Papers

The surface $r=\infty$ of the Kerr–Bertotti–Robinson (KBR) spacetime is not the collection of the endpoints of infinitely extended light rays, and the Coulomb type component of gravitational field strength represented by $\Psi_2$ remains nonvanishing there, indicating that this surface is not a real boundary. We construct a natural extension across this surface, which connects the exterior of one KBR region to the interior of a neighboring one and, upon iteration, produces an infinite chain of regions connected by wormhole-like bridges. The extension also exposes the neighboring ring singularity without an intervening horizon, challenging the weak cosmic censorship conjecture and raising the question of whether the extended geometry is stable under perturbations. We therefore study the quasinormal modes (QNMs) of a test massless scalar field on a two-universe scattering segment. For axisymmetric perturbations, the existence of purely imaginary unstable QNMs is analytically proved for every $\ell$, with their origin tied to the chronology-violating region. In the $(\ell,m)=(2,2)$ sector, unstable QNM branches driven by a black-hole-bomb mechanism are found. Finally, the wormhole geometry produces a double-barrier cavity and families of weakly damped QNMs, suggesting echo-like responses.

These notes develop the diagnostics of chaos from their classical definitions through to holography, in six lectures given at the ST4 Workshop, Chennai Mathematical Institute, in 2026. No prior exposure to chaos theory is assumed. Lectures 1 and 2 define classical chaos geometrically and build the instruments that measure it, every one of which reads a phase-space trajectory. Quantization removes that trajectory, and Lectures 3 and 4 rebuild the instruments from spectra and eigenstates instead: level statistics, random matrix theory, the spectral form factor and eigenstate thermalization, as much the working toolkit for thermalization in many-body physics as for holography. What they return is a classification and a count rather than a rate. Lecture 5 recovers the rate from the squared commutator, treats scrambling as operator growth, follows the commutator to the out-of-time-order correlator, marks where that diagnostic stops being reliable, and reaches the butterfly velocity and the chaos bound. Lecture 6 constructs the eternal black hole, the thermofield double and the shock-wave geometry behind that bound, then follows two probes whose exponents discriminate between backgrounds where the horizon result cannot, one tracking a thermodynamic phase transition, the other resolving an anisotropy induced by a background field. That last part is research rather than review, and every figure throughout is computed from the scripts supplied with the source rather than redrawn from the literature. Every lecture closes with exercises. Claude Opus 5, an AI assistant developed by Anthropic, worked as a research assistant under the author's direction.

We investigate uncertainty quantification for the neutron-star equation of state (EOS) by comparing point-estimation and distributional inference approaches using the same Chebyshev and piecewise-linear parameterizations. We combine neutron-star mass–radius and gravitational-wave tidal-deformability information within Bayesian, multilayer-perceptron (MLP), and normalizing-flow frameworks. Although the methods yield similar mean EOS behavior, the deterministic MLP produces substantially narrower uncertainty bands at high densities. We show that this behavior is associated with the point-estimation objective, which maps degenerate solutions toward the conditional mean rather than representing the full parameter posterior. By contrast, the normalizing flow yields distributions more consistent with the Bayesian inference. Our results demonstrate that reliable uncertainty quantification of the high-density EOS requires methods that represent conditional probability distributions rather than only point estimates.

We construct perturbative black-brane solutions carrying neutral scalar hair in the $G_{2}$-invariant truncation of four-dimensional dyonic $\mathcal{N}=8$ ISO(7) gauged supergravity, an exact consistent truncation of massive type~IIA supergravity on $S^{6}$. Expanding around the $G_{2}$-symmetric AdS$_{4}$-Schwarzschild black brane in powers of a dimensionless scalar-charge parameter $\eps$, we show that the perturbation hierarchy reduces, order by order, to a nested sequence of inhomogeneous Legendre equations (demonstrated through second order). The first-order scalar satisfies a Pöschl–Teller equation with exact parameter $\nu(\nu+1)=2/3$, giving $\nu=-\frac{1}{2}+\frac{\sqrt{33}}{6}$, and the first-order metric is fixed algebraically by the same function, $A^{(1)}=-\tfrac{\sqrt{7}}{2}\phi^{(1)}$, so the entire $\mathcal{O}(\eps)$ sector is one Legendre function. An identity $\nu+1=\Delta_{+}/3$, with $\Delta_{+}=(3+\sqrt{33})/2\approx 4.372$ the conformal dimension of the dual scalar operator, relates the Pöschl-Teller parameter to the AdS/CFT dictionary via the nonlinear Fefferman-Graham map $z \propto z_{\mathrm{FG}}^{3}$. The $\mathcal{O}(\eps^{2})$ back-reaction yields explicit corrections to the renormalized free energy and Bekenstein-Hawking entropy density at fixed temperature, consistent with the first law; equivalently, at fixed entropy or energy density it lowers the Hawking temperature by $\delta T/T=\mathcal{O}(\eps^{2})$. For the leading transport coefficients the shear viscosity saturates $\eta/s=1/(4\pi)$, while the Eling-Oz horizon formula gives a nonzero bulk viscosity $\zeta/\eta=\bigl(\tfrac{\Delta_{+}-3}{2}\bigr)^{2}\eps^{2}=\tfrac{3(7-\sqrt{33})}{8}\eps^{2}\approx0.471\,\eps^{2}$. With $c_{s}^{2}=\tfrac12-\mathcal O(\eps^{2})<\tfrac12$, this indicates a softening of the equation of state.

We argue that the non-invertible symmetry of $\mathcal{N}=4$ super Yang-Mills theory, obtained by combining S-duality, half-space gauging, and R-symmetry twist, is realized as an outer-automorphism of the associated vertex operator algebra. Hence, the Schur and Macdonald index of $\mathcal{N}=4$ super Yang-Mills theory with non-invertible symmetry twist corresponds to the (refined) vacuum character of the associated vertex operator algebra with the outer-automorphism twist. We provide a few checks of our proposal from the large-$N$ limit and by matching the fusion rules, which involve charge conjugation on local operators. The `charged' Schur index indeed matches the outer-automorphism-twisted vacuum character.

We compute quark and meson spectral functions in low-energy QCD, including all-order scatterings and decays of pions and the scalar $\sigma$-mode. For low energies the gluons decouple and the dynamics of two-flavour QCD is well-described by a Quark-Meson model. The computations are performed directly in Minkowski space, using the spectral functional renormalisation group. The full mesonic realtime dynamics is captured with the emergent composite approach which is extended here to realtime processes. The inclusion of all-order scatterings and decays is achieved through the self-consistent treatment of the scattering tails and the momentum-dependent resummation of the four-meson vertex via its Bethe-Salpeter equation. We illustrate the importance of higher-order scatterings using the example of the $\pi \to 3\pi$ scattering threshold - the lowest kinematically accessible channel for the pion.

We derive the quantum state of the asymptotic linearized gravity field of a compact energy source, using BRST quantization. Barnich has previously derived the quantum state of the U(1) field in QED using the Gauss law constraint. The role of Gauss's law is now played by the linearized Hamiltonian and momentum constraints in gravity. For a static mass source, only the scalar Hamiltonian constraint is shifted. The quantum state corresponding to the static Newton field originates in the non-radiative scalar constraint sector, not in the propagating transverse-traceless graviton sector. We discuss implications for holography and for black hole information: 1. the exact quantum state of the compact source is encoded in the asymptotic quantum state of its gravity field 2. Hawking amplitudes are dependent, via the gravity state outside the horizon, on the black hole internal state.

We construct non-abelian uni-vector deformations of solutions in non-abelian Einstein\–Maxwell theories and gauged supergravities, obtained as Scherk–Schwarz reductions of general relativity and double field theory, respectively. We provide examples of deformed backgrounds for both cases. We show that non-abelian deformations in Einstein–Maxwell theories can be presented as coordinate transformations in the parent theory, general relativity, extending the result of arXiv:2508.09637 for abelian uni-vector deformations.

We study the quantum dynamics of supersymmetric Galilean Yang-Mills (SGYM), with a focus on three dimensions where there is a classical scale symmetry. This is believed to be the worldvolume theory describing D2-branes within type IIA non-relativistic string theory and arises as a non-relativistic limit of maximally supersymmetric (relativistic) Yang-Mills. Unlike the relativistic case, we find that SGYM only admits perturbative dynamical excitations on the Coulomb branch. We show that in three dimensions the classical scale symmetry is broken at one-loop due to logarithmic divergences. These cannot be regulated without introducing additional marginal deformations, and we present evidence that the theory is asymptotically free once these are included. In particular our analysis suggests that three-dimensional SGYM flows to a strongly coupled non-relativistic M2-brane theory.

We investigate the fermion spin dynamics and induced dipole interactions in an external Yang-Mills gauge field that represents a non-Abelian plane wave. We use the exact solutions of the Dirac equation in the external gauge field, the exact fermion Green's function, and the renormalized one-loop fermion-gluon vertex in the axial gauge. The exact propagator produces spin-dependent structures proportional to $\sigma ^{\mu \nu }F_{\mu \nu }^{a} $. The external field induces spin precession at tree level through a non-Abelian extension of the Bargmann-Michel-Telegdi equation, coupled with Wong color transport. The exact vertex produces a phase-dependent Pauli form factor $F_{2}^{a}\left( 0;\varphi ,\varphi ^{\prime }\right) $ beyond tree level, which serves as an anomalous chromomagnetic dipole coupling. For monochromatic two-color noncommuting plane waves, explicit weak-field expressions are derived for the induced Pauli coefficient and the anomalous spin-precession frequency. The commutator term $gf^{abc}A_{\mu }^{b}A_{\nu }^{c}$ in this case, generates additional field-strength components and dynamically induced precession axes not present in Abelian backgrounds, resulting in coupled spin-color precession. The exact one-loop coefficient is provided as a harmonic expansion suitable for periodic Yang-Mills waves. Explicit expressions for spin-dependent amplitudes, spin-flip probabilities, and polarization asymmetries are provided. These results establish a direct connection between exact background-field propagators, renormalized vertex functions, anomalous dipole interactions, and observable spin effects in strong non-Abelian gauge fields.

We study how a recently introduced causal-set propagator defined in terms of links responds to curvature. On sprinklings embedded in a flat 1+1-dimensional Minkowski space, this propagator is known to reproduce the massless retarded continuum Greens function on large scales. In conformally flat embeddings with constant curvature $\mathcal R$, the causal order is unchanged, but the link relationship is sensitive to the physical volume of the spanned causal diamond. We show that this volume dependence generates a leading curvature correction to the propagator. On large distances, this correction is equivalent to a continuum scalar propagator with an effective curvature coupling of the form $\xi \mathcal R$ with the coupling constant $\xi_{\rm eff}=1/6$, equivalently corresponding to an effective mass-squared parameter $m_{\rm eff}^2=\xi_{\rm eff}\mathcal R$. This coupling is not inserted by hand but emerges from the link-based path sum itself, reflecting the microscopic structure of the causal set. Numerical simulations on sprinklings embedded in $\mathrm{AdS}_{1+1}$ and $\mathrm{dS}_{1+1}$ support the predicted curvature response and indicate that the effective coupling persists as the sprinkling density is increased.

Recent studies have explored whether physical constants and symmetries can be selected by extremizing bipartite entanglement entropy. We extend this approach to a three-particle setting. Specifically, we consider a $2+1$-particle system in which particles $B$ and $C$ are initially entangled, and then particles $A$ and $B$ scatter without the direct participation of particle $C$ for both gluon-gluon and graviton-graviton scatterings. Using angular expansions around the forward and backward scattering limits, we analyze bipartite entanglement in the corresponding two-particle system of $A$ and $B$ and two measures of genuine tripartite entanglement (GTE), the concurrence fill and the generalized geometric measure (GGM), in the three-particle system. As is already known, the bipartite entanglement provides no universal criterion for selecting the symmetry-preserving theory: depending on the initial helicities, it may either maximize or minimize the entanglement entropy. By contrast, we find that the GTE-based analysis uniquely identifies the gauge-invariant and diffeomorphism-invariant theories as those that suppress the GTE in gluon-gluon and graviton-graviton scatterings, respectively. This result suggests a nontrivial connection between tripartite entanglement and the gauge and diffeomorphism symmetries of fundamental interactions.

We investigate the Page curve and information recovery in a near-extremal regular black hole inspired by T-duality, in which the central singularity is resolved by a minimal length scale. By integrating the first law of thermodynamics at fixed minimal length, we obtain a black-hole entropy containing an intrinsically quantum logarithmic correction, while the area-law contribution vanishes in the extremal limit. Consequently, the extremal remnant carries a finite entropy of purely quantum origin. Using the island prescription in the near-horizon regime, we evaluate the generalized entropy of radiation and compare the standard area functional with an alternative functional constructed from the corrected thermodynamic entropy of the black hole. In the near-extremal limit, the latter reduces analytically to the classical extremization problem with an effective coupling, and it places the physical island closer to the outer horizon. Although both prescriptions produce a Page transition, they predict different saturation values and Page times. While the standard functional yields a plateau controlled by the area term, the corrected prescription saturates at the full thermodynamic entropy of the outer horizon and, in the extremal limit, at the logarithmic remnant entropy. Including evaporation and backreaction, the Page curve develops the expected descending branch and asymptotes to the entropy of the cold extremal remnant rather than to zero. Our results indicate that a thermodynamically consistent description of information recovery from regular near-extremal black holes requires incorporating the intrinsic quantum correction to the gravitational entropy.

We obtain a local first law of dynamics within the interior geometries that are governed by cosmological solutions connecting the event horizon and the Kasner-like singularity. Evaluating the Iyer-Wald identity between the horizon and near-singularity geometries, we define a Kasner potential and transported response coefficients. We test the first law by both exact and numerical solutions. Our formalism provides a self-consistent approach to study the interior dynamics without having to appeal to the outside geometry and the asymptotic charges. It may provide a new approach to study the phase structures of the interior dynamics.

Machine learning and data analysis techniques have recently emerged as powerful tools for identifying patterns and formulating conjectures in mathematical research, most notably in the field of low-dimensional topology. In this paper, we initiate a systematic approach to handling mathematical data structured as (truncated) infinite $q$-series, or equivalently, infinite series of integers. To apply this data analysis pipeline, we construct a comprehensive dataset of $\widehat{Z}$-invariants (homological blocks) for plumbed 3-manifolds. We demonstrate that neural networks can reliably extract essential topological information, such as homology class and underlying graph structure, directly from the $q$-series coefficients. A central feature of our methodology is a focus on interpretability; by contrasting local gradient sensitivity with global feature relevance, we reveal that the networks learn to bypass complex topological rules in favor of specific spectral and geometric proxies. Finally, we apply this pipeline to probe homology cobordism, discovering a high-accuracy predictive relationship between the $\widehat{Z}$-invariant exponents and the Heegaard Floer $d$-invariant (correction term). These results suggest that $\widehat{Z}$-invariants capture subtle geometric information regarding cobordism equivalences, warranting a new direction for the study of quantum invariants.

We study a modified charged BTZ black hole in \((2+1)\)–dimensional anti-de Sitter spacetime, where the standard geometry is surrounded by quintessence-like anisotropic matter and a cloud of strings. The corresponding metric function incorporates the effects of the electric charge and the two matter sources. The optical analysis reveals that their combined influence gives rise to a stable circular photon orbit, leading to distinctive modifications of photon dynamics and the associated energy emission. The thermodynamic analysis in a finite cavity further shows that the electric charge and exotic matter sources significantly affect the local thermal stability and phase structure of the black hole. The combined effects of these matter sources give rise to a stable photon sphere and novel thermodynamic behavior absent in the standard charged BTZ solution.

A closed Friedmann-Lemaitre-Robertson-Walker Wheeler-DeWitt model is formulated as an exact reflection problem. The derivative of its reflection phase defines a relational crossing duration whose first two clock moments follow from a covariant scalar-clock observable. An analytic expression is obtained for this duration, its classical recollapse limit, and the leading quantum correction. A finite spectral packet also gives an operational, equal-prior minimum error probability for distinguishing the two orientations of recollapse with a geometric reading. The construction is extended to a bounded finite quantum detector. Its multichannel reflection matrix yields probe transitions, spectral-probe correlations, and a matrix duration. A direct weak-coupling calculation verifies the predicted transition and duration scalings. Scalar-clock conditioning and oriented geometric sections are shown to be local representations of the same positive-frequency Dirac sector. These results define duration and transition observables entirely through correlations and scattering records of a stationary constrained state, without introducing a background clock.

In four-flavor massless QED with a magnetic monopole, a single incident fermion is scattered into a state carrying a half-integral current vector in each flavor channel, called the semiton. Recently, the semiton has been interpreted as a twisted-sector fermion, but its microscopic Hilbert-space representation has not been fully developed. In this Letter, using a bosonized fermion–rotor model, we construct the semiton state explicitly. This is achieved by first identifying the correct vacuum of the system, and then constructing a unitary operator that shifts the rotor and creates a particle-hole cloud in each flavor channel. The semiton state is obtained by applying this operator to an ordinary fermion state. The resulting state has a localized fermion flavor density in the core region and a half-integral semiton front at the wavepacket. We also show that the state carries the correct $U(1)_M$ charge expectation values and energy-density profile. Our result provides a concrete Hilbert-space realization of the varying-Fock-space interpretation of monopole–fermion scattering, advocated in previous works by the author and collaborators.

We initiate a systematic study of compactifications of type II string theory to two dimensions on Ricci-flat spaces with fluxes and sources. We derive universal constraints on such compactifications, and then develop type IIA compactifications on $\mathrm{Spin}(7)$-holonomy spaces with bulk fluxes whose tadpole is cancelled by $\mathrm{OF1}$-planes. We derive the resulting two-dimensional $\mathcal N=(1,1)$ supergravity for a toroidal $\mathrm{Spin}(7)$ orbifold, and we extend the metric and universal sectors geometrically to general compact $\mathrm{Spin}(7)$ manifolds. In candidate supersymmetric Minkowski vacua, all untwisted shape modes appear in the flux scalar potential and can in principle be classically stabilised. However, in an explicit toroidal orbifold example flux quantisation together with the tadpole bound may obstruct the existence of candidate vacua supported entirely within the untwisted sector. The string-frame volume in string units is bounded by $7\chi/192$, so suppressing $\alpha'$ corrections requires $\mathrm{Spin}(7)$ manifolds with large Euler characteristic.

We conduct numerical simulations to map out the phase diagram and critical behavior of a lattice Higgs model composed of two massless staggered fermion fields forming a doublet under a global $SU(2)$ and coupled to a scalar field in the adjoint representation of the group. The scalar action consists of a potential comprising quadratic and quartic terms and a scalar kinetic term. At fixed quartic coupling we explore a two-dimensional parameter space finding a massless symmetric phase at weak coupling and a massive symmetric phase (SMG phase) at strong coupling. An intermediate anti-ferromagnetic phase separates these two regimes. These results are consistent with leading order weak and strong coupling expansions. We find that the critical lines bounding the intermediate phase merge at a unique point where all fermion bilinear condensates vanish but fermion susceptibilities diverge as non-trivial powers of the lattice size. We conjecture that this merged point corresponds to a multicritical point and may describe a phase consisting of a condensate of certain topological defects.

We calculate the five-point tree-level Compton amplitudes for minimally coupled spinning matter in gauge theory and gravity in the massive spinor-helicity formalism, using BCFW recursion and KLT relations. The results are written in terms of symmetrized spinor products and are simultaneously valid for a range of matter spins, which directly generalizes the amplitudes of Arkani-Hamed, Huang, and Huang to the five-point case. Unlike the three- and four-point cases, each amplitude can be a sum of two or more distinct spin structures. Spurious poles are manifestly canceled through a mechanism that holds for all massive spins $s\leq 1$ in gauge theory and $s\leq 2$ in gravity. While we keep the amplitudes exact, their classical limits are expected to be relevant for binary systems of Kerr black holes in the post-Minkowskian expansion.

The definition of a Lorentzian gravitational path integral requires specifying which causal struc- tures are admitted off shell. Of particular importance are light-conical singularities, codimension- two irregularities of the light cone structure that render the gravitational action complex and can therefore lead to exponential enhancement or suppression of configurations. We investigate the prevalence and structure of such singularities in Lorentzian Regge gravity. We develop an efficient algorithm for randomly generating realizable Regge geometries satisfying the generalized triangle inequalities, and use it to sample the configuration space of four-dimensional triangulations. We find that light-conical singularities are generic: for bones shared by many four-simplices, configurations with irregular light cone structure overwhelmingly dominate, with trouser-type singularities gen- erally entropically favored. Restrictions on the causal character of subsimplices, such as requiring all tetrahedra to be spacelike, can substantially alter these statistics. We further show that both yarmulke- and, in four dimensions, trouser-type singularities can occur on unbounded regions of configuration space associated with simple refinement moves. Beyond light-conical singularities, we analyze the causal structure of Lorentzian triangulations more generally, for example by introducing the notion of pairwise embeddable chrono-topologies and a set of discrete causality data. Our re- sults demonstrate that causal irregularities are an intrinsic and abundant feature of the unrestricted Lorentzian Regge configuration space, sharpening the question of which causality conditions should be imposed in Regge gravity, spin-foam models, and ultimately the continuum gravitational path integral.

We begin by introducing a new parameterization for the Ovcharenko-Podolský family of exact solutions of the Einstein-Maxwell equations describing uncharged and non-accelerating static black holes immersed in an external Bertotti-Robinson electromagnetic field. The new spacetime is, in general, different from the original Ovcharenko-Podolský one, in particular on the branches solving the field equations, for which the original parameterization can lead to black holes with a negative mass, whereas the new one always leads to black holes with a positive mass. Moreover, by introducing additional gauge constants, we remove the various pathologies and ensure consistency with the laws of black hole thermodynamics. We then obtain a new family of Einstein-Maxwell solutions by generalizing this new parameterization, through Harrison transformation within the Ernst method, to also include an external electromagnetic field of the Bonnor-Melvin type. We again identify the appropriate gauge constants required to remove the possible pathologies and ensure that the resulting solutions satisfy the laws of black hole thermodynamics. Finally, we investigate the vacuum subcase of the new family and establish its relation to previously known metrics.

We develop a real twistor-space formulation of four-dimensional CFT Wightman correlators. We write the conformal generators and little-group constraints in ambitwistor space. The quadric condition $Z\cdot W$=0 arises naturally from the little-group constraints. We then solve for both the parity-even and parity-odd correlators in the ambitwistor variables. We further establish the connection between the ambitwistor correlators and the ones in the helicity-basis Grassmannian via a half-Fourier transform, and also develop the Penrose transform to recover the corresponding position-space correlators. We notice that for correlators with multiple tensor structures, the Penrose transform yields the full space of conformally invariant structures, but regularizing the associated Schwinger integrals selects the physical correlator. Finally, starting from the covariant Grassmannian formulation, we construct massive twistors that make the little-group symmetries manifest and develop their Penrose transform, which again reproduces the correct position-space correlators.

We solve the Bethe Ansatz equations (BAEs) to evaluate the superconformal index of four-dimensional $\mathcal{N}=4$ super-Yang–Mills with rank-two gauge algebra, at equal angular momentum fugacities $p=q$. The solutions for $A_2$ are known, and those for $D_2\cong A_1\oplus A_1$ follow readily from the (equally known) $A_1$ ones. The first genuinely new case is $B_2\cong C_2$, for which we obtain the complete solution set in closed form; for the exceptional case $G_2$ our results are numerical, with a single fully rational solution obtained in closed form. To the best of our knowledge, this is the first time any solution, analytic or numerical, has been found in non-$A$-type gauge algebra (for $\mathcal{N}=4$ or any other $\mathcal{N}=1$ theory). Along the way we uncover several phenomena absent in type $A$: isolated Weyl-fixed solutions contributing nontrivially to the index; isolated solutions in which at most half of the holonomies have rational coefficients (in contrast with the $A$-type fully rational Hong–Liu family); and solutions whose $|\omega|\to0$ limit (with $p=q=:e^{2\pi i\omega}$) evades assumptions standardly made in the Cardy-like limit literature, landing on saddles that the usual analysis does not capture, for all types $BC\!D$. We also clarify the Bethe origin of the finite logarithmic correction to the Cardy expansion: it arises from the orbit size of a BAE solution under the gauge symmetries of the equations, rather than from the one-form center symmetry of the theory, to which the index is insensitive.

We construct a generating functional for infrared-finite scattering amplitudes in massive quantum electrodynamics (QED) by including Faddeev-Kulish dressings in a holomorphic coherent-state representation, in the spirit of the construction of Aref'eva-Faddeev-Slavnov (AFS) for the undressed Dyson S-matrix. At tree level, the dressed generating functional reduces to an AFS path integral with dressed boundary conditions.

Variations in the gauge couplings of the Standard Model have been searched for experimentally and proposed as solutions to open questions in fundamental physics. Varying gauge couplings can arise from a dynamical scalar field coupled to the gauge kinetic term. This mechanism has been invoked extensively, assuming a minimal linear coupling of the scalar to the gauge bosons. In this work, we investigate how this operator is generated from the ultraviolet perspective. We argue that in weakly coupled, renormalizable completions in four spacetime dimensions, gauge invariance forces the leading dependence of the effective gauge coupling on the scalar to be logarithmic rather than linear. The gauge coupling evolution in these scenarios can be entirely described by the renormalization-group running with dynamical mass thresholds. Beyond leading order or four dimensions, we provide examples showing that more general behavior is possible. Finally, we discuss the phenomenological implications of dynamically evolving gauge couplings, particularly in early-universe settings.

We define detector operators in the nonrelativistic conformal field theory describing fermions at unitarity. We reduce the problem of computing the momentum distribution and correlation between final particles produced by a local source ("point-produced") to the computation of correlation functions involving the detector operators. The general formalism is applied to the point production of three unitary fermions, where we find the momentum and angular distribution of final particles. We discuss a nonrelativistic version of celestial holography, which maps the asymptotic out-state to a quantum wave function in the so-called "oscillator frame."

We investigate signatures of chirality at Chern insulator transitions in the presence of a boundary. The transition between a trivial insulator and a Chern insulator with Chern number $C=1$ is described by a massless Dirac fermion whose parity anomaly gives a critical Hall conductivity $\sigma_{xy}=\frac{1}{2}\frac{e^2}{h}$. Using a Dirac mass domain wall construction, we show that the chiral edge mode delocalizes into the bulk at criticality, but the boundary fermion correlation function retains a chiral structure and acquires the scaling dimension of the bulk fermion. We compute current correlation functions and demonstrate that the anomaly of the bulk Hall response is matched by delocalized chiral modes near the boundary. Using only the residual conformal symmetry of a (2+1)d conformal field theory (CFT) in a half-space, we identify parity-odd terms in current and energy-momentum tensor correlation functions that encode these modes and determine their electromagnetic and gravitational anomaly coefficients. Our analysis therefore applies to general time-reversal breaking (2+1)d CFTs, beyond the free Dirac transition. In particular, the analysis of the gravitational anomaly is also applicable to the free Majorana CFT governing the transition between a trivial superconductor and a topological superconductor. We also extend our results to more general Chern number changing transitions and to the transition between a (3+1)d topological insulator and a trivial insulator.

Conformal symmetry strongly constrains correlation functions. In momentum space, the conformal Ward identities are solved for three-point functions by integrals of a product of three Bessel functions ("triple-$K$ integrals"); more generally, integrals of this type ("multiple-$K$ integrals"), serve as the building blocks of a wide class of higher-point conformal correlators. These integrals belong to the class of generalised hypergeometric functions, and while their series representations are known in principle, none converges throughout the entire physical region of kinematic space selected by momentum conservation. In this work, we construct series representations adapted to exactly this physical domain. Using the method of brackets, which turns the evaluation of definite integrals into solving a linear system of algebraic equations, we derive various Lauricella-type series representations for multiple-$K$ integrals. For triple-$K$ integrals, conventionally expressed in terms of the Appell $F_4$ function whose series converges only outside the triangle-inequality region, we find instead a compact, two-branch series that converges throughout the entire physical region and for arbitrary scaling dimensions. We then extend this construction to general multiple-$K$ integrals: by introducing a new set of kinematic variables, we build an iterative series representation that converges for all physical kinematic configurations.