Papers

We point out a puzzle about BPS black holes in AdS$_4\times S^7/\mathbb{Z}_k$: the entropy grows when $k$ increases although the number of gauge-invariant operators in the dual conformal field theory that can contribute to the entropy grows as $k$ decreases, due to the presence of monopole operators. Analyzing the superconformal index of ABJM, we show that the same puzzle exists for finite $N$ in the large charge limit. We propose a resolution based on an increasing number of trace relations as $k$ decreases, which dominates over the other effect. We argue that this mostly affects fortuitous operators and we provide an explicit example of a fortuitous operator that disappears at small $k$.

The classical soft graviton theorem aims to determine certain terms in the early- and late-time tails of the gravitational waveform produced in a scattering process, using only the asymptotic momenta of the scattered objects. In this paper we find an infinite series of leading-logarithmic tail terms in the Newtonian limit, where the incoming and outgoing particles move slowly, but the scattering itself can be violent. The all-orders conjecture of Alessio, Di Vecchia and Heissenberg agrees with our results for two-particle scattering, but does not hold in general when the initial or final state contains three or more particles. We also propose an all-orders conjecture for these tails beyond the Newtonian limit, but do not prove it. Nevertheless, the conjecture passes several nontrivial consistency checks.

Elastic scattering can generate entanglement and non-local magic between the spins of the two outgoing particles. The amount of entanglement and magic produced is encoded in the S-matrix; a unitary quantum gate that acts on the joint Hilbert space of the two spin-J particles. Previous work derived the entanglement power in terms of the scattering phase shifts for rotationally invariant S-matrices with J=1/2,1,3/2. In this paper, a closed form solution is derived for all J. We further prove that a broad class of non-local quantum resources depend to leading order only on the squared projective Hilbert-Schmidt distance from the nearest non-entangling gate. This holds for the entanglement power and non-local magic power, and the choice of resource simply changes the overall normalization. This universality establishes a common geometric origin for the leading resource-generating capacity of scattering, independent of the details of the interaction.

We present an explicit formula for the $n$-gon graph contribution to the integrand of the cosmological wavefunction for conformally coupled $\phi^3$ theory in de Sitter space. Our result relies on the fact that its symbol satisfies total compatibility with respect to the $B_{2n-1}$ cluster algebra. We introduce $B$-type quadrangular polylogarithms, constructed from centrally symmetric quadrangulations of a $4n$-gon, and show that the $n$-gon wavefunction coefficient is expressed by a natural weight-$n$ function of this class. We prove our formula by directly relating the recursive differential equation satisfied by these wavefunction coefficients to a recursive coproduct formula for $B$-type quadrangular polylogarithms, with the remaining additive constant fixed by the soft limit.

We present a phenomenological study of the longitudinal unpolarized structure function $F_{UU,L}$ in semi-inclusive deep inelastic scattering (SIDIS) and its ratio with the transverse structure function, $R_{\, \rm SIDIS}=F_{UU,L}/F_{UU,T}$. We study their transverse-momentum dependence, combining for the first time low- and high-transverse-momentum contributions across the kinematic range relevant to SIDIS measurements. At low transverse momentum, $F_{UU,L}$ is power suppressed, making it sensitive to next-to-next-to-leading-power dynamics in the unpolarized sector. We find that this contribution can be sizable, with $R_{\, \rm SIDIS}\simeq 0.3$ at low transverse momentum and hard scale, producing a correction of the order of $10\%$ to transverse-momentum-integrated observables. Although these estimates remain subject to theoretical uncertainties, our results indicate that accounting for $F_{UU,L}$ is likely to be important for the interpretation of SIDIS multiplicities and motivate dedicated measurements of $R_{\, \rm SIDIS}$ to better constrain the longitudinal contribution.

We examine, in the Landau background gauge, the effective action of a quantum theory of gravitational and matter fields. We analyse the BRST invariance of this theory and study the Ward identities which implement the symmetry of the background effective action. The off-shell and on-shell counterterm Lagrangians are explicitly calculated in this gauge at one-loop order. We argue that the matter self-energy in the Landau background gauge, which is finite to this order, may be finite also at higher loops.

The large-$D$ membrane paradigm provides a perturbative, non-gravitational description of black hole dynamics. While the local second law of thermodynamics has been established for neutral membranes, the explicit determination of the local entropy production equation within the charged large-$D$ membrane framework remains unsolved. In this paper, we compute the subleading-order correction to the divergence of the membrane velocity for a charged black hole in a flat background. We evaluate the corresponding scalar Einstein-Maxwell constraint equation at the physical event horizon via the non-affine null Raychaudhuri equation, and relate the physical horizon expressions to the auxiliary membrane kinematics, demonstrating that the membrane velocity produces a non-negative divergence at the first non-vanishing order in the membrane worldvolume, which is driven by two quadratic forms: the membrane shear and a novel contribution generated purely by the Maxwell field. Using this result, we construct the membrane entropy current and prove that its divergence is non-negative through this order, thereby establishing the local second law for the charged large-$D$ membrane paradigm.

We formulate a nonlocal spin-$1/2$ field theory in curved spacetime in which the form factor is a nonpolynomial entire function of the covariant Dirac operator. We derive the corresponding nonlocal Dirac equation and analyze its coupling to gravity and to a $\rm U(1)$ gauge field. As an application, we study the theory in the open chart of de Sitter spacetime and construct the associated nonlocal Bunch–Davies modes. The nonlocal dynamics can be mapped, branch by branch, onto the local Dirac equation with an effective mass $M_n$. For the exponential form factors considered here, the effective masses are expressed in terms of branches of the Lambert-$W$ function. We show that the Euclidean analyticity structure of the Bunch–Davies construction is preserved. On the physical positive-residue branch, the Bogoliubov coefficients, squeezing parameter, occupation number, and entanglement entropy acquire explicit nonlocal corrections, while the absence of physical fermionic supercurvature modes is preserved. The local theory is smoothly recovered in the limit $\Lambda\to\infty$.

An oscillon in scalar field theory in $(1+1)$-dimensions is an oscillating, localised, almost periodic classical solution that slowly radiates and decays to the vacuum solution. We calculate its momentum flux integrated over a period, which would be zero for an exact breather. For an oscillon of small amplitude $\eps$, truncated at order $\eps^N$, we show that the integrated momentum flux is of leading order $\eps^{N+4}$ for even $N$ and $\eps^{N+5}$ for odd $N$, except for $N=1$, where it is of order $\eps^4$. It is therefore very small.

We prove a closed formula for DT4 invariants of one dimensional stable sheaves on the total space of $\mathcal{O}_{\mathbb{P}^2}(-1,-2)$ for all degree curve classes. During the proof, orientations of moduli spaces are chosen in a canonical way. Combining with Gopakumar-Vafa type invariants proposed and computed by Klemm-Pandharipande, we verify a conjecture proposed by the author, Maulik and Toda.

We study two-Higgs-doublet models, extensions of the Standard Model (SM) containing two Higgs doublets rather than one. Tree-level Higgs-mediated interactions that change quark or lepton flavor are conventionally suppressed by a discrete $\mathbb{Z}_2$ symmetry. Motivated by string compactifications, we instead impose a non-invertible Fibonacci selection rule at a matching scale $\Lambda$ for that purpose. This rule also forbids the independent mass parameter that normally allows the additional Higgs particles to become arbitrarily heavy, while renormalization-group evolution regenerates it below $\Lambda$. An allowed $\mathbb{Z}_2$-breaking quartic interaction simultaneously avoids stable cosmological domain walls. In the exact alignment limit, requiring perturbative unitarity and boundedness from below up to the matching scale bounds the masses of the three non-SM Higgs states; for $\Lambda=10^{19}\,\mathrm{GeV}$, they lie below about $150\,\mathrm{GeV}$. The scanned parameter space predicts a suppressed Higgs diphoton signal strength within the projected reach of the HL-LHC. This illustrates how a stringy non-invertible selection rule can leave testable signatures in Higgs physics.

In this work we investigate the behaviour of early inflationary cosmic correlation functions under the flow of the infrared scale for interacting spectator scalar field theories. We start with the derivation of the one-loop effective potentials for $\phi^4$, $\phi^4$-Yukawa and broken phase $O(N)$ model in the inflationary de Sitter spacetime, first using the local Schwinger-DeWitt expansion propagator, and next using the exact de Sitter propagator. The Schwinger-DeWitt expansion being valid in a sub-Hubble length scale, furnishes suitable framework to determine the leading curvature corrections to high energy phenomena having sub-Hubble Compton wavelengths. These effective potentials show spontaneous symmetry breaking pattern, and are next used as ultraviolet boundary conditions in the Wetterich equation generating the desired IR flow. The flow is shown to drive the theory towards symmetry restoration. Using these IR-improved effective potentials, we next compute the stochastic power spectra and spectral indices for the temporally coincident two point correlation function of the spectator scalar. We show that the results corresponding to the aforementioned two kind of propagators are roughly in qualitative agreement with each other, but may differ quantitatively significantly. Both of them show increase in the spectral index in the deep IR for $\phi^4$ and $\phi^4$-Yukawa. The effect of fermionic interaction has been emphasised. Also for the case of the $O(N)$ scalar, we demonstrate that the effective potential's shape becomes more and more insensitive to the flow as $N$ gets large. Implication of this on the correlation function has been discussed.

We investigate the relation between gravitational entropy and holographic entanglement entropy in braneworld gravity with gravitational dynamics localized on the brane. We first consider the DGP brane model, in which an Einstein-Hilbert term is included in the brane action, and derive the condition under which the gravitational entropy of a braneworld black hole agrees with the holographic entanglement entropy. We show that the two entropies coincide when a specific condition involving the extrinsic curvature of the brane is satisfied. In particular, this condition is automatically satisfied for static configurations. We then generalize the brane-localized Einstein-Hilbert term to higher-curvature gravitational theories. For Lovelock gravity and $f(R)$ gravity on the brane, we derive the corresponding conditions for the equality between gravitational entropy and holographic entanglement entropy and show that they take the same form as that obtained for Einstein gravity. Although the equality is not guaranteed for general stationary configurations, the required condition is satisfied for static braneworld black holes, leading to an agreement between the two entropies. These results clarify the relation between gravitational entropy and holographic entanglement entropy in braneworld gravity and contribute to a deeper understanding of holography in braneworld settings.

We consider monoenergetic particles propagating in three spatial dimensions and undergoing elastic scattering off an external medium. For an arbitrary rotationally invariant scattering kernel, we derive an exact expression for the frequency- and wave-number-dependent diffusivity, thereby resumming the hydrodynamic gradient expansion to all orders. The result is an infinite continued fraction, with each successive level determined by the relaxation rate of a higher angular harmonic of the collision operator. Thus, while ordinary diffusion probes only the first angular harmonic, the resummed constitutive relation retains information about the full scattering kernel. We show how this microscopic information manifests itself in the analytic structure of the diffusivity and in the response to a localized external source. In particular, deviations from Fick's law decay exponentially with distance, at a rate determined by the nearest complex-wavenumber singularity of the inverse diffusivity. Thus, exponentially small departures from hydrodynamics retain detailed information about the underlying scattering process.

Inspired by a recent gauge field description of shallow water equations [SciPost Phys. 14 (2023) 102], in this work we build a simple Lagrangian formulation for the case of constant potential vorticity. An attractive feature of the construction is that it does not appeal to a Clebsch-like parametrization or the use of auxiliary fields. It is also emphasized that on the equator, where the Coriolis parameter vanishes, the shallow water equations exhibit extra SO(2,1) conformal invariance. Using the group-theoretic method, a new exact solution is found, which is associated with the scaling symmetry transformation entering the conformal group.

We investigate gravitational-wave echoes in a horizonless compact object model, a non-perturbative solution of the semi-classical Einstein equation that reproduces the entropy-area law from the bulk structure. By employing a model of anisotropic fluids in classical general relativity (GR), we show that odd-parity gravitational waves are not reflected at the object's surface but are trapped inside due to the exponentially strong redshift, and that the echo-delay time extremely exceeds the age of the Universe. Consequently, the echo signals are effectively unobservable, which is consistent with current null results from echo searches. We finally discuss how this quantum black hole and the conventional classical black hole could be distinguished observationally.

We investigate global consistency conditions for five-dimensional warped braneworlds in Palatini and metric-affine gravity. Starting from the generalized Gibbons-Kallosh-Linde framework, we derive integrated sum rules for cosmological warped geometries and examine how independent connection degrees of freedom modify the standard Einstein-gravity obstruction to smooth compact braneworlds. The analysis separates geometric, matter, and non-Riemannian contributions and clarifies the conditions under which the usual no-go result can be modified. We show that the generalized consistency relations provide necessary global constraints on admissible configurations and identify the terms responsible for possible departures from the Riemannian case. The results establish a systematic framework for testing the global consistency of smooth five-dimensional braneworlds in gravitational theories with independent affine structure.

We investigate whether a minimally coupled Abelian Maxwell sector can relax the Gibbons–Kallosh–Linde (GKL) obstruction to smooth five-dimensional warped braneworlds. For a canonical scalar field coupled to a bulk Maxwell field, we derive the one-parameter family of global consistency conditions while keeping the tangential F_{{\mu}{\nu}} and mixed F_{{\mu}5} sectors separate. This decomposition is essential because the Maxwell stress tensor is not traceless in five dimensions. We show that electromagnetic terms formally modify the integrated GKL constraints, but this modification does not provide a consistent evasion of the no-go theorem under the standard assumptions. Exact four-dimensional Poincaré invariance excludes a nonzero deterministic classical Maxwell field from the background, since its field strength would select preferred four-dimensional directions. The Maxwell sector therefore vanishes in the symmetric background and the usual smooth-brane obstruction is recovered. We also identify which assumptions must be relaxed for a nontrivial electromagnetic contribution and distinguish this background result from local consistency conditions for bulk gauge-field localization.

We construct the operator product expansions (OPEs) of the ${\cal N}=8$ ${\rm w}_{1+\infty}$ currents (obtained from the antiholomorphic light transform) with the matter fields in the ${\cal N}=8$ supergravity. The corresponding (anti)commutators are obtained from the double contour integrals using the above OPEs. By computing the triple (anti)commutators consisting of current-current-matter (i.e., the Jacobi identity for fixed last element), the previously known ${\cal N}=8$ celestial soft symmetry algebra between these currents (in the antiholomorphic sector) is reproduced. We repeat the above procedure by taking the holomorphic light transform and the OPEs of the ${\cal N}=8$ ${\rm \bar{w}}_{1+\infty}$ currents with the matter fields and the corresponding (anti)commutators are obtained. We also obtain another type of ${\cal N}=8$ celestial soft symmetry algebra between these currents (in the holomorphic sector). As a subalgebra in the antiholomorphic and holomorphic sectors, there exists the ${\cal N}=8$ extended BMS (Bondi, van der Burg, Metzner, and Sachs in 1962) algebra. We analyze the above results from the ${\cal N}=8$ conformally soft ${\rm w}_{1+\infty}$ (or ${\rm \bar{w}}_{1+\infty}$) currents in the antiholomorphic (or holomorphic) sector. The nontrivial soft theorems for the graviton, the gravitinos and the graviphotons are described. Finally, using the shadow transform (the consecutive composition of holomorphic and antiholomorphic light transforms), we obtain the OPEs of the ${\cal N}=8$ shadow ${\rm w}_{1+\infty}$ (or ${\rm \bar{w}}_{1+\infty}$) currents with the matter fields where the stress-energy tensor, the supercurrents and the supertranslation generator of the ${\cal N}=8$ extended BMS algebra are identified.

Consider a quantum state vector proportional to any given column of the character table of the symmetric group ($S_n$), corresponding to a superposition over the irreps weighted by the character values. It was recently argued that sampling from this state is classically hard under reasonable complexity-theoretic assumptions, while there exists an efficient quantum algorithm for preparing this character state using the quantum Fourier transform (QFT) over $S_n$ [arXiv:2501.12579]. We give an alternative, simpler algorithm to prepare this character state with a sequence of Hamiltonian evolutions, as well as an efficient implementation with reconfigurable qubits that uses only nearest-neighbor gates in 1D. The gate complexity, accounting for Hamiltonian simulation errors, depends on the choice of the column: For fixed target state error, the estimated sufficient total gate count ranges from as few as $O(n)$ gates for the $n$-cycle column, to $\widetilde O(n^{1.75})$ gates near the conjecturally classically hard parameter regime, to provably $\widetilde O(n^{2.5})$ gates in the worst case. In contrast, the QFT subroutine in the prior approach uses $\widetilde O(n^3)$ gates. Our gate count estimates come from leading-order state-dependent Trotter error analysis, and the proven worst-case bound relies on a QSVT-based Hamiltonian simulation method; numerical benchmarks of the Trotter-based protocol suggest that the estimates are conservative. Furthermore, we generalize our protocol to give an explicit algorithm for the quantum character transform (QCT) over $S_n$ using $\widetilde O(n^{2.5})$ gates and $O(n)$ qubits. An application to entanglement entropy of symmetric orbifold conformal field theories is also discussed.

We construct an exact NUT-charged generalization of the Ellis-Bronnikov wormhole in the Einstein-phantom-scalar system. The solution is stationary, asymptotically locally flat at both ends and free of curvature singularities. It inherits characteristic features of the Taub-NUT geometry, including the Dirac-Misner string and closed timelike curves, while the absence of Killing horizons eliminates the horizon-induced infinite winding of geodesics and renders the spacetime geodesically complete. By allowing the phantom field to have a nontrivial scalar potential with an infinite sequence of AdS vacua, we also obtain an exact asymptotically locally AdS wormhole with a NUT charge, which interpolates between different vacua at its two asymptotic ends. Using the covariant phase space formalism, we show that the Hamiltonian charges associated with the two asymptotic regions are finite under appropriate scalar boundary conditions. By performing a double Wick rotation to Euclidean signature, we demonstrate that the conformal class of the resulting metric admits an ambi-Kähler structure.

We show how extremal near horizon geometries can be recovered from the non extremal near horizon expansion for smooth stationary black hole families admitting a smooth extremal limit. In Gaussian null coordinates common to the family, we apply the near extremal scaling of the surface gravity together with a radial scaling and a rescaling of Killing time. Although the physical surface gravity vanishes, the scaled horizon can retain a finite surface gravity. The extremal near horizon geometry is obtained when this scaled surface gravity is taken to zero. Under the near extremal scaling and the time rescaling, the quadratic radial term in the temporal metric component can survive, while higher radial orders vanish. Expansion through second order is therefore sufficient and is necessary whenever the quadratic term has a nonzero limiting coefficient in the chosen chart. We derive a common pullback prescription for extracting the near horizon coefficients and recover the known throats of Reissner-Nordström, Kerr, Kerr-Newman, and charged rotating Bañados-Teitelboim-Zanelli (BTZ) black holes. Neutral rotating BTZ provides a case in which the string Carroll expansion already suffices.

Finite-temperature quantum-gravitational corrections to the cosmological constant introduce two additional density parameters, $\Omega_{\Lambda_2}$ and $\Omega_{\Lambda_3}$, the associated terms of which scale as $a^{-4}$ and $a^{-2}$, respectively. In previous work (Papers I and II), these parameters were studied using machine-learning methods that identified a best-fit point. Here we complete the picture with two complementary analyses. First, we perform a Bayesian MCMC analysis using a modified version of class against the Planck 2018 likelihoods. This tells us what the data can constrain once all parameter uncertainties have been marginalized over. Both parameters are consistent with zero, with $\Omega_{\Lambda_3}$ constrained only from one side. Second, we build a CMB pipeline from scratch, following Weinberg's kinetic-theory formulation and validating it against class. This allows us to study the underlying physics. The response analysis carried out with it shows that $\Omega_{\Lambda_3}$ is almost entirely absorbed by a shift in $h$ over the well-measured acoustic multipoles. The part of the signal that cannot be absorbed resides at low $\ell$, where cosmic variance dominates. Meanwhile, the $\Omega_{\Lambda_2}$ signature is nearly degenerate with the overall amplitude of the spectrum. The effects of both parameters on the CMB power spectrum are small, since they represent higher-order thermal corrections to the cosmological constant. The resulting constraints are correspondingly weak: the induced shifts are largely absorbed by degeneracies with $h$ and with the overall normalization of the spectrum. This is the reason why neither parameter is detected.

We investigate inflationary magnetogenesis with massive vector fields, including charged scalar particle production via the Schwinger effect and its backreaction on the background electric energy density. Initially neglecting the Schwinger effect, we quantize the vector field and solve the equations of motion for transverse and longitudinal modes, computing the energy momentum tensor. We then incorporate the backreaction of the electric energy density and show that the longitudinal energy density is subleading compared to the electric background. Since the electric energy density dominates the magnetic contribution, we can treat the former separately from the perturbative contributions. We adopt this approach when studying particle production via the Schwinger effect, assuming that its backreaction acts only on the background electric energy density. The latter is computed using a classical background electromagnetic field minimally coupled to a quantum complex scalar field. The magnetic and longitudinal components are instead described through quantum fields. In this framework, the Schwinger production affects only the electric energy density, as long as it remains dominant compared to the other components. As a consistency check, we verify that for a null coupling constant we recover the same electric energy density as obtained within the fully quantum electromagnetic description. Finally, we show that for non-null coupling constants Schwinger particle production drastically reduces the electric energy density during the final stages of inflation.

We study how the critical behavior of holographic superconductors controls the interior geometry of charged black holes. Although the exterior condensate vanishes at criticality, the critical zero mode develops logarithmic oscillations near the Reissner–Nordström Cauchy horizon, rendering the interior critical limit nonuniform. We show that the transmitted critical-mode amplitude, together with local Cauchy-horizon quantities, fixes both the leading Einstein–Rosen collapse $\Delta z_{\rm ER}$ width and the phase $\Theta_J$ in the subsequent Josephson regime. If the condensate obeys $t\equiv1-T/T_c\propto\epsilon^{2r}$, then $\Delta z_{\rm ER}\propto t^{1/r}$ and $\Theta_J\propto t^{-1/r}$. This Josephson phase is transferred to the first Kasner epoch, producing oscillatory Kasner exponents and an accumulation law $t_n\sim n^{-r}$ for successive zeros of the scalar velocity. The critical zero mode fixes the leading collapse and phase coefficients, which are quantitatively confirmed by fully nonlinear solutions in both the ordinary and tricritical cases. At low temperature, the interior instead depends on the infrared completion of the zero-temperature geometry. Thus near-critical scaling is governed by critical-mode transmission, whereas the low-temperature behavior is controlled by the infrared endpoint.

We perform an exhaustive study of the parameter space of the chiral magnetic model with a Dzyaloshinskii-Moriya interaction, an in-plane anisotropy term and an out-of-plane Zeeman magnetic field. We map out the phase diagram for the model and find a chiral soliton lattice hosted on a chiral soliton lattice as the ground state in a particular corner of the phase diagram. There is a smooth crossover, as opposed to a phase transition, to the skyrmion crystal lattice phase. Finally, we also discover excited composite soliton states such as an anti-skyrmion coexisting with a bimeron or an anti-bimeron.

By applying the Iyer-Wald formalism to the modified Lagrangian form for the gravitational system, where the coupling constants appearing in the original Lagrangian form, such as the cosmological constant, are promoted to the dynamical scalar fields by introducing the corresponding auxiliary gauge fields, we derive the first law of the extended black hole thermodynamics, where the resulting thermodynamic volumes as the conjugate thermodynamic variables to the coupling constants are nevertheless independent of the auxiliary gauge fields and exactly the same as those obtained before via varying the coupling constants by brute force. In addition, it is further shown that such volumes can also be expressed in terms of the auxiliary gauge fields and evaluated on the black hole horizon, validating the proposal made in the previous literature.

We develop an equivariant K-theoretic vertex formalism for a colored, abelianized higher-rank sector on smooth projective toric threefolds $X$. Nonzero first Chern classes are encoded as magnetic fluxes and absorbed into line-bundle twists, after which the virtual character is redistributed into finite perturbative, vertex, and edge contributions. We compute the resulting (unreduced and trace-free) virtual dimensions and analyze the interaction terms relevant to a possible reduction to rank one. We emphasize that, on a compact toric threefold, colored partitions do not classify all torus-fixed stable higher-rank sheaves: the resulting series is therefore a proposed abelianized/framed vertex series unless an additional comparison theorem with the full stable-sheaf moduli problem is supplied. When the threefold obstruction theory is not available in the desired form, we pass to the local Calabi–Yau fourfold $Y=\operatorname{Tot}(K_X)$ and formulate the corresponding colored DT4 vertex. A $K_X$-twisted tautological insertion removes fixed points with nontrivial fiber layers, reducing solid partitions to plane partitions at the level of the local vertex. In rank one we give a scheme-theoretic explanation: the Hilbert functor of subschemes of $X$ is canonically isomorphic to the constrained Hilbert functor of subschemes of $Y$ lying scheme-theoretically in the fixed zero section. Under the hypotheses of the DT4 K-theoretic Lefschetz/virtual-pullback formalism, the associated support insertion reduces the rank-one fourfold theory to the threefold theory. We keep this global rank-one statement logically distinct from the higher-rank colored-vertex reduction.

We investigate the effective action of a 7-dimensional topological $BF$ theory under a 4-dimensional reduction. We discuss the topological implications of the resulting theory and analyze the corresponding field equations. In particular, we derive exact solutions to these equations of motion in both Minkowski and Schwarzschild spacetimes.

We study anomaly-induced semiclassical backreaction on an analytic family of discretely self-similar solutions of Einstein–Klein–Gordon theory at large spacetime dimension. Horizon regularity fixes a null-coordinate Floquet exponent and selects a unique regular quantum state. The Polyakov auxiliary field is affine-periodic rather than periodic, with a monodromy that is determined independently by a global zero-mode condition and by the conformal-coordinate scaling. Spherical reduction introduces a quantum coupling that grows toward the center, so the periodic first-order expansion is nonuniform. We construct the resulting nonlinear inner problem and find a critical dilaton surface. Generic crossings of this surface are singular, whereas matching to the regular zero-mass exterior selects a narrow, high-curvature throat. The anomaly also separates the effective null-energy boundaries from the classical matter flux and zero-curvature curves. These results show that the minimally coupled Polyakov model does not yield a generic smooth, exactly self-similar completion of the classical center.