Papers

We investigate the stretching of a spherically symmetric solution in the Skyrme model. The three-dimensional generalization of the Cassini ovals is considered as ansatz of extension. The linear dependence of static energy on the longitudinal deformation parameter is obtained with a slope $\sigma \approx 0.14\ \mathrm{ GeV}^2$, which is in good agreement with the string tension known from various models of confinement.

We investigate holographic entanglement entropy and its generalization to pseudo-entropy in a boosted BTZ black hole geometry, which is holographically dual to a moving strongly coupled plasma. By solving the extremal surface equation, we derive an exact closed-form expression for the holographic entanglement entropy as a function of the boundary interval length, boost velocity, temperature, and conserved quantities. The resulting exact analytic formula correctly reproduces the standard thermal BTZ entanglement entropy in the vanishing-boost limit and smoothly reduces to the vacuum conformal result at zero temperature, thereby providing robust consistency checks. Motivated by the off-diagonal structure of the boosted metric arising from the Lorentz transformation, we then study two families of analytic continuations to Euclidean signature. Both require a simultaneous continuation of the time coordinate and the boost parameter so that the Euclidean metric remains real. If only these two parameters are continued, the boundary interval becomes imaginary and the analytically continued holographic entanglement entropy is complex-valued; we interpret this complex entropy as the holographic dual of pseudo-entropy for the boosted thermal state. If, in addition, a conserved charge is also continued, the boundary interval remains real and the resulting entropy is real. Our construction establishes a direct connection between boosted HEE and holographic pseudo-entropy, extending previous analytic frameworks developed for rotating BTZ geometries to Lorentz-boosted spacetimes.

We examine the round Euclidean no-boundary saddle proposed by Liu, Quintin, and Afshordi in "Ultraviolet Completion of the Big Bang in Quadratic Gravity." For a free, massless, conformal realization of their large matter sector, the omitted type-A trace anomaly contributes at leading order in the matter multiplicity. Within the one-loop renormalization-group-improved pure quadratic truncation, both the sphere effective action and the local trace equation show that a stationary point of the running $R^2$ coupling supports a round sphere only if the total Euler anomaly coefficient vanishes. Using the proposed beta function, we further find that no real stationary coupling ratio exists when this coefficient exceeds $79/420$, a bound violated parametrically by the assumed matter sector when the remaining gravitational contributions are perturbatively controlled and of order unity. The local obstruction also applies to a round half-sphere and cannot be removed by shifting its radius. Under these assumptions, the proposed round starting geometry and its stated continuation to Lorentzian inflation are excluded, without ruling out inflation from other initial conditions.

We show that anomaly-free second-order supersymmetry (SUSY) in quantum mechanics admits a projective-geometric formulation in terms of a projective connection and a vector field preserving it. Once the physical coordinate and energy origin are fixed, these data determine the partner Hamiltonians and supercharges; the field's quadratic invariant fixes the central parameter and candidate singlet energies. The fixed-point structure of this field yields explicit criteria for smooth continuation of the quantum operators through superpotential zeros. The classically fictitious similarity transformation selects affine data whose covariant factors generate the Schwarzian quantum correction. Dimensional reduction of a Riemannian model reproduces these affine data through the volume measure, while a coordinate adapted to the selected field implements coupling-constant metamorphosis. A one-gap Lame pair illustrates the relation between the SUSY auxiliary geometry and the Gelfand-Dikii-Virasoro structure governing physical solution products. Its Lax-Novikov integral acts on Schrodinger solutions through the first-order action of a projective stabilizer; in the reflectionless limit it becomes proportional, in the physical coordinate, to the auxiliary third Bol operator.

We determine the electrostatic Dirichlet-to-Neumann map of an arbitrary four-dimensional Majumdar–Papapetrou spacetime. For a minimally coupled spectator Maxwell field, the potentials of the disconnected extremal horizons may be prescribed independently, and the resulting flux charges are related by an exact capacitance matrix. If the centers have masses $M_A$ and coordinate separations $R_{AB}$, its entries are $C_{AA}=M_A+\sum_{B\ne A}M_AM_B/R_{AB}$ and $C_{AB}=-M_AM_B/R_{AB}$. The matrix is a positive diagonal term plus the weighted Laplacian of the complete graph of horizons, giving an exact common-mode and differential-mode decomposition of the field energy. The grounded Dirichlet Green function shows that a point charge partitions its flux among infinity and the horizons according to elementary harmonic measures. A direct translation of the multicenter Green function of Frolov and Zelnikov gives the diagonal completion used in their published Eq.~(68), whose component-resolved Gauss flux preserves the total horizon charge but generally transfers charge between horizons. The symmetric Green function that preserves every horizon charge instead contains the inverse of the full capacitance matrix. For a binary, the difference gives a finite, negative-semidefinite shift of the electrostatic self-energy and a corresponding exact self-force difference without an additional ultraviolet subtraction. We finally extend the response matrix, flux partition, and fixed-charge Green function to $D=n+3$ dimensions, where the graph weights acquire the universal factor $\pi^{n/2}/\Gamma(n/2)$.

In this paper we study the Borel summability and quantum modularity of the free energy computed by refined topological recursion on the Weber-type refined spectral curve. On the one hand, we verify expected properties, namely, Stokes rays come in a pair and the Borel-Laplace sum of the free energy coincides with the Barnes double gamma function. Somewhat unexpectedly, on the other hand, we prove that the generating function of the Stokes constants along each ray is a Jacobi quantum modular function. Our proof suggests that quantum modularity may be a general feature of refined topological recursion, applicable beyond the Weber-type refined spectral curve.

We investigate the question of equivalence between the Einstein and Jordan frames for generic scalar-tensor and $f(R)$ gravity theories in the context of minisuperspace quantum cosmology. We consider, particularly, the minisuperspace of a homogeneous and isotropic universe with the Friedmann-Lemaître-Robertson-Walker metric. We show that the equivalence between the frames depends on the operator ordering scheme of the quantum Hamiltonian: if the quantum theory is covariant under point canonical transformations in the minisuperspace, then the equivalence between the two frames holds. We find that for a large class of operator orderings, including the Laplace-Beltrami operator in the minisuperspace, the wave functions of the universe obtained in the two frames are related by the same transformations that connect the two frames, i.e., the frames are quantum mechanically equivalent. Further, we show that adding a fluid clock with an arbitrary equation of state parameter $w$ to the system breaks the equivalence, except for the case of a radiation fluid ($w=1/3$). All our arguments are presented at a formal level in both canonical and path integral approaches, and for any scalar-tensor, $f(R)$ theories without assuming any particular model.

At large $N$, the analytic structure of holographic thermal correlation functions contains non-trivial information about the near-singularity region of dual black holes. In particular, complex-time singularities can be related to geodesics that probe the black hole interior and bounce off the singularity. We analyse the asymptotic behaviour of holographic OPE data entering scalar two-point functions in states dual to black branes whose blackening factors truncate at finite order in inverse powers of the radial coordinate. For this class of geometries, we find a direct relation between the power-law exponent of these bouncing singularities and the Kasner exponent associated with the time direction near the black hole singularity. We discuss the regime of validity of this relation and comment on possible extensions to the transverse spatial directions.

We propose a novel method for constructing ghost-free multi-field higher-derivative scalar-tensor theories. The idea is to start from spatially covariant gravity and couple it to an additional scalar field. The resulting spatially covariant scalar field theory can be understood as a bi-scalar-tensor theory in the unitary gauge of one scalar. Within a broad class in which metric velocities enter through the extrinsic curvature and the additional scalar enters through temporal derivatives up to second order, we perform a Hamiltonian constraint analysis. We show that the theory generally propagates five physical degrees of freedom (DOFs), i.e., two tensor modes and three scalar modes. One of the scalar modes is the unwanted mode associated with the additional scalar. We then derive a degeneracy condition and an additional consistency condition. When both conditions hold, the unwanted mode is eliminated and the theory propagates four degrees of freedom. The resulting theories fall into two cases, depending on whether the preservation of the tertiary constraint closes without a new constraint or generates a quaternary constraint. We also study the two-field disformal transformation, derive its invertibility condition, and express it in the unitary gauge. Our construction provides a systematic approach to constructing bi- and multi-scalar-tensor theories with higher derivatives.

Strong and weak symmetries and their spontaneous breaking distinguish different forms of symmetry and order in mixed states. We show that complete one- and two-point correlation functions, equivalently all two-site reduced density matrices, determine exact strong symmetry for continuous onsite unitary actions of connected groups: states with identical correlations have the same strong symmetry and character. We also present an algorithm to determine the maximal connected strong-symmetry sub-Lie group and its character from two-point correlations. Because such correlations are more accessible than full-state tomography, the algorithm is readily applicable to experiments ranging from condensed-matter systems to quantum simulators and circuits. We also identify group and representation-specific cases in which fewer correlations suffice.

We construct gauge-origami systems on the toric Calabi–Yau fourfold $\mathcal{C}\times\mathbb{C}$ from its brane brick model. Brick matchings and phase boundaries determine elementary D6- and D4-brane framings, respectively, realizing the magnificent-four, tetrahedron-instanton, and spiked-instanton sectors in a unified framed quiver quantum mechanics. We define their partition functions by Jeffrey–Kirwan residues and describe the fixed points in terms of colored BPS crystals. General gauge-origami configurations are obtained by combining these elementary sectors through their common fractional D0-brane quiver. This construction provides a first step toward gauge origami on general toric Calabi–Yau fourfolds.

In this work, we construct models compatible with ACT constraints on inflation in the framework of $f\left(Q, \tilde R\right)$ gravity, with $f(R)$ gravity coupled to a scalar field. The $f\left(Q, \tilde R\right)$ theory is equivalent to $f(Q, B)$ gravity models or $f(Q, C)$ gravity where $B$ or $C$ is the difference between $Q$ and the scalar curvature $\tilde R$ in Einstein's gravity, so, $B=Q-\tilde R$ or $C=\tilde R - Q$. Using appropriate reconstruction techniques compatible with $f\left(Q, \tilde R\right)$ theory, we propose two functional behaviors of the Hubble rate $H$ that satisfy the constraints, and we construct models that realize these behaviors of the Hubble rate $H$ using $f\left(Q, \tilde R\right)$ gravity and $f(R)$ gravity coupled with a scalar field. We also discuss the reheating stage after inflation. Although the scenario of $f(R)$ gravity coupled with a scalar field is well-known, since the model can be rewritten in the form of Einstein's gravity coupled with two scalar fields, the non-trivial problem occurs in the $f\left(Q, \tilde R\right)$ gravity case, because the number of dynamical degrees of freedom in $f(Q)$ gravity is not settled.

We investigate the correspondence between quasinormal modes (QNMs) and grey-body factors (GBFs) of higher-dimensional Reissner-Nordström black holes. Considering gravitational perturbations of these black holes, we systematically compute the QNMs and GBFs for scalar, vector, and tensor perturbations, including the $(+)$ and $(-)$ types of scalar and vector perturbations, to examine the validity of the correspondence. The QNMs are computed using the continued fraction method, and the integration-through-midpoints method is applied when required by the singularity structure of the Frobenius series for each perturbation type and set of black-hole parameters. We evaluate the accuracy of the correspondence for each spacetime dimension $D$ and black-hole charge by comparing the GBFs obtained from the correspondence with those obtained by numerical integration. The correspondence provides a good approximation to the GBFs for the scalar($+$), vector($+$), vector($-$), and tensor perturbations. However, for the scalar($-$) type, the correspondence exhibits pronounced deviations at low to intermediate charges in $D > 6$. At higher charges, it recovers reasonable accuracy despite the effective potential exhibiting a double-peak structure inconsistent with the assumptions underlying the Wentzel-Kramers-Brillouin (WKB)-based derivation of the correspondence. In contrast to the other types, for which the correspondence deteriorates as $D$ increases, the scalar($-$) type is more accurate in $D=8$ than in $D=7$. Furthermore, the correspondence performs well for the vector($-$) perturbation of near-extreme black holes, even though the potential develops a double-peak structure in $D>6$. These results indicate that the correspondence is applicable over a broader range than expected from the assumptions of the WKB approximation.

We study the hydrodynamics of a charged fluid with spin degrees of freedom. We argue that, in a conformal field theory in 3+1 dimensions, the spin density is related to a linear combination of standard second-order transport coefficients, and an additional term quadratic in the gauge- and mixed gauge-gravitational anomaly coefficients. We compute the spin density in several free and holographic theories and compare the results.

In this paper we prove in full detail the isomorphism between the solvable group $\mathcal{S}_{2,2+p}$, metric equivalent to the Calabi Vesentini symmetric space $\mathrm{SO(2,2+p)/SO(2)\times SO(2+p)}$, and the solvable group $\mathcal{S}_{\mathrm{L}(-1,p)}$ supporting the Kähler metric of the homogeneous non symmetric special manifold L$(-1,p)$. From an Alekseevskyan point of view the two spaces simply correspond to two different quadratic forms on the same solvable Lie algebra that we present and compare in detail. Furthermore considering the full group of isometries $\mathrm{Iso}_{\mathrm{L}(-1,p)}$ of L$(-1,p)$ we show that $\mathrm{Iso_{L(-1,p)}}\subset \mathrm{SO}(2,2+p)$ is a non-semisimple subgroup of the simple isometry group of Calabi-Vesentini manifolds. Altogether the Special Kähler manifold L$(-1,p)$ can be seen as a coset manifold $\mathrm{Iso_{L(-1,p)}}/\mathrm{H}$ where $\mathrm{H}=\mathrm{U(1)}_L\times \mathrm{SO(p)}$, the generator of $\mathrm{U(1)}_L$ being in $\so(2,2+p)$, yet not in the canonical $\so(2)\oplus\so(2+p)$ subalgebra. The action of $\mathrm{Iso}_{\mathrm{L}(-1,p)}$ on the solvable coordinates of the CV manifold can be constructed directly. This identification provides a valuable tool for Cartan Neural Networks, introducing additional non linear transformations in every map from one layer to the next one of an $r=2$ Neural Network based on the CV Tits Satake universality class; in perspective, this new tool increases expressivity.

We revisit the combinatorial loop-equation method for computing planar disc amplitudes in multi-matrix models with polynomial potentials. We develop a procedure which organises the Schwinger-Dyson constraints by the level of their boundary insertions and, when the resulting system closes, produces an algebraic equation for the planar disc function. Using this method we recover the disc partition functions associated with fixed, mixed, and free boundary conditions in the three-state Potts model, previously obtained by saddle-point methods. As a novelty we derive an algebraic equation for the fixed-boundary resolvent of the three-state Potts model with unequal quadratic and cubic couplings. We further show one can compute the disc partition function for several unsolved two- and three-matrix models. These examples demonstrate that loop equations provide a practical route to identifying and solving sectors of multi-matrix models beyond the standard one- and two-matrix cases.

I present new results for the massive cusp anomalous dimension at four loops in QCD. I combine partial exact and conjectured analytical expressions with an approximation that is based on the asymptotic behavior of the cusp anomalous dimension to produce the newest estimates. Detailed comparisons are made with previous approximate results that highlight the robustness of the calculational approach.

We derive the renormalized local equation of motion for the gravitational self-force on a compact object moving through matter in a black-hole spacetime. We integrate out the environment and metric perturbations on a Schwinger–Keldysh contour, obtaining a causal effective action given by a sum of curved-spacetime Feynman diagrams organized by powers of the mass ratio and environmental density. At leading order, an environmental wake gravitationally backreacts on the worldline. The corresponding diagram contains a logarithmic ultraviolet divergence that renormalizes the leading dissipative fluid–particle coupling. We isolate the divergence with a local large-momentum expansion, in which the curved-spacetime retarded Green's function reduces to its flat tangent-space form. We translate the resulting singularity into a Barack–Ori mode-sum scheme, deriving an analytic puncture and regularization parameters for an ideal-fluid environment. Mode subtraction fixes the divergent part but not the finite Wilson coefficient. For collisionless dust, we determine this coefficient by matching to an exact near-zone calculation for a black hole moving through the medium. This gives a complete renormalization prescription suitable for numerical self-force calculations.

We show that the equations of motion for magnetically charged $p$-branes in $(n+p+2)$-dimensional AdS reduce to coupled one-dimensional Toda-type equations, and use this structure to construct static toroidal black branes with $T^n$ symmetry and $p$ translation-invariant directions. In the pure-Maxwell case we obtain an exact $\sinh$ solution–a $(p+2)$-dimensional black brane times an $n$-torus, asymptotically $\mathrm{AdS}_{p+2}\times T^n$–whose dilatonic analogue, with a scalar coupled through $e^{a\phi}$, has a running torus modulus and a non-trivial dilaton. A perturbative expansion reduces the linearized system to a pair of Pöschl-Teller equations, one elementary and one solved by associated Legendre functions of non-integer degree, so that the general solution is non-elementary; at weak coupling $\epsilon=a^2$ we construct the $O(a^2)$ backreaction explicitly. From the surface gravity, horizon area, and holographic renormalization we obtain the Hawking temperature, entropy density, and mass density, verifying the first law $d\mathcal{M}=T_H\,ds$ and the traceless conformal stress tensor of the dual $(p+1)$-dimensional theory. At $O(a^2)$ the first law persists, but the dilaton source shifts the Stefan-Boltzmann exponent to $p_{\rm eff}=p+a^2/2$, breaking the clean power law $s\propto T_H^p$ while the normalizable dilaton hair decouples from the entropy.

It is well known that in (2+1)-dimensional general relativity, black hole solutions exist only in the anti-de Sitter case–the so-called Banados-Teitelboim-Zanelli (BTZ) black hole. In this letter, we construct, for the first time, three independent LQG-inspired frameworks for (2+1)-dimensional black holes: the $\bar{\mu}$-scheme holonomy corrections, the quantum Oppenheimer-Snyder collapse, and a covariant effective model. Remarkably, all three approaches yield the same qualitative picture: when LQG-inspired corrections are included, we find three distinct types of solutions corresponding to asymptotically flat, anti-de Sitter, and de Sitter geometries. Our results demonstrate that the asymmetry between classical (2+1)-dimensional and (3+1)-dimensional solutions is eliminated upon incorporating effective quantum gravity effects.

We investigate an inflationary scenario within a modified gravity theory that extends the Einstein-Hilbert action through a non-minimal coupling between the inflaton $\phi$ and the trace of the energy-momentum tensor $T$. This $f(\phi)T$ framework is designed to naturally recover the standard general relativity in the post-inflationary universe. Within this theory, we apply the Woods-Saxon potential. The slow-roll dynamics yields a luminal sound speed $c_s^2=1$ for our specific choice of the coupling function. For the model parameters $a \lesssim 1$ and the modest coupling $\beta=0.1$, we obtain a scalar spectral index $n_s\approx0.959$–$0.966$ and a tensor-to-scalar ratio $r<0.01$, which reveal a good agreement with the latest Planck and BICEP/Keck data on the $(n_s, r)$ constraints. Crucially, we shall demonstrate that the $f(\phi)T$ coupling is essential for the observational viability of the model with respect to the CMB two-point statistics. This work establishes a viable inflationary scenario within the scope of the inflationary observables which are considered here.

We develop an operator-level framework for reconstructing the angular structure of small-$x$ diffractive amplitudes from energy-flow measurements. Focusing on coherent diffractive dijet production in the leading-eikonal approximation, we establish the relation between the angular multipoles of the dipole amplitude, the gluon Wigner distribution, and the diffractive scattering amplitude, while keeping their distinct radial transforms explicit. We show that, in the real common-phase sector and away from diffractive zeros, the normalized scattering amplitude can be reconstructed directly, up to a global sign convention, from the square root of the normalized energy-flow distribution. For generic complex amplitudes, the measured intensity instead determines autocorrelations of the amplitude spectrum and phase retrieval is not unique without additional information. We further discuss corrections arising from harmonic-dependent phases, practical conditions for an EIC extraction, and effects beyond the leading-eikonal approximation.

The problem of time in quantum gravity is often presented as a consequence of applying quantum theory to general relativity. These notes adopt a broader perspective. We first examine how time enters classical mechanics, relativity, quantum theory, and quantum field theory, distinguishing three principal aspects: causal order, temporal duration, and the present. This analysis shows that the conceptual difficulties associated with time do not arise only when gravity is quantised. Tensions between causal structure, clocks, observables, and measurement are already present in the theories from which quantum gravity is constructed. We then survey responses to the problem of time across the principal approaches to quantum gravity. These include canonical and Wheeler-DeWitt quantisation, path-integral and histories formulations, perturbative and background-dependent approaches, causal sets, and twistor theory. The programmes are compared according to whether they retain an external time, render time problematic through quantisation, or incorporate causal or temporal structure into their foundations. Particular emphasis is placed on questions arising from quantum foundations and relativistic quantum information. We conclude by examining linearised gravity and weak-gravity quantum systems. This regime shows that the problem of time is not confined to Planck-scale physics, but arises already in attempts to provide a predictive and operationally meaningful quantum description of spacetime near flat geometry.

We study charged Klein-Gordon modes on a $(2+1)$-dimensional Ellis traversable wormhole in an external magnetic field defined as uniform in an auxiliary Euclidean embedding space. The spatial geometry is intrinsically catenoidal, while the embedding is used only to construct the external gauge connection. Pullback of the ambient potential yields a regular azimuthal gauge field whose associated intrinsic magnetic scalar reverses sign across the throat. Separation of the Klein-Gordon equation leads to a radial problem that is unitarily equivalent to a one-dimensional Schrödinger operator with a regular effective potential. For nonzero magnetic coupling, the quadratic asymptotic term produces radial confinement on the complete wormhole. The radial equation can be reduced to the confluent-Heun class. Polynomial truncation is possible only on constrained parameter submanifolds determined by simultaneous termination conditions and therefore does not represent the generic confined spectrum. We derive the corresponding relativistic energies and analyze the lowest polynomial sector, while distinguishing the formal continuous-$\ell$ solvability correlations from the integer values of the azimuthal separation constant required by the standard angular periodicity condition.

We construct the Yukawa couplings of the quark sector as overlap integrals of Gaussian wave functions in extra spatial dimensions. Assuming that the wave function of each chiral fermion is localised at a point in the extra dimensions, while that of the Higgs doublet is flat, the Yukawa matrix elements are determined by the relative positions of the three left-handed quark doublets, $q_1$, $q_2$, $q_3$, and the six right-handed quark singlets, $u_1$, $u_2$, $u_3$, $d_1$, $d_2$, $d_3$. We employ numerical optimisation techniques from machine learning to identify field configurations that satisfy all current experimental constraints on the quark Yukawa sector. In this framework, the large hierarchy of quark masses arises naturally, since the overlap integrals of Gaussian wave functions are exponentially sensitive to ${\mathcal O}(1)$ separations in the extra dimensions.

We study a five-dimensional realization of endothermic dark matter motivated by the recent high-recoil LUX-ZEPLIN event. A pseudo-Dirac Standard-Model singlet fermion is localized in the extra dimension and couples to a bulk $U(1)_D$ gauge field. The same localization that suppresses a distant-brane Majorana operator and generates the inelastic splitting also controls the couplings to the Kaluza–Klein (KK) dark-photon tower, producing a localization-filtered mediator propagator. We organize the phenomenology around two representative mass–splitting points: an LZ benchmark, $m_\chi=1.10~\TeV$ and $\delta=365~\keV$, and an IceCube-motivated comparison point, $m_\chi=1.08~\TeV$ and $\delta=566~\keV$. The latter is used only to compare terrestrial and solar kinematics; the published IceCube bound is Higgsino-specific and is not directly imposed on our singlet KK model. At the LZ benchmark with $R^{-1}=0.30~\GeV$ and $m_0=0.10~\GeV$, the KK tower enhances the high-recoil xenon rate by a factor of about four relative to a single dark photon at fixed microscopic coupling. The LZ point lies just below the terrestrial halo speed ceiling, whereas the $566~\keV$ comparison point is inaccessible to ordinary xenon scattering and instead lies in a kinematic regime accessible to solar capture; any neutrino constraint on the present singlet model remains dependent on its capture and annihilation dynamics. The characteristic prediction of the extra-dimensional construction is therefore a correlated interplay of localization, KK spectral distortion, and inelastic kinematics.

We develop a probabilistic construction of Toda field theory correlators from the Coulomb gas representation. Reinterpreting $n$-point functions of primaries as $w$-moments of random variables with respect to a Gaussian measure, we derive constraints on the Toda momenta and coupling constant. $L^2$ and $L^p$ convergence conditions for the associated random variables, triviality constraints on the correlators, and local integrability conditions for the cross-moments are derived. These constraints define an extended region of convergence for the Toda correlators and provide a systematic, probabilistic interpretation of the Seiberg bounds. We proceed to analytically continue the moment indices by employing a Mellin-Barnes representation of the random variables. The local convergence boundaries become meromorphic local integrability divisors, and, in the semiclassical $b\to0$ limit, these divisors reconstruct the Liouville charge pole lattice. For $A_n$ Toda theory, we show that natural $W_n$-closure conditions in this semiclassical limit select rank-one, Virasoro-like submodules inside the full Toda theory. On these slices, the scalar integrability conditions reduce to Liouville-like divisors and recover the corresponding Toda charge pole lattices.

Black holes ring when perturbed, whereas their response to a stationary gravitational-wave background is a real-frequency scattering problem, not a source of additional quasi-normal-mode lines. We make this standard distinction quantitative by treating a black hole as a frequency-, angle-, and polarization-dependent filter. For an isotropic stationary background around Schwarzschild holes, elastic scattering produces no net monopole signal, so horizon absorption is the only population-level spectral distortion. We calculate this transfer function for Schwarzschild holes in detail, identify its absorptive and phase-delay signatures, and connect the stationary response to the causal ringdown excited by a finite wave packet. We then promote the single-hole result to an angular and polarization transport kernel for a cosmological population. The resulting optical depth is negligible for realistic black-hole populations, including asteroid-mass primordial black holes comprising all dark matter. We then extend the analysis to Kerr holes, for which superradiance allows genuine amplification in selected co-rotating modes, but isotropic incidence and random spin orientations strongly dilute the diffuse signal. Observable effects are therefore more likely in rare, nearby, aligned, rapidly spinning, or transiently illuminated systems than through cumulative cosmological propagation.

We calculate the $\theta$ dependence of the vacuum energy in an infinite family of cousins of QCD, where the vacuum structure can be analyzed exactly. These theories are $\mathcal{N}=1$ EFTs that descend from $\mathcal{N}=2$ $SU(N_c)$ gauge theory with $N_f$ flavors of fundamental hypermultiplets, that are then deformed by anomaly mediated supersymmetry breaking (AMSB). We reproduce results found in previous work that used the full $\mathcal{N}=2$ machinery but via a much simpler $\mathcal{N}=1$ method. Our method easily extends to infinitely many more theories for which the full $\mathcal{N}=2$ approach becomes intractable. We identify values of $\theta$ for which the theory undergoes a phase transition, and find that they can occur at both fractional and irrational values of $\theta/\pi$.

We compute the cohomology of the projective special linear Lie superalgebra $\mathfrak{psl}(n|n)$ over a field of characteristic zero in trivial and in adjoint coefficients in full detail through its associated Hochschild–Serre spectral sequences. Applications in physics exist for its real form, the projective special unitary Lie superalgebra $\mathfrak{psu}(n|n)$ — in particular, $\mathfrak{psu}(2,2|4)$ for AdS/CFT. We also find the cohomology of $\mathfrak{sl}(m|n)$ in trivial coefficients along the way and we point out a discrepancy with a previous statement in the literature. We present the explicit cohomology groups of low degree for $n\leqslant4$.