Papers

For any conformal net, not necessarily rational, we construct the associated Cardy-type conformal field theory on the Minkowski spacetimes $(\mathbb R/2\pi\mathbb Z)\times\mathbb R$ for closed strings and $[0,\pi]\times\mathbb R$ for open strings within the framework of algebraic quantum field theory. In addition to verifying some of their basic properties, we prove three forms of Haag duality for multi-double-cones and boundary intervals, interpreted respectively as the Minkowskian versions of modular invariance, the Cardy consistency condition, and the Morita equivalence of boundary field algebras.

We propose an asymptotic bootstrap method to evaluate moment integrals that arise in problems related to hermitian matrix models. These are normalized integrals of the form $a_n=\int^{\infty}_{-\infty} x^n \exp(-V(x)) dx$ where $V(x)$ is a polynomial of $x$. The method is applicable even when the coupling constants in $V(x)= x^{2\ell}/(2\ell) + g_{2\ell-2} x^{2\ell-2}/(2\ell-2) + \dots$ are complex. We prove that the method converges asymptotically exponentially fast on a cone region determined by the first subleading coupling $g_{2\ell-2}$, which must have a positive real part and an absolute value of the argument less than $\pi/{\ell}$. We use our method to study the phase structure of Hermitian matrix models by constructing the orthogonal polynomials associated to these measures.

Can a closed quantum system generate time-crystal-like nonequilibrium behavior, self-similar scaling structures, and non-Markovian memory without external driving or coupling to a macroscopic environment? We address this question within the quantum Bateman oscillator formulated in a nonrelativistic (2 + 1)-dimensional phase-space noncommutative framework generated by spin-induced spatial deformation. The resulting doubled quantum dynamics is governed by a time-independent Hermitian Hamiltonian and exhibits an underlying SU(1, 1) structure with amplified and damped collective modes. We show that these modes satisfy an exact discrete scaling covariance, leading to self-similar temporal evolution without external driving. Upon tracing over one oscillator sector, the reduced dynamics becomes intrinsically non-Markovian and is governed by a history-dependent memory kernel. The same scaling structure admits a geometric representation in terms of logarithmic-spiral trajectories associated with the amplified and damped branches of the Bateman system. Because the mechanism relies on nonequilibrium reduced dynamics rather than equilibrium expectation values of local observables, it lies outside the assumptions underlying conventional no-go theorems for equilibrium time crystals. Our results identify spin as the common physical origin of the amplified and damped Bateman dynamics, self-similar scaling periodicity, logarithmic-spiral structures, and non-Markovian memory, also suggesting a natural extension of the mechanism to relativistic anyonic systems.

We use numerical relativity to study the scalar field evolution sourced by hyperbolic encounters of black holes in quadratic scalar Gauss-Bonnet gravity. In this theory, single black holes are known to acquire a scalar hair through scalarization for certain values of their mass and spin. We work in the decoupling limit and evolve the scalar field on top of a background metric. Seeding binary black holes with an initial scalar field, we find that configurations which initially cannot sustain a scalar hair temporarily scalarize during an encounter and thereby exhibit dynamical scalarization. This is possible for both positive and negative couplings between the scalar field and curvature in black hole binaries with zero and non-zero initial spins, respectively. Furthermore, we find that the change in the spin magnitude of black holes during certain hyperbolic encounters can lead to permanent spin-induced scalarization (or descalarization), which we refer to as spin-up (de)scalarization.

This paper presents a curated selection of research questions at the intersection of quantum gravity and quantum information, chosen to highlight issues that we regard as particularly important for researchers in both fields. We organize the discussion into four main themes: the operational characterization of observables, the role of observers, quantum error correction, and the infinite-dimensionality of Hilbert spaces. We hope that addressing these questions will engage researchers across both communities and further strengthen the profound interplay between the two disciplines.

We classify AdS$_3$ solutions of the U(1) invariant sector of minimal $d=7$ supergravity. We find two classes of solutions preserving ${\cal N}=(2,0)$ supersymmetry for which the internal space M$_4$ is either a negative curvature Kahler-Einstein manifold or a circle fibration over $\Sigma\times \mathbb{R}$. For the later, in the case that $\Sigma$ has constant curvature, we reduce finding a solution to solving a single ODE that admits polynomial solutions. Among these are interesting solutions whose uplifts to $d=11$ describe M5 branes wrapping various $d=4$ orbifolds. These include a topological $\mathbb{CP}^2$ with 2 orbifold fixed points that we identify as the weighted projective space $\mathbb{WCP}^2_{[k,k,\ell]}$. We are also able to construct solutions with M5 branes that wrap a spindle fibred over constant curvature Riemann surfaces of arbitrary genus. Such solutions should provide holographic duals to the ${\cal N}=(2,0)$ SCFT associated to the M5 brane compactified to $d=2$ on these orbifolds. We match the holographic central charges of these solutions to a field theory computation in terms of anomaly polynomials and c-extremisation.

We explain how to extract thermal OPE coefficients of double-twist operators in scalar two-point functions at infinite spatial volume from suitably regularized integrals of thermal response functions in momentum space. As a specific application, we implement the proposed approach to four-dimensional holographic CFTs with an Einstein bulk action, where the finite-temperature state is captured by an AdS$_5$ black-brane geometry. In that case, the thermal response function can be computed on the gravitational side by solving numerically a radial ODE reduction of the Klein-Gordon equation. We obtain accurate values of double-twist data from this solution, completing previous holographic and bootstrap studies of thermal two-point functions. Some of the reported spin-resolved data that we compute are new.

It is widely believed that self-gravitating radiation cannot reach thermal equilibrium with a black hole in asymptotically flat spacetime. We describe a marginally stable continuum exception to the standard instability argument. Our main observation is that the photon sphere controls the Israel junction conditions (IJCs), the Tolman-Oppenheimer-Volkoff (TOV) equation, and finite-radius black hole thermodynamics: the IJCs and TOV equation are equivalent at zero radial pressure, the inverse specific heat at the photon sphere is proportional to $-\Lambda$, and adding shells at fixed total mass lowers the asymptotic Hawking temperature if and only if the local specific heat is positive. Using these results, we show how to compute thermodynamic entropies without the Euclidean path integral. The exception described here results from companion work with M.J. Strassler, where we found that a "hillingar black hole" (HBH) mimics an ordinary Schwarzschild black hole of mass $M$, sharing its Hawking temperature, photon ring, and, in equilibrium, its coarse-grained entropy $S = 4 \pi M^2$. Here, we show these features are not tuned; they follow uniquely from joint mechanical and thermodynamic constraints. Conditions for thermodynamic mimicry and for the formation of extended photon spheres are found; for self-similar solutions they coincide. A one-parameter family of self-similar systems – all of which, excepting the HBH, require massless walls at the edges of their extended photon spheres – satisfies both conditions. This family includes "stiffest stars" and "frozen stars".

In a companion paper we considered a Schwarzschild black hole of mass $m$ enveloped by a thick "ocean" of massless particles that extends the black hole's photon sphere into a region of finite depth. There we showed that this "hillingar black hole", of ADM mass $M$, optically mimics an ordinary black hole of the same mass. Here we find it also mimics the black hole thermodynamically: the formal assumption of thermal equilibrium implies the system has the same temperature and entropy as an ordinary black hole of mass $M$. We check this result carefully using multiple methods; a further method and indications of metastability are given by one of us in a companion paper. In AdS space, the mimicry does not hold, and the solutions have a richer structure. While it is far from clear that these systems are models for more realistic ones, we note possible connections with black hole evolution. In particular, assuming thermal equilibrium can be established and maintained, an HBH in a cavity of radius $\geq 3M$ can evaporate, potentially posing the information puzzle in a small finite volume.

We consider spherically symmetric static solutions to Einstein's equations describing a Schwarzschild black hole enveloped by a thick shell of orbiting massless particles with zero radial pressure. The orbiting gas is ultra-compact and ultra-relativistic, and can be viewed as the marginally stable limit of a stable Einstein cluster. These solutions, which we refer to as "hillingar black holes", extend the photon sphere into a region of arbitrary depth. We compare these objects to black holes surrounded by other gases and note they have numerous special properties at the classical level; in particular, they appear optically indistinguishable from ordinary black holes to observers at infinity. We speculate concerning the possibility that these objects (or others much like them) might exist in nature, and whether they might be observable despite their similar outward appearance to ordinary black holes. We examine their thermodynamics and stability in companion papers.

The dark sector need not be restricted to simple field content. Indeed, simple bosonic configurations, such as scalar-tensor or dark photon models, contrast with the much richer picture painted by many ultraviolet scenarios. Polology is the study of propagator poles, which correspond to particle states in any given theory. We outline a numerical polology framework for discovering perturbative, ghost-free models with consistent interactions, which produces theoretical model priors by sampling the coupling space. The method is tested on tensor field theories of up to rank three. Subsequent observational constraint pipelines are illustrated for black hole superradiance (M33 X-7), dynamical dark energy (DESI DR2, Pantheon and SH0ES) and gravitational waves (GWTC-3).

In this paper we review the super $T\bar{T}$ deformation of $\mathcal{N}=(1,1)$ theories in the superspace formulation, alongside its interpretation in the context of noncritical string theory. By combining the superspace approach with concepts from the study of super-Riemann surfaces, we demonstrate that super $T\bar{T}$ deformations in superspace can be naturally interpreted as a noncritical RNS superstring theory. We also propose a possible interpretation of the super $T\bar{T}$ deformations as 2D supergravity in the superspace through some field redefinitions.

We identify the 3d theory that realizes the RG-wall interface of 4d $\mathcal{N}=2$ $SU(N)$ Super-Yang-Mills, interpolating between the UV Lagrangian and the IR Seiberg-Witten effective description. The same theory also describes the low-energy boundary condition that corresponds to giving half-BPS Dirichlet boundary condition in the Lagrangian description of 4d $\mathcal{N}=2$ $SU(N)$ SYM. The theory is a 3d $\mathcal{N}=2$ SCFT that can be obtained as a massive deformation of the $T[SU(N)]$ theory, which is the S-duality interface of 4d $\mathcal{N}=4$ $SU(N)$ SYM. As the main validating tests, we match half-indices and discuss non-trivial consistency conditions when colliding such interfaces.

We study static, spherically symmetric screening solutions in the $\mathfrak{so}(N)$-invariant multi-field de Sitter Galileon theory, with matter couplings preserving the internal symmetry. Unlike in flat space, the de Sitter radial problem is affected both by the cosmological horizon and by the possible appearance of a finite strong-coupling radius, where the radial equation becomes singular and the effective description breaks down. Near the source, the quartic interaction governs the nonlinear screened branch, while farther away the quadratic terms dominate, defining a de Sitter analog of the Vainshtein radius at their crossover. A screened solution is viable only if this crossover occurs before the strong-coupling radius is reached. We study perturbations around the radial background and identify branches for which perturbations are stable and subluminal, within a restricted radial domain set by the strong-coupling point. Finally, we compare the de Sitter and Anti-de Sitter cases, isolating effects due to curvature from those specific to the cosmological horizon. Our results indicate that curvature can alleviate the superluminality issues that arise in Galileon theories for appropriate choices of matter couplings, though this comes at the price of a finite region of validity for the effective theory.

We develop a particle-on-a-group formulation of super-JT gravity aimed at computing supersymmetric correlators. We show that the (super)JT gravity can be described by a particle moving on the isometry group satisfying constraints from boundary conditions of (super)JT gravity. In this language the $\mathcal N=2$ and $\mathcal N=4$ theories are described by constrained particles on $SU(1,1|1)$ and $PSU(1,1|2)$. Solving the constraints gives the super-Schwarzian actions. We also quantize the reduced superparticle, with careful treatment of the fermionic constraints. We then derive the physical worldline supercharges from the requirement that the transformations preserve the constraints. These charges allow us to construct supersymmetric interval propagators in invariant variables and to formulate boundary-anchored Wilson-line operators for both superconformal primary and descendant insertions. Finally, we use these ingredients to build an algorithm for correlators. We obtain the $\mathcal N=2$ and $\mathcal N=4$ three-point composition kernels and zero-energy scalar three-point functions. For four-point functions, the same method reproduces the standard bosonic JT OTOC, gives an explicit zero-energy OTOC in $\mathcal N=2$ and $\mathcal N=4$ SJT, which is potentially useful for studying Berry curvature and BPS chaos.

We show that in holographic 2d CFTs, the entanglement entropies across several choices of global state and subregion can be written in a way that at once has a microscopic interpretation and matches the leading large$-c$ organization of the Ryu-Takayanagi formula. This representation is obtained by applying crossing symmetry to the replica manifolds. From the boundary point of view, each rewritten entropy looks like an algebraic entanglement entropy for the Virasoro algebra restricted to the region, with center labels obtained by coarse-graining heavy primaries of the BCFT on the regulated region into bins labeled by Liouville momenta. At large $c$, a resulting sum over bins is dominated by a saddle, and the $O(c)$ part of the entropy comes from the Cardy density of heavy primaries in the dominant bin. We identify this $O(c)$ part of the entropy with the Ryu-Takayanagi area. Physically, this suggests a concrete statistical origin for the Ryu-Takayanagi area as coming from coarse-grained Virasoro intertwiners across the entangling cut. The result also provides quantitative criteria for the amount of coarse-graining allowed for the consistency of the interpretation.

Complex conformal field theories (CFTs) have recently emerged as essential frameworks for understanding non-Hermitian criticality, weakly first-order phase transitions, and walking renormalization group flows, while their general structures remain largely unknown. In this work, we propose a systematic construction of complex CFTs by analytically continuing exactly marginal couplings into the complex plane. This procedure applies uniformly to bulk, boundary, and defect deformations, preserving conformal symmetry while generically complexifying operator spectra and other universal data. Using the compact free boson as a solvable laboratory, we uncover the global structure of the complexified Gaussian conformal manifold. More generally, we demonstrate that genuinely complex rational CFTs do not exist: rational points remain confined to the real regime, providing a sharp distinction between real and complex theories. In the defect case, we investigate the one-parameter family of conformal defects in the Ising CFT and derive exact expressions for the defect spectrum, energy transmission coefficient, and effective central charge from analytic continuation. The theoretical predictions are precisely verified in non-Hermitian critical Ising and free fermion chains using bulk-defect correlators, entanglement entropy, and complex energy transport, providing concrete evidence for the complex defect conformal manifold. Finally, we study complex boundary renormalization-group flows through the AdS/BCFT correspondence. Our results establish complex conformal manifolds as a controlled bridge between solvable lattice models, complex CFTs, and holography, while providing stringent analytic benchmarks for the nonunitary conformal bootstrap.

The two-dimensional Breitenlohner-Maison (BM) model is a classically integrable subsector of $D=4$ general relativity endowed with two commuting Killing isometries, obtained via Kaluza-Klein reduction to $D=2$. Integrable deformations of such a theory have recently been constructed via auxiliary fields in the so-called $\nu$-frame. In this work we first extend this point of view by deriving the complementary auxiliary field perspective known as $\mu$-frame, and then explicitly construct the uplift to $D=4$ of both descriptions, relying on an ansatz inspired by duality-invariant Lagrangian formulations of Einstein theory. The resulting four-dimensional deformed model thus obtained is a higher-derivative theory which lacks manifest diffeomorphism invariance in both frames. We comment on possible resolutions of this puzzling feature and on the physical interpretation of the model in $D=4$.

We study three-dimensional CFTs on compact Euclidean manifolds in two complementary limits: small deformations of the round $S^3$ and the small-fiber, large-squashing limit of Seifert manifolds. Near the round sphere, conformal perturbation theory expresses the free-energy response through integrated stress-tensor correlators. We derive a harmonic-space formula for the universal quadratic response to arbitrary metric squashing and show that it is proportional to the $c_T$ coefficient of the stress-tensor two-point function. For unitary CFTs this establishes that the sphere free energy is a local maximum in the space of metric deformations. This result extends to conserved spin-$s$ currents, whose quadratic response alternates in sign. For the specific case of squashing the Hopf fiber, we find an explicit form for the cubic response, and in addition obtain the leading correction to the two-point function of scalar operators. In the small-fiber limit, corresponding to the large temperature regime of the CFT, the partition function is governed by a two-dimensional thermal effective action constructed out of the Weyl-rescaled base metric and a Kaluza–Klein field strength. The thermal effective action relates CFT free energies on different backgrounds, including squashed Lens spaces, which we discuss in detail. We also explicitly determine the Wilson coefficients of this effective action, to various orders in the high-temperature expansion, for free fields, the large-$N$ critical O($N$) model, and holographic CFTs.

We compute the galaxy power spectrum at two-loop order in cosmological perturbation theory (effective field theory, EFT). We derive galaxy bias operators through the fifth order and obtain two-loop renormalization conditions for their bias coefficients. We also derive the relevant higher-derivative and stochastic contributions, and implement IR resummation using time-sliced perturbation theory. We obtain the complete dark-matter and galaxy renormalization conditions for the higher-derivative operators at the power spectrum, bispectrum, and trispectrum level. Having identified the redundant operators, we find that the two-loop galaxy power spectrum requires 21 additional bias, higher derivative and stochastic parameters per galaxy sample relative to the one-loop model. We compare our computation with the galaxy-galaxy and galaxy-matter power spectra from the PT Challenge N-body simulation at $z=0.61$ and find a per mille-level agreement up to $k=0.85~h$Mpc$^{-1}$. We show that even with conservative priors on all EFT parameters, the two-loop model produces an unbiased measurement of the mass fluctuation amplitude $\sigma_8$ with three times narrower error-bars than the linear theory model. The improvement over the one-loop model is $\simeq 40\%$. This suggests significant gains in the two-loop EFT analyses of galaxy clustering and galaxy–lensing two-point functions (“$2\times2$ pt”) from CMB lensing maps and imaging surveys like Euclid, LSST, and Roman. In addition, our two-loop computation offers a probe of new physics scenarios that modify the shape of the matter power spectrum at wavenumbers $(0.4-0.8)~h$Mpc$^{-1}$ such as the presence of ultra-light axion dark matter sub-components with masses $m_a\sim 10^{-24}$ eV.

We provide a testing ground for dualities of non-supersymmetric type 0 orientifolds by finding the global forms of their gauge groups. As an example, we consider the Bergman–Gaberdiel proposal for a duality of theories with $\mathfrak{so}(32) \oplus \mathfrak{so}(32)$ gauge algebra. On the orientifold side, we perform a scan of brane states, similar to the case of the D0-brane in type I string theory, and identify spinorial states that constrain the gauge group. Comparing with the bosonic string side, where the gauge group is determined by the internal momentum lattice as in heterotic strings, our analysis shows that the gauge groups match, hence supporting the duality beyond the perturbative string spectrum.

We present LinApart3, an efficient multivariate partial fraction decomposition algorithm for rational functions with linear denominators. Our decomposition algorithm guarantees that each term contains at most as many distinct denominators from the original set as partial fraction variables, introduces no spurious singularities, is independent of variable ordering, and is insensitive to the presence of spectator variables. While general multivariate approaches based on Gröbner bases or Leinartas' method handle arbitrary polynomial denominators, they suffer from intermediate expression swell. LinApart3 replaces polynomial-ideal computations with linear algebra and residue extraction by exploiting the geometry of the hyperplane arrangement defined by the denominators, circumventing this issue just as LinApart did in the univariate case. Because the individual basis contributions are independent, the algorithm is moreover naturally parallelizable. To showcase the utility of our algorithm we implemented the algorithm both in Wolfram Mathematica and FORM.

Classical nonabelian Hodge theory identifies Dolbeault and de Rham moduli spaces by providing a real-analytic isomorphism. In this paper, motivated by the Kapustin–Witten theory, we study this correspondence in the more general framework of perfect complexes on proper varieties, paying special attention to the surface case. We establish a Lagrangian correspondence which relates the shifted symplectic geometries by Pantev–Toën–Vaquié–Vezzosi (PTVV) between the derived stacks of flat and Higgs perfect complexes. We investigate the existence of derived twistor structures of hyperkähler type on the moduli stack of perfect complexes endowed with $\lambda$-connections by Deligne–Hitchin–Simpson. We establish a version of the AKSZ/PTVV transgression, Lagrangian intersection, and (hyperkähler) symplectic reduction theorems in this context. Moreover, we prove that the derived Riemann–Hilbert correspondence of Porta and Holstein–Porta, which states an equivalence of derived analytic stacks of perfect complexes on $X_{\mathrm{Betti}}$ and $X_{\mathrm{DR}}$, is compatible with the natural shifted–symplectic structures. We then study the relation between the shifted (pre-)twistor structures and the shifted symplectic forms on the fibers, and prove that the analytic Deligne–Hitchin–Simpson moduli stack on a smooth projective variety $X$ has a canonical $2(1-\dim X)$ shifted pretwistor structure over $\mathbb{P}^1_{\mathbb{C}}$, a result which has been anticipated for some time. In particular, the moduli stack of solutions to the Kapustin–Witten equations modulo gauge equivalence on a smooth proper complex algebraic surface exibits a $(-2)$-shifted (pre)twistor structure as a family over $\mathbb{P}^1_{\mathbb{C}}$.

Quantum chaos is commonly associated with the emergence of random-matrix statistics in the spectra of quantum systems. A useful diagnostic is provided by the spectral form factor (SFF), which for random matrix ensembles displays a universal linear-growth regime (`ramp'). In the last decade, a landmark result by Bertini, Kos and Prosen (BKP) identified for the first time a class of geometrically local quantum dynamics of finite-dimensional particles where the SFF provably exhibits a random-matrix ramp: periodically driven (Floquet) qudit chains whose evolution is described by `dual-unitary' circuits. Here, building on the BKP result and on a recently proposed variant of the Feynman-Kitaev clock construction, we obtain a spectral ramp in a class of static, geometrically local Hamiltonians. Our strategy is to embed the Floquet quasienergy spectrum of a dual-unitary circuit into the energy spectrum of a static local Hamiltonian, and to prove that the latter's connected SFF inherits the BKP ramp within a symmetry sector. This is to our knowledge the first proof of a spectral ramp in a time-independent, geometrically local many-body system with finite local Hilbert space dimension.

Using Hamiltonian methods in open bosonic string field theory, we derive the mass of a D0-brane as the central charge of the spontaneously broken Poincaré algebra in 26 dimensions. Though the formulas originate from string field theory, the calculation can be carried out completely on-shell. This illustrates a general procedure for computing spacetime central charges in string theory.

We describe the Poisson bracket of a Lagrangian field theory expressed in the framework of $L_\infty$ algebras. If the symplectic structure on phase space is defined following recent work, we show that the Poisson bracket can be computed with the Peierls formula, confirming the well-known result of the covariant phase space formalism. Of particular interest is the Poisson bracket in nonlocal theories. In $p$-adic string theory we find that a straightforward construction of the Poisson bracket is obstructed by higher derivative instabilities. We also discuss the inverse relation between the Poisson bracket and symplectic structure in the language of homological algebra, extending some ideas in the mathematical physics literature.

I discuss the recently discovered representation of conformal four-point ladder integrals in terms of the thermal free energy of free massive scalar fields. These integrals satisfy a novel second-order differential equation in even dimensions $D$ at arbitrary loop order $L$. I also present a simple derivation of the all-loop resummation of conformal ladder integrals for arbitrary $D$. Possible connections to the thermal bootstrap, multiloop calculations, integrability, AdS/CFT and string theory are briefly discussed. This is a proceedings contribution to the Athens Workshop in Theoretical Physics: 10th Anniversary, held at the National and Kapodistrian University of Athens on December 17–19 2025.

Normalizing Flows excel at modeling a single fixed density, yet many problems across the sciences, such as high energy physics, instead require modeling how that density deforms as a function of continuous parameters: the strength of a physical effect, a calibration constant, or a source of systematic uncertainty. Learning a separate flow for every parameter configuration quickly becomes intractable, since the number of joint settings grows exponentially with the number of parameters. We introduce Factorizable Normalizing Flows (FNFs), which represent the parameter-dependent density as a fixed, high-fidelity flow for a reference configuration composed with a learnable transformation that is polynomial in the parameters and factorized over them. This structure has a practical consequence: each parameter's effect is learned in isolation, from samples in which that parameter alone is varied. The combined response of many parameters is then recovered by summation at inference, without ever sampling their combinatorially large joint space. On a controlled problem with two interpretable deformations applied jointly to the data, the learned transformation reproduces the true deformations and matches the optimal likelihood, while optional interaction terms capture residual correlations when several parameters vary strongly at once. The resulting model is interpretable, scales linearly with the number of parameters, and keeps the likelihood tractable. This provides a general tool for any inference workflow requiring continuous density morphing, and directly enables the next generation of unbinned likelihood fits in high energy physics.

It is well-known in integral geometry that a maximally symmetric Riemannian manifold, such as a static slice of vacuum AdS spacetime, can be perfectly covered by the geodesics in the Kinematic space, which we call the partial-entanglement-entropy (PEE) threads. In this context, the area of a codimension-one surface in the manifold can be computed by counting its intersections with the PEE threads, which is the celebrated Crofton formula. In this paper, we analyze the Kinematic space for a generic subregion in vacuum AdS space, and propose that the PEE threads emanate from a co-dimension one surface can perfectly cover a subregion in the manifold. Furthermore, we build holographic tensor network models on the network of the PEE threads confined in a subregion, thereby providing a concrete framework that realizes the surface-state correspondence and the generalized entanglement wedges for gravitational subregions.

We use the Banerjee–Mandal–Sahoo dipole-current Ward identity for the one-loop logarithmic soft-photon theorem as input and determine its finite-energy action on Mellin-difference hard currents. The commutator with such hard currents has a scheme-independent hard-hard residue that survives every one-particle redefinition. With the meromorphic continuation stated explicitly below, a two-particle Plancherel transform identifies this residue with an analytic two-particle primary module, and the coefficient map is a hard-current one-cocycle. The cocycle defines a minimal filtered abelian extension. It has a canonical two-particle primitive and integrates to an affine action. For scalar hard legs, the fixed-leg operator agrees coefficient by coefficient with the symmetry-governed long-range logarithmic tower of Choi, Kadhe, and Puhm. Applied to a tree-level scalar-QED photon-exchange block, the finite-energy analysis determines the logarithmic two-particle coefficient functional from the ordinary hard amplitude and the Banerjee–Mandal–Sahoo ordered-pair soft kernel. This gives a finite-energy relation between the Banerjee–Mandal–Sahoo dipole-current Ward identity and the exponentiated long-range celestial OPE.