor groups, and Minahan-Nemeschansky CFTs with E6, E7 and E8 flavor groups. We use the Seiberg-Witten curves of these theories to compute the mass deformed sphere free energy F (m) at large N in terms of novel matrix models with non-polynomial potentials. We show how F (m) can be used along with the analytic bootstrap to fix the large N expansion of flavor multiplet correlators in these CFTs, which are dual to scattering of gluons on AdS5 × S3, and in the flat space limit determine the effective theory of sevenbranes in F-theory. As a first step in this program, we use the matrix models to compute the log N term in F (m) and thereby fix the logarithmic threshold in the AdS5 × S3 holographic correlator, which matches the flat space prediction.

Schwarzschild geodesics from scattering amplitudes to all orders in GN

The dynamics of the leading self-force (0SF) order, corresponding to the geodesic motion of a light body in the exact background of a heavy body, are explicitly derived for the first time using a flat space scattering amplitude-based approach. This is achieved by utilising the cubic formulation of Einstein’s general relativity coupled to the effective worldline action of massive point particles, which was employed to derive the Schwarzschild black hole metric in [1].

Soft limits of gluon and graviton correlators in Anti-de Sitter space

We derive formulae for the soft limit of tree-level gluon and graviton correlators in Anti-de Sitter space, which arise from Feynman diagrams encoding the Weinberg soft theorems in flat space. Other types of diagrams can also contribute to the soft limit at leading order in the soft momentum, but have a different pole structure. We derive these results at four points using explicit formulae recently obtained from the cosmological bootstrap and double copy, and extend them to any multiplicity using bootstrap techniques in Mellin-momentum space.

The common solution space of general relativity

We review the solution space for the field equations of Einstein's General Relativity for various static, spherically symmetric spacetimes. We consider the vacuum case, represented by the Schwarzschild black hole; the de Sitter-Schwarzschild geometry, which includes a cosmological constant; the Reissner-Nordström geometry, which accounts for the presence of charge. Additionally we consider the homogenenous and anisotropic locally rotational Bianchi II spacetime in the vacuum. Our analysis reveals that the field equations for these scenarios share a common three-dimensional group of point transformations, with the generators being the elements of the D⊗sT2 Lie algebra, known as the semidirect product of dilations and translations in the plane. Due to this algebraic property the field equations for the aforementioned gravitational models can be expressed in the equivalent form of the null geodesic equations for conformally flat geometries. Consequently, the solution space for the field equations is common, and it is the solution space for the free particle in a flat space. This approach open new directions on the construction of analytic solutions in gravitational physics and cosmology.

Eguchi–Hanson metrics arising from Kähler–Einstein edge metrics

Calabi–Hirzebruch manifolds are higher-dimensional generalizations of both the football and Hirzebruch surfaces. We construct a family of Kähler-Einstein edge metrics singular along two disjoint divisors on the Calabi–Hirzebruch manifolds and study their Gromov–Hausdorff limits when either cone angle tends to its extreme value. As a very special case, we show that the celebrated Eguchi–Hanson metric arises in this way naturally as a Gromov–Hausdorff limit. We also completely describe all other (possibly rescaled) Gromov–Hausdorff limits which exhibit a wide range of behaviors, resolving in this setting a conjecture of Cheltsov–Rubinstein. This gives a new interpretation of both the Eguchi–Hanson space and Calabi’s Ricci flat spaces as limits of compact singular Einstein spaces.

Integrability in gravity from Chern-Simons theory

This paper presents a new perspective on integrability in theories of gravity. We show how the stationary, axisymmetric sector of General Relativity can be described by the boundary dynamics of a four-dimensional Chern-Simons theory. This approach shows promise for simplifying solution generating methods in both General Relativity and higher-dimensional supergravity theories. The four-dimensional Chern-Simons theory presented generalises those for flat space integrable models by introducing a space-time dependent branch cut in the spectral plane. We also make contact with twistor space approaches to integrability, showing how the branch cut defects of four-dimensional Chern-Simons theory arise from a discrete reduction of six-dimensional Chern-Simons theory.

Quantizing Carrollian field theories

Carrollian field theories have recently emerged as a candidate dual to flat space quantum gravity. We carefully quantize simple two-derivative Carrollian theories, revealing a strong sensitivity to the ultraviolet. They can be regulated upon being placed on a spatial lattice and working at finite inverse temperature. Unlike in conventional field theories, the details of the lattice-regulated Carrollian theories remain important at long distances even in the limit that the lattice spacing is sent to zero. We use that limit to define interacting continuum models with a tractable perturbative expansion. The ensuing theories are those of generalized free fields, with non-Gaussian correlations suppressed by positive powers of the lattice spacing, and an unbroken supertranslation symmetry.

MADE: Multicurvature Adaptive Embedding for Temporal Knowledge Graph Completion.

Temporal knowledge graphs (TKGs) are receiving increased attention due to their time-dependent properties and the evolving nature of knowledge over time. TKGs typically contain complex geometric structures, such as hierarchical, ring, and chain structures, which can often be mixed together. However, embedding TKGs into Euclidean space, as is typically done with TKG completion (TKGC) models, presents a challenge when dealing with high-dimensional nonlinear data and complex geometric structures. To address this issue, we propose a novel TKGC model called multicurvature adaptive embedding (MADE). MADE models TKGs in multicurvature spaces, including flat Euclidean space (zero curvature), hyperbolic space (negative curvature), and hyperspherical space (positive curvature), to handle multiple geometric structures. We assign different weights to different curvature spaces in a data-driven manner to strengthen the ideal curvature spaces for modeling and weaken the inappropriate ones. Additionally, we introduce the quadruplet distributor (QD) to assist the information interaction in each geometric space. Ultimately, we develop an innovative temporal regularization to enhance the smoothness of timestamp embeddings by strengthening the correlation of neighboring timestamps. Experimental results show that MADE outperforms the existing state-of-the-art TKGC models.

Electromagnetic field in an expanding universe

An expanding universe filled with a massive electromagnetic field is studied. The electromagnetic field (photon) is found to acquire mass during an inflationary period. The photon mass varies with the Hubble parameter as m γ = 3ℏ H/c 2, where H, ℏ and c are the Hubble parameter, Planck constant, and speed of light. A leftover magnetic field since the inflationary era permeates the space between galaxies and interstellar media. The strength of the electromagnetic field changes with the scale factor, E ∝ a −3. The photon gains mass when interacting with the gravitational field while remaining massless in flat space. The massive electromagnetic field adds energy density to the Universe proportional to m 2.

Approaches for predicting the soil-water characteristic curves of compacted quartz sand based on particle packing theory

The soil-water characteristic curve (SWCC) is an important basis for describing hydraulic properties, which is closely related to environmental problems such as the migration of toxic substances from groundwater and soil. The SWCC models vary considerably with respect to the void ratio or bulk density. However, the void ratio and bulk density of the same soil change at different depths or positions within a flat space. Accurate simulation or calculation requires a precise SWCC, but measuring every SWCC under each density or void ratio condition is difficult. In this study, two particle packing models, the Kwan model and the modified linear packing density model (MLPDM), were introduced into the van Genuchten (VG) model to predict the SWCC for soil at the minimum void ratio (the maximum dry density). Then, the prediction method of the SWCCs for the same soil with different densities or void ratios was given. A series of laboratory tests using quartz sand mixtures were conducted to verify the accuracy of two extended models (the K-VG model and the M-VG model). For the K-VG model, the average RMSE was 0.020. For the M-VG model, the average RMSE was 0.017. In general, the SWCCs predicted by the two extended models were consistent with the measured values.

Real-time anomaly detection of the stochastically excited systems on spherical ([formula omitted]) manifold

Advanced analytical tools have become crucial in today’s constantly changing and complex systems. Real-time Principal Geodesic Analysis (RPGA) is a novel technique that provides a unique perspective for analyzing nonlinear data on differentiable manifolds. Traditional linear methods are often inadequate when exploring the complexities of such data. Orthogonal transformation techniques such as Principal Component Analysis (PCA) and Principal Geodesic Analysis (PGA) are highly desirable for condition monitoring stochastically excited systems in domains like mechanical, aerospace, and civil engineering. However, uncertainties and dynamic fluctuations necessitate robust analytical methods for early change detection to ensure safety, performance, and cost-effectiveness. Limitations posed by linear orthogonal transformation techniques such as PCA and its recursive counterparts make it imperative to adapt these techniques to nonlinear situations where data does not evolve in a flat Euclidean space. Significant advancements have been made in this field over recent decades, with data-driven real-time algorithms such as RPCA, RCCA, and RSSA providing reliable solutions for complex multidimensional problems. However, for curved space, the nonlinear RPGA technique takes center stage. It is known for its effectiveness in extracting meaningful information from the complex data stream. This paper explores the foundational concepts and methodologies underlying the transition from linear to nonlinear data analysis. By examining examples such as stochastic geometric oscillator on S2, and the inverted spherical pendulum cart system navigating a rough surface, we illustrate the significance of reliable, real-time damage detection techniques provided by tools such as RPGA.

Hagedorn temperature in holography: world-sheet and effective approaches

We provide general results on the Hagedorn temperature of planar, strongly coupled confining gauge theories holographically dual to type II superstring models on curved backgrounds with Ramond-Ramond and Kalb-Ramond fluxes and non-trivial dilaton. For exact backgrounds the Hagedorn temperature is determined up to next-to-next-to-next-to-leading order (NNNLO) in an expansion in α′; in all the other cases the results can be safely trusted up to NNLO. To reach these goals we exploit two complementary approaches. On the one hand, we perform an extrapolation to the Hagedorn regime of world-sheet results obtained from the semiclassical quantization of string configurations winding around the compact Euclidean time direction. En passant, we provide a detailed derivation of the fermionic part of the world-sheet spectrum, which is hard to find in the literature. On the other hand, we perturbatively solve the equations of motion for the thermal scalar field corresponding to the lightest mode of the winding string, which in flat space becomes tachyonic above the Hagedorn temperature. The interplay between different approaches is surely convenient, but we provide insights about a possible derivation of the whole NNLO correction to the Hagedorn temperature from a pure world-sheet perspective; furthermore, we determine the effective mass of the thermal scalar from the world-sheet in full generality.

On a Simple Characterization of Conformally Flat 4-Dimensional Spaces of Neutral Signature

In this work, the full characterization of 4-dimensional conformally flat spaces of neutral signature is given by using methods based on holonomy structure. Possible holonomy types are obtained for the spaces in question and several remarks are made. Various examples are presented related to this investigation.

Optimal Trajectory Planning for Wheeled Robots (OTPWR): A Globally and Dynamically Optimal Trajectory Planning Method for Wheeled Mobile Robots

In recent years, with the widespread application of indoor inspection robots, efficient motion planning has become crucial. Addressing the issue of discontinuous and suboptimal robot trajectories resulting from the independent nature of global and local planning, we propose a novel optimal path-planning method for wheeled mobile robots. This method leverages differential flatness to reduce dimensionality and decouple the problem, achieving globally optimal, collision-free paths in a two-dimensional flat output space through diagonal search and polynomial trajectory optimization. Comparative experiments in a simulated environment demonstrate that the proposed improved path search algorithm reduces search time by 46.6% and decreases the number of visited nodes by 43.1% compared to the original algorithm. This method not only ensures the optimal path and efficient planning but also ensures that the robot’s motion trajectory satisfies the dynamic constraints, verifying the effectiveness of the proposed optimal path planning algorithm for wheeled mobile robots.

Nonequivalence of Carroll limits in relativistic string theory

We construct new Carroll strings in flat space by considering the Carroll limit of equivalent relativistic string theories at the classical level. In the limit these Carroll strings are no longer equivalent and have different degrees of freedom. This fact is due to a general phenomenon that the configuration space and canonical Lagrangians are no longer equivalent after a singular redefinition of the variables and of the parameters of the starting Lagrangians. Published by the American Physical Society 2024

Electromagnetic wave propagation in Eddington-inspired Born–Infeld gravity space-time with topological defects

In this study, we focus on examining the characteristics of electromagnetic fields within a curved space-time background under the framework of Eddington-inspired Born–Infeld (EiBI) gravity, in the presence of a global monopole. We derived Maxwell’s vacuum field equations in this curved spacetime and obtained a set of linear differential equations for the electric and magnetic fields. After decoupling these equations, we solved for the analytical solutions of both the electric and magnetic fields using special functions. We then extended our analysis to the same EiBI-gravity framework, this time incorporating a cosmic string. Following a similar approach, we derived the first-order differential equations governing the electric and magnetic fields and obtained their analytical solutions using special functions. Our findings demonstrate significant influences of the global monopole, cosmic string, and the Eddington parameters on the behavior of electromagnetic waves in this curved space-time configuration with topological defects, resulting in notable deviations from the Minkowski flat space case.

A foray on SCFT3 via super spinor-helicity and Grassmann twistor variables

In this paper, we develop a momentum super space formalism for N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1, 2 superconformal field theories in three dimensions. First, we solve for super-correlators in the usual momentum superspace variables. However, we found that expressing quantities in super space spinor helicity variables greatly simplifies the analysis. Further, by performing a “half” Fourier transform of the Grassmann coordinates which is analogous to the Twistor transform, an even more remarkable simplification occurs. Using this formalism, we first compute all three point correlation functions involving conserved super-currents with arbitrary spins in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1, 2 theories. We discover interesting double copy relations in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 super-correlators. Further, we discovered super double copy relations that take us from N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 to N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 super-correlators. We also extend our results to correlators involving scalars as well as to higher points. Further, we comment on the connection of our results with the flat space super amplitudes in one higher dimension.

A Study of Time Evolution of Some Cosmological Parameters in The Framework of an Anisotropic Kaluza-Klein Metric Using an Empirical Exponential Scale Factor

The present study attempts to determine the time dependence of some cosmological parameters in flat space (i.e., a space of zero spatial curvature), in the framework of an anisotropic Kaluza-Klein metric. The field equations for this work have been derived from the metric by assuming a power-law relation between the normal scale factor and the scale factor corresponding to the extra (i.e., the fifth) dimension. An empirical scale factor, having the expression of a = B exp(αtβ), has been used here in order to derive the expressions for some cosmological parameters as functions of time. The reason for choosing this scale factor is that it generates an expression for the deceleration parameter which undergoes a change of sign, as time goes on, from positive to negative, indicating a transition of the universe from an initial state of decelerated expansion to that of an accelerated expansion (which is its present state), as has been inferred from astrophysical observations. We have graphically depicted the evolution of some cosmological parameters with respect to what one may call the relative time, expressed as t/t0, where t0 is the present age of the universe. The present study finds the dynamical cosmological constant (Λ) to be negative, and it becomes less negative with time, changing at a gradually decreasing rate. The dependence of pressure of the all-pervading cosmic fluid upon density, corresponding to the fifth dimension, has been described in terms of a skewness parameter (δ) which comes out to be decreasing with time. The anisotropy factor has been calculated in this study, whose numerical value has been found to be decreasing with time, indicating a journey of the universe towards phases of gradually smaller anisotropy.