Papers

We present the first formulation of linearised gravity in four dimensions which is manifestly Lorentz covariant and democratic, i.e. treats the two frames related by electric-magnetic duality on equal footing. It is well-known that four-dimensional linearised gravity belongs to a class of singleton representations of the four-dimensional conformal algebra $\mathfrak{so}(2,4)$. Our key insight is viewing this algebra as the isometry of $\text{AdS}_5$ and realising the massless spin-2 field as edge modes of a five-dimensional topological field taking values in a specific finite-dimensional representation of $\mathfrak{so}(2,4)$. The desired four-dimensional action is then found by a covariant boundary reduction procedure.

We study the phenomenological consequences of higher symmetry structures in axion effective field theories. Higher-group and non-invertible symmetries impose parametric constraints on the energy scales at which different symmetries can emerge in the infrared, providing a guide to the ultraviolet physics. We clarify and analyze these emergence constraints in axion EFTs coupled to abelian and non-abelian gauge bosons, with and without charged matter. We show that emergence constraints are universally saturated by anomaly inflow onto topological defects, while in perturbative UV completions they are supererogatory owing to the parametric separation of scales.

We investigate equatorial periodic orbits and their gravitational wave radiation in the spacetime of an Euler–Heisenberg (EH) black hole surrounded by perfect fluid dark matter (PFDM). The combined effects of quantum electrodynamic corrections and dark matter are incorporated through an effective metric, and the dynamics of timelike geodesics are analyzed using the effective potential formalism. We derive the conditions for marginally bound and innermost stable circular orbits, classify periodic trajectories using the rational parameter and topological indices, and identify a rich hierarchy of zoom–whirl motions in the strong-field regime. Gravitational wave signals from periodic orbits are computed using the numerical kludge method, revealing characteristic burst-like features associated with whirl phases. Our results show that perfect fluid dark matter systematically modifies the stability thresholds and suppresses the waveform amplitude, while QED corrections enhance high-frequency components generated near the horizon. These findings demonstrate that periodic orbits in the EH–PFDM spacetime provide a sensitive probe of quantum corrections and dark matter effects in strong gravitational fields.

We propose that the enhanced Higgs quartic coupling required by radiatively broken electroweak symmetry (RBEWS) emerges naturally from SO(10) grand unification. Our previous analysis demonstrated that a coupling enhancement factor $k = \lambda_{\rm enhanced}/\lambda_{\rm SM}$ leads to absolute vacuum stability with a UV Landau pole near the GUT scale for $k \gtrsim 1.03$. The RBEWS prediction $e_{125} = 7.2$ of Steele and Wang, when properly translated from the Coleman-Weinberg scheme at the electroweak VEV to the $\overline{\rm MS}$ scheme at $M_t$ via scheme conversion and scale-dependent ratio evolution, yields $k(M_t) \approx 6.0$–$6.4$, corresponding to a UV pole at $\Lambda_{\rm UV} \sim {1.5\text{--}2} \times 10^{16}$~GeV – remarkably close to the GUT scale $M_{\rm GUT} \sim 2 \times 10^{16}$~GeV. We argue this coincidence is not accidental: the UV pole signals the scale where the Standard Model effective description must be embedded into the full SO(10) structure. We derive threshold corrections from SO(10) scalar sectors containing $\mathbf{10}_H$, $\mathbf{\overline{126}}_H$, and $\mathbf{45}_H$ representations, showing that portal couplings between the light Higgs doublet and heavy GUT scalars can generate enhancement factors of order $k \sim 5$–$10$ at the matching scale. The Coleman-Weinberg mechanism operating within a classically scale-invariant GUT scalar potential provides a dynamical origin for both RBEWS and the hierarchy between $M_{\rm GUT}$ and the electroweak scale.

We investigate Maxwell-Chern-Simons theory on a three-dimensional noncommutative spacetime endowed with a constant spacelike Poisson structure. By exploiting the residual rotational symmetry, we construct exact classical solutions corresponding to pointlike electric and magnetic charges. We demonstrate that noncommutativity acts as a natural regulator, ensuring a finite total electromagnetic energy and thereby resolving the classical self-energy divergence. Furthermore, some of these solutions exhibit a non-perturbative dependence on the noncommutativity parameter and allow for the generation of an arbitrary magnetic flux. We also present a noncommutative generalization of Gauss's law, providing a robust framework for the physical interpretation of these exact solutions.

We present the TQ4Q2.0 fragmentation functions for the production of all-heavy (fully heavy) $S$-wave tetraquarks ($T_{4Q}$) with scalar ($0^{++}$), axial-vector ($1^{+-}$), and tensor ($2^{++}$) quantum numbers in high-energy hadronic collisions. This work extends the previous TQ4Q1.1 framework by incorporating nonconstituent heavy-quark contributions and introducing a replica-based uncertainty-quantification strategy derived from multi-scale variations (MHOUs). The construction follows a nonrelativistic QCD factorization approach, combining gluon- and heavy-quark-initiated fragmentation channels at leading power. Initial-scale inputs are modeled through updated potential-inspired wave functions, while the subsequent DGLAP evolution is performed via the threshold-aware HF-NRevo scheme. A comprehensive systematic analysis of uncertainties is carried out, with contributions from color-composite long-distance matrix elements (LDMEs) and perturbative multiscale inputs. The resulting TQ4Q2.0 grids, publicly released in LHAPDF6 format, provide the first complete phenomenological set for all-heavy exotics, enabling precise studies of all-charm tetraquark production and jet-associated observables within the JETHAD environment. This article completes the high-energy resummation-driven generation of the TQ4Q program and establishes a definitive baseline for future collider-oriented analyses of all-heavy multiquark dynamics.

The $S$-algebra originally arose as a chiral algebra of asymptotic symmetries of Yang-Mills theory. We show that in the self-dual sector of Yang-Mills, the $S$-algebra gets upgraded to an infinite-dimensional algebra of $1$-form symmetries in the bulk. The associated 2-form currents encode the integrability and hierarchies of self-dual Yang-Mills. As an application, we prove the equality of Carrollian corner charges with modes of the celestial chiral algebra by expressing them as integrals of the same 2-form currents over homologous 2-cycles.

A spectrally isolated quartet can admit a local two-qubit description at each point in parameter space and still acquire a loop holonomy outside the local subgroup $\mathrm{U}(2)\otimes\mathrm{U}(2)$. We study this question in three localized topological settings, a BHZ ribbon, a spinful SSH chain, and a BBH corner quartet. On a fixed quartet, changing only the loop can move the holonomy from almost local to entangling. In BHZ, co-rotating and counter-rotating edge-field loops have nearly the same eigenphase data, but only the counter-rotating loop yields an Ising-like entangler. SSH gives a controlled-rotation example in a numerically stable edge quartet. BBH shows the same issue in a higher-order corner quartet. Standard Berry data, including Berry phases, Chern numbers, determinant phases, and eigenphase spectra, do not separate these cases. The main diagnostic is the distance from the loop holonomy to the extracted local subgroup. Canonical two-qubit coordinates are used only after reduction failure has been identified. The quartet is the smallest setting in which this question can be tested explicitly. The same subgroup-reduction problem extends to any isolated multiplet with pointwise product type $D=\prod_{\alpha}d_\alpha$, where the relevant local subgroup is the embedded product group $G_{\mathbf d}=\mathrm{Im}[\prod_{\alpha}U(d_\alpha)\to U(D)]$. In the terminology of Ref.[arXiv:2601.13764], these examples realize loop-dependent entangling gluing.

We propose a novel way of detecting gravitons using emission of photons from charged array of quantum harmonic oscillators placed inside of a cavity while the cavity is being pumped with low frequency photons. We observe that when the detector is in its ground state, a single graviton is absorbed by the detector while it jumps a single energy level by simultaneously emitting a photon. We also observe that while the detector de-excites from an higher energy level, it spontaneously emits a high frequency graviton, by absorbing a single photon. This analytical outcome encourages us to propose a very simple tabletop graviton detector model as the transition probabilities can be significantly enhanced by pumping photons in the initial state of the system. This mechanism gives us a physical way to `visualize' the effect of gravitons with a relativistic system. We also show that Dyson's original argument on the non-detectability of gravitons can be completely circumvented using our proposed graviton detector.

We construct a class of stationary, axisymmetric, horizonless spacetimes whose curvature is generated entirely by smooth, localised differential rotation $\Omega(r)$, while the spatial geometry remains exactly flat. Despite vanishing ADM mass, these helicity-supported configurations exhibit non-trivial curvature, finite tidal forces, and a gravitomagnetic field arising from the radial shear of the rotation. The twisted stationary Killing congruence produces global frame-dragging, including a gravitational Sagnac effect, and the effective potential admits stable circular orbits for null and timelike particles. The tidal tensor gives oscillatory restoring forces, ensuring stability against radial perturbations. Linearising the Einstein equations yields a wave equation for axisymmetric perturbations of $\Omega(r)$; the effective potential is positive and localised, the operator is self-adjoint and positive definite, and the frequency spectrum is real, implying linear stability. Perturbations propagate as shear waves analogous to Alfvén waves. These results show that differential rotation alone can sustain a regular, asymptotically flat gravitational field with rich dynamics. This class of spacetimes provides a tractable platform for exploring gravitomagnetism, tidal and wave phenomena in smooth rotating backgrounds, with direct applications to rotating astrophysical structures.

We present GlobalCY, a JAX-based framework for globally defined and symmetry-aware neural Kähler-potential models on projective hypersurface Calabi–Yau geometries. The central problem is that local-input neural Kähler-potential models can train successfully while still failing the geometry-sensitive diagnostics that matter in hard quartic regimes, especially near singular and near-singular members of the Cefalú family. To study this, we compare three model families – a local-input baseline, a globally defined invariant model, and a symmetry-aware global model – on the hard Cefalú cases $\lambda=0.75$ and $\lambda=1.0$ using a fixed multi-seed protocol and a geometry-aware diagnostic suite. In this benchmark, the globally defined invariant model is the strongest overall family, outperforming the local baseline on the two clearest geometric comparison metrics, negative-eigenvalue frequency and projective-invariance drift, in both cases. The gains are strongest at $\lambda=0.75$, while $\lambda=1.0$ remains more difficult. The current symmetry-aware model improves projective-invariance drift relative to the local baseline, but does not yet surpass the plain global invariant model. These results show that global invariant structure is a meaningful architectural constraint for learned Kähler-potential modeling in hard quartic Calabi–Yau settings.

In this work, we systematically investigate the null geodesics of electrically charged black holes in a gravitational framework that incorporates Lorentz violation induced by a background Kalb-Ramond (KR) field, in the presence of perfect-fluid dark matter. The properties of the photon sphere, black hole shadow, and photon trajectories are analyzed in detail. Furthermore, to explore the combined effects of Lorentz violation and dark matter on the motion of neutral test particles, we examine the innermost stable circular orbit (ISCO) in this spacetime. In addition, the epicyclic frequencies of test particles are studied to gain further insight into the dynamical behavior of particle motion around these black holes. The main analytical results are complemented by a phenomenological QPO analysis, a thermodynamic investigation, and a discussion of the sparsity of Hawking radiation, allowing us to connect optical, dynamical, and thermodynamic signatures within a single framework.

We propose a radiative Dirac neutrino mass model stabilized by a non-invertible fusion rule originating from a $Z_3 \times Z_3'$ gauging. The imposed symmetry forbids tree-level Yukawa couplings and ensures that neutrino masses are generated only at the one-loop level through the exchange of exotic fermions and inert scalars. This minimal framework simultaneously accommodates neutrino masses and mixings consistent with current oscillation data, while providing a viable dark matter candidate. We analyze lepton flavor violating processes and lepton anomalous magnetic moments, finding that all contributions remain well below present experimental bounds. In the dark matter sector, the bosonic singlet emerges as a promising candidate with relic density compatible with cosmological observations, whereas the fermionic option is strongly disfavored due to suppressed annihilation cross sections. Our study demonstrates that non-invertible fusion rules can serve as a powerful organizing principle for constructing minimal and phenomenologically consistent extensions of the Standard Model, linking neutrino physics and dark matter within a unified radiative framework.

We update the holomorphic modular bootstrap incorporating a recent result that computes the exact S-matrix within the Modular Linear Differential Equation (MLDE) setting. Further, using knowledge of the allowed exponents modulo one, we obtain admissible solutions to all MLDE's with up to six characters and Wronskian index < 6 and one accessory parameter with c_eff <= 24. We then identify which of the admissible solutions have good fusion rules – we call such solutions tenable. When possible, we identify the CFT and in the unitary cases the MTC class they belong to.

In this contribution we present recent developments in the formulation and solution of Initial Boundary Value Problems (IBVPs). Building upon a modern variational action formulation of classical dynamics, we treat Initial Boundary Value Problems directly on the action level, bypassing governing equations. We show that by including coordinate maps as dynamical degrees of freedom together with propagating fields two key results emerge. Space-time symmetries remain protected even after discretization, leading to an exact conservation of Noether charges even for discrete IBVPs. The dynamical nature of the coordinate maps leads to an adjustment of space-time resolution, guided by Noether charge conservation, realizing a form of automatic adaptive mesh refinement. We stress that as long as SBP operators are used for the discretization, our results are independent of whether the dynamics are solved on the action or governing equation level and hold in particular also at high order. As proof-of-principle for our approach we present its application to scalar wave-propagation in 1+1 dimensions.

This paper extends our previous work (arXiv:2504.14313) of a single timelike subregion to two, in the framework of AdS$_3$-Vaidya holography. We confirm the positivity of timelike mutual information and the statement of weak monotonicity when the subregions are non-overlapping. We also study entanglement inequalities such as Araki-Lieb inequality and strong subadditivity when the intervals start to overlap. In line with the recent findings in the literature, we provide explicit working examples showing that the timelike version of the strong subadditivity is generally violated in these setups, even though the statements of subadditivity and Araki-Lieb inequality hold true.

We investigate uni-vector deformation in the Type IIA setup and show that the D0-brane background is mapped into itself (sedimentation), and other extremal backgrounds get bound with a dissolved D0-brane charge. Explicitly we generate F1-D0 and D2-D0 bound states background from uni-vector deformations. For the former we show that deformation of the non-extremal string gives the correct thermal F1-D0 bound state. We discuss relations between critical uni-vector deformations and DLCQ of M-theory.

Direct canonical quantization of Jackiw–Teitelboim gravity at finite cutoff leads to a critically singular Pöschl–Teller problem whose endpoint admits inequivalent self-adjoint domains. We solve the full one-parameter family, obtain its exact spectral data, and identify the extension parameter with a renormalized short-distance matching datum in geodesic-length space. Self-adjointness, positivity, and unitary Brown–York evolution leave a stable family, whereas a source-free local completion without an endpoint contact term selects the Friedrichs realization. With the Brown–York clock and a local cap prescription fixed, the disk amplitude distinguishes the domains. In the large-boundary limit the bulk operator becomes universally Liouville, while a cap tied to the receding endpoint can retain domain dependence. Analytic continuation also maps radial resonances to poles of the Brown–York resolvent. These results isolate the short-distance datum that distinguishes the reduced canonical realizations and that a microscopic completion would have to determine.

We present a semiclassical account of the Veneziano-Yankielowicz (VY) superpotential in four-dimensional $N=1$ super Yang-Mills theory. Motivated by two-dimensional gauged linear sigma models, where superpotentials arise from vortex dynamics, we reinterpret domain walls as fundamental objects associated with higher-form gauge fields. In this formulation, the vacuum structure is encoded in a compact three-form gauge field, whose four-form flux labels topological sectors. In the presence of charged matter with total charge $N$, these sectors exhibit a natural $\mathbb{Z}_N$ structure, leading to a decomposition into $N$ semiclassical contributions. These contributions arise from Euclidean point-like configurations in the higher-form sector, analogous to fractional instantons. We show that these configurations provide the relevant non-perturbative contributions to the effective superpotential. Integrating out the associated degrees of freedom reproduces the VY superpotential in the infrared. This gives a semiclassical origin of the VY superpotential in terms of higher-form gauge dynamics.

We show that a relative holographic energy coincides with the relative Hamiltonian energy.

In 2021, Lei et al. claimed the equivalence between the two Lagrangians $\mathcal{L}_1 =-mc\sqrt{-g_{\mu\nu}{\dot{x}}^\mu{\dot{x}}^\nu}-V$ and $\mathcal{L}_2 = \frac{1}{2}mg_{\mu\nu} {\dot{x}}^\mu{\dot{x}}^\nu-V$ for describing particle dynamics in combined gravitational and matter fields. In the present work, we rigorously demonstrate that their equivalence depends critically on the external potential V. Both Lagrangians yield identical Hamiltonians that strictly satisfy the mass shell constraint, and are therefore equivalent when V vanishes or corresponds to an electromagnetic potential. However, they are generally not equivalent for generic external potentials excluding the electromagnetic ones. This discrepancy arises because L1 and L2 correspond to different Hamiltonian formulations. The Hamiltonian derived from L1 inherently enforces the mass shell constraint, whereas the Hamiltonian from L2 does not. When the Schwarzschild metric supplemented with an artificial mechanical potential is taken as a toy model, numerical investigations reveal that L1 leads to chaotic behavior, which signifies non-integrable dynamics. By contrast, L2 can be shown analytically to produce integrable dynamics free of chaos. In many scenarios, L1 is strongly recommended due to its theoretical superiority and universality. L2 is generally suitable for classical approximate problems involving low energy and weak gravity. Nevertheless, it is the preferred choice for strong field problems concerning the dynamics of charged (or neutral) particles near black holes with (or without) external electromagnetic fields, owing to its mathematical simplicity and computational efficiency. Moreover, it can still satisfy the mass shell constraint when an additional constraint is imposed on its corresponding Hamiltonian.

We study the geodesic motion and epicyclic oscillations of massive test particles around a static, spherically symmetric black hole (BH) solution of the Einstein–Skyrme (ES) theory in Anti-de Sitter (AdS) spacetime. The lapse function of this BH depends on the Skyrme coupling $\eta$, a charge-like parameter $Q$ inherited from the Skyrme term, and the cosmological constant $\Lambda<0$. We first map out the horizon structure and identify three regimes-non-extremal BH (NEBH), extremal BH (EBH), and naked BH (NBH)-showing that the NEBH $\to$ EBH $\to$ NBH transition is governed by $Q$ rather than $\eta$, which enters $f(r)$ only as a constant shift. We then derive the effective potential (EP), locate the innermost stable circular orbit (ISCO), and compute the radiative efficiency, finding that $\mathcal{E}_{\rm ISCO}>1$ in AdS renders the standard Novikov-Thorne formula negative. The corrected radial epicyclic frequency $\Omega_r$ reveals a distinctive AdS signature: $\nu_r$ grows at large $r$ and overtakes the orbital frequency $\nu_\phi$, causing the periastron precession frequency $\nu_p = \nu_\varphi - \nu_r$ to change sign-a feature absent in asymptotically flat geometries. Adopting the relativistic precession (RP) model for quasi-periodic oscillations (QPOs), we perform a Markov chain Monte Carlo (MCMC) analysis using twin-peak QPO data from XTE~J1550-564, GRO~J1655-40, Sgr~A$^*$, and M82~X-1. The posteriors converge to $Q\approx 0.6$ across all sources, with orbital radii near $r\approx 4.2\,M$ and masses consistent with independent estimates, demonstrating that the ES-AdS BH accommodates the observed frequency pairs within physically motivated parameter ranges.

We study analytic properties of thermal spectral functions of holographic CFTs, examining both their (a) exact properties at finite momentum and (b) asymptotics at large momentum. For even-dimensional holographic CFTs on Minkowski spacetime and for scalar primaries with integer dimensions, we demonstrate that the exact spectral function at finite momentum factorizes into a perturbative/OPE piece and a non-perturbative piece. The former is controlled by stress tensor exchange and fixed by a near-boundary analysis. The latter encodes information about the bulk interior, including the black hole horizon and singularity. Utilizing the exact factorization, we obtain the full transseries expansion of the non-perturbative piece at large timelike momentum. This is achieved by employing exact WKB techniques to compute the monodromy of the bulk wave equation. Finally, we use these results to work out the singular loci of a spatially averaged thermofield double correlator in the complex time plane. These singular loci have been argued to provide imprints of the black hole curvature singularity in the dual CFT observables. Our result, which includes the case of non-vanishing momentum, gives a clear link between the non-perturbative spectral function and the black hole singularity.

H-EFT-VA established a physics-informed solution to the Barren Plateau (BP) problem via a hierarchical EFT UV-cutoff, guaranteeing gradient variance in Omega(1/poly(N)). However, localization restricts the ansatz to a polynomial subspace, creating a reference-state gap for states distant from |0>^N. We introduce Adaptive H-EFT-VA (A-H-EFT) to navigate the trainability-expressibility tradeoff by expanding the reachable Hilbert space along a safe trajectory. Gradient variance is maintained in Omega(1/poly(N)) if sigma(t) <= 0.5/sqrt(LN) (Theorem 1). A Safe Expansion Corollary and Monotone Growth Lemma confirm expansion without discontinuous jumps. Benchmarking across 16 experiments (up to N=14) shows A-H-EFT achieves fidelity F=0.54, doubling static H-EFT-VA (F=0.27) and outperforming HEA (F~0.01), with gradient variance >= 0.5 throughout. For Heisenberg XXZ (Delta_ref=1), A-H-EFT identifies the negative ground state while static methods fail. Results are statistically significant (p < 10^-37). Robustness over three decades of hyperparameters enables deployment without search. This is the first rigorously bounded trajectory through the VQA landscape.

We investigate how the junction law for multipartite entanglement is realized in confining holographic backgrounds, using genuine multi-entropy (GM) as our main diagnostic. We first study an AdS$_3$ hard-wall toy model as an analytic benchmark, where multi-way cuts and junction geometries can be analyzed explicitly. In this setup, we classify the relevant saddles, determine the dominant phases, and show that the genuinely multipartite contribution diagnosed by GM is localized near the junction. We also examine how this structure depends on subsystem sizes, asymmetry, and the confinement scale, including phase transitions between competing saddles. We then move beyond the hard-wall benchmark to smooth confining geometries, focusing on the D4-soliton and D3-soliton backgrounds and formulating the corresponding framework also for the Klebanov–Strassler background. In the smooth-cap examples, we find that the junction picture persists, while the detailed phase structure differs from the hard-wall case: in particular, the hard-wall plateau does not survive, and GM instead decreases monotonically and vanishes at a finite critical scale. We also find that the short-distance behavior is background-dependent, with $\mathrm{GM}^{(3)}\sim L^{-4}$ in the D4-soliton background, $\mathrm{GM}^{(3)}\sim L^{-2}$ in the D3-soliton background, and $\mathrm{GM}^{(3)}\sim L^{-2}\cdot (\log L)^{2}$ in the Klebanov–Strassler background. These results clarify which features of the junction-law picture are robust in confining holography and which features of the phase structure and short-distance scaling are background-dependent.

We construct a generalized Lagrangian that unifies the Gross-Neveu-Yukawa, Nambu-Jona-Lasinio-Yukawa, and Wess-Zumino models, allowing for arbitrary scalar and fermion flavors in $D$-dimensional regularization. This framework clarifies how emergent supersymmetry arises at critical points and reveals structural connections between these theories. The unified formulation provides additional supersymmetry Ward identities that simplify loop calculations, even for non-supersymmetric models. As an application, we show how this technique can reduce the computational cost of determining anomalous dimensions of twist-two operators.

F-theory compactifications with a nontrivial Mordell-Weil group realize abelian gauge symmetry through rational sections, but their consistency is ultimately a statement about the quantum effective action. We show that compactification on a circle makes this statement concrete: the quantized, parity-odd Chern-Simons couplings of the resulting three-dimensional theory provide a one-loop exact and scheme-independent encoding of all local four-dimensional abelian anomalies, including the mixed gauge-gravitational terms, together with their Green-Schwarz cancellation. We determine these Chern-Simons couplings in two logically independent ways, first from flux-induced terms in the M-theory dual description, and second from an explicit one-loop integration over the complete massive spectrum, including Kaluza-Klein towers and Coulomb-branch states. The agreement fixes all normalizations and clarifies how large gauge transformations reorganize the spectrum. We then show compatibility with type IIB modular duality once the known ten-dimensional duality counterterm is included, and we present a fully explicit rank-two example over projective three-space.

We construct an analytic model of static semiclassical backreaction for a Schwarzschild black hole in the Hartle–Hawking state enclosed within a finite spherical cavity. Using a minimal renormalized stress–energy tensor consistent with conservation, thermal asymptotics, and horizon regularity, we integrate the reduced semiclassical Einstein equations under Dirichlet boundary conditions at the cavity wall. This yields explicit expressions for the corrections to the mass function, redshift factor, horizon location, and surface gravity. We obtain a closed-form first-order correction to the Hawking temperature in terms of a dimensionless backreaction parameter and the cavity radius. The temperature shift decomposes into redshift renormalization, geometric horizon displacement, and a local energy-density contribution at the horizon. The perturbative expansion is controlled by a parameter of order $M_P^2/M^2$, ensuring validity for macroscopic black holes. The near-horizon geometry retains its universal Rindler$^{2}\times S^{2}$ structure, indicating that semiclassical effects renormalize rather than modify the geometric origin of Hawking radiation.

Crystalline symmetries give rise to topological invariants that can distinguish quantum phases of matter. Understanding these in strongly interacting systems is an ongoing research direction requiring non-perturbative methods. Recent developments have demonstrated that even classic models, like the Harper-Hofstadter model of free fermions on a lattice in a magnetic field, yield a host of crystalline symmetry protected topological invariants. Here we review some of these developments, focusing mainly on how to characterize, classify, and detect invariants arising from lattice translation and rotation symmetries along with charge conservation in two-dimensional systems, including integer and fractional Chern insulators.

Black hole spacetimes contain several geometrically distinguished hypersurfaces, including event and Cauchy horizons, stationary-limit surfaces, and curvature singularities. These structures are usually identified by different geometric or causal criteria. We show that, for the Kerr–Newman black hole, the membrane-paradigm pressure of a stretched horizon admits an analytically continued scalar representative whose fully factorized form has zeros, poles, and singular factors aligned with these standard Kerr–Newman loci. Its zeros occur at the outer and inner horizons, its poles occur at the outer and inner stationary-limit surfaces, its higher-order divergence contains the ring-singularity factor, and its large-$r$ behavior records the asymptotic decay of the scalar. The construction is not proposed as a new invariant characterization of the spacetime, nor as a physical extension of the membrane fluid into the black-hole interior. Rather, it gives a compact membrane-pressure-based diagnostic whose analytic structure reorganizes several familiar surfaces of the Kerr–Newman geometry. We also note a secondary, purely algebraic analogy with generalized multi-component van der Waals-type equations of state.