Papers

This work investigates the deformed-derivative formalism introduced by Borges, with emphasis on the relation between the linear operator $D_{(q)}$ and its nonlinear dual counterpart $D^{(q)}$. Directly inserting the dual derivative into the kinetic term leads to a nonlinear Schrödinger equation and obscures the usual interpretation of superposition and probability. We show that this nonlinearity can be removed by a simultaneous transformation of the coordinate and of the wave function. The transformed problem is an ordinary linear Schrödinger equation in a deformed coordinate, and its representation in the physical coordinate is equivalent to a Hermitian position-dependent-mass (PDM) Hamiltonian. In this formulation, the deformation parameter $q$ determines both the effective mass profile and the associated metric. The formalism is applied to the free particle, the infinite square well, the rectangular barrier, and the harmonic oscillator in the weak-deformation regime. Comparison with the nonadditive-translation approach of Costa Filho et al. shows that the Borges dual-$q$ framework provides an alternative route to the same effective geometric structure. For $q<1$, the effective confinement length is reduced, which raises the bound-state spectrum and enhances tunneling; for $q>1$, the effective length is increased, which lowers the spectrum and suppresses tunneling relative to the undeformed limit $q=1$.

We study the renormalization group flow of non-local form factors in four-dimensional quantum gravity within the proper-time formalism at quadratic order in the curvature expansion. We show that the flow equations can be integrated down to $k=0$, allowing the reconstruction of the full momentum dependence of the form factors. Within this framework, we construct asymptotically safe solutions at this order. We find that asymptotic safety of the flow does not automatically ensure a finite cutoff-independent $\Lambda\to\infty$ limit for the integrated solutions, which in general develop a logarithmic divergence $\ln(q^2/\Lambda^2)$, so that a renormalization condition is still required. A finite $\Lambda\to\infty$ limit compatible with asymptotic safety is obtained only when the ultraviolet boundary condition selects the non-Gaussian fixed point. This yields finite dimensionful form factors, removes UV logarithmic contributions, and ensures independence from the renormalization scale $\mu$. The resulting renormalized asymptotically safe form factors display a power-law decay $\sim1/q^2$ in the ultraviolet and reproduce the expected logarithmic structure in the infrared, with the Planck scale replacing the renormalization scale.

The dressed propagator of a ghost coupled to ordinary fields develops a pair of complex conjugate poles in the first Riemann sheet above the multi-particle threshold. We study the implications of this pole structure for the asymptotic field and its negative-norm one-particle state. Within the operator formalism of local quantum field theory, we show that certain quadratic couplings between the ghost field and the composite field of the multi-particle state persist at asymptotic times. These induce quantum interference effects that render the negative-norm one-particle state non-orthogonal to a superposition of positive-norm multi-particle states. Consequently, no free asymptotic one-particle ghost state exists, while the true free asymptotic states have zero norm and hence vanishing localization probability. The real and imaginary parts of the complex mass admit a clear physical interpretation; in particular, the inverse imaginary part sets the timescale for the onset of non-orthogonality. A freely propagating ghost is therefore confined to time intervals much shorter than its inverse width, so that a detector can never observe an isolated ghost particle asymptotically. Open questions and potential applications are discussed in the conclusions.

We propose that any compact $d$-manifold with elliptic data, $\mathcal{J}$, prepares a quantum state $|\mathcal{J}\rangle$ on its $(d-1)$-boundary $\sigma$. Elliptic data consists of metric and field values, or their conjugates, but not both. No asymptotic structure is required. Inner products and traces are evaluated by the gravitational path integral with closed boundary conditions obtained by gluing elliptic data manifolds. In particular, we give a prescription for the Rényi entropies $S_n$ of a subregion of $\sigma$. In a class of examples, we find that $S_n$ is nonnegative and nonincreasing with $n$, as required for consistency. We obtain the von Neumann entropy by analytic continuation and find agreement with the minimal surface prescription of Bousso and Penington.

We bootstrap the four-point next-to-maximally helicity-violating (NMHV) form factor of the chiral stress-tensor supermultiplet in planar maximally supersymmetric Yang-Mills theory through three loops at the symbol level. At two loops, an ansatz built from NMHV leading singularities and the relevant five-point one-mass integral function space is fixed uniquely by physical constraints; the resulting ratio function symbol contains 78 letters, all drawn from the 88-letter alphabet previously identified in the four-point MHV sector. At three loops, using this 88-letter alphabet as input and imposing extended Steinmann relations satisfied by the minimally-subtracted hard function, together with other physical constraints, we determine the three-loop symbol uniquely. Both results pass soft, double-soft and directional dual conformal invariance (DDCI) checks, provide the first multi-loop non-MHV form-factor data, and support the universality of the 88-letter alphabet for four-point form factors beyond the MHV sector.

We propose a new framework for solving the strong CP problem via a heavy axion, using mirror symmetry and grand unification. The mirror GUT sector remains unbroken and dynamically generates a calculable heavy mass scale via confinement without fine tuning. Models in this class feature a heavy axion, whose potential is less sensitive to Planck scale corrections, as well as a rich hidden sector from the confined mirror GUT. The solution to the strong CP problem remains unspoiled by the presence of additional phases in the GUT Yukawas, yet allowing the possibility of electric dipole moments within the reach of future experiments. Our proposal offers new directions in GUT model building, axion phenomenology, dark matter and cosmology.

We extend the frequency-domain analysis of quasinormal modes in a dynamical, spherically symmetric black hole spacetime undergoing constant-rate mass evolution. In particular, we report a novel feature of the spectrum: the presence of purely imaginary eigenvalues in addition to the usual light-ring modes. We study the frequencies of these modes both analytically and numerically. The analytical calculation uses a novel formalism based on recent advances in connection coefficients of Heun functions. We then compute the frequencies numerically using a spectral method on hyperboloidal slices and find excellent agreement between the two approaches. Finally, we validate the frequency-domain results against an independent set of time-domain simulations. Our analysis shows that the purely imaginary modes govern the late-time signal through exponentially decaying tails. In the Schwarzschild limit, both frequency- and time-domain studies consistently show that the purely imaginary modes give rise to the familiar Schwarzschild power-law tail.

We review the notion of Yang-Baxter integrability for spin chains that have Hilbert spaces with constraints, such as a Rydberg blockade. We focus on anyonic chains, whose constraints arise from the fusion rules of the fusion categories on which they are based. We discuss the emergence of Temperley-Lieb algebras and present a new result on which types of anyonic chains exhibit them. We then give an overview of known results for integrable anyonic chains and extend them to several fusion categories up to rank $7$. Using a modification of the boost operator formalism, we find several new integrable anyonic chains and discuss some of their properties. These include spin-$\frac32$ models for $\mathfrak{su}(2)_k$ fusion categories, anyonic chains based on the Tambara-Yamagami fusion categories TY$(\mathbb{Z}_n)$, and product fusion categories Fib$\times$Fib and Fib$\times$Ising. We review recent results for spin chains based on the Haagerup-Izumi fusion category HI$(\mathbb{Z}_3)$, and present preliminary numerics for a HI$(\mathbb{Z}_5)$ model.

We introduce a framework for emulating graphs and, through them, curved spaces of arbitrary dimension, using arrays of superconducting wires. The array consists of two stacked layers of wires, horizontal and vertical, such that wires are parallel within each layer and perpendicular between layers. By discretizing a space into a graph, assigning a superconducting wire with a rigid phase to each vertex, and coupling pairs of wires through Josephson junctions according to the graph edges, arbitrary geometries and topologies can be engineered in a controlled setting. The superconducting phases then realize scalar field theories on the emergent geometry. We establish experimentally realistic conditions for implementing these architectures and develop a dictionary relating measurable circuit observables to quantities in the emulated field theory. As an application, we develop the implementation of hyperbolic (Anti-de Sitter) spaces of constant negative curvature and use them as an experimentally accessible platform to explore holographic duality in arbitrary dimensions. We investigate the effects of disorder in the Josephson couplings, which translate into metric variations in the bulk-boundary correspondence, and analyze their impact on boundary scaling exponents both analytically and numerically, finding that holographic duality remains robust even in the presence of strong disorder. Beyond holography, the framework opens a broad range of architectural possibilities, including the exploration of physics on highly nontrivial graphs and toy models of dynamical spacetimes.

We study the time evolution of the entanglement structure of holographic conformal field theories after a local quench. Using the mutual information between two spatial intervals as a probe, we find that $1+1$-dimensional conformal field theories exhibit a rich pattern of dynamical phase transitions. In the large-central-charge limit, mutual information develops sharp non-analyticities at critical times, providing a concrete entanglement-based realization of dynamical quantum phase transitions. We find that the dynamics organize into six distinct phases of mutual information, each controlled by the dominance of a different conformal block, or equivalently, a different holographic geodesic configuration. This phase structure goes beyond the standard quasi-particle picture, explaining non-analytic features that are not captured by simple light-cone propagation from the quench points. We further identify a dynamical $D_4$ symmetry acting on the interval endpoints that controls the presence or absence of mutual information. The onset of mutual information is governed by the breaking of this symmetry to a $\mathbb{Z}_2 \times \mathbb{Z}_2$ subgroup, suggesting a symmetry-based characterization of non-equilibrium entanglement dynamics analogous to the role of symmetry in equilibrium critical phenomena. Finally, numerical studies of critical spin chains indicate that finite-$c$ effects smooth out the sharp large-$c$ transitions between different mutual-information phases, while the transitions between phases with and without mutual information appear to remain non-analytic. These results offer a unifying perspective on real-time entanglement dynamics and their critical features in conformal many-body systems.

We analyse the rôle of approximate axionic shift symmetries in gravity-decoupling limits arising in type II Calabi-Yau compactifications. Associated with each shift symmetry there is an axionic string whose tension constrains the gradient of the Kähler potential, as expected in regimes where gravity becomes weakly coupled. The gradients of these tensions define vector fields on moduli space, analogous to those associated with BPS particle masses. Together, they characterise the evolution of different effective field theory sectors along asymptotic limits, encoding both their coupling to gravity and their kinetic mixing. By deriving upper bounds on the inner products of these vector fields, we show that they split into mutually orthogonal subsets, one of which decouples from gravity. Finally, we relate the Laplacian of certain axionic string tensions to a divergent moduli space curvature.

We introduce FunKit, a Mathematica package for the derivation and tracing of functional equations from arbitrary master equations. FunKit provides an expression vocabulary and a set of rules that allow for derivations in any given field theory and master equation. It also allows users to add extensions for more specific equation systems. Therefore, it can be used in a wide range of situations, for example Dyson–Schwinger or functional RG equations, flowing reparametrisations, nPI equations, (modified) STIs and WTIs, functional Polchinski and Wegner flows, functional master equations with sources, and many others. Besides interfacing with the \FORM language to trace large tensor expressions efficiently, FunKit also provides facilities to export arbitrary Mathematica expressions to C++, Julia or Fortran code, including the results of derivations, which can then be evaluated numerically. Both the tracing and code generation can also be used independently and in combination with other packages.

We investigate $O(N)$ boundary conformal field theories (BCFTs) with boundary interactions in $d=4-\epsilon$ and $d=3-\epsilon$ employing the analytic bootstrap. By deriving universal constraints on conformal data, we show that infinitely many operator expansions can be expressed in terms of a finite set of inputs. Complementing the analytic bootstrap with a perturbative renormalization-group analysis, we identify totally new boundary fixed points in $d=4-\epsilon$, including non-unitary ones, generated by a boundary cubic coupling, and compute their conformal data to leading order. Moreover, we leverage our solution in $d=3-\epsilon$ to extract, for the first time, the boundary conformal data for the tricritical $O(N)$ model. Altogether, our approach provides a unified prescription for BCFTs with interacting boundaries and streamlines the determination of bulk and boundary operator expansions.

Finite-size effects limit the accuracy with which conformal data can be extracted from lattice simulations of critical systems. While action improvement suppresses some corrections to scaling, it does not address operator-dependent effects arising from imperfect lattice representations of continuum conformal fields. In this work, we propose a data-driven method for improving lattice operators themselves, constructing estimators with enhanced overlap with the corresponding primary operators of the continuum conformal field theory. We identify improved lattice representations of leading spin and energy operators in three two-dimensional critical systems: the Ising model, the q = 3 Potts model, and the dilute q = 3 Potts model. In all cases, the resulting operators exhibit reduced corrections to scaling and yield more accurate estimates of scaling dimensions compared to conventional lattice choices. The code and analysis workflows used to produce these results are made available in an accompanying GitHub repository.

Scattering amplitudes in planar ${\cal N}=4$ supersymmetric Yang-Mills theory are dual to expectation values of null polygonal Wilson loops. The Amplituhedron provides a geometric construction for the all-loop integrand as the canonical form on the geometric region in the Grassmannian defined by a certain set of inequalities. For a closely related object, the logarithm of the scattering amplitude, the integrand is reproduced in a similar way using negative geometries. When integrated over all loop momenta except one, the result is infrared (IR) finite and equal to the expectation value of a certain Wilson loop with a Lagrangian insertion. At four points, this quantity, ${\cal F}(g,z)$ only depends on a single cross ratio $z$ and the 't Hooft coupling $g$. At weak coupling, it is known up to three loops from perturbative Wilson loop computations and at strong coupling through the AdS/CFT correspondence at leading order. In this paper, we explore this object further through the lens of the Amplituhedron and negative geometries, which provide very natural IR finite building blocks. We perform an explicit three-loop computation of all negative geometries and show that the number of internal cycles in the diagram is closely linked to the depth of polylogarithms. We calculate the cusp anomalous dimension $\Gamma_{\rm cusp}$ by integrating ${\cal F}(g,z)$ over $z$. We show that the higher-cycle diagrams are suppressed if we consider separate odd and even zeta contributions. Furthermore, we focus on certain convergent infinite series of one-cycle diagrams, perform all-loop order resummations of such contributions, and discuss various features of the result.

The hydrodynamic formulation of the Dirac equation has historically been hindered by the inability to close the system of physical variables without resorting to infinite moment hierarchies. We resolve this longstanding issue by developing a fully covariant chiral-hydrodynamic formulation of the Dirac field in curved spacetime. Working in the Weyl representation, we introduce two independent null vectors, $P_L^\mu$ and $P_R^\mu$, which decouple the left and right chiral components. This allows us to define chiral geodesic and stochastic velocities, yielding a closed system of exactly eight real equations that corresponds directly to the Dirac field degrees of freedom. Remarkably, this formulation naturally isolates the spin-orbit coupling $(q/2)\sigma^{\mu\nu}F_{\mu\nu}$ while demonstrating the vanishing of the spin-gravity coupling in torsion-free general relativity. To demonstrate the analytical power of this framework, we specialize to the Schwarzschild geometry. We obtain exact radial solutions in terms of confluent Heun functions and directly compute the quasi-bound state spectrum (fermionic resonances), quasinormal mode frequencies, and greybody factors. Furthermore, by establishing an exact energy balance equation – representing the first law of thermodynamics for Dirac fields – we derive the Hawking radiation flux purely from chiral flux conservation at the event horizon. This work not only provides a closed hydrodynamic theory for spin-1/2 fluids but also establishes a unified framework for analyzing quantum information and fermion dynamics in strong gravitational backgrounds.

We demonstrate how the two-dimensional gravity emerges within “GR from RG” program initiated in \cite{Adami:2025pqr, Sheikh-Jabbari:2026uol}. To achieve this, we consider a generic 2d CFT with a 3d holographic description, which we assume to be well-described by pure Einstein-AdS$_3$ gravity in the bulk. We study the holographic RG flow for the 2d CFT action and show that the renormalization group (RG) corrected action at an arbitrary energy scale contains a 2d scalar-tensor gravity theory. In the simplest case, the flow induces Jackiw-Teitelboim (JT) gravity, where the bulk radial lapse function seeds the dynamical dilaton field of the JT gravity. We show that the standard T$\bar{\text{T}}$ deformation of the 2d CFT is recovered as a special case in the Fefferman-Graham limit where the lapse is fixed. We further establish the robustness of the RG induced gravity picture by verifying its consistency under holographic renormalization and by generalizing the result to a one-parameter family of boundary conditions. Our results provide a first-principles derivation of the JT gravity at a finite cutoff as an intrinsic manifestation of the holographic RG flow in a non-Fefferman-Graham gauge

{We revisit the resulting bispectrum of inflation with kinetic-coupled vector fields by organizing the dynamics in terms of $h$, which measures the vector kinetic contribution relative to that of the scalar field. We derive a low-energy effective field theory (EFT) and evaluate the bispectrum of the curvature perturbation for the large-$h$ regime. For $h\gg1$, the entropic perturbation becomes heavy and can be integrated out; the remaining curvature mode has an imaginary sound speed and undergoes transient growth before horizon crossing. In contrast to the $h\ll1$ regime, where transfer from the vector sector persists outside the horizon and produces a local-type contribution enhanced as $h^2N_K^3$, we find that the bispectrum signals are controlled by the matching parameter $x\simeq1.63Ah$, where $A<1$ is a numerical factor remaining dependent on the UV-to-EFT matching prescription. As a result, in addition to the known flattened-enhanced signals scaling as $x^3$, a flattened-enhanced signal scaling as $x^2$ and a pronounced local projection scaling as $x$ are present. The bispectrum therefore distinguishes vector-supported inflationary dynamics even for an exactly isotropic background.

The wormhole contribution to the gravitational path integral may be interpreted as smooth remnant of correlations among the erratic large-$N$ behaviors of dual CFTs. In this work, we investigate this idea in (2+1)-dimensional gravity. We show that one-sided boundary gravitons are intrinsically incomplete in the sense that the associated observable algebra has a nontrivial center regardless of choices of boundary conditions. Based on asymptotic symmetries, we bootstrap a general Poisson bracket to construct completions of the boundary gravitons. In the simplest completion, the commutant of the boundary graviton observable algebra is given by an observable algebra of monodromy data which we interpret as an effective description of one-sided black holes. We show that, to describe Lorentzian multi-boundary wormholes, only the monodromy data with a positivity restriction is needed. The positivity restriction results in emergent erratic large-$N$ behaviors for some observables. We filter out the erratic observables by restricting to a subspace on which they act trivially. The monodromy observables generate nonlocal symmetries lack of corresponding local currents. We show that gauging the nonlocal symmetries is equivalent to filtering out the erratic observables. For one CFT, gauging the nonlocal symmetries at the quantum level removes all black hole states. Filtering the partition function of CFTs leads to an apparent ensemble averaging. For two CFTs, a Hilbert subspace describing wormholes survives after gauging global part of the nonlocal symmetries. The filtered partition function of the two CFTs is an ensemble average over quantum gates entangling the monodromy degrees of freedom of the two CFTs. The correlation between the erratic observables of the two CFTs is preserved, which contributes to the filtered partition function as a wormhole term.

The correct gravitational representation of the supersymmetric index is a smooth complex solution, rather than the naïve Euclidean continuation of the Lorentizan BPS black hole. We construct such saddles for black holes with \(\mathrm{AdS}_3\times S^3\) boundary conditions in the STU model. We arrive at the same finite-temperature BTZ \(\times S^3\) geometries in two independent ways: from two-center four-dimensional BPS solutions with complex dipole charges, and from six-dimensional black strings after imposing the BPS relation among conserved charges. We analyze the resulting smoothness and periodicity conditions and show that they are precisely those required for globally well-defined supersymmetry and for the thermodynamic potentials of the supersymmetric index.

We study a nonequilibrium coherent effect generated by a finite chemical potential in a complex scalar field with a conserved $U(1)$ charge. The scalar excitation is treated as a probe coupled to an equilibrium thermal reservoir, so the self-energy is an equilibrium kernel and there is no backreaction on the bath. Solving the Schwinger-Keldysh-Kadanoff-Baym equations in the normal phase, when the chemical potential is smaller than the dispersion relation, we keep the particle and antiparticle quasiparticle poles separate. The source-driven inhomogeneous statistical propagator is fixed by the reservoir and relaxes to the usual decoherent equilibrium form. By contrast, the homogeneous solution carries initial-condition memory; finite chemical potential turns this memory into a transient particle-antiparticle interference pattern by splitting the two charge-sector phases. The effect is not a new equilibrium mode, but a phase-sensitive remnant of the initial data that is erased by damping as $t\to\infty$. We define a normalized interference contrast extracted from the mixed charge-sector terms, illustrate the relaxation using the plasmon damping rate of hot scalar $\phi^4$ theory, and show that the same normal-phase solution displays the infrared enhancement that precedes Bose-Einstein condensation.

We use finite group topological lattice gauge theory, also known as the quantum double model, as a lens to explore a notion of topological order enriched by a non-invertible symmetry. For invertible symmetry enriched topological order, there is an established axiomatisation in terms of a G-crossed braided fusion category. We lay the foundations for a generalisation of this notion. By condensing an arbitrary algebra of charges in a quantum double model, we demonstrate that the category of localised excitations in the resulting theory forms a hypergroup-graded extension of the category of deconfined excitations. For every element in the hypergroup, the associated domain wall acts in a typically non-invertible way on these localised excitations. Both this action and the monoidal structure are compatible with the hypergroup grading. The actual categorical action is encoded in a Hopf monad on the category of localised excitations, and gauging the non-invertible symmetry amounts to computing the category of modules over this Hopf monad. Finally, we outline how this framework naturally extends to theories obtained by condensing algebras in a generic string-net model.

This paper establishes that Krylov complexity contains the entire information about the dynamics of a quantum operator, extending the list of equivalent quantities that can serve this purpose, such as the Lanczos coefficients, the return amplitude, and the spectral density. To demonstrate this equivalence, an explicit recursive algorithm is constructed to calculate Lanczos coefficients from the Taylor expansion of the Krylov complexity around $t=0$. Furthermore, the paper discusses the distinction between Krylov and spread complexity, clarifying why a similar recursive algorithm cannot exist for the latter without additional dynamical input. These results provide a “proof of principle” for using Krylov complexity as a complete characterization of operator evolution in quantum systems.

In the present paper, we first describe the principal basis of non-integral geometry. Non-integral geometry is a new field of generalized function (distribution) theory where the effects breaking the symmetry of integration measure have been investigated. In turn, the non-symmetric integration measure (the non-invariant measure) leads to the complex form of the universal, dimension-independent inverse operator with the additional contributions compared to the methods of integral geometry. The additional term with the complex integration measure serves to the extension that improves the image reconstruction procedure. Then, we prove that this additional term $f_A$ in the universal inverse Radon transforms plays a role of the regularizing contribution. In particular, we show that owing to the presence of $f_A$ the corresponding complex singularities can be eliminated in the image reconstruction process.

We discuss the production of a lepton pair in quark-antiquark annihilation in the electroweak Standard Model. We present the ultraviolet-renormalised and infrared-subtracted finite contribution of the complete set of second-order virtual corrections with a closed fermionic loop, to the cross section of the process $u\bar u\to \mu^+\mu^-$. The evaluation of these contributions is based on an automated methodology, whose validity is discussed with explicit tests. These corrections are one of the building blocks necessary for the simulation of the inclusive lepton-pair production cross section at next-to-next-to-leading order in the electroweak interaction.

We interpret the $B$-lattice flavor framework as a compositeness hierarchy: all three fermion generations are elementary chiral fields, but third-generation Yukawa couplings are undressed ($Q=0$), while lighter generations acquire their Yukawa couplings through chains of spin-$0$ subconstituents (“hops”) whose depth is counted by the $\mathbb{Z}_9$ discrete gauge symmetry. The $\mathbb{Z}_9$ charge admits a two-index decomposition $q_9\mapsto(a,b)$ that identifies two hop species ($\alpha$, $\beta$) and organizes all fundamental scales from $v_{\rm EW}$ to $M_{\rm Pl}$ on a “ninths ladder” $\Lambda\times\epsilon^{n/9}$. The lattice structure yields the CKM and PMNS mixing parameterizations (with all mixing exponents expressible as charge differences $\Delta Q$ dressed by a universal Fritzsch–Xing phase shift of $\pm 1/9$), a seesaw benchmark $m_3\simeq 51$~meV, the axion mass window $m_a\sim 7$–$12\;\mu$eV, and the prediction $\tan\beta\simeq 10$–$16$ (from the chain internal factor combined with the DFSZ-II two-Higgs-doublet structure), all from two parameters ($\Lambda$ and $\epsilon = 14/75$). Generation-dependent Peccei–Quinn charges restore the axion–photon coupling ($C_{a\gamma}\simeq 0.6$–$1.0$) from “back to invisible” suppression. An illustrative UV realization in terms of hypercolor-confined scalars communicated to the SM by a gauge-invariant messenger chain is presented as an existence proof.

We investigate perturbative dynamics, tidal effects, and relativistic frequency shifts in a dyonic Kalb-Ramond black hole generated by a Lorentz-violating antisymmetric tensor background. The geometry is controlled by the mass $M$, the electric charge $Q$, the magnetic charge $p$, and the Lorentz-violating parameter $\ell$, with the dyonic sector entering through the effective combination $P_{\ell}^{2}=Q^{2}/(1-\ell)^{2}+p^{2}/(1-2\ell)$. First, we analyze the gravitational Doppler effect for radial signal exchange between freely falling and static observers, showing how the dyonic charges weaken the redshift by shifting the frequency ratio toward unity. We then compute the radial and angular tidal forces in a freely falling frame and determine the characteristic radii at which the usual stretching and compression patterns are reversed. The gravitational time delay is also evaluated for null trajectories, showing that the electric and magnetic sectors reduce the delay relative to the reference configuration. In the perturbative sector, we derive the scalar, vector, tensor, and spinor effective potentials and compute the corresponding quasinormal frequencies through the sixth-order WKB method. The numerical spectra indicate that the Lorentz-violating parameter gives the dominant correction, increasing the oscillation frequencies and modifying the damping rates, while the dyonic charges produce milder shifts. Finally, the time-domain profiles confirm the presence of damped quasinormal ringing followed by late-time power-law tails.

In this paper, we compute the leading-order and first-order radiative corrections to the Casimir energy of a massive Lorentz-violating scalar field governed by a $\phi^4$ interaction on a network. For simplicity, the network is chosen to consist of three edges connected at a single central junction, with the scalar field defined in 1+1 dimensions on each edge. Dirichlet boundary conditions are imposed at the outer ends of the edges, thereby confining the field on the network. Beyond addressing the massive case of a Lorentz-violating scalar field, a key novelty of this work lies in the calculation of the radiative corrections to the Casimir energy using position-dependent counterterms. These counterterms emerge from a systematic renormalization program that consistently incorporates the effects of boundary conditions into the renormalization procedure. To eliminate divergences arising from vacuum energy contributions, we employ the box subtraction scheme in conjunction with cutoff regularization. Our results show that both the leading-order and first-order Casimir energies are negative, regardless of the presence or absence of Lorentz-violating effects, and are in agreement with general physical expectations.

We present a quantum mechanical approach to understanding the Hilbert space and the defect Hilbert spaces associated with line operators of BF theory combined with level-$k$ Chern-Simons theory. The defect Hilbert spaces are closely related to the category of $*$-representations of the $C^*$-algebra of the compactly supported sections of the Fell line bundle over the conjugation action groupoid $G//_{\mathrm Ad} G$, and the structure of this category and the groupoid action on the objects of this category is interpreted quantum mechanically. We show that the action of the line operators on the Hilbert space of the $BF+kCS$ TQFT is given concretely by a convolution between the kernels that represent the line operators, and that the codimension-$2$ twist and the codimension-$1$ prequantum line bundle arise as two transgressions of the same universal level $k\in H^4(BG,\mathbb{Z})$. For finite gauge group, the resulting convolution-eigenvalue formula is identified with the Verlinde formula for the (twisted) Drinfeld double $D^\omega(G)$ via an explicit phase-by-phase match with the known finite modular data. For compact Lie group, the convolution-kernel eigenvalues coincide in the regular sector with the semiclassical Hopf-link $S$-kernel, identifying two complementary derivations of the same modular data.

We study the modular symmetry in magnetized compactifications. The zero-modes with different Scherk-Schwarz phases transform each other. A generic model does not include modes with all the Scherk-Schwarz phases. Incomplete multiplet representations appear. Thus, the modular symmetry is violated as group-like symmetry. However, the modular symmetry still controls coupling terms in those models. Modular forms of the full symmetry appear as coupling constants.