Papers

We study the quantised version of the Kadomtsev-Petviashvili equation in the formulation proposed in [4]. We analyse two aspects. One is the conservation of the first higher charge $H_3$ (besides the total mass $H_0$, the momentum $H_1$ and the Hamiltonian $H_2$). We find the expression of $H_3$ and prove commutativity, provided we regularise a specific ultraviolet divergence. The result then does not depend on the choice of a regulator, under fairly general assumptions. We then study the Bethe ansatz in finite volume and obtain a general solution to the two-body problem. We have used AI supervised by us to aid the proofs.

Black holes are conjectured to be maximally chaotic, with their quantum spectra expected to exhibit universal random-matrix correlations. This expectation, however, has so far rested largely on holographic descriptions or on many-body models in which randomness is introduced rather than derived from the microscopic degrees of freedom of the horizon itself. In the horizon matrix model, the fuzzy sphere supports highly degenerate parton states whose multiplicity accounts for the black-hole entropy. We show that quantum fluctuations of the horizon geometry lift this degeneracy and generate an ensemble of one-parton Hamiltonians. Both the operator structure and the rank-dependent covariance of this ensemble are determined by the microscopic model, rather than fixed by quenched couplings. Despite this structured covariance, the resulting spectra exhibit GOE, GUE, and GSE statistics dictated by their antiunitary symmetry classes, with the adjacent-gap statistic becoming increasingly self-averaging with matrix size. The model also realizes the universal Pandey–Mehta orthogonal-to-unitary crossover, with the crossover scale determined by the microscopic geometric covariance. A rank-resolved variance analysis shows that the fluctuation strength remains broadly distributed across tensor ranks up to the microscopic cutoff, providing a microscopic basis for the observed Wigner–Dyson universality and effectively nonlocal dynamics. The antiunitary symmetry acts directly on the fuzzy-sphere geometry, so the Dyson class of a conditional Hamiltonian is determined by the antiunitary symmetry of the underlying quantum geometry. This provides a microscopic counterpart of the Stanford–Witten relation between geometry and random-matrix universality in JT gravity. These results establish a microscopic route from quantum horizon geometry to black-hole spectral chaos.

We uncover the quantum geometric structure of graviton Wigner functions and stress-energy tensors via quantum field theory in curved spacetime, revealing the phase-space geometry of spacetime and relativistic quantum-state manifolds. We show that the polarization mode expansion of the underlying graviton field operator is fully captured by quantum geometry, as encoded in the spinor-helicity formalism, thereby establishing the geometric framework from the outset. Applying this to a weak gravitational background, we demonstrate the roles of the quantum metric and connection in describing graviton transport beyond the chiral vortical effect. We also clarify the role of the quantum metric in normalizing the perfect-fluid graviton stress tensor within this approach. This work paves the path for explorations of multistate Hilbert-space geometry in high-energy and gravitational physics. In addition, this framework naturally encompasses lower-spin excitations, allowing for a unified quantum geometric treatment of bosonic and fermionic many-body systems in condensed matter and particle physics.

We investigate whether the recently reported high-energy nuclear recoil event by the LUX-ZEPLIN (LZ) collaboration can be interpreted as a signal of freeze-in dark matter (DM) in a non-standard cosmological history. We consider DM production from the Standard Model thermal bath during an era of modified expansion preceding Big Bang nucleosynthesis. Requiring consistency with the observed DM abundance, cosmological and collider constraints, we identify viable parameter regions that can accommodate the LZ event. Our results demonstrate the potential of direct detection experiments to probe non-standard cosmological histories and, more broadly, alternative theories of gravity.

We construct $S^1$-invariant solutions of the $SU(3)$ pure Yang-Mills theory on four-dimensional Euclidean space using the Cho-Faddeev-Niemi decomposition and a harmonic map from $S^2$ into the flag manifold $F_3=SU(3)/U(1)^2$. The resulting ansatz reduces the full Yang-Mills equations to coupled radial equations. Solving the coupled equations, we derive two types of solutions: analytic self-dual solutions interpreted as noninteracting clusters of hyperbolic monopoles and non-self-dual numerical solutions that can be interpreted as hyperbolic monopole-antimonopole bound states. In the large-mass limit of the hyperbolic monopole-antimonopole bound states, their normalized action approaches the energy of the corresponding non-Bogomolny $SU(3)$ monopole in flat space.

The LUX-ZEPLIN (LZ) experiment has reported a high-energy nuclear recoil candidate, LZ230616, at $E_R = 248 \pm 23_{\rm stat} \pm 23_{\rm sys}\,$keV, which is difficult to reconcile with elastic scattering of halo dark matter (DM). Endothermic inelastic scattering offers a natural explanation, but it rests on a nucleon coupling that is off-diagonal rather than diagonal. We show that this feature arises automatically within a radiative Dirac neutrino mass framework. The Standard Model is extended by three right-handed neutrinos (RHNs), a pair of vector-like neutral fermions for each generations, and a scalar sector comprising an inert doublet and a real singlet, governed by a $\mathcal{Z}_2 \times \mathcal{Z}_4$ symmetry. The $\mathcal{Z}_2$ symmetry stabilizes the DM, while the $\mathcal{Z}_4$ symmetry forbids the tree-level Dirac Yukawa coupling together with all renormalizable Majorana mass terms, thereby allowing the generation of Dirac neutrino masses at the one-loop level through a softly broken scalar trilinear coupling `$\kappa$'. Since the neutral dark-sector fields can be expressed in terms of real scalar mass eigenstates, the $Z$ boson couples to them purely off-diagonally and elastic $Z$-mediated scattering is absent identically rather than simply suppressed. The singlet–doublet mixing induced by `$\kappa$' governs both the inelastic rate and the one-loop Dirac neutrino mass, which therefore vanish as $\kappa \to 0$. We find that DM masses in the few hundred GeV to TeV range, with splittings of $\mathcal{O}(340-360)\,$ keV, simultaneously reproduce the observed relic abundance, account for the LZ event and yield neutrino masses of the correct order, while respecting elastic direct-detection limits and all relevant theoretical and experimental constraints. A substantial part of the surviving parameter space lies within reach of DARWIN.

We develop Geometric Invariant Theory for affine superschemes under the action of reductive algebraic supergroups, formulating the theory in terms of coordinate Hopf superalgebras in order to accommodate anticommuting and nilpotent variables. Within this algebraic setting, we construct affine superquotients and establish their basic structural properties under suitable hypotheses. We then prove a supergeometric analogue of the Hilbert-Mumford criterion, giving a numerical test for the semistability and stability of points under natural constraints on the supergroup coaction. We show that this numerical criterion can be used to construct the GIT superquotient of these superschemes, and we give examples illustrating how the supergeometric GIT superquotient differs from its classical counterpart.

The LUX-ZEPLIN experiment has reported a single nuclear recoil candidate at $E_R = 248 \pm 23(\mathrm{stat}) \pm 23(\mathrm{sys})$~keV, disfavouring the background-only hypothesis at a global significance of $2.6\sigma$. The difficulty such an event poses is not the recoil energy itself but the absence of any accompanying excess at low energy as elastic scattering of halo dark matter yields a monotonically falling spectrum, and supplying the required momentum transfer $q \simeq 246$~MeV already demands $m_\chi \gtrsim 79$~GeV. We confront the event with the two kinematically distinct mechanisms that evade this limitation, endothermic inelastic dark matter, in which a mass splitting $\delta \sim \mathcal{O}(100)$~keV forbids low-energy recoils and boosted dark matter, in which a light relativistic flux supplies the momentum, treating both with the same model-independent scalar–scalar, pseudoscalar–scalar and pseudoscalar–pseudoscalar effective operators. The two scenarios prove spectrally distinguishable. The inelastic spectra sit near the observed energy for every operator, placing only $6$–$17\%$ of events below 150~keV, whereas the boosted spectra depend critically on the operator: the scalar and pseudoscalar–scalar interactions place $99\%$ and $92\%$ of their events below 150~keV, while the pseudoscalar–pseudoscalar interaction places $75\%$ above it. Momentum dependence is thus essential to the boosted interpretation, but in the inelastic case it trades against the splitting, the preferred $\delta$ decreasing monotonically from $\mathcal{O}_{ss}$ to $\mathcal{O}_{ps}$ to $\mathcal{O}_{pp}$. Because the scenarios differ across the whole high-energy window, a handful of additional events would separate them, placing the question within reach of the full LZ exposure.

We consider collisions between particles moving freely in the Schwarzschild background. We suggest classification of scenarios that lead to unbounded energy output in the center of mass frame. Although this is forbidden in the exterior of a black hole, the phenomenon becomes possible with account of white hole region and mirror universe in the complete space-time diagram. Also, we scrutinize collisions in the vicinity of the bifurcation point and reveal the role of resting observers in the inner region beyond the horizon. In some scenarios $E_{c.m.}$ not only diverges but becomes seemingly infinite that is, as a matter of fact, impossible according to the principle of so-called kinematic censorship. We resolve a corresponding paradox and argue, why this principle remains valid.

In this paper, we investigate the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity formulated on a three dimensional Riemannian manifold. We derive the gap equation in the path integral picture by employing a Hubbard-Stratonovich transformation followed by a saddle point approximation. Under the assumption that the length scale set by the metric is much larger than the healing length, we solve the gap equation to find a slowly varying gap function that exhibits the effects of curvature. To rigorously address the ultraviolet divergences inherent in the gap equation, we utilize heat kernel expansion techniques, which provide a systematic and mathematically transparent renormalization scheme. We explicitly evaluate the renormalized gap equation at both zero and finite temperatures, computing the relevant integrals asymptotically in the weak-coupling limit ($\Delta \ll \mu$). Furthermore, we establish a universal, regularization-independent relation connecting the finite-temperature gap to the zero-temperature gap and the temperature itself. Based on this fundamental relation, we analytically prove the monotonic decrease of the energy gap with increasing temperature ($\partial \Delta / \partial T < 0$), successfully recovering the standard universal BCS behavior near the critical temperature within a curved space framework.

In Einstein-Cartan spinor gravity under the irreversible vierbein postulate, only the spinor among the first-order variables has a kinetic term of its own at tree level. Any nonzero background vierbein breaks the local-Lorentz (LL) symmetry spontaneously. Our previous work showed that the spinor loop generates the kinetic and mass terms of the LL gauge field, manifesting LL as a hidden local symmetry. Here we show that the spinor loop also generates the vierbein two-point function, while respecting the background gauge invariance and satisfying the Ward-Takahashi identities of the true gauge invariance, for both the general-coordinate and LL symmetries. The scheme-independent logarithmic divergence assembles the familiar induced-gravity ingredients: a cosmological constant, an Einstein-Hilbert and a Weyl-squared term for the metric sector, and a mass and a higher-derivative kinetic term for the totally antisymmetric torsion. In the vierbein sector, this torsion is carried by the derivative of the antisymmetric part of the linear vierbein fluctuation. This part is the would-be Nambu-Goldstone boson of the spontaneously broken LL symmetry, and its couplings in the minimally coupled action are all fixed by general-coordinate and LL gauge invariance. The five induced terms form a single effective action, manifestly invariant under both symmetries, completing the vierbein sector of the hidden-local-Lorentz emergent-gravity scenario.

There is a 4N-dimensional moduli space of SU(2) BPS monopoles with charge N, but introducing a Higgs potential removes this degeneracy and yields a repulsive force between monopoles. However, if a point in the BPS monopole moduli space is sufficiently symmetric, then it flows to a similar non-BPS saddle point solution once the Higgs potential is introduced. Monopole fission, into constituent monopoles with lower charges, results from symmetry breaking perturbations of such saddle point solutions. This fission is investigated numerically, by evolving the gradient flow equations of the Yang-Mills-Higgs theory, with initial conditions and symmetry breaking perturbations created using rational maps between Riemann spheres. The results resemble components of BPS monopole scattering processes obtained within the geodesic approximation for low speed monopole dynamics.

We study extremal mass branches in four-dimensional Einstein gravity coupled to one dilaton and two Maxwell fields with opposite-sign dilaton couplings. For generic couplings the extremal mass is governed by a nonlinear first-order equation with no known closed-form solution. We prove that this equation has a unique positive $C^2$ solution on the full real charge-ratio line and that the solution is real analytic, so positivity on the entire positive charge cone selects a unique global mass branch. Along this branch, any two nonproportional positive charge vectors attract for $ab>-1$, have vanishing leading force at $ab=-1$, and repel for $ab<-1$. The same transition fixes the transverse convexity of the extremal mass surface and the sign of the mass difference under arbitrary finite positive-charge partitions. The sign theorem extends to higher dimensions through the known dimensional rescaling. A radial reconstruction then shows that every point of the mass branch defines an asymptotically flat exterior with a finite-area inner endpoint, while smooth or analytic horizon extension is a separate condition that restricts the couplings and scalar flow.

We study the quantization of an M2 brane wrapping $\mathrm{AdS}_2\times S^1$ inside $\mathrm{AdS}_4\times S^7/\mathbb{Z}_k$, dual to the $\frac{1}{2}$-BPS circular Wilson loop in ABJM theory. The classical action and the one-loop contribution were previously shown to reproduce the gauge-theory localization result. Here we extend the analysis to the two-loop order. In an adapted static and $\kappa$-symmetry gauge the world-volume action contains no cubic interaction vertices, so that the two-loop correction is determined entirely by quartic interactions and is manifestly free of logarithmic UV divergences. We find that the bosonic and fermionic contributions combine into a perfect square $ {\mathscr X}^2$ where ${\mathscr X}$ is a linear combination of the bosonic and fermionic 3d Green's functions and their derivatives. The evaluation of $\mathscr X$ requires fixing the $\mathrm{AdS}_2$ boundary quantization of the massless fermionic modes. With the boundary conditions selected by the preserved supersymmetry, $\mathscr X$ vanishes and hence the complete two-loop correction is found to be zero. This result would not be consistent with identifying the M2-brane loop expansion directly with the large-$N$ expansion of the canonical, fixed-$(N,k)$ localization prediction. Instead, it agrees precisely with the recent proposal that the semiclassical M2-brane partition function should be matched to the gauge-theory grand-canonical ensemble, in which the Wilson loop expectation value is perturbatively one-loop exact. The vanishing of the two-loop correction thus provides a non-trivial test of this grand-canonical identification.

In this paper, we study the decomposition, in the caterpillar or weak-coupling limit, of a spectral curve with two irregular poles of order two, namely a spectral curve of type $(2,2)$, into two spectral curves each having one regular singularity(order-$1$ pole) and one irregular pole of order two, namely spectral curves of type $(1,2)$ or $(2,1)$. Such spectral curves can be identified with a double Gelfand–Zeitlin (DGZ) system. We show that the Duistermaat–Heckman measure on the DGZ side can be identified with the Weyl measure on the gauge-theory side arises from gluing two quivers. Furthermore, the Fourier expansion of the isomonodromic tau function can be interpreted as a Peter–Weyl expansion in the real polarization of the DGZ system. On the other hand, we introduce the nodal caterpillar spectral network. Applying the caterpillar spectral network, we then obtain an explicit expression for the Stokes matrix up to a normalization factor. This expression agrees with the existing analytic formulas. Moreover, we identify the variation of endpoint normalization arising in the Stokes computation with the equivariant Euler class of the fluctuation complex over the Coulomb moduli space. Finally, using the gluing technique for spectral curves and the DGZ system, we conjecturally predict the perturbative coefficients of the higher-rank $P_{\mathrm{III}}$ isomonodromic tau function from the viewpoint of spectral geometry.

We study how the ground-state properties of the rho meson respond when the number of light-quark flavors N_f is driven towards the conformal window. The interaction is a symmetry-preserving, confining vector–vector contact interaction whose effective coupling carries a flavor dependence designed to encode the screening produced by fermion loops in the deep infrared. That interaction feeds the Schwinger–Dyson equation for the dressed-quark propagator and the homogeneous Bethe–Salpeter equation in the rainbow-ladder truncation, regularized in the Schwinger proper-time scheme so that quark production thresholds are absent while chiral symmetry remains broken. Raising N_f weakens the coupling, suppresses the dynamically generated mass and restores chiral symmetry together with deconfinement at a critical flavor number N^{c}_{f}. The vector channel responds differently from the pseudoscalar one. Because the $\rho$ is not a Goldstone boson, its mass is not protected by the symmetry that is being restored, and it simply follows the shrinking constituent scale, while the canonically normalized amplitude E_rho and the leptonic decay constant f_rho soften along the way. In this framework the rho sits above the nominal q\bar{q} threshold for every flavor number examined, so the state survives only as long as confinement removes that threshold. Dissociation is therefore governed by the divergence of the confinement length scale rather than by a crossing of the bound-state mass, and we contrast both criteria. The three elastic electromagnetic form factors G_E, G_M and G_Q, computed in the generalized impulse approximation, flatten as N_f grows, the zero crossing of G_E migrates towards the infrared and the charge radius expands, whereas the magnitude of the quadrupole moment decreases. The rho thus becomes a larger and rounder object as the theory approaches conformality.

Entanglement harvesting provides an operational means of probing the nonlocal correlation structure of quantum fields through localized detector-field interactions. Although harvesting has been investigated in a wide variety of spacetime backgrounds and quantum field states, existing studies largely focus on individual detector configurations or specific physical settings. In this work, we develop a unified perturbative framework for entanglement harvesting with localized Unruh-DeWitt detectors that is applicable to arbitrary detector trajectories and quantum field states through the corresponding Wightman function. By expressing the detector response in terms of dimensionless variables and analytically reducing the perturbative integrals to numerically efficient representations, the framework enables a systematic exploration of multidimensional harvesting landscapes, phase boundaries, and harvesting robustness. As representative applications, we investigate inertial detectors in the Minkowski vacuum, thermal field states, and uniformly accelerated observers. We show that finite temperature enhances both the local detector response and the nonlocal exchange amplitude, leading, within the perturbative Gaussian-switching framework considered here, to an enlargement of the harvesting region in parameter space. The resulting harvesting landscapes provide a global characterization of detector-field interactions and demonstrate how temperature and observer motion reshape the competition between local detector noise and nonlocal field correlations. The framework developed here provides a unified and readily extensible methodology for investigating entanglement harvesting across a broad class of relativistic quantum systems.

We study the free field realization of the $\mathcal{W}$-algebra associated with the exceptional Lie algebras $E_6$, $E_7$, $E_8$, and $F_4$. We develop a recursive construction in which a $\mathcal{W}$-algebra of rank $r$ is obtained from a $\mathcal{W}$-algebra of rank $r-1$ together with a free boson. The $\mathcal{W}$-currents are constructed from the zero commutation relation with the screening charges. The $\mathcal{W}E_6/\mathcal{W}E_7$ algebra is constructed from the $\mathcal{W}D_5/\mathcal{W}D_6$ algebra and is shown to be the same as that realized from the $\mathcal{W}A_5/\mathcal{W}E_6$ algebra, up to a change of the free field basis. The spin-$8$ generator of the $\mathcal{W}E_8$ algebra is built from the $\mathcal{W}D_7$ algebra. The recursive construction of the $\mathcal{W}BC_r$ algebras is also studied. We then realize the $\mathcal{W}F_4$ algebra based on the $\mathcal{W}BC_3$ algebra. Furthermore, the $\mathcal{W}$-charges of the generators of the $\mathcal{W}E_{6,7}$, $\mathcal{W}BC_{2,3}$, and $\mathcal{W}F_4$ algebras are calculated and expressed in terms of the Casimir invariants.

In this paper we study the relations between the Hamiltonian hierarchy generated by trigonometric Cherednik-Dunkl operators and the one generated by rational Dunkl operators. We develop two nested structures relating these two hierarchies. The first nested relation reconstructs the higher rational spin Calogero-Sutherland Hamiltonians exactly from the trigonometric ones. The second nested relation reconstructs the higher trigonometric spin Calogero-Sutherland Hamiltonians as leading terms from the rational ones. In addition, we provide an explanation of the eigenvalues of the trigonometric spin Calogero-Sutherland Hamiltonians in terms of $N$-colored Young diagrams and Maya diagrams.

We study a thick braneworld model in asymptotically anti-de Sitter spacetime, supported by a non-topological lump-type scalar field. Unlike in boson stars, which are also non-topological solutions, the complex scalar field in our setup must have zero oscillation frequency. Consequently, the background scalar field is static and real, which breaks the global $U(1)$ symmetry. This is similar to spontaneous symmetry breaking, but it is not the same. We then investigate the localization of vector and fermion fields within this framework. By introducing a specific non-minimal coupling, we demonstrate that the zero modes of both fields can be successfully localized. Notably, this mechanism allows for the simultaneous localization of both left- and right-handed fermion zero modes.

Pseudo entropy extends entanglement entropy from a single quantum state to a pair of nonorthogonal states and is generally complex. Taking advantage of Cauchy-Riemann equations, Kramers-Kronig relations, and analytic continuation in state parameters, we show how and to what extent real and imaginary parts of pseudo entropy can be derived from ordinary entanglement entropy. For families with holomorphic coefficients in finite-dimensional Hilbert spaces, the reduced transition matrix equals the ordinary reduced density matrix formula evaluated at complex parameters. Then a convergent Taylor series gives the real and imaginary parts of pseudo entropy from even and odd derivatives of entanglement entropy at the real midpoint. The matrix identity also gives formulae for excess pseudo entropy and Renyi entropies, and interpolation formulae for families with polynomial coefficients. We apply these results to boundary-state quenches and thermal states in conformal field theory, and to fermionic and bosonic Gaussian states and quenches. In conformal field theory, Kramers-Kronig relations give the first moment of the imaginary part in terms of the central charge and one-point functions for boundary states.

This paper investigates the holographic Casimir effect in AdS/BCFT with a brane-localized scalar field. The scalar field takes different values at the strip's two boundaries, leading to mixed boundary conditions. The massive brane-localized scalar field typically breaks the boundary conformal symmetries, allowing us to study the holographic Casimir effect in non-CFTs. In two-dimensional spacetime, we apply the holographic g-theorem to prove the holographic bound on the Casimir effect for general brane-localized matter fields. In higher dimensions, we find that the brane-localized scalar field reduces the Casimir amplitude when the mass squared \(m^2=0\), but can increase it when \(m^2 < 0\). Additionally, there is a no-hair theorem for the cases where \(m^2>0\). These findings indicate that relevant boundary deformations can enhance the Casimir effect. As a byproduct, we obtain a gravitational dual of the repulsive Casimir force under mixed boundary conditions and argue that it aligns with the cosmic censorship conjecture.

Ringdown gravitational waves from binary black hole mergers can be modeled as superpositions of quasinormal modes (QNMs), whose frequencies and excitation factors encode properties of the remnant Kerr black hole. Reliable extraction of multiple QNM components is challenging because of mode overlap and noise. We develop an autoencoder-based framework for multi-component QNM analysis, in which the latent space is trained to represent the physical parameters of individual modes, enabling waveform denoising and parameter estimation within a common framework. Using controlled model waveforms constructed as finite sums of Kerr QNMs with recently established high-precision frequencies and excitation factors, including their nontrivial spin dependence near resonant excitation, we assess the method across partitioned spin intervals. The model achieves good in-domain waveform reconstruction and parameter recovery for the two longest-lived components of eight-component input waveforms, while its performance degrades when the validation spins lie far outside the training range. In a selected spin interval, the framework also recovers the 32 parameters of an eight-component waveform with good overall agreement. These results demonstrate the feasibility of physics-informed autoencoder-based inference for a prescribed multi-component ringdown waveform family and motivate further tests with progressively more realistic signals.

We investigate primordial black hole (PBH) formation during an early matter-dominated era using fully nonlinear numerical relativity. The initial condition is set by a functional form of the curvature perturbation including ellipticity, which makes the configuration triaxial. Two kinds of matter descriptions are considered: dust fluid and collisionless particles. In the dust fluid description, numerical computation crashes associated with the appearance of a singularity at which the fluid density diverges, unless the singularity is hidden well inside the apparent horizon. We found that, for the dust fluid description, to observe the horizon formation before calculations crash, the initial amplitude must be larger than the previous analytic estimation by a factor of 2. On the other hand, with the particle system description, calculations do not crash, and we may observe black hole formation after subsequent evolution of the system. Then the threshold of black hole formation is significantly smaller than the previous analytic estimation by an order of magnitude.

We propose a new class of sigma models that extend AKSZ sigma models, which we refer to as gauged AKSZ sigma models. These models are constructed using QP-manifolds. The compatibility between the AKSZ structure and gauging is characterized by geometric conditions on the target graded manifold. As an example, we explicitly analyze the gauging of higher Courant sigma models in higher dimensions. When fluxes and boundaries are introduced into gauged higher Courant sigma models, generalizations of momentum maps and Hamiltonian $G$-spaces arise as consistency conditions for the models.

In the context of the AdS/CFT correspondence, the Jafferis-Lewkowycz-Maldacena-Suh (JLMS) formula relates the boundary modular Hamiltonian to the HRT area operator and the corresponding bulk modular Hamiltonian. In standard discussions of a holographic code built from perturbative excitations on a fixed classical background, the $A/4G$ term dominates over the bulk modular Hamiltonian in the limit $G \to 0$. However, this setup is insufficient to describe the action of boundary modular flows that change the classical background. In contrast, our recent discussion of `large' codes provides a useful framework to obtain a JLMS formula that relates bulk and boundary modular flows acting on appropriate states in the code. Here, one sees that the bulk modular Hamiltonian is not in fact subdominant when the modular flow defined by one state acts on another state peaked around a sufficiently different classical background. Instead, the bulk modular Hamiltonian can be of order $1/G$ and can dominate over the $A/4G$ term. Nevertheless, we show that if the two states have the same classical background except for data conjugate to the HRT area then, under mild assumptions and over a large range of flow parameters, the modular-flowed state is described by a classical background given by the HRT-area flow alone.

The ringdown of a black hole following a merger is a potential candidate for revealing the signatures of quantum gravity in the emerging gravitational waves. In quantum theory, black holes are expected to have a discrete area and energy spectrum, which conflicts with the classical notion of an horizon that absorbs all infalling perturbations. We propose that the quantum black hole dissipates the energy it cannot absorb by emitting “soft” gravitons that carry away the energy difference between the energy of the infalling perturbation and the quantum transition energy. We find that the ringdown spectrum is consequently enriched with low-frequency components, and that there exists a weak low-frequency flux that persists for a timescale much longer than the ringdown itself.

A collection of evidence is presented showing that the conjectured reverse isoperimetric inequality (RII) for asymptotically anti-de Sitter (AdS) as well as de Sitter (dS) black holes, ${\cal R}_\text{RII} \ge 1$, is of fundamental importance and its violation (${\cal R}_\text{RII} < 1$) gives rise to either thermodynamic instabilities, or physically unreasonable solutions, or naked singularities. We show that AdS black holes violating reverse isoperimetric inequality, known as superentropic black holes, satisfy neither mechanical stability requirement of ${\kappa _T} \geqslant {\kappa _S} \geqslant 0$ nor thermal stability requirement of ${C_P} \geqslant {C_V} \geqslant 0$. This property makes them thermodynamically unstable for the whole range of parameter space and explains all the diverse behaviors so far reported. We conjecture this statement holds for all superentropic black holes. We confirm that there is no counterexample to the thermodynamic instability conjecture of superentropic black holes and extend this to dS space by presenting the first example of superentropicity in dS space. By bringing evidence, we then propose two other conjectures: 1) there is no asymptotically flat limit of superentropic black holes, and 2) there exist two stronger versions of Penrose isoperimetric inequality (PII) for asymptotically flat black holes; the first is the asymptotically flat limit of the RII and the second is a new inequality in terms of the ADM mass and the thermodynamics volume, that we call thermo-volumetric inequality, obtained via insights from the $\Lambda \to 0$ limit of extended black hole thermodynamics in AdS. We show that the Penrose isoperimetric inequality (PII) is weaker than the reverse isoperimetric inequality (RII), as demonstrated explicitly for $D=4,5$ and for all black hole families we have examined.

The conventional grid-based approach used for storing and evaluating ordinary parton distribution functions (PDFs), as implemented for example in LHAPDF, becomes much more demanding for double parton distributions (DPDs). While a collinear PDF depends on one longitudinal momentum fraction and one factorization scale, an unequal-scale DPD depends on two momentum fractions and two independent factorization scales for every parton-flavor combination. Direct tabulation of this four-dimensional dependence therefore requires considerably more disk space and runtime memory. In this work, we develop a hybrid neural-network method for compact storage and fast evaluation of $y$-independent unequal-scale DPDs. The network compresses the reference grid by reproducing most of its values, while the original values are stored only at points where the neural prediction does not reach the required accuracy. In this way, the neural model captures the bulk of the grid, with a sparse table covering the remaining difficult points. For the GS09-based dense grid studied here, only $0.694\%$ of the active flavor values need to be stored in the sparse table. Compared with the compressed dense grid, the hybrid method reduces the disk footprint by a factor of $10.6$, peak memory use by a factor of $31.6$, and initialization time by about a factor of $10.6$. At the same time, its evaluation throughput reaches about $85\%$ of the dense-grid rate. The method is implemented natively in C++ within PDFxTMDLib and requires neither Python nor PyTorch at runtime. The hybrid scheme therefore offers much lower memory and startup costs, with only a modest reduction in evaluation speed.

We propose a new product entropy, defined as the Rényi (or von Neumann) entropy of a normalized product operator constructed from two density matrices. We establish a duality showing that the SVD entanglement entropy of a subsystem for two pure states is exactly equivalent to the product entropy of the complementary subsystem. This connection provides both a transparent physical interpretation in terms of the spectral diversity of the subsystem state product and a computationally efficient route that bypasses the reduced transition matrix. For low-lying eigenstates, we derive analytical expressions for the subsystem product entropy between the ground state and primary excitations in two-dimensional conformal field theories, explicitly verified against the critical Ising chain. In quantum quench dynamics, the quasiparticle picture yields time evolution in the scaling limit: following a global quench, the subsystem product entropy exhibits distinct sequences of thermalization and revivals, whereas under a local operator quench, it develops characteristic plateaus whose constant values are determined by the inserted operator. Extensive numerical calculations on the critical Ising chain confirm the analytical predictions with excellent accuracy.