Papers

We develop a systematic method for computing the braiding matrices of four-point functions in RCFTs. This is implemented for the case of Virasoro minimal models. The first few terms of the conformal block series are directly computed through the Shapovalov form. The Fuchsian (BPZ) ODEs for order three and higher have accessory parameters that are fixed by this direct computation. We thus bypass the derivation from the null-state condition which gets tedious for higher-level null states. The braiding F-matrices are connection matrices between the Frobenius solutions at two singular points of the ODE. The F-matrices are computed numerically at first, agreeing with the results of Dotsenko–Fateev, which uses the Coulomb-gas formalism. We find that after a change of basis, the F-matrix is rendered unitary, which determines (products of) the three-point structure constants. We conjecture that the squares of the entries of the unitary F-matrix lie in a cyclotomic extension of the rational numbers. This enables us to convert our numerical estimates for the F-matrix into exact ones. The formalism is illustrated through numerous examples in Virasoro minimal models. We also discuss how these methods can be extended to cases involving symmetries that extend the Virasoro symmetry as well as for tenable examples that arise from the holomorphic modular bootstrap program.

In this paper, we study the asymmetric phases of the quartic type (0, 1) and (1, 0) Dirac en- sembles via three approaches: the Riemann-Hilbert approach, bootstrapping with positivity, and Hamiltonian Monte Carlo (HMC) simulations. The focus of this work is on asymmetric solutions to the Schwinger-Dyson and saddle point equations of these models, whose solution spaces prove deeply intricate. Via the Riemann-Hilbert approach, we are able to give explicit formulae for the eigenvalue distributions and free energy of various solutions. Using Hamiltonian Monte Carlo simulations, we are able to reconstruct the phase structure. Lastly, using bootstrapping with positivity, we are able to reconstruct the eigenvalue distribution of these models from their bootstrapped moments. All three methods show excellent agreement for a large matrix size.

This work describes a variational technique to solve non-Hermitian quantum systems, in particular those with negative coupling quartic self-interaction. Differences of applying variational methods from standard quantum mechanics to non-Hermitian quantum mechanics – including potential pitfalls – are highlighted. A successful implementation of the variational approach for the non-Hermitian case is presented. The technique is used to analyze scalar quantum field theory in d=2 with negative quartic self-interaction in the Hamiltonian formulation in lattice discretization.

We discuss classical and quantum features of spacetimes with a positive cosmological constant. After introducing basic aspects of de Sitter cosmology and the classical geometry of de Sitter space, we consider de Sitter space both as a rigid background, on which quantum fields propagate, and as a fluctuating spacetime in its own right. We review the wavefunction of an expanding universe, black holes in de Sitter space, and the de Sitter entropy, and compare the cosmological horizon with the black hole horizon. The final three sections are dedicated to concrete models of de Sitter space in two, three and four spacetime dimensions.

We revisit Deser's current reconstruction in first-order Yang–Mills theory with an independent two-form. We fix the quadratic auxiliary sector and the first gauge correction, and ask whether a prescribed curvature interaction can be added without higher-order action terms. For a compact semisimple gauge algebra, requiring the mixed Noether identity to close without a $gh$ action term fixes the algebraic auxiliary representative to the tangent representative, up to linear curvature and dual-curvature terms on each simple ideal. For the analytic radial class $U=\Phi(f\cdot f)$, with first nonlinear term $c_p(f\cdot f)^p$, no allowed representative avoids a third-order action correction when all second-order action coefficients vanish. A free-field scaling argument inside a root $\mathfrak{su}(2)$ subalgebra establishes this result for every compact semisimple gauge algebra. Ordinary covariantization and an auxiliary-field shift give an explicit completion. For $p=2$, its seven-gluon contact term cancels a nonzero tree-level Ward contraction at real nonsingular kinematics. We also study an affine star-gauge sector motivated by Seiberg–Witten maps, where strict current feedback and second-order action truncation select different representatives. Together these results show how the prescribed auxiliary data determine which interactions are required by current reconstruction.

To study the BPS black hole states in typical $\mathcal{N}=1$ AdS$_5$/CFT$_4$ models, we formulate a classical cohomology problem in the Klebanov-Witten theory, which is defined at a strongly coupled RG fixed point. We set up its weakly coupled cohomology problem in the UV theory, suggesting a refined notion of monotonous and fortuitous cohomologies so that simple baryonic operators belong to the former. We construct infinitely many fortuitous cohomologies in the $SU(2)\times SU(2)$ theory and interpret some of them as $N=2$ versions of hairy black hole states dressed by baryonic condensates. We also comment that the recently constructed fortuitous cohomologies in ABJM exhibit generalized monotonicity.

We construct a new class of smooth, finite-energy solitons in the bosonic $SU(2)\times U(1)$ Weinberg-Salam theory, which we dub $electroweak$ $balls$. Their localization mechanism is analogous in spirit to that of $Q$-balls: the charged vector fields possess a harmonic time dependence while the energy-momentum tensor remains time independent. We explicitly construct both spherically symmetric electric-type solutions and axisymmetric magnetic-type solutions, and show that they form families characterized by a finite frequency interval, a mass gap, and a two-branch structure. These configurations provide electroweak counterparts of Proca-Higgs balls, with the vector-boson masses generated by the Higgs mechanism rather than introduced explicitly. The construction is not tied to the measured parameters of the Standard Model. More generally, it applies to bosonic electroweak-type sectors with different gauge couplings, Higgs self-coupling and symmetry-breaking scale, and hence potentially very different characteristic particle and soliton mass scales. For the families studied here, we do not find solutions at the measured Standard Model couplings and mass ratios.

Conformal interfaces provide an important setting for studying universal transmission phenomena in one-dimensional quantum critical systems. While energy and information transmission across such interfaces are characterized by universal quantities in closed systems, the corresponding role of conformal interfaces in dissipative dynamics is less understood. In this work, we study relaxation in locally dissipative quantum critical chains with a conformal interface and show that a universal characterization emerges at the level of individual relaxation modes. We first show that the relaxation coefficient defined in the recent work [1] from the Liouvillian gap can become non-universal for certain boundary conditions because the mode determining the smallest decay rate can change as the interface transmission is varied. To resolve this ambiguity, we introduce a mode-resolved relaxation coefficient $c_{\rm relax}$ by continuously tracking the same Liouvillian rapidity mode as a function of the interface transmission. Using analytical and numerical calculations for a critical harmonic chain and a critical free-fermion chain, we find that, in the weak-dissipation regime, $c_{\rm relax}$ follows the same universal dependence on the interface transmission in all cases considered, independent of microscopic details such as the boundary conditions, dissipation strength, and location of the local dissipation. For boundary dissipation, this universal behavior persists even at finite dissipation strength. Our results establish a universal mode-resolved characterization of relaxation across conformal interfaces in open quantum critical systems.

We consider spin-$s \geq \frac{3}{2}$ complex, strictly massless fermionic gauge fields on the four-sphere, $S^4$, which is the Euclidean continuation of de Sitter spacetime, dS$_4$. For $s=\frac{5}{2}$, after briefly discussing different options for the quantisation of the theory on dS$_4$, we compute the one-loop sphere path integral. We explain how it can be expressed in terms of functional determinants, generalising previous results for $s=\frac{3}{2}$. We proceed to rewrite the result in terms of `bulk' unitary Harish-Chandra characters in the discrete series of the de Sitter isometry group, $Spin(4,1)$, and `edge' characters. We then generalise the character expression of the sphere path integral for any complex, strictly massless spin-$s\geq\frac{3}{2}$ fermionic gauge potential. Additionally, we compute the coefficient of the logarithmic divergence of the $S^4$ path integral for all spin-$s \geq \frac{3}{2}$, and we show that it is fully encoded by bulk and edge characters. We further show that at one loop no imaginary phase appears for strictly massless fermionic gauge potentials, contrary to the case of bosons. Lastly, we confirm the exact one-loop cancellation between bulk and edge contributions for the case of an infinite tower of complex massless fermions of spins $s=\frac{1}{2}, \frac{3}{2},\dots$, observed recently.

We construct bosonic stars in the Einstein-Weinberg-Salam theory, namely the bosonic sector of the electroweak Standard Model minimally coupled to Einstein's theory of gravity. These configurations are everywhere regular, spherically symmetric and asymptotically flat, consisting of a static condensate of the massive $W$ and $Z$ bosons, whose masses arise from the Higgs mechanism. Although the corresponding fields exist in nature, gravitational effects are negligible at the physical electroweak scale and the configurations would be microscopic. Keeping Newton's constant and the electroweak mass ratios at their physical values, we therefore interpret the model as the bosonic sector of a dark electroweak theory with a radically different energy scale. Choosing the ultralight vector boson mass $m_{\mathrm{W}}c^{2}=8.7\times10^{-13}\,$eV, as suggested by the Proca star interpretation of GW190521, our bosonic stars can attain masses in the intermediate-mass black hole range.

We develop an operator algebraic generalization of the Yoshida-Kitaev information recovery protocol that applies to von Neumann algebras of arbitrary type. The construction is based on finite-index inclusions, with the Jones basic construction and canonical endomorphisms providing the central algebraic tools. Unlike the finite-dimensional qubit description, the infinite-dimensional theory exhibits new structural phenomena that play an essential role in information recovery. In particular, the diary information is represented non-locally by a choice of Pimsner-Popa basis associated with the inclusion. Exploiting the Temperley-Lieb relations satisfied by the Jones projections, we introduce an algebraic notion of computational complexity and show that it is naturally measured by the Jones-Kosaki index. For irreducible depth-two inclusions, we demonstrate that the information transfer is implemented by an algebraic Fourier transform. Finally, we discuss a potential spacetime interpretation of the construction, including the emergence of an island algebra in terms of (non-)isometric embeddings and an operator algebraic realization of black hole complementarity.

We consider type IIB string theory on $\mathrm{AdS}_3\times \mathrm{S}^3\times \mathbb{T}^4$ in a large AdS radius expansion at arbitrary values of the relative RR and NS-NS fluxes. We develop a method for computing scaling dimensions in the dual CFT in this limit by combining a string field theory calculation, valid at finite quantized NS-NS flux but perturbative in RR flux, with a finite mixed-flux ansatz, which is constrained by worldsheet parity and motivated by classical string solutions. We apply the method to three families of states. In the pure-RR limit, our results agree both with a family of states computed using the worldsheet bootstrap and with another family computed using integrability.

Finiteness appears to play a fundamental role in the mathematical structures underlying the laws of physics. Tame geometry, defined by the theory of o-minimality, provides a framework for making this idea precise. In this thesis we study the concept of tameness and finiteness of complexity across applications in quantum field theory and quantum gravity, and demonstrate that many classes of physical functions and theories are tame. To quantify these ideas, we use the novel theory of sharp o-minimality to measure the complexity of physical objects. The analysis covers wavefunctions in quantum mechanics, classical field configurations, non-perturbative observables in zero-dimensional quantum field theories, cosmological correlation functions, and Lagrangians of effective field theories. Finally, we argue that the emergence of tameness in physics aligns with manifestations of finiteness in quantum gravity, converging towards the idea that effective theories of quantum gravity admit a description of finite complexity. The thesis contains an introductory review for readers who wish to get familiar with tame geometry and its applications to physics.

We obtain the Kerr black hole in an external uniform magnetic field in the Kaluza-Klein (KK) theory via a standard solution-generating technique. Using this as our starting point, we construct exact static solutions describing a Schwarzschild black hole immersed in the external uniform magnetic field in general string-inspired Einstein-Maxwell-dilaton (EMD) theories, thereby embedding the static Kerr-Bertotti-Robinson (KBR) solution into the general framework of EMD theories. We also construct electrovacuum solutions involving both electric and magnetic fields in EMD theories and generalize the system to include mutltiple Maxwell fields and dilatons. These equations of motion can be reduced to Toda-like equations, allowing us to obtain analytic electrovacua associated with Toda equations of all rank-2 Lie groups. We further construct exact solutions of a Schwarzschild black hole immersed in these electrovacua.

We show that SYK-like melonic solvability and maximal chaos survive in a sparse deterministic model, without disorder averaging. Our deterministic hypergraph SYK model (hSYK) consists of $N$ Majorana fermions on a $q$-uniform, $d$-regular interaction hypergraph with only $O(dN)$ interaction terms and uniform positive couplings. Under explicit geometric and spectral expansion conditions, its two-point function satisfies the standard melonic Schwinger-Dyson equation up to $O(d^{-1})$ corrections, while the leading bilocal effective action coincides with that of ordinary SYK at the dominant large-$d$ saddle. Its connected four-point function factorizes into a temporal SYK ladder kernel and a line-graph random walk. A finite line-graph spectral gap controls the visibility of the maximally chaotic uniform mode in generic localized OTOCs and, by gapping non-uniform spatial modes in the bilocal spectrum, excludes additional light fluctuation modes beyond the Schwarzian. Exact diagonalization further shows melonic two-point scaling, single-instance Wigner-Dyson statistics, and the predicted graph-controlled spatial decay of OTOCs. The model therefore offers a disorder-free realization of the essential melonic and chaotic features of SYK, in a form more amenable to quantum simulation.

The primordial statistics underlying the CMB and large-scale structure are encoded in late-time cosmological correlation functions (constructed from the coefficients of the wavefunction of the universe). They are the closest analogue cosmology has to an $S$-matrix, yet the standard $S$-matrix toolkit does not directly apply. This review explains how the modern scattering-amplitudes program (spinor helicity, on-shell recursion, the double copy, positive geometry, and generalized unitarity) has nevertheless been extended to cosmology and, through a single analytic continuation, to Euclidean anti-de Sitter space, where the same objects appear as holographic boundary correlators and are often cleanest to compute. The organizing framework is momentum space on the late-time boundary, where the primordial statistics are simplest and where the total energy of the external legs, the sum of their momentum magnitudes, controls the analytic structure. At the total-energy singularity, the flat-space scattering amplitude emerges. We develop this toolkit starting from the wave equation and use it to compute tree-level scalar, gauge-field, and graviton correlators, exposing relations across spins and dimensions. We then show how bulk integrals can be bypassed using cosmological polytopes, on-shell recursion, and curved-space versions of the double copy, before extending the discussion to loops, soft limits, transition amplitudes, and (A)dS cutting rules. Finally, we connect these structures to inflationary observables and to three allied programs: the cosmological optical theorem, the cosmological bootstrap, and the cosmological collider. We conclude with a map of the current frontier and a set of open and interesting problems.

We derive an entropy formula for a recently proposed tensor network model of states in 3D gravity which satisfy the constraint equations of diffeomorphism invariance, and including the effects of matter. After accounting for the invertibility of the spacetime metric, we argue that this formula matches the quantum extremal surface formula (for small but finite $G_N$), and is not restricted to time symmetric spacetimes. We derive the result via canonical quantization without relying on path integrals, and discuss an extension to compact bulk subregions.

We study how $SL(2,\mathbb Z)$ transformations act on pole-truncated linear-response effective theories in $(2+1)$ dimensions. For correlators meromorphic within a finite low-frequency domain, the EFT data include both derivative-expanded transport coefficients and the retained non-hydrodynamic poles. Since the $S$-transformation inverts the optical conductivity and exchanges its poles and zeros, it does not necessarily commute with the infrared truncation. We derive explicit maps for static susceptibilities, the Einstein relation and AC transport coefficients. In the purely hydrodynamic sector the map closes on the original transport data, while in quasihydrodynamic regimes the transformed pole positions and, in general, residues constitute additional infrared data. We test the construction in the alternatively quantised D3/D5 probe-brane system, where an EFT retaining one original gapped pole can correspond to transformed EFTs retaining zero, one or two. The holographic results verify the analytic maps and exhibit an inverse-Drude conductivity at large values of the control parameter.

We investigate the geometry of the Fermion mixing matrix in the Standard Model. The principal structure studied is the CKM matrix, $V$, viewed as an element of the double coset space $M$, where $M=T\backslash SU(3)/T$, and $T=U(1)\times U(1)$. The space $M$ is given a Riemannian metric, and has both point singularities, which it is shown correspond to the known renormalization group fixed points, and line singularities connecting the fixed points. A key discrete object inducing many of these properties is the Weyl group, $W$, of $SU(3)$, $W$ being the symmetric group $S_3$, which appears in the guise of the vertices of a hexagon in $V$.

Many examples of integrable sigma models admit families of integrable deformations that involve coupling the undeformed “seed” theory to auxiliary fields with algebraic equations of motion. In this work, we apply this paradigm to the case where the seed theory is the lambda model, which interpolates between the Wess-Zumino-Witten model and the non-Abelian T-dual of the principal chiral model. We present an auxiliary field deformation of the lambda model and exhibit a Lax connection whose flatness is equivalent to the remaining equations of motion after imposing the algebraic field equations. We also carry out a Hamiltonian analysis of the deformed theory, verifying that it obeys the assumptions of theorems in arXiv:2605.18213 which guarantee both (i) that the model enjoys the Maillet structure that ensures involution of its conserved charges, and (ii) that the theory possesses a classical Yangian. Thus, as in other cases, the auxiliary field deformation of the lambda model changes its classical dynamics while leaving the integrable structure essentially unchanged.

We construct theories of Carrollian electromagnetism coupled to an axion field. Obtained by performing a $c \to 0$ expansion of axion electrodynamics, these are of two types: electric and magnetic. We derive the Lagrangians and work out the general classical solutions for both. While the electric theory has the axion effectively decoupled from the electromagnetic fields, the magnetic theory is endowed with a nontrivial axion coupling that leads to interesting phenomenology. In particular, a Carrollian analogue of the Witten effect, wherein a magnetic monopole in the presence of a constant non-vanishing axion acquires an electric charge and becomes a dyon, is present in the magnetic theory. At the same time, the linear magnetoelectric effect vanishes entirely, a conclusion we establish by performing a thorough Dirac-Bergmann analysis of the constraint structure of the magnetic theory and a proper identification of its physical phase space.

We investigate a family of $k$-essence cosmological models selected by the requirement that the field equations admit a nontrivial variational symmetry. Inside this general family of theories, we concentrate on the simplest modification to quintessence, induced by power-law terms of the kinetic energy. The symmetry generator introduces a conservation law which we use to obtain an exact vacuum solution. We generalize our analysis and investigate the structure of the phase-space of the cosmological field equations by performing a dynamical systems analysis, both in the absence and the presence of a pressureless matter source. We employ Hubble-normalized variables and the Poincaré compactification, so as to determine the asymptotic behavior of the model and identify all admissible cosmological epochs, from early-time to late-time behavior. We discuss in detail the existence and stability conditions of each fixed point as functions of the free parameters, and characterize the full asymptotic structure of the cosmological history as provided by this symmetry-driven $k$-essence model. We explore the conditions under which the model can describe cosmic acceleration. We find that this $k$-essence model successfully describes the inflationary epoch with a natural exit to the matter-dominated era, with the inflaton ultimately evolving to a dark matter source.

The Scattering Facet (SF) of the Cosmological Polytope is a positive geometry defined from a Feynman diagram and encoding its combinatorics. Its canonical differential form reproduces the result of the integration over the energy components of loop momenta in scalar Feynman integrals, and is therefore closely related to the Loop-Tree Duality (LTD). In this article, we present a novel description of the SF as a base-fiber geometry. The fibers are a family of polytopes called deformed graphical zonotopes, which have a natural interpretation in terms of the causal evolution of the underlying Feynman diagram, and capture interesting information on scattering amplitudes. This decomposition of the SF induces a description of the canonical form leading to the so-called causal representation. Moreover, it provides new mathematical tools that allow us to characterize all its regular triangulations. Canonical forms obtained from these triangulations correspond to spanning tree representations, matching the original LTD amplitudes expressed as a sum over spanning trees. This geometrical viewpoint provides us with new methods to obtain the causal and spanning tree representations, without the need of performing any integration.

Recent EHT polarimetric observations provide a powerful probe of near-horizon physics and modified gravity. We investigate the shadow and polarization images of a static Schwarzschild-MOG black hole illuminated by geometrically thick, optically thin accretion flows. Adopting a phenomenological model and the Hou disk model, and solving null geodesics and relativistic radiative transfer, we study how the MOG parameter, inclination angle, anisotropic synchrotron emission, and flow geometry affect the total intensity and linear polarization. It is found that the MOG parameter mainly enlarges and brightens higher-order photon rings and broadens the shadow, while the inclination angle breaks axial symmetry and enhances polarized emission inside the shadow. The Hou disk model suppresses off-equatorial radiative interference and localizes polarization signals within higher-order rings. Electric vector position angle (EVPA) patterns are mostly azimuthal but vary rapidly near lensed images. These results provide theoretical diagnostics for constraining MOG with EHT/ngEHT polarimetric observations.

We present a physics-informed machine learning framework based on Neural Ordinary Differential Equations that solves the holographic inverse problem: reconstructing the bulk spacetime and gauge field of a charged AdS black hole directly from boundary fermionic spectral functions. Encoding the UV asymptotics, horizon regularity, and zero temperature extremality as hard constraints in the neural network architecture, our framework reliably reconstructs the extremal Reissner-Nordström AdS geometry across three quantum critical regimes set by the $U(1)$ probe charge—non-Fermi liquid, marginal Fermi liquid (strange metal), and Fermi-liquid-like states—and can jointly infer the probe charge itself to sub-percent accuracy. Relaxing the near-AdS boundary constraint uncovers a geometrical degeneracy: bulk profiles that differ throughout the radial direction but share the same near-horizon $AdS_2 \times \mathbb{R}^2$ data reproduce identical spectral functions near the Fermi surface. This isospectral non-uniqueness is precisely the bulk degeneracy expected on general holographic grounds at zero temperature, and its spontaneous emergence across independent training runs shows that the network isolates the IR CFT universality rather than overfitting a single UV completion.

We study the near-horizon charged-scalar instability associated with Schwinger pair production in the charged regular black hole geometry of the zero-point-length T-duality prescription. Using the same effective geometry and regularized gauge potential, we construct the extremal branch, its ${\rm AdS}_2 \times S^2$ near-horizon throat, and the charged-scalar instability threshold for a general form factor. For the explicit T-duality form factor the extremal branch terminates at $r_\mathrm{ext} = \sqrt2 \, l_0$, where the extremal charge and the near-horizon electric field vanish while the ${\rm AdS}_2$ radius remains finite at $\sqrt3 \, l_0$. At this endpoint the exact near-horizon instability parameter is negative. In the semiclassical regime the corresponding charge-to-mass threshold diverges as the endpoint is approached. Thus, for each fixed massive species with finite $q/m$, a sufficiently near-endpoint portion of the extremal branch is free of the local charged-scalar instability. For comparison, the Ayón-Beato-García (ABG) Einstein-nonlinear-electrodynamics solution has a nonvanishing extremal near-horizon electric field. For an additional minimally coupled charged-scalar probe, the corresponding local Schwinger threshold remains finite. Thus the divergent near-endpoint Schwinger barrier found in the T-duality branch is not a generic consequence of regularity.

We study higher-order differential equations corresponding to integrals of the Dotsenko–Fateev type and associated with four-point correlation functions with degenerate fields $\Phi_{n,1}$. We determine the general structure and coefficients of the equations in terms of the integral parameters $ \{ A,B,C,g \} $. In particular, special cases of these equations in CFT are used for verification. In addition, the relationship between higher-order differential equations and hypergeometric operators is established.

We construct turbulent gravitational wakes in asymptotically AdS$_4$ spacetimes by solving the full non-linear Einstein equations in the characteristic formulation. Specifically, we show that a localized boundary deformation acting on a boosted black brane generates a wake whose boundary dynamics exhibit the formation of coherent vortical structures that are advected downstream by the background flow. Analysing a wake propagating along the x direction, we show that it displays the characteristic self-similar scaling of classical turbulent wakes. We then identify the geometric imprint of the wake on the horizon area density, the traceless horizon extrinsic curvature, and bulk metric perturbations, finding coherent structures that extend from the asymptotic boundary to the horizon. These results provide the first study of turbulent gravitational wakes in four-dimensional asymptotically AdS spacetimes.

Interacting dark energy (IDE) models, featuring interactions between dark matter (DM) and dark energy (DE), have received significant renewed interest. However, little attention has been devoted to understanding their genuine cosmological signatures, with comparisons against $\Lambda$CDM often carried out at fixed cosmological parameters. Considering the widely studied model with energy exchange rate proportional to the DE density, we carry out a three-level sequence of comparisons, starting from the naïve fixed-parameter comparison, then compensating for shifts in $\theta_s$ and $z_{\text{eq}}$, and then removing effects due to the modified background. We show that fixing $\theta_s$ and $z_{\text{eq}}$ leads to pre-recombination evolutions of the Weyl potential, photon monopole, and baryon velocity which are nearly identical to their $\Lambda$CDM counterparts, and hence virtually indistinguishable Cosmic Microwave Background (CMB) temperature power spectra. Comparing IDE to a non-interacting $w$CDM model with the same background, we find small scale-dependent signatures in CMB lensing. We find larger differences in the matter power spectrum, reflecting the different values of $\Omega_m$ resulting from the different partition of the dark sector into DM and DE, with only percent-level differences remaining once we compare $\Omega_m^2P_m(k)$, indicating the importance of high-fidelity determinations of $\Omega_m$. Our controlled comparison strategy can be a powerful tool for a wide range of models beyond $\Lambda$CDM.

In this paper, we investigate the localization of various $q$-form fields on a codimension-two brane. In particular, the $0$-form scalar field, the $1$-form $U(1)$ gauge vector field, and the $2$-form Kalb-Ramond field are considered with gravitational coupling, where a coupling function $F(R)$ is introduced into the six-dimensional actions of these fields. The function $F(R)$ depends on the scalar curvature of the bulk. Within this framework, we find that the massless modes of different $q$-form fields can be localized on the thick brane for positive values of the coupling parameters $t_1$ and $t_2$. For the massive modes, the different $q$-form fields exhibit similar localization properties determined by the coupling parameter $t_2$. When $0<t_2<v^2/24$, the massive modes of these fields cannot be localized on the brane, while they may exist as the resonant modes. In the case $t_2=v^2/24$, a finite number of massive modes can be localized on the thick brane, and the number of localized modes increases with the coupling parameter $t_1$. Finally, when $t_2>v^2/24$, the effective potentials associated with the Kaluza-Klein modes of these $q$-form fields form infinitely deep potential wells, so that all massive modes can be localized on the brane. Moreover, the tachyonic massive modes can always be excluded for different $q$-form fields.