Papers

We establish a correspondence between anomaly detection in high-noise regimes and the renormalization group flow of non-equilibrium field theories. We provide a physical grounding for this framework by proving that the detection of phase transitions in interacting non-equilibrium systems maps to the study of an effective equilibrium field theory near its Gaussian fixed point, which we identify with the universal Marchenko-Pastur distribution. Applying the Functional Renormalization Group to the two-dimensional Model A, we demonstrate that the noise-to-signal ratio acts as a physical temperature, where the signal emerges as ordered domains within a thermalized background of fluctuations. Using the exact Onsager solution as a benchmark, we show that this approach identifies critical thresholds with an error below 4%, significantly outperforming standard information-theoretic metrics such as the Kullback-Leibler divergence. Our results provide a universal strategy for resolving structures in complex datasets near criticality, bridging the gap between statistical mechanics and statistical inference.

Understanding stochastic inflation, and in particular the systematic computation of controlled corrections from first principles, remains an important open problem. In this work, we address this problem from two complementary perspectives. First, we derive the effective field theory governing long-wavelength modes from the reduced density matrix of a coarse-grained description. In this framework, locality in time follows from the thin-shell approximation, while locality in space is recovered dynamically in the super-Hubble regime. The resulting open effective field theory contains both dissipative and diffusive operators, with diffusion dominating as the coarse-graining scale is pushed into the infrared. This construction reproduces the usual Fokker-Planck equation at leading order and allows us to compute its corrections, including subleading contributions to the stochastic dynamics. Second, we study the evolution of the density matrix under changes of the coarse-graining scale. We show that this flow is governed by a Polchinski-type renormalisation group equation formulated directly for the density matrix. Dissipative and diffusive operators are generated dynamically along the flow, and the resulting effective action matches the Schwinger-Keldysh description. We derive a generalised Fokker-Planck equation directly from the renormalisation group flow, systematically incorporating subleading corrections and recovering the results obtained in the open effective field theory approach.

Today, the observable cosmos exhibits a remarkable degree of isotropy and plausibly began in a nearly isotropic initial state. The properties of the Lorentzian Chern-Simons-Kodama (CSK) functional can provide an understanding of this initial state. In gravity with a positive cosmological constant, the Chern-Simons-Kodama (CSK) wavefunctional is an exact, chiral solution of the quantum gravitational constraints. We suggest that the normalizability and other issues with this functional, if interpreted as a proper state of quantum gravity, instead suggest an embedding into a larger quantum gravitational completion, and recast the CSK functional as a gravitational sphaleron with observationally desirable properties. By perturbing around the dominant de Sitter saddle of the wavefunctional with appropriate quantum gravitational boundary conditions, we find that for a closed universe the system is dynamically driven to spatial isotropy, while all anisotropic modes acquire positive quadratic curvature and are Gaussian-suppressed. The decay of this sphaleron therefore proceeds along an isotropic channel, providing an intrinsic quantum-gravitational mechanism for dynamical isotropization. This isotropization effect is robust under the inclusion of a slow-roll inflaton, and no analogous isotropic sphaleron exists for spatially flat or hyperbolic geometries. Taken together, these results recast the Lorentzian CSK functional as a chiral sphaleron that naturally prepares an approximately isotropic de Sitter background for inflation. Beyond this phenomenological study, we further suggest that the CSK functional can be understood as a boundary functional for a class of anomaly-free objects, including a complexified generalization of the Hartle-Hawking state.

We analytically prove that the Veneziano amplitude is the unique outcome of a dual bootstrap based on dispersive sum rules, unitarity, and a small amount of additional stringy input. This stringy input can be either the string monodromy condition or the recently uncovered splitting condition. A key ingredient in our proofs is to interpret the dispersive sum rules as sequences of moments. Also important is the precise incorporation of the extra stringy input into the amplitude ansatz. Together, these ingredients make the bootstrap analytically tractable and uniquely fix the Veneziano amplitude.

We study zero-temperature false vacuum decay in $D$ compact spatial dimensions and show that for volumes below a critical value a new bounce solution, different from Coleman's celebrated $O(D)$ bubble, mediates the decay process, and typically leads to an exponentially enhanced decay rate. The bounce, when analytically continued to Lorentzian signature, nucleates a homogeneous field configuration for spatial volumes below a critical value, and quasi-homogeneous configurations for slightly larger volumes, and is not of the form of a thin or thick-walled bubble embedded in a false vacuum background. We explicitly show that the new bounce has the necessary features associated with false vacuum decay, following from its eigenvalue spectrum of fluctuations. The cross-over from homogeneous to quasi-homogeneous solutions as the spatial volume is increased is discussed, as is a real-time interpretation of the bounce. We apply this bounce to the study of a scalar field model, as well as a close cousin of the Schwinger effect that applies to $(1+1)d$ axion electrodynamics in compact space.

Random circuit models often describe local dynamics using generic two-qubit gates, which have proven successful in capturing entanglement growth and operator spreading in many contexts. This approach naturally leads to the expectation that detailed gate structure plays only a limited role in coarse-grained entanglement and scrambling diagnostics. We show that the internal structure of multipartite circuit primitives can significantly influence these dynamical rates, even within a fixed random-circuit architecture. To investigate this, we study an exactly simulable family of Clifford quantum circuits built from fixed $n$-qubit graph-state preparation unitaries, which we treat as elementary building blocks. Specifically, we consider a one-dimensional chain of $N$ qubits initialized in a product state and evolved by layers in which nonoverlapping length-$n$ blocks are placed at uniformly random positions with sparsity $\alpha$. We find that different choices of graph-state building blocks lead to strongly varying dynamical rates. Graph states that are inequivalent under local Clifford (LC) transformations generate sharply different entanglement velocities $v_E$ and butterfly velocities $v_B$, even though the circuits are drawn from the same ensemble with identical architecture and randomness parameters. We further show that this hierarchy is captured by two complementary block-level characteristics: the distribution of entanglement across internal bipartitions of the graph state, which correlates with $v_E$, and a graph-theoretic connectivity profile across bipartitions, which correlates with $v_B$. Neither descriptor alone fully determines the dynamics; rather, entanglement growth and operator spreading are controlled by distinct structural features of the local circuit blocks. Notably, AME states appear among the fastest scrambling building blocks within the ensembles studied here.

It is well known that a strictly massless spin-$3/2$ particle can interact consistently only within supergravity. Recently, positivity arguments have shown that an effective field theory of a massive Majorana spin-$3/2$ particle admits an arbitrarily small mass only if a graviton is present and the four-fermion contact interactions are tuned to the values dictated by $\mathcal{N}=1$ supergravity. In this work, we investigate how this limit is approached at finite mass. Assuming that the graviton $t$-channel pole can be discarded, we derive non-forward, tree-level dispersive bounds on massive spin-3/2 contact operators and determine the region of effective couplings consistent with unitarity and analyticity. For sufficiently small $m$, we find that the allowed parameter space forms a bounded, Planck-suppressed neighbourhood of the supergravity point, defined by the supergravity values of the four-fermion couplings. The supergravity point lies on the boundary of this region. In the regime $m \ll M_{\rm Pl}$, the volume of the allowed region scales parametrically as \[ \mathrm{Vol} \sim \frac{m^{6}}{M_{\rm Pl}^{6}} \, , \] and shrinks to zero as $m \to 0$, smoothly reproducing the massless-limit results. The allowed region becomes unbounded when mass approaches the Planck scale. We further analyze the effect of including additional light scalar and pseudo-scalar degrees of freedom, motivated by the Polonyi model, and find that their couplings are also bounded in a way similar to the contact couplings and that it doesn't enlarge the allowed contact coupling space.

It has been suggested that Hořava-Witten theory could provide a concrete realization of the Dark Dimension Scenario. In this context, the observable Standard Model sector is naturally localized in the micron-sized large dimension, which is the interval in the eleventh direction. Considering Calabi-Yau manifolds supporting generic vector bundles including also abelian factors, we point out that symmetric tadpole cancellation on the $E_8$ walls has the potential to ameliorate some of the issues of such a realization, including too fast proton decay. By taking not only the hierarchically small value of the dark energy but also the size of the Standard Model gauge couplings into account, one is driven to a special infinite distance limit, which is the Hořava-Witten analogue of a limit recently at the focus of the M-theoretic Emergence Proposal. Extrapolating results obtained for BPS-saturated amplitudes, we speculate about the possibility of obtaining the moduli dependence of the scalar potential, the gauge couplings and the Planck scale by simple one-loop Schwinger integrals over towers of states.

Hints of cosmic microwave background polarization rotation ($\Delta\vartheta \sim 10^{-3}$ rad) are commonly attributed to late-time dynamics of ultralight axions. We show that such ultralight degrees of freedom are not required. Polarization rotation naturally arises as a geometric interface phase acquired when photons cross interfaces between topologically distinct dark sector vacua. The effect is a discrete phase shift fixed by the normalization of a wall-supported electromagnetic Chern–Simons interaction and protected by an emergent $1$-form symmetry of the low energy effective theory. This mechanism reproduces the familiar adiabatic rotation induced by light axion domain walls, but persists for arbitrarily thin walls where the axion is heavy or absent. In this regime the rotation manifests as a Pancharatnam phase localized at vacuum interfaces, independent of redshift and photon frequency below a natural ultraviolet cutoff. Cosmic birefringence thus emerges as a probe of vacuum structure in the dark sector, rather than of light-field dynamics.

We investigate the Raychaudhuri dynamics of charged spin-nondegenerate Lorentz-violating particle congruences under minimal electromagnetic coupling. The coupling is introduced through the gauge-covariant momentum $P_{\mu}=\pi_{\mu}-qA_{\mu}$, so that the branch dispersion relation keeps its free functional form, while the electromagnetic field enters through the evolution of $P_{\mu}$. For a generic branch $\mathcal D^{(\pm)}(P)$, the tangent $k^{\mu}_{(\pm)}$ and the momentum Hessian $M^{\mu\nu}_{(\pm)}$ determine the covariant acceleration, $a^{\mu}_{(\pm)}=-qM^{\mu\nu}_{(\pm)}F_{\nu\rho}k^{\rho}_{(\pm)}$. As a consequence, the Raychaudhuri equation acquires the branch-dependent electromagnetic source $-q\nabla_{\mu}\!\left(M^{\mu\nu}_{(\pm)}F_{\nu\rho}k^{\rho}_{(\pm)}\right)$. We apply this construction to the $b_{\mu}$, $H_{\mu\nu}$, and $d_{\mu\nu}$ sectors, obtaining the corresponding branch tangents, Hessians, accelerations, and focusing equations. In flat spacetime, the electromagnetic field modifies the expansion through the divergence of the effective branch force. Therefore, uniform fields may bend the trajectories, whereas local focusing requires field gradients or, in the magnetic case, a coupling to an already deformed congruence. We also develop the analogous description for semiclassical quasiparticle beams, where the band Hessian plays the role of an effective electromagnetic response tensor. For anisotropic parabolic, Dirac-like, and Weyl-type dispersions, the same geometric structure relates electromagnetic textures to beam focusing. In two-branch systems, the opposite Hessians of the branches can produce focusing in one congruence and defocusing in the other, giving a quasiparticle realization of branch-dependent birefringence.

We investigate which dilaton gravity models can reproduce the typical two dimensional analogue black holes realized in platforms such as superconducting quantum circuits. We identify the most reasonable assumptions these models must satisfy, and determine the dilaton models for which the state-dependence of the Hawking temperature, T, can be switched on and off, a feature that is absent in four dimensional black holes. When the analogue black hole exhibits state-independent temperature, as in the cases considered here, the kinematics governing T decouples from the dynamics underlying S. Our numerical analysis reveals that the given analogue black holes do not correspond to known dilaton gravity models, limiting their usefulness for extracting theoretical insights. We then show that the logic can be easily reversed: starting from established well known dilaton models, one can derive the conditions that laboratory implementations must satisfy. This shifts the challenge from the theoretical perspective to the experimental realization.

It is well known that two parallel photon beams do not deflect under the effect of their energy-momentum tensor. In this work, we propose a novel model where two spatially separated Bose-Einstein condensates are outcoupled to create two parallel atom laser beams. We find out that apart from the classical deflection, a purely quantum gravity induced tidal deflection is observed which results in an irreducible noise in the geodesic separation of the two beams. Based on this simple but novel theoretical outcome, we propose an experimental model for detecting this quantum gravity induced standard deviation in the geodesic separation of the two parallel matter-wave beams.

We study the parafermionization of the Monster CFT with respect to its $\mathbb{Z}_{pA}$ subgroups, with $p$ an odd prime. Under certain assumptions, we show that the parafermionization is equal to a non-invertible gauging of $\mathcal{P}(p) \times \mathcal{P}(p)^\vee$, where $\mathcal{P}(p)$ is the theory of $\mathbb{Z}_p$-parafermions and $\mathcal{P}(p)^\vee$ is an appropriate dual theory, with global symmetry characterized by the centralizer of $\mathbb{Z}_{pA}$. By tracking the symmetries of $\mathcal{P}(p) \times \mathcal{P}(p)^\vee$ through the non-invertible gauging, we argue that the diagonal Monster CFT has $\mathrm{Rep}(\mathfrak{so}(3)_p) \boxtimes \mathrm{Rep}(\mathfrak{so}(3)_p)^\mathrm{op}$ symmetry, and hence that the holomorphic Monster theory has symmetry $\mathrm{Rep}(\mathfrak{so}(3)_p)$. We then compute the defect McKay-Thompson series associated to these symmetries, and prove that their invariance subgroups are $\Gamma_1(p+2)$.

We explore Krylov complexity in the BMN matrix model following a systematic reduction of it, known as the pulsating fuzzy sphere model. We present an analytical setup that allows us to calculate Lanczos coefficients in both large and small deformation limits of the matrix model.

The gravitational field of two-body system, a high energetic particle and a massive particle at rest, is studied in the linearized Einstein gravity. The ultrarelativistic particle yields a plane-fronted gravitational shockwave which perturbes gravitational field of the particle at rest. The problem can be also considered as a fixed-target high energy collision. We show that this collision is accompanied by the gravitational radiation, as is expected from the earlier results on the high-energy scattering. The new effect is a secondary spherical gravitational shockwave when the initial shockwave hits the massive particle. In the considered approximation the flux of gravitational radiation and the amplitude of the spherical shockwave are found in an analytic form. The suggested approach is also applicable when the null particle is replaced by plane null shells of a general profile. Implications of these effects for astrophysics are shortly discussed.

We discuss the cosmology of axion-scalar systems in asymptotic limits of type IIB/F-theory flux compactifications. These results allow us to test a putative extension of the Distance Conjecture in a dynamical setting, which posits that towers of states should become exponentially light in the distance measured along the trajectory (as well as in the geodesic one). In the case of infinite distance limits, we review a known classification of late-time asymptotic solutions, which always verify the extension of the conjecture whenever all relevant effects are taken into account. We also extend the analysis to the case of finite distance limits, where the analogous statement would require trajectories approaching the singularity to have a finite length. Surprisingly, we find this is not the case for the class of models under consideration. Moreover, the new solutions we find exhibit asymptotic accelerated expansion when approaching the boundary of moduli space.

We propose a numerical method based on the master field for large-$N$ reduced matrix models. While the master field is originally an infinite-dimensional matrix, in this method it is regularized to a finite dimension, with the requirement that it satisfies the loop equations as much as possible. This formulation can be directly implemented for numerical computation, and since there is no sign problem at the fundamental level, the method can be applied regardless of whether the model is of Euclidean or Minkowski type. In numerical calculations for one- and two-matrix models, the exact solution is well reproduced in the Euclidean case, while perturbative results are well reproduced in the Minkowski case. This demonstrates the effectiveness of the method and supports the idea that the matrix models studied in this paper admit a regularized master-field description.

Understanding the nature of dark matter (DM) is a fundamental challenge in particle physics. In this paper, we investigate the potential of vector boson fusion (VBF) processes at the Large Hadron Collider (LHC) to demonstrate, as a proof of principle, the feasibility of distinguishing between different dark matter scenarios, focusing on Higgs portal DM (HPDM) and neutralino DM in the $2j + \not\!\! E_T$ final state and exploiting the distinctive kinematic features of the VBF jets and the missing transverse energy. Our study reveals that the polarization of weak bosons in VBF plays a crucial role in shaping the transverse momentum distributions of the tagged jets, with the jets being less energetic in the transverse direction for the Higgs portal scenario compared to the neutralino scenario. In addition, the kinematic variables $\Delta\eta$ and $\Delta\phi$ exhibit characteristic differences between the Higgs portal and neutralino DM signals, providing significant discriminating power between these scenarios. We further apply a Kolmogorov–Smirnov test using linear discriminant analysis to quantify the distinguishability of the signals and find that the Higgs portal signals can be differentiated from neutralino DM signals with a C.L. exceeding $5\sigma$, thereby establishing the viability of collider-based discrimination between dark matter models.

We study the Casimir effect for a parallel plate setup with one plate with dynamical edge mode (DEM) boundary conditions, and one plate with perfect electromagnetic conductor (PEMC) boundary conditions. In order to restore BRST invariance, new edge fields are introduced on the DEM plate. We then lift the boundary conditions into the action using Lagrange multiplier fields, and integrate out the bulk fields to obtain a non-local effective boundary theory from which we compute the Casimir energy. The resulting Casimir force is identical to a PMC-PEMC setup, implying that, from the point of view of the Casimir effect, a DEM plate is equivalent to a PMC plate. We also include a detailed derivation of the general functional method used to compute the Casimir energy from the partition function.

In multifield inflation driven by $d$ scalar fields, $O (d)$ symmetry renders the number of fields irrelevant at classical level. This ceases to be the case once stochastic effects are accommodated. The statistical quantities such as the mean number and the variance of $e$-folds as well as the primordial power spectrum and its scale dependence are perturbatively calculated in a small-noise regime. In particular, a general formula is derived for arbitrary higher-order statistical moments of the stochastic number of $e$-folds at all perturbative orders, keeping the dependence on the number of fields fully analytical. It is also discussed that the requirement for inflation to be successfully terminated puts a theoretical bound on the number of fields from above. Those general results are demonstrated for several $O (d)$-symmetric models.

We employ the polar decomposition of the Dirac field to describe it as an effective spinorial fluid. We then construct a $(1+1+2)$ covariant formalism for the Dirac field that avoids the introduction of tetrad fields and Clifford matrices. Within this framework, we analyze the conditions under which a self-gravitating Dirac field can be consistently embedded in Locally Rotationally Symmetric (LRS) space-times of types I, II, and III. In accordance with the LRS symmetry requirements, we extend a previous work by assuming that the velocity and spin vector fields of the Dirac field lie in the planes defined pointwise by the generators of the time-like and space-like congruences, which underlie the $(1+1+2)$ decomposition. We present some analytical and numerical solutions to illustrate the applicability of the proposed framework.

Altermagnets are a novel class of magnetic systems characterized by their momentum-dependent spin splitting without net magnetization. In this work, we extend established Euclidean tight-binding models of altermagnets to regular hyperbolic lattices in two spatial dimensions defined on a discretized Poincaré disk. Using hyperbolic crystallography and Abelian hyperbolic band theory, we show that the inclusion of next-nearest neighbor hopping is sufficient to induce spin splitting in minimal tight-binding models of bipartite hyperbolic lattices. While certain families and special cases of hyperbolic lattices remain antiferromagnetic, we identify an entire family and a special case that generically permit spin splitting in this framework. Hence, altermagnetism is inherent to certain hyperbolic lattices. Since Abelian hyperbolic band theory yields a momentum space that is at least four-dimensional, we classify the leading spin-splitting harmonics using four-dimensional atomic orbitals. As an outlook, we apply non-Abelian hyperbolic band theory to the $\{10,5\}$ lattice. Although its Abelian spectrum is spin degenerate, two selected higher-genus supercells exhibit spin splitting. This finding suggests, that spin degeneracy does not necessarily persist within all translation representation sectors.

We study Euclidean wineglass wormholes, which mediate the nucleation of inflationary spacetimes from an existing spacetime with asymptotically flat or Anti-de Sitter regions. These wormholes are distinguished by the presence of a local maximum of the scale factor, which allows the analytically continued Lorentzian spacetime to expand after materialization. We present explicit numerical wormhole solutions supported either by an axionic field or a magnetic gauge field, in both cases in conjunction with a self-interacting scalar field. More exotic solutions, with multiple extrema of the scale factor, are also described. As we discovered recently, in the limit of small axionic or magnetic charge, wineglass wormhole solutions split into two separate geometries, one being the background spacetime and the other a disconnected no-boundary instanton. We study the associated topology changing transition in detail and provide an extensive discussion of both the properties and puzzles exhibited by this common family of wineglass/no-boundary instantons.

In this article, we solved the time–independent one–dimensional Klein–Gordon equation in the presence of $\alpha$–attractor potential using the Log derivative method. We calculated the reflection coefficient $\mathcal{R}$ and the transmission coefficient $\mathcal{T}$, showing that the superradiance phenomenon is present. In order to demonstrate the accuracy of our method, we performed a comparison with the analytical solution for the hyperbolic tangent potential.

The persistent challenge of creating stable de Sitter vacua within string theory undermines the observational validity of the $\Lambda$ cold dark matter (CDM) model. This difficulty suggests that the concordance model of cosmology, characterized by a constant dark energy $\Lambda$, may reside in the swampland of inconsistent quantum gravity theories rather than the string landscape of consistent ones. Recent observational data, particularly from the Dark Energy Spectroscopic Instrument (DESI), have significantly challenged $\Lambda$CDM cosmology. Specifically, the combination of DESI baryon acoustic oscillation measurements with cosmological surveys seem to indicate a preference for a dynamic, time-evolving dark energy rather than a constant, with roughly 10% reduction in density over the last several billion years. This review summarizes significant advancements made over the past two years in linking DESI findings to string-inspired scenarios.

Several simple asymptotically-free chiral gauge theories are studied. The only “free parameters” of our models are the choice of the gauge group and the matter Weyl fermion representations, and the relative magnitudes of the renormalization-group-invariant scales $\Lambda_i$ associated with each gauge group. None of our models has nontrivial nonabelian global symmetries (“family”–like fermion representations). We rely on some recent theoretical developments on the dynamics of strongly–coupled chiral gauge theories, based on the generalized symmetries and associated new types of anomaly-matching consideration, but also on the solid knowledge on vectorlike gauge theories such as QCD and supersymmetric Yang-Mills theories. The structures of the infrared effective theories, the RG flows, and the light spectra found in these models are surprisingly rich and intriguing.

We study an $O(N)$ scalar multiplet nonminimally coupled to gravity and follow its renormalization-group (RG) flow in the vicinity of an interacting, nonperturbatively UV-complete scaling regime of scalar-tensor theory. In the broken phase, the radial mode mediates a universal Yukawa correction to Newtonian gravity, parametrized by a strength $\alpha$ and range $\lambda$. Imposing UV completeness – regular RG trajectories that reach the UV scaling regime – restricts the infrared data to a finite wedge, which maps to a narrow region in the $(\alpha,\lambda)$ plane. Its complement is, therefore, ruled out by UV completeness alone. Remarkably, part of this theory-excluded domain lies below current experimental exclusion envelopes, so improved fifth-force searches can directly test and potentially falsify this class of UV-complete scalar-tensor models.

We study even non-unital Dirac data on the two-dimensional Moyal plane with fixed deformation parameter \(\vartheta\neq0\), whose Dirac operator is affine in the left and right regular coordinate multipliers, with coefficient matrices \(T_L\) and \(T_R\). Two invariants organize this class. The right block controls algebra–Dirac transversality: local compactness holds if and only if \(\det T_R\neq0\), with \(|\det T_R|\) determining the local volume coefficient and \(\operatorname{sgn}\det T_R\) the oriented Gaussian \(K\)-homology pairing. Independently, \(\Gamma=\vartheta(\det T_L-\det T_R)\) controls the spectral regime and, together with \(|\det T_R|\), the exact localized heat trace: \(\Gamma\neq0\) gives infinitely degenerate Landau levels, while for \(\Gamma=0\) in the transverse regime the spectrum is \(\mathbb R\) with empty point spectrum. Applied to Wallet's Landau–Moyal family, the criterion shows that local compactness survives at the metric-degenerate point \(\xi_{\mathrm W}=1\) but fails at \(\xi_{\mathrm W}=-1\), thereby resolving the local-compactness behaviour at both exceptional one-sided points. We then derive a distinguished affine family from a fixed nondegenerate sector \((\hbar_0,\vartheta_0,B_0)\) of the kinematical symmetry group of two-dimensional noncommutative quantum mechanics. The sector fixes the effective Moyal scale \(\vartheta_{\mathrm{eff}}=\vartheta_0/(1-\vartheta_0B_0/\hbar_0)\) and \(T_R\), while an affine \(U(1)_\star\) background coupling deforms only the left affine part of the Dirac operator. Consequently, local compactness, noncommutative volume and \(K\)-homology class remain unchanged, whereas \(\Gamma=\tfrac12(1-eF_{xy}/B_0)\) drives a transition at \(eF_{xy}=B_0\) from Landau levels to spectrum \(\mathbb R\) with empty point spectrum.

More profound than bulk topological order of quantum materials is only its unwinding via gapless excitations along boundaries of the sample. We recast this bulk-edge correspondence – for the experimentally relevant case of fractional quantum Hall (FQH) systems – in terms of effective relative higher gauge theory, controlled by choices of classifying fibrations. For FQH systems, we identify the complex Hopf fibration as classifying the bulk/boundary topological effects, and find that it yields a non-Lagrangian reconstruction of Floreanini-Jackiw/Wess-Zumino-Witten chiral edge currents. Remarkably, the resulting effective FQH higher gauge theory turns out to be "geometrically engineered" on M2/M5-branes probing A-type orbi-singularities in 11D supergravity, globally completed by flux-quantization in twisted equivariant differential (TED) Cohomotopy: Here the M-string ends of M2-branes on M5-branes engineer the FQH liquid's boundary. This geometric engineering on M-branes might naturally elucidate the curious combination of $W_\infty$-symmetry and of super-symmetry that is known to govern the collective excitations of FQH liquids at long wavelengths.

We study the free particle (FP), harmonic oscillator (HO) and inverted harmonic oscillator (IHO) as parabolic, elliptic and hyperbolic realizations of one conformal/metaplectic structure. Within this framework, we develop a common free-particle construction of HO bound and coherent states and IHO resonance and scattering data. The construction extends the FP–HO conformal bridge to both hyperbolic orientations using real metaplectic rotations through $\pm\pi/4$. These rotations map $2\Omega D$ to $\mp H_{\rm IHO}$ and carry the FP threshold Jordan module into the two Gamow families. Applied to Fourier characters, they give light-cone generating states whose Taylor and Mellin decompositions recover resonance and real-energy connection data, respectively. The familiar IHO scattering amplitudes and their Gamma-function poles thereby enter the same construction. We identify the crossed roles of the elliptic and hyperbolic generators in the two stationary bridges and develop the hyperbolic Cayley–Niederer map as their time-dependent counterpart. The $\mathfrak{osp}(1|2)$ extension keeps track of the two parity sectors and the linear generating operators. A phase-space analysis reconciles exact Liouville transport with nonzero stationary transmission, distinguishing asymptotic quantum currents from classical trajectories across separatrices. We also briefly discuss applications of the common quadratic structure to barrier scattering, time-dependent oscillators and quantum fields in gravitational backgrounds.