Papers

In the Wilson formulation of lattice gauge theory, the fundamental degrees of freedom are group-valued link variables, while the action is a sum over the trace of the plaquettes, the smallest Wilson loops. For generative sampling methods such as normalizing flows, this poses a challenge: the distribution of an individual plaquette is easy to model, but mapping sampled plaquettes to the links is the obstruction. We explore a statistical way around this in two dimensions for the $\mathrm{SU}(N)$ gauge theory with the Wilson plaquette action. The action can be written in terms of holonomy variables, which determine the links once a consistency condition at the four corners of an extended lattice is satisfied. Using a boundary condition that only affects the holonomies, we trade this consistency condition for a conditional sampling problem, which reduces to sampling $X,Y\in\mathrm{SU}(N)$ given the group commutator $Z=XYX^\dagger Y^\dagger$, where $Z\in\mathrm{SU}(N)$ depends on the four corners. Plaquettes are sampled independently, and the resulting configurations carry weights due to the additional conditional sampling. We obtain the density of $Z$, which determines these weights, in closed form for $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$. A normalizing flow models the single-plaquette distribution for $\mathrm{SU}(2)$ and $\mathrm{SU}(3)$; for the group-commutator problem we use a closed-form sampler for $\mathrm{SU}(2)$ and a trained normalizing flow for $\mathrm{SU}(3)$. In our tests on $2\times2$ and $32\times32$ lattices at two couplings each, per-plaquette acceptance rates exceed $99\%$ and the effective sample size of the final weights is moderate, typically above one half.

We analyze spacetime shortcuts generated by longitudinal contraction and transverse expansion of the intrinsic spatial metric at fixed reference volume. Smooth bounded ultrastatic corridors are globally hyperbolic and geodesically complete and produce an exact travel-time advance between exterior clocks. An axial maximum makes the axial light ray globally minimizing; a strict maximum on an interior interval establishes Olum's endpoint-surface criterion for exactly matched corridors. Exact Einstein-source identities distinguish integrated Eulerian matter energy from asymptotic mass: infinite tubes have negative integrated energy, whereas finite corridors admit competing gradient contributions. A transverse Ricci identity proves local null-energy violation in every nontrivial localized finite corridor. The axial averaged null energy is strictly negative for localized axial-maximum profiles, and quantitative bounds relate its magnitude to the reference time advance and peak strain. We also obtain the complete finite-profile Ricci spectrum and the axial passenger tidal tensor. These results provide explicit causal and source constraints on a prepared geometry without specifying its physical formation or matter dynamics.

We nonperturbatively construct a gauge-invariant measure for overlap fermions on four-dimensional Euclidean lattices in a class of nonabelian chiral gauge theories that includes the Standard Model. On admissible gauge fields, comparing chiral projectors at successive lattice spacings yields an exponentially local current and establishes the global properties needed for Lüscher's reconstruction of a smooth, gauge-invariant fermion measure. This provides a candidate nonperturbative lattice definition of the Standard Model.

We investigate the cosmological dynamics of the nonlinear $\sigma$-model in a locally rotational symmetric Bianchi V background geometry. We employ the $SU(2)$ and the $SL(2,\mathbb{R})$ internal symmetries, for which the kinetic metric of the multi-scalar fields is a space of constant curvature. We assume an inhomogeneous ansatz for the scalar fields, such that the $\sigma$-model, which allows the existence of nontrivial solutions in the Bianchi V geometry. For an exponential potential, we perform a detailed analysis of the phase space, we determine the stationary points, and we study their stability properties. The scalar field inhomogeneities prevent both the slow-roll and hyperbolic inflationary epochs. The existence of a cosmological constant is required to describe cosmic acceleration.

An expanding bubble in AdS$_5$ has a dS$_4$ geometry; the dark bubble scenario explores whether this can be our Universe. Motivated by this, we consider new non-supersymmetric AdS$_5$ compactifications of M-theory and investigate their non-perturbative instabilities. We focus on a compactification on the flag manifold. In certain regions of the flux landscape, the non-supersymmetric vacua become unstable under the nucleation of M5-brane bubbles wrapping an internal two-cycle, thus realising the dark bubble scenario. The cosmological constant on the bubble can be made parametrically small by tuning the internal fluxes, making the M5-branes close to the extremality bound. The solution relates the induced cosmological constant on the brane and the typical largest length scale of the extra dimensions through $R^{-1}\sim \Lambda_4^{1/4}$ in 4d Planck units, allowing for extra dimensions of the order of tens of microns. We also consider various alternatives, which for various reasons are currently not as successful.

We derive exact massless S-matrices from algebraic root-system geometry and derive thermodynamic Bethe ansatz (TBA) equations that generate minimal renormalization group (RG) flows. We show how P. Dorey's geometric construction, originally developed for massive S-matrices, can be extended to massless cases and prove that these new $S$-matrices straightforwardly satisfy the bootstrap equations. We focus on minimal D_n coset CFTs, which are c=1 free-boson CFTs with compactification radius \sqrt{n/2}, and find their UV completions by analyzing the TBA.

We explore the idea that the mass parameter $m$ can be used as a diagnostic tool for identifying the critical points of thermal two-point functions of massive free scalar fields in odd $d$ dimensions. For general values of $m$ these correlators admit an expansion in thermal conformal blocks associated with symmetric traceless operators of definite scaling dimension. For each spin the higher-twist contributions can be resummed into a closed-form dressing factor of the leading-twist coefficient. At special values of $m$ for which the thermal trace of the energy-momentum tensor vanishes they are identified as two-point functions of thermal CFTs. Through a dimensional uplift, valid for all $m$, we map these correlators to thermal two-point functions of higher-derivative massive theories in $d+2$ dimensions. We show that the thermal trace of the energy-momentum tensor of the $d+2$-dimensional theory is proportional to the free energy of the original $d$-dimensional theory. When that thermal trace vanishes for $m\neq 0$ the uplifted correlators are identified as two-point functions of a higher-derivative nonunitary thermal CFT$_{d+2}$, but they differ from the uplifted nontrivial thermal two-point functions of the original unitary CFT$_d$.

We introduce novel scalar generating potentials to compute three and four point cosmological correlators in orthogonal Grassmannian space. These correlators are obtained by applying appropriate directional derivatives on these potentials, combined with helicity-dependent minors. We emphasize that, this is not a spin-raising analog, but a much more powerful construction. The four point generating potential is associated with the geometry of a projective triangle. First-order directional derivatives generate gluon correlator, while the Hessian of the same scalar potential produces the four-graviton correlators, thus realizing a differential double copy. Thus, double copy is a natural consequence of the underlying scalar potential as the origin of the gluon and graviton four point correlators. This framework extends naturally to any spin-$s$. We exhibit a geometric origin for the gluon colour identities: the projective homogeneity of the scalar potential implies Kleiss-Kuijf identity, while the kinematic Jacobi identity follows as a consequence of the closure of the triangle under directional derivatives. We also uncover a connection between the BCFW bridge deformation of the scalar potential and directional derivatives that produces the spin-s correlator. Thus the generating potential provides a universal geometrical interpretation for any spin-$s$ four point Grassmannian correlator, unlike \cite{Bala:2026lvw} where geometric interpretation was only shown for the gluons.

We compute the higher-rank translation ($\mathcal{T}$) and rotation ($\mathcal{R}$) matrices associated with a genus-one spectral curve. Using these data, we reconstruct the worldsheet correlators of vertex operators in the corresponding supersymmetric Landau–Ginzburg (LG) theory. This provides a generalization of the genus-zero, one-cut spectral-curve construction to higher-genus geometries. The matrix-model correlators are mapped to integrals over the moduli space of Riemann surfaces for the symmetric sextic double-well potential, which has a double-cut solution. We further identify the rational and elliptic contributions to the worldsheet integrated vertex operators and show that each of the sector precisely matches the corresponding correlators of single-trace operators in the matrix model. Altogether, our construction showcases an explicit dictionary between the higher-genus spectral-curve data, the associated Cohomological field theory (CohFT), and the worldsheet description of the corresponding matrix-model observables.

Spacetime singularities displayed by classical black hole solutions are expected to be cured by high-energy effects, beyond the regime of validity of General Relativity. On the other hand, there are well-known examples of classical theories allowing for nonsingular black hole solutions. The most prominent is obtained by complementing the gravitational sector of General Relativity with a minimally coupled nonlinear-electrodynamics sector. It has recently been shown that purely electric/purely magnetic nonsingular black holes in an Einstein-nonlinear-electrodynamics setup are always plagued by a gradient instability. In this work, we derive the quadratic action for linear perturbations of static, spherically symmetric dyonic solutions of Einstein-nonlinear-electrodynamics theories with a general dependence on both invariants. We show that regularity of the center, absence of ghosts and angular gradient stability cannot hold simultaneously: arbitrarily close to a regular center, at least one photon polarization has a negative squared angular propagation speed. We further show that regularity drives one electromagnetic kinetic eigenvalue to zero at the center, which may limit perturbative control near the core. Evading the obstruction requires extending the theory, for example by introducing additional degrees of freedom.

We consider the extension of the $\delta$-shift connecting Tr$(\phi^3)$ and pion amplitudes to observables in cosmology. We start in flat-space and explain how one can extract the pion wavefunction and correlators from those of Tr$(\phi^3)$ theory, via a simple kinematic deformation. As opposed to the amplitude, the $\delta$-shift for the wavefunction is no longer a linear deformation on the kinematics since the wavefunction depends on magnitudes rather than the squares of momenta. We "linearize" this deformation by using the frequency representation - allowing us to recast the contribution from any graph as a sum over residues. We then show how this construction extends to de Sitter, and find that, remarkably, in the frequency representation, the pion de Sitter wavefunction and correlators can be formulated in precisely the same way as in flat-space. Amongst other things, this unified formulation reveals a simple universal soft-limit for flat-space and de Sitter, where, in the soft-limit, the wavefunction turns into a sum of wavefunctions in a mixed theory of pions and scalars.

We discuss 2-dimensional $QCD$ with a fundamental quark of mass $m$ in the 't Hooft limit ($N \to \infty, \, g^2 \to 0, \, g^2N =$ const). The spectrum of this theory includes a chain of meson states. The masses of these states can be evaluated by solving the proper integral Bethe-Salpeter equation. An alternative simple procedure consists in solving the Schrödinger equation for the relativistic Hamiltonian $$ \hat H\ =\ 2\sqrt{\hat p^2 + m^2} + \sigma |x| $$ with the string tension $\sigma = g^2N/4$. We compare the results obtained by these two methods. In the leading semiclassical approximation, they coincide. We show, however, that the preasymptotic WKB corrections to the masses calculated in these two ways are different. For the ground state, they play an essential role. But already for $n=3$, the relative difference between the two predictions does not exceed 2.5% and rapidly falls off as $n$ increases.

We calculate the conservative deflection of a photon by a massive spinless source, such as a black hole, at third post-Minkowskian order using scattering amplitudes. We work in the heavy-source and long-range limits and adopt the geometric-optics approximation. The required five-point photon-graviton tree amplitude is constructed by on-shell recursion and sewn to three scalar-graviton vertices across a five-fold cut. Polynomial division organizes the cut information into sixteen contributions, including the residues with fewer propagators. We perform the source energy integrations and retain the near-on-shell photon regions in the remaining spatial integrals. Within the static region in which all exchanged momenta are soft, we relate a photon pole prescription to the subtraction of the cubic tree iteration from the conservative kernel. The sum of the sixteen contributions determines the classical logarithmic term at two loops. Its transverse Fourier transform gives the interaction radial action and yields the third-order deflection angle. The result agrees with the weak-deflection expansion of Schwarzschild null geodesics and demonstrates how nonlinear classical light bending is encoded in loop amplitudes around flat spacetime.

We argue that some of the puzzling features associated with the normalization of gravitational wave functions can be substantially alleviated by restricting the path integral to a sum over connected, allowable geometries. This prescription gives sensible results when applied to classical transitions and it provides non-trivial no-boundary probabilities, in line with old expectations. We highlight the additional impact of both boundary conditions and integration contours on these issues.

We study $SO(2n)$ gauge theories with a degenerate curvature kinetic operator defined by an orthogonal complex structure. The kinetic image is $\su(n)$, while the adjoint vacuum has stabilizer $U(n)$. We distinguish fixed projectors, unrestricted adjoint Higgs fields, and Higgs fields constrained exactly to the orbit $SO(2n)/U(n)$. For the constrained theory with a positive Higgs kinetic coefficient, the coset connection is auxiliary and the central connection is null. A nonlinear Dirac analysis establishes a regular weak-curvature branch with $2(n^2-1)$ physical degrees of freedom in four dimensions, locally equivalent to $SU(n)$ Yang–Mills theory. Normal Higgs displacements in the unrestricted theory can change the kinetic rank, so this count does not extend to that field space. We also characterize possible algebraic branches. We express the Stelle–West Higgs–gravity action through a common Clifford involution, and introduce projected Pontryagin theories and a full-rank equivariant completion.

The frequency-dependent tidal response of a neutron star is constrained by causality, reality and passivity, which turn a suitably rescaled response into a holomorphic self-map of the upper complex frequency half-plane after some suitable conditions. The Schwarz–Pick theorem then bounds the leading dynamical correction to the tidal deformability from above. In this work we confront this bound with the most recent calculations of the dynamical tidal response in the literature which have also shown that the static tidal deformability and its correction at second order in frequency are tightly correlated. We find that the bound is saturated to within a few per cent, a quantitative confirmation of f-mode dominance in the relativistic regime. The combination of the static deformability with an independent determination of the f-mode frequency thus may provide an equation-of-state-independent upper limit on the dynamical tide.

We present the first holographic study of gravitational form factors (GFFs) of scalar, pseudoscalar and canonical-spin-averaged tensor glueball targets. These observables test gravitational cubic couplings whose physical validity is not established by spectra and static-force calibration. We construct a dynamical Einstein–scalar–axion representation inspired by the Babington–Crooks–Evans adjoint-mass deformation of $\mathcal N=4$ super-Yang–Mills theory and its Type-IIB supergravity dual. Leading ultraviolet source data are retained, while a soft-wall-like infrared reconstruction incorporates asymptotically linear radial mass-squared trajectories. Spectra and Wilson observables fix the parameters at a common Sommer scale without GFF input. The full constrained cubic vertices fail to describe 27 correlated scalar lattice observations, giving $\chi^2/\mathrm{dof}=73.60/27=2.73$. We propose normalized heavy-target (HT) effective cubic couplings for pure-glue matching. HT adopts leading radial shapes suggested by infinitely heavy targets at fixed probe dynamics and normalizes their overlap weights on the physical ground states. With the background, modes and propagators unchanged, HT gives $\chi^2/\mathrm{dof}=16.98/27=0.63$, without numerical adjustment to GFF data. Tensor and scalar source overlaps determine $A$ and the physical normalized trace; the exact finite-mass energy–momentum-tensor relation determines $D$. We predict the pseudoscalar and tensor form factors and three-target Breit pressure and shear profiles. On the reference infrared-decaying axion branch, the HT Breit mass radii are $0.276$, $0.300$ and $0.311\,\mathrm{fm}$, respectively. The results identify gravitational cubic matching as a distinct step beyond background calibration and provide a prescription for further lattice tests.

Whether the spin-1/2 Aharonov-Bohm Hamiltonian supports bound states has remained controversial since Hagen concluded that no such states exist for a Dirac particle. Here, we show that Hagen's solution corresponds to a particular member of the one-parameter family of self-adjoint extensions that describe the singular Zeeman interaction, and we identify which member the microscopic physics selects. Matching the extension parameter to a finite-radius flux tube yields $\nu=0$ precisely at $g=2$, and it does so in every flux sector. The critical $g$ factor is therefore universal, $g_c=2$, and depends on neither the flux sector nor the regularization radius. The flux sector controls the depth of the bound state energy, and we obtain a closed-form expression showing that binding deepens markedly with the integer part of the flux. Hagen's conclusion is thus confirmed and sharpened: it is not a statement about one arbitrary member of a family of extensions, but about the member that the physics selects when the magnetic moment takes its Dirac value, and an anomalous moment is a necessary and sufficient condition for binding in the singular channel.

We study the dimensional reduction of non-Abelian Yang-Mills theory from five to four dimensions. In this framework, Hadamard descent is formulated as a restriction to gauge-equivalence classes that are invariant, up to gauge transformations, under translations along the reduced direction. The central result is that, prior to gauge fixing, the adjoint scalar of the reduced theory is not the connection component $A_u$ itself, but the gauge-covariant combination $\widetilde{\Phi}=\epsilon-\iota_YA$, which follows naturally from the covariant Cartan identity. The usual identification with $A_u$ is recovered, up to the sign convention adopted here, in the $u$-independent gauge, while global holonomy data remain additional information to be specified separately. We show that the reduced theory takes the form of a Yang–Mills–Higgs theory with the Higgs field in the adjoint representation. As an example, we focus on the $U(2)$ gauge theory and analyze the Higgs mechanism, together with the associated symmetry breaking pattern, when the reduced theory is expanded around the ground state of the parent theory. We then include fermions and show that, in the $u$-independent gauge, the extra-dimensional component of the gauge connection generates a pseudoscalar Yukawa coupling which, on the vacuum configuration, can be converted into a fermion mass term by a chiral field redefinition, while the gauge interaction remains vectorial.

We develop a computational framework for the quantum relative entropy based on the replica trick and apply it to analyze the evaporation process method of Jackiw-Teitelboim black holes. We show that the relative entropies between the exact radiation state and the Hawking saddle state and the replica wormholes saddle state quantify information-theoretic costs inherent to coarse-grained semiclassical descriptions. During evaporation, the relative entropy for the Hawking saddle increases monotonically, reflecting a progressively higher thermodynamic cost of sustaining the Hawking description. Conversely, the relative entropy associated with the replica wormholes saddle decreases, which indicates that the description of connected geometries becomes progressively more accurate. Moreover, the sign of the difference between the two quantities reverses precisely at the Page time, providing an information-theoretic diagnosis for the transition between competing gravitational saddles. Our results recast the Page transition as a flow of relative entropy costs between competing semi-classical descriptions, revealing an information-theoretic structure underlying the emergence of effective irreversibility in coarse-grained black hole evaporation.

The density dependence of four-quark condensates is a major source of uncertainty in operator product expansion analyses of hadrons in nuclear matter. Their determination is particularly relevant to the $N$–$J/\psi$ interaction, whose long-range attraction appears to be well described by two-pion exchange in lattice QCD. We evaluate the pion-coupled four-quark condensates using one-nucleon intermediate-state insertions and incorporate them into the QCD sum rule analysis for the $J/\psi$ meson in nuclear matter. At normal nuclear matter density, we find a mass shift of $-1.75$ MeV, which is of the same order as the approximately $-5$ MeV attraction inferred from a two-pion exchange potential fitted to lattice QCD results. A previous factorization-based treatment required the introduction of unnaturally large phenomenological parameters to obtain a comparable shift. Our result shows that one-nucleon intermediate states provide a significant contribution to the in-medium four-quark condensates and improve the connection between the operator product expansion and the long-range $N$–$J/\psi$ interaction.

Skyrmion lines and semi-local strings are examples of infrared-finite cosmic strings that arise in theories with spontaneously broken global symmetries. Unlike global $U(1)$ strings, they do not support the long-range Nambu-Goldstone boson (NGB) profile responsible for a logarithmically infrared-divergent string tension. We argue that NGB emission from their low-frequency oscillations is strongly suppressed. Consequently, their gravitational-wave signals may be comparable to those from local strings.

We describe a Segre-Veronese construction of polynomial solutions to the interpolation conditions of the numerators of tree-level MHV gravity amplitudes given by Hodges' formula. The five- and six-point Hodges numerators arise from Segre determinants, while for every $n\geq7$ suitable Segre determinants lead to solutions linearly independent of the Hodges numerator. The existence of such independent elements for $n=7,8$ was recently established by Mai and Zhang.

HOMFLY-PT polynomials describe analytic continuation of Wilson averages in Chern-Simons theory to arbitrary complex values of two independent parameters $q$ and $A$. They also depend on the knot and representation of the gauge group. However, when $q^2$ is a primitive $m$-th root of unity, e.g. $q=\pm e^{i\pi/m}$, these polynomials exhibit a universal (knot-independent) factorization: $H_R$ becomes the product of the polynomials for the $m$-core of the Young diagram $R$ and of one and the same factor $H_{[m]}$ for every $m$-ribbon of $R$. In all the examples which we computed this factor is just the special polynomial at $A^m$. At $q=\pm1$ factorization follows from the singularity of the cabled HOMFLY, which is the same for all representations. At $m>1$ the relevant singularity is in the projector, and what survives of it is a finite sum of braids $X_m$ – the $m$-th power sum inside the cable, which can be freely moved through the other strands. For the proof we use the Murnaghan-Nakayama rule and the Adams operation. Discovery of this hidden structure explains the old factorization puzzle.

We show that for bulk discrete series fields whose dual operators have integer conformal dimensions, the Generalised Energy Conservation (GEC) condition for de Sitter (dS) scattering amplitudes defined in even-dimensional extended Poincaré spacetime corresponds precisely to the absence of boundary conformal anomalies, including beta-function contributions, at the level of tree-level contact interactions. In contrast, holographic consistency in AdS/CFT enforces only the locality of the conformal anomaly. We find novel, unique four-point interactions involving three identical conformally coupled scalars and one discrete series scalar by performing both GEC and holographic bootstraps. This demonstrates that the solutions to GEC form a measure-zero subset of the holographically consistent solutions. Our holographic analysis agrees precisely with the expected renormalisation conditions for conformal field theory correlators. In addition, the constraint for a particular four-point configuration may be relevant for testing holographic consistency of scale-separated non-supersymmetric AdS$_4$ vacua.

Regular black holes in general relativity coupled to nonlinear electrodynamics have been reported to exhibit an apparent mismatch in their thermodynamics: the temperature obtained by differentiating the mass along the family of regular metrics differs from the Hawking temperature by a factor $\Xi$, and the first law seems to need a modified entropy or a corrected internal energy. We examine this question for an electrically charged family of regular black holes in $D\geq4$ dimensions whose metrics reduce to the Bardeen and Hayward geometries in four dimensions. The source is written as a Hamiltonian density $H(P)$ with fixed couplings and no solution parameters. This theory has a two-parameter family of static solutions, of which the regular metrics form a one-parameter subfamily. We prove analytically that on the whole solution space the standard first law of thermodynamics holds with the Hawking temperature, the area entropy and the horizon potential, in agreement with the general first law for nonlinear electrodynamics. The factor $\Xi$ appears when the mass is differentiated along the regular metric family at fixed length parameter, a variation that moves through the space of theories and also changes the conserved charge. We show that $\Xi^{-1}$ is the Jacobian of this projection, equal to the fraction of the mass inside the horizon, give its general form for an arbitrary regular mass function, and show that $\Xi^{-1}dM_{ADM}$ is not a closed one-form on the two-parameter metric family, so this correction cannot be the differential of a local state function there. We also point out that the Legendre map between the $F$ and $P$ frames degenerates at a radius that lies outside the horizon of near-extremal solutions in $D\geq5$.

There are reasons to believe that the region of uniform curvature that we observe within our horizon is perhaps a tiny patch, much smaller than the length scales of inhomogeneity and global connectivity of an extremely complicated manifold. However, the recent supernova (SN) results suggest that the horizon scale could be comparable to or even much larger than curvature radius. Non trivial global structure too tends to be of the order of the local curvature scale. Probing (slightly) beyond the cosmic horizon can potentially reveal nontrivial global structure lurking around or just beyond the horizon scale. The cosmic microwave background (CMB) anisotropy is a sensitive probe of the universe on length scales up to and somewhats beyond the horizon scale and is perhaps poised to detect or put interesting limits on nontrivial features the global structure. This point is exemplified by high CMB anisotropy signal of the Elliptical topology in a large fraction of the revised parameter space of the closed FRW universe (spherical geometry) currently preferred by SN observations.

A pronounced pyramid-shaped enhancement in the axion density power spectrum induced by the QCD phase transition is identified. The enhancement occurs near the horizon scale at the transition, with a characteristic comoving length scale comparable to the size of the Solar System, and amplifies the axion density perturbation by factors exceeding $3.5 \times 10^{13}$ relative to the CDM case. Under the hilltop initial condition, coupling to radiation triggers this enhancement. The initial condition is related to the axion mass, suggesting that observation of the axion pyramid in cosmology may constrain the QCD axion mass.

We investigate scalar induced gravitational waves (SIGWs) at third-order during a matter-dominated (MD) era. We classify the sources into four parts—built from three first-order scalars, and from a first-order scalar times the second-order tensor, vector, and scalar perturbations—and obtain all four kernel functions analytically in closed form. Unlike their radiation-dominated counterparts, the MD kernel functions grow secularly, translating into a rapid growth of the spectrum, $\Omega^{(3)}_{\mathrm{GW}}\propto x^{6}$ versus $\Omega^{(2)}_{\mathrm{GW}}\propto x^{2}$ at $x\equiv k\eta\gg1$. We derive the third-order power spectrum as a five-fold integral for a general primordial spectrum, and evaluate it numerically for a monochromatic one, where all contributions from the second-order vector source vanish identically. As an application, we consider the early MD era driven by primordial black holes (PBHs) before their evaporation. Requiring that the induced gravitational waves do not backreact on the cosmological perturbations, $\int\mathrm{d}\ln k\, \Omega^{\mathrm{tot}}_{\mathrm{GW}}<1$ then bounds the initial PBH abundance as $\Omega_{\mathrm{PBH},f}\lesssim4.6\times10^{-7}\, (m_{\mathrm{PBH}}/10^{7}\,\mathrm{g})^{-0.48}$, about three orders of magnitude stronger than the second-order-only bound. Our results indicate that higher-order SIGWs inevitably become nonperturbative during a sufficiently long MD era, with important implications for scenarios involving an early matter domination.

We propose a new formula relating the rationalized Drinfeld associators obtained from the Knizhnik–Zamolodchikov equation to the quantum $6j$-symbols of the algebra $\mathcal{U}_q(\mathfrak{g})$: $ U_{q} = U \cdot \Phi^{KZ}_{\mathbb{Q}} $. We prove this relation in the case of the $\mathfrak{sl}_N$ Lie algebra for the tensor cube of the fundamental and the first symmetric representation. We also check it perturbatively for the $\mathfrak{sl}_N$ algebra for representations $[2]\otimes [1] \otimes [1]$, $[2]\otimes [2] \otimes [1]$, the non-trivial cases with multiplicities — $[2,1]^{\otimes 3}$, $[2,2]^{\otimes 3}$, and in the case of the $\mathfrak{so}_5$ Lie algebra for the tensor cube of the fundamental representation.