Singular Spaces Research Articles

Janus Bound States in the Continuum with Asymmetric Topological Charges.

We propose a novel topological defect called Janus bound states in the continuum (BICs), featuring asymmetric topological charges in upward and downward radiation channels. Our approach involves a photonic crystal slab (PCS) that initially exhibits both out-of-plane and in-plane mirror symmetry, and this PCS possesses one BIC at the Γ point and two BICs off the Γ point. By introducing certain perturbations that break the out-of-plane mirror symmetry, the two off-Γ BICs decompose into four circularly polarized states (C points) with identical topological charges (each with half the topological charge of the original BIC) while the at-Γ BIC is preserved. Then, we selectively manipulate the four C points associated with the downward radiation channel to converge at the at-Γ BIC, forming a Janus BIC with distinct topological charges for upward and downward radiation. By further introducing in-plane mirror symmetry perturbation, we can bring two of the C points with the same handedness and identical topological charges for upward radiation to merge into the Janus BIC. This process results in a Janus chiral BIC which exhibits large intrinsic chirality and an infinite Q factor. Janus BICs can induce distinct Pancharatnam-Berry phase singularities in momentum space for different incident channels, providing a new approach to control orbital angular momentum. Janus chiral BICs hold promise in enhancing direction-dependent and spin-dependent asymmetric light-matter interaction, opening new pathways for improving chirality-dependent operation for on-chip devices.

Quantum probabilistic space analysis for enhanced two-wheeler traffic safety: from classical limitations to advanced quantum circuits

In the increasing traffic scenario and the presence of automated four-wheeler vehicle-like situations, we need to consider the possible safe driving for the two-wheelers also. As two-wheelers need manual balances, embedding the autopilot systems is a tougher task. To resolve this issue, we attempt to address this problem with probability spaces. Then, these classical probability spaces are promoted into quantum probability spaces. The quantum computing aspects such as quantum gates are embedded in the analysis of the quantum probability space. With those quantum computing protocols, various traffic scenarios are calculated, such as the probability of accidents and the avoidance probability of accidents. A set of mathematical lemmas with certain conditions and their proof for mapping classical to quantum probability space and the singularities in this quantum probability space are also discussed in this work.

On proper intersections on a singular analytic space

Given a reduced analytic space Y we introduce a class of nice cycles, including all effective Q-Cartier divisors. Equidimensional nice cycles that intersect properly allow for a natural intersection product. Using ∂¯-potentials and residue calculus we provide an intrinsic way of defining this product. The intrinsic definition makes it possible to prove global formulas. In case Y is smooth all cycles are differences of nice cycles, and so we get a new way to define classical proper intersections.

Historicising Mauritian self-determination at the International Court of Justice

Between 2018 and 2019, the International Court of Justice considered whether the formation and current existence of British Indian Ocean Territory violated international law. This article reveals how three temporalities – decolonial nation-state, colonial and indigenous – shaped, and were shaped by, differing conceptions of Mauritian self-determination within the Case. In doing so, this study provides an account of previously unrecognised notions of self-determination that were formative of the recent Chagos Archipelago dispute. I argue that the British delegation recasts the meaning of Mauritian self-determination as an expression of intra-colonial rights, past and present. Furthermore, I contend that Chagossian responses to the Case advanced a notion of indigeneity as having multiple moments and sites of emergence – with corresponding rights claims that extricate self-determination from a singular period and space of injustice. These colonial and indigenous conceptions of self-determination are shown to challenge, yet ultimately become subordinated to, the familiar nation-state debates on complete/incomplete Mauritian decolonisation. A focus upon representations of time does more than uncover aspects of the recent Chagos Archipelago dispute, which were structured by novel attempts to conceptualise self-determination. This article holds wider significance for post-colonial IR. The competing registers of shaped time strove to reinforce or revise what counts as post-1960 norms of self-determination.

Magnetically tunable bound states in the continuum with arbitrary polarization and intrinsic chirality

Bound states in the continuum (BICs), which are exotic localized eigenstates embedded in the continuum spectrum and exhibit topological polarization singularities in momentum space, have recently attracted great attention in both fundamental and applied physics. Here, based on a magneto-optical (MO) photonic crystal (PhC) slab placed in external magnetic fields with time-reversal symmetry (TRS) breaking, we theoretically propose magnetically tunable BICs with arbitrary polarization covering the entire Poincaré sphere and efficient off-Γ chiral emission of circularly polarized states (C point). More interestingly, by further breaking the in-plane inversion symmetry of the MO PhC slab to generate a pair of C points spawning from the eliminated BICs and tuning the external magnetic field strength to move one C point to the Γ point, an at-Γ intrinsic chiral BIC exhibits chiral characteristics on both sides of the PhC slab with near-unity circular dichroism exceeding 0.99 and a high-quality factor of 46,000 owing to the preserved out-of-plane mirror symmetry. Moreover, the chirality of the chiral BICs can be inverted by flipping the magnetic bias. Our work opens an unprecedented avenue to explore the unique topological photonics of BICs with broken TRS and promises multiple applications in chiral-optical effects, structured light, and tunable optical devices.

Behavior of a Pendulum with a Singular Configuration Space

Behavior of a Pendulum with a Singular Configuration Space

Transfer and the spectrum-level Siegel–Sullivan KO-orientation for singular spaces

Integrally oriented normally nonsingular maps between singular spaces have associated transfer homomorphisms on KO-homology at odd primes. We prove that such transfers preserve Siegel–Sullivan orientations, defined when the singular spaces are compact pseudomanifolds satisfying the Witt condition, for example pure-dimensional compact complex algebraic varieties. This holds for bundle transfers associated to block bundles with manifold fibers as well as for Gysin restrictions associated to normally nonsingular inclusions. Our method is based on constructing a lift of the Siegel–Sullivan orientation to a morphism of highly structured ring spectra which factors through L-theory.

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.

Dynamic gain and frequency comb formation in exceptional-point lasers

Exceptional points (EPs)—singularities in the parameter space of non-Hermitian systems where two nearby eigenmodes coalesce—feature unique properties with applications such as sensitivity enhancement and chiral emission. Existing realizations of EP lasers operate with static populations in the gain medium. By analyzing the full-wave Maxwell–Bloch equations, here we show that in a laser operating sufficiently close to an EP, the nonlinear gain will spontaneously induce a multi-spectral multi-modal instability above a pump threshold, which initiates an oscillating population inversion and generates a frequency comb. The efficiency of comb generation is enhanced by both the spectral degeneracy and the spatial coalescence of modes near an EP. Such an “EP comb” has a widely tunable repetition rate, self-starts without external modulators or a continuous-wave pump, and can be realized with an ultra-compact footprint. We develop an exact solution of the Maxwell–Bloch equations with an oscillating inversion, describing all spatiotemporal properties of the EP comb as a limit cycle. We numerically illustrate this phenomenon in a 5-μm-long gain-loss coupled AlGaAs cavity and adjust the EP comb repetition rate from 20 to 27 GHz. This work provides a rigorous spatiotemporal description of the rich laser behaviors that arise from the interplay between the non-Hermiticity, nonlinearity, and dynamics of a gain medium.

The relationship between singular value decomposition orthogonal space vector and sound field reconstruction in equivalent source method

In acoustic holography, the equivalent source method represents a typical ill-posed inverse problem, characterized by the significant impact of measurement errors on the sought solution. To minimize the interference of noise with the sought solution, we analyze the relationship between orthogonal space vectors obtained through singular value decomposition and acoustic field reconstruction. The research findings indicate that the primary information of the acoustic field exhibits strong correlation with a small number of space vectors, while noise information demonstrates random correlation with all space vectors. Therefore, we retain the space vectors with higher correlation to the acoustic field while filtering out redundant ones, thereby stabilizing and accurately reconstructing the acoustic field. Numerical analysis of the acoustic field reconstruction demonstrates that compared to regularization methods such as Tikhonov and truncated singular value decomposition (TSVD), this approach achieves higher computational accuracy. Furthermore, the text combines practical examples to further explain the reasons behind the superior accuracy of this method.

End of the world brane networks for infinite distance limits in CY moduli space

Dynamical Cobordism provides a powerful method to probe infinite distance limits in moduli/field spaces parameterized by scalars constrained by generic potentials, employing configurations of codimension-1 end of the world (ETW) branes. These branes, characterized in terms of critical exponents, mark codimension-1 boundaries in the spacetime in correspondence of finite spacetime distance singularities at which the scalars diverge. Using these tools, we explore the network of infinite distance singularities in the complex structure moduli space of Calabi-Yau fourfolds compactifications in M-theory with a four-form flux turned on, which is described in terms of normal intersecting divisors classified by asymptotic Hodge theory. We provide spacetime realizations for these loci in terms of networks of intersecting codimension-1 ETW branes classified by specific critical exponents which encapsulate the relevant information of the asymptotic Hodge structure characterizing the corresponding divisors.

Some Aspects of Differential Topology of Subcartesian Spaces

In this paper, we investigate the differential topological properties of a large class of singular spaces: subcarteisan space. First, a minor further result on the partition of unity for differential spaces is derived. Second, the tubular neighborhood theorem for subcartesian spaces with constant structural dimensions is established. Third, the concept of Morse functions on smooth manifolds is generalized to differential spaces. For subcartesian space with constant structural dimension, a class of examples of Morse functions is provided. With the assumption that the subcartesian space can be embedded as a bounded subset of an Euclidean space, it is proved that any smooth bounded function on this space can be approximated by Morse functions. The infinitesimal stability of Morse functions on subcartesian spaces is studied. Classical results on Morse functions on smooth manifolds can be treated directly as corollaries of our results here.

Obstacle avoidance planning for industrial robots based on singular manifold splitting configuration space

SummaryObstacle avoidance planning is the primary element in ensuring safe robot applications such as welding, assembly, and drilling. The states in the configuration space (C‐space) provide the pose information of any part of the manipulator and are preferentially considered in motion planning. However, it is difficult to express the environmental information directly in the high dimensional C‐space, limiting the application of C‐space obstacle avoidance planning. This paper proposes a singular manifold splitting C‐space method and designs a compatible obstacle avoidance strategy. The specific method is as follows: first, according to the specific structure of industrial robots, arm‐wrist separation obstacle avoidance planning is proposed to fix the robot as a 3R manipulator to reduce the dimension of C‐space. Next, the C‐space is segmented according to the singular manifolds, and the unique domain is delineated to complete the streamlining of the volume of the C‐space. Then, with the help of the point cloud, the obstacles are enveloped and mapped to the unique domain to construct the pseudo‐obstacle map. Industrial robots' obstacle avoidance planning is completed based on the pseudo‐obstacle map combined with an improved Rapidly‐Exploring Random Trees (RRT) algorithm. This method dramatically improves the efficiency of obstacle avoidance planning in the C‐space and avoids the effect of singularities on industrial robots. Finally, the effectiveness of the method is verified by physical experiments.

Lipschitz regularity of sub-elliptic harmonic maps into CAT(0) space

Abstract We prove the local Lipschitz continuity of sub-elliptic harmonic maps between certain singular spaces, more specifically from the 𝑛-dimensional Heisenberg group into CAT ⁡ ( 0 ) \operatorname{CAT}(0) spaces. Our main theorem establishes that these maps have the desired Lipschitz regularity, extending the Hölder regularity in this setting proven in [Y. Gui, J. Jost and X. Li-Jost, Subelliptic harmonic maps with values in metric spaces of nonpositive curvature, Commun. Math. Res. 38 (2022), 4, 516–534] and obtaining same regularity as in [H.-C. Zhang and X.-P. Zhu, Lipschitz continuity of harmonic maps between Alexandrov spaces, Invent. Math. 211 (2018), 3, 863–934] for certain sub-Riemannian geometries; see also [N. Gigli, On the regularity of harmonic maps from RCD ⁢ ( K , N ) \mathrm{RCD}(K,N) to CAT ⁢ ( 0 ) \mathrm{CAT}(0) spaces and related results, preprint (2022), https://arxiv.org/abs/2204.04317; and A. Mondino and D. Semola, Lipschitz continuity and Bochner–Eells–Sampson inequality for harmonic maps from RCD ⁡ ( k , n ) \operatorname{RCD}(k,n) spaces to CAT ⁡ ( 0 ) \operatorname{CAT}(0) spaces, preprint (2022), https://arxiv.org/abs/2202.01590] for the generalisation to RCD spaces. The present result paves the way for a general regularity theory of sub-elliptic harmonic maps, providing a versatile approach applicable beyond the Heisenberg group.

Polarization-Controlled Exciton-Polaritons in WS2 Strongly Coupled with Low-Symmetry Photonic Crystal Nanostructures.

Atomically thin transition metal dichalcogenides (TMDs) with ambient stable exciton resonances have emerged as an ideal material platform for exciton-polaritons. In particular, the strong coupling between excitons in TMDs and optical resonances in anisotropic photonic nanostructures can form exciton-polaritons with polarization selectivity, which offers a new degree of freedom for the manipulation of the light-matter interaction. In this work, we present the experimental demonstration of polarization-controlled exciton-polaritons in tungsten disulfide (WS2) strongly coupled with polarization singularities in the momentum space of low-symmetry photonic crystal (PhC) nanostructures. The utilization of polarization singularities can not only effectively modulate the polarization states of exciton-polaritons in the momentum space but also facilitate or suppress their far field coupling capabilities by tuning the in-plane momentum. Our results provide new strategies for creating polarization-selective exciton-polaritons.

On the Degrees of Freedom Count on Singular Phase Space Submanifolds

On the Degrees of Freedom Count on Singular Phase Space Submanifolds

Beyond large complex structure: quantized periods and boundary data for one-modulus singularities

We study periods in an integral basis near all possible singularities in one-dimensional complex structure moduli spaces of Calabi-Yau threefolds. Near large complex structure points these asymptotic periods are well understood in terms of the topological data of the mirror Calabi-Yau manifold. The aim of this work is to characterize the period data near other boundaries in moduli space such as conifold and K-points. Using results from Hodge theory, we provide the general form of these periods in a quantized three-cycle basis. Based on these periods we compute the prepotential and related physical couplings of the underlying supergravity theory. Moreover, we elucidate the meaning of the model-dependent coefficients that appear in these expressions: these can be identified with certain topological and arithmetic numbers associated to the singular geometry at the moduli space boundary. We illustrate our findings by studying a wide set of examples.

Time Functions on Lorentzian Length Spaces

AbstractIn general relativity, time functions are crucial objects whose existence and properties are intimately tied to the causal structure of a spacetime and also to the initial value formulation of the Einstein equations. In this work we establish all fundamental classical existence results on time functions in the setting of Lorentzian (pre-)length spaces (including causally plain continuous spacetimes, closed cone fields and even more singular spaces). More precisely, we characterize the existence of time functions by K-causality, show that a modified notion of Geroch’s volume functions are time functions if and only if the space is causally continuous, and lastly, characterize global hyperbolicity by the existence of Cauchy time functions, and Cauchy sets. Our results thus inevitably show that no manifold structure is needed in order to obtain suitable time functions.

Poles, shocks, and tygers: The time-reversible Burgers equation.

We construct a formally time-reversible, one-dimensional forced Burgers equationby imposing a global constraint of energy conservation, wherein the constant viscosity is modified to a fluctuating state-dependent dissipation coefficient. The system exhibits dynamical properties which bear strong similarities with those observed for the Burgers equationand can be understood using the dynamics of the poles, shocks, and truncation effects, such as tygers. A complex interplay of these give rise to interesting statistical regimes ranging from hydrodynamic behavior to a completely thermalized warm phase. The end of the hydrodynamic regime is associated with the appearance of a shock in the solution and a continuous transition leading to a truncation-dependent state. Beyond this, the truncation effects such as tygers and the appearance of secondary discontinuity at the resonance point in the solution strongly influence the statistical properties. These disappear at the second transition, at which the global quantities exhibit a jump and attain values that are consistent with the establishment of a quasiequilibrium state characterized by energy equipartition among the Fourier modes. Our comparative analysis shows that the macroscopic statistical properties of the formally time-reversible system and the Burgers equationare equivalent in all the regimes, irrespective of the truncation effects, and this equivalence is not just limited to the hydrodynamic regime, thereby further strengthening the Gallavotti's equivalence conjecture. The properties of the system are further examined by inspecting the complex space singularities in the velocity field of the Burgers equation. Furthermore, an effective theory is proposed to describe the discontinuous transition.

Disconnected gauge groups in the infrared

Gauging a discrete 0-form symmetry of a QFT is a procedure that changes the global form of the gauge group but not its perturbative dynamics. In this work, we study the Seiberg-Witten solution of theories resulting from the gauging of charge conjugation in 4d 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 theories with SU(N) gauge group and fundamental hypermultiplets. The basic idea of our procedure is to identify the ℤ2 action at the level of the SW curve and perform the quotient, and it should also be applicable to non-lagrangian theories. We study dynamical aspects of these theories such as their moduli space singularities and the corresponding physics; in particular, we explore the complex structure singularity arising from the quotient procedure. We also discuss some implications of our work in regards to three problems: the geometric classification of 4d SCFTs, the study of non-invertible symmetries from the SW geometry, and the String Theory engineering of theories with disconnected gauge groups.