Papers

We study exact recovery for non-Abelian topological surface codes based on the quantum double $D(G)$ of any finite group $G$. We determine an orthogonal error basis, that is comprised of group multiplication and irreducible representation operators. Whenever the errors are supported on closed loops on the lattice or dual lattice, there is a redundancy in the basis, due to stabilizers. To remedy this, we introduce a gauge-fixing that results in a complete and orthogonal error basis on the code space. Assuming as input predetermined, neutral correctable error clusters, we construct charge and flux transfer circuits that move the errors onto ancillas that then get projected out. This is an exact and deterministic recovery protocol, applicable to surface codes for any finite, in particular non-Abelian, group $G$.

The Redfield equation with time-dependent coefficients (or TCL2 equation) gives one of the simplest perturbative descriptions of open quantum system dynamics derived from a microscopic theory. Its status as an approximation is nevertheless subtle: while perturbation theory fixes the reduced dynamics order-by-order in the system-environment coupling, solving the Redfield equation yields a particular resummation that need not preserve complete positivity. Here we compare Redfield evolution with an exponentiated (Magnus) resummation of the same second-order perturbative map, which is completely positive and trace preserving (CPTP) for a broad class of physical conditions. We also construct a simple CPTP resolvent resummation, illustrating the non-uniqueness of resummations of the same perturbative map. The Redfield and Magnus prescriptions agree to second order but differ through time-ordering corrections at higher orders. We exhibit a weak-coupling criterion which guarantees that Redfield evolution, even when non-Markovian, remains close to a CPTP map. We illustrate the criterion in two examples, focusing on finite-dimensional (qubit) systems. Our results clarify when, and why, Redfield evolution can remain reliable even when it is highly non-Lindbladian.

We argue that BF theory with $SO(d+1,1)$ gauge group provides a SymTFT ”skeleton” for building AdS$_{d+1}$/CFT$_d$ pairs via gluing consistency as the bootstrap principle, loosely analogous to how compact Chern-Simons theory is a TQFT skeleton for rational CFTs. This is a non-perturbative $bulk$ framework for quantum gravity. The bootstrap is formulated as presentation independence of physical amplitudes within a universal, defect-closed BF sewing calculus, which includes Wilson lines, codimension-two monodromy defects, and their junctions. The flat BF connection encodes the conformal geometry of the boundary, while Wilson-line path integrals construct candidate quantum-gravity microstates and charge sectors. When the bootstrapped spectrum has a stress tensor, Ward identities naturally connect it to the $B$-field, a relation we call generalized simplicity. In large-$N$, large-gap regimes admitting a gravitational EFT, this motivates the introduction of an effective notion of the $B$-field. In the Einstein limit, this becomes the ”simplicity” relation used in spin-foam models. Conventional spin-foam models try to obtain gravity by constraining BF theory. We suggest instead that BF theory is the Platonic sewing skeleton, from which theories are to be bootstrapped. Despite the topological nature of the skeleton, perturbative local bulk fields can emerge via Wilson-towers of multi-trace primaries, reproducing their one-loop determinants. We note that the BF Fourier kernel is a suggestive higher-dimensional analogue of the modular-$S$ structure. Because the bulk gluing principle in itself does not require a boundary description, the framework is conceptually broader than AdS/CFT. We sketch how de Sitter space may fit into this picture, with a natural role for observers. Our approach elevates gluing consistency to a universal bulk principle underlying theories of quantum gravity.

We study extremal and next-to-extremal four-point correlators involving general quarter-BPS operators in $\mathcal{N}$=4 Super Yang-Mills. Using a combination of $R$-symmetry considerations, operator product expansion content, and analytic superspace techniques, we show that such correlators get contributions only from operators belonging to short and semi-short multiplets. This result together with the non-renormalizability of the relevant three-point functions guarantees that all such four-point correlators are protected from quantum corrections.

We derive sharp lower bounds on the energy cost of Page-like entanglement entropy excursions in two-dimensional conformal field theory. For states satisfying the primary quantum null energy condition (QNEC), the positive null energy $E_+$ obeys a sharp lower bound with two branches separated by the threshold $\Delta S=(c/6)\ln 2$. We also derive a sharp lower bound on the net energy $E_{\rm net}$, with a coefficient larger by a factor of two than the value quoted by Abdolrahimi and Page [Phys. Rev. D 92, 083005 (2015)] and, in the QNEC-saturating sector, we obtain a sharp lower bound on the negative energy $E_-$. The constants are sharp as infima over smooth conformal vacua. We determine the entropy profiles that minimize $E_+$ for general QNEC states and the one that minimizes $E_-$ in the saturating sector. Finally, for an excursion containing an infinite sequence of small sub-excursions within a finite duration, we derive a necessary condition for the total positive energy to remain finite, which is stronger than requiring the entropy to have finite total variation.

We propose a formulation of a quantum field theory within the framework of the unfolded dynamics approach, originally developed in higher-spin gauge theory. We construct an unfolded system that describes a hierarchy of Schwinger-Dyson equations for vertex functions of a scalar QFT in terms of unfolded first-order equations. A distinctive feature of the constructed system is the appearance of two additional evolution parameters responsible for the WKB and RG behaviors. As an illustration, a one-loop analysis of the theory with quartic self-interaction is carried out.

We demonstrate that form factors evaluated at complex-valued momenta can be directly accessed from position-space Euclidean correlators without analytically continuing the latter and without solving an inverse problem. The approach directly applies a bilateral Laplace transform to the lattice data, which provides the analytically continued form factor wherever it converges. Focusing on the pion form factor in Quantum Chromodynamics (QCD), this gives access to the observable as a function of invariant \(Q^2\) throughout an extended analytic domain consisting of all complex \(Q^2\) with real part larger than \(-4m_\pi^2\). This could, for example, be used to improve charge-radius determinations or provide more complete data for fits based on conformal mapping of the data. After developing the approach for infinite-volume correlators, we discuss various options for finite-volume estimators suitable for realistic calculations, including position-space fits or application of a spatial chemical potential represented by boundary conditions with an imaginary twist angle. Finally, we test the method in the two-dimensional \(O(3)\) nonlinear sigma model using Monte Carlo data.

In this article, we give a global construction of the Virasoro conformal blocks on Teichmüller space of Riemann surfaces of genus $g$ with $m$ marked points, when the central charge is $c>25$. We prove that they are global holomorphic sections of a holomorphic line bundle and they satisfy the Ward identity, which encodes their conformal invariance. For the sphere with $3$ points, the space of conformal blocks is one-dimensional. For a given marked Riemann surface and a marked pair of pants decomposition, it is in general an infinite dimensional Hilbert space, isomorphic to $L^2(\R_+^{3g-3+m})$ with respect to some measure involving the DOZZ structure constants. We show that it carries a projective unitary representation of the mapping class group. More generally, there are unitary isomorphisms for each Moore-Seiberg move, changing a marked pair of pants decomposition into another one.

This paper is the second part of the proof of the conformal bootstrap for Liouville conformal field theory on compact surfaces with boundary. It is devoted to the spectral theory of the half-annulus semigroup and to the resulting bootstrap formula. Building on the Segal axioms and gluing properties established in Part~1~\cite{GRW1}, we identify the generator of the half-annulus semigroup with the boundary Hamiltonian and establish its spectral decomposition by scattering methods. This yields a direct-integral decomposition of the boundary state space into irreducible Virasoro representations, governed by the boundary spectral measure. Combining this decomposition with Ward identities for boundary deformations, we express general correlation functions as integrals over spectral parameters attached to cutting curves, with integrands given by products of bulk and boundary structure constants and the corresponding conformal blocks. These results also underpin the analyticity and symmetry properties of conformal blocks~\cite{GRSSbloc}, and contribute to the construction of mapping class group representations on spaces of conformal blocks~\cite{PapierBlocs}.

We develop a Fock-space formulation that reduces expectation values of vacuum-family quasi-primaries in qKdV eigenstates of arbitrary highest-weight conformal families to a free-bosonic problem, and employ it to test the generalized eigenstate thermalization hypothesis (GETH), in 2D conformal field theories with large central charge $c$. We demonstrate that the first two quantum Korteweg–de Vries (qKdV) charges $Q_1$ and $Q_3$ lift all degeneracies except along at most countably many curves in the $(c,\Delta)$ plane, where $\Delta$ is the highest weight of the conformal family, so that the full-qKdV simultaneous eigenstates are unambiguously determined by $Q_1$ and $Q_3$. Based on this, a near-complete toolkit for testing in the vacuum conformal family or falsifying the general form of GETH, i.e. equating qKdV eigenstates and thermal states of the generalized Gibbs ensemble in the sense of expectations of local observables, is provided in this paper, through the infinite tower of qKdV charges. A stronger form of the GETH, namely that all expectations of local observables of a definite level should be expressible of qKdV charges but not contain microscopic eigenstate information, is directly falsified at subleading order in the large-$c$ expansion. At levels 6 and 8 in the vacuum family, expectation values of quasi-primaries in simultaneous eigenstates of the qKdV charges are not fixed by first finitely many qKdV charges, and microscopic information of the eigenstate is required.

Null string theory (NST) is a two-dimensional (2d) sigma-model on a Carrollian worldsheet with 2d diffeomorphisms, Weyl and Carroll-Weyl (CW) scalings as gauge symmetries whereas the Isberg-Lindström-Sundborg-Theodoridis (ILST) theory on a generic target space only realizes the first two and not the CW gauge symmetry. We show in this work that NST on the Carrollian target spaces (CTS) is well-defined if the background admits a homothetic kernel generator and that, it is described by the ILST theory on the same CTS. To this end, we formulate a minimal null string theory on CTS, starting from first principles and then discuss beyond minimal theories as deformations away from minimality. We study some classes of such deformations involving the clock one-from for ruled Carrollian spaces and/or connection on the Carrollian space. We discuss implications of this for the recently introduced null string holography. As examples CTS, we briefly discuss null strings probing black hole horizons and the null string holography in this context.

We discuss the perturbative stability of warped anti-de Sitter (WAdS$_3$) spaces in three-dimensional gauged supergravity. To this end, we first derive the analogue of the Breitenlohner-Freedman bound for a single scalar field on such a curved background. We then apply this analysis to the half-maximal $D=3$ gauged supergravity obtained from the compactification of the $D=6$, $\mathcal{N}=(1,1)$ supergravity on S$^3$. Families of three-dimensional configurations, uplifting to spaces of the form $\mathrm{WAdS}_3 \times \mathrm{WS}^3$, are found by turning on appropriate vector and scalar fields, supporting the AdS$_3$ and S$^3$ warpings, respectively. Finally, we show that the spectra of scalar fluctuations satisfy the stability criteria for each solution.

Henneaux-Teitelboim gravity is a diffeomorphism-invariant formulation of unimodular gravity that is locally equivalent to general relativity but includes an extra boundary variable, unimodular time, whose difference between hypersurfaces measures the enclosed 4-volume. In general, unimodular versions of quantum gravity can be expected to deviate from theories based on general relativity, but a discrete path-integral formulation for Henneaux-Teitelboim gravity has not yet been established. In this paper, we discretise Henneaux-Teitelboim gravity using the methods of Euclidean Regge calculus. Testing this framework in a symmetry-reduced model of Regge cosmology, we reproduce results in previous literature and demonstrate that the continuum limit can be naturally defined using unimodular time rather than proper time. Because unimodular time appears as boundary data as opposed to being a degree of freedom in the bulk, this approach allows for the comparison of general simplicial triangulations, beyond highly symmetric discretisations, to the continuum. Our work provides a first step towards a discrete path-integral formulation of Henneaux-Teitelboim gravity and adds a new perspective on time as a boundary variable in quantum gravity.

We investigate anisotropic domain wall and cosmological solutions in $D$-dimensional gravitational theories of arbitrary spacetime signature coupled to scalar fields with a scalar potential. We identify a class of scalar potentials for which the second-order equations of motion admit a first-order formulation. The associated bosonic flows allow non-trivial Ricci-flat worldvolumes, while their Killing-spinor realization further requires the worldvolume to admit a non-vanishing parallel spinor. Five-dimensional ${\cal N}=2$ gauged supergravity provides a concrete realization of this construction, with the worldvolume geometry determined by its signature and giving rise to pp-wave, hyper-Kähler, and hypersymplectic geometries. We further construct a general class of anisotropic solutions depending on a single variable, with the anisotropy encoded in a constant traceless matrix. A particular branch arises when a subset of scalar fields contribute through their kinetic terms but do not enter the scalar potential; these fields follow geodesics on the scalar manifold. As an explicit application, we recover the Kasner-dilaton-AdS solutions and show that, in the appropriate limit, they satisfy the first-order flow equations.

We construct a finite cyclic $L_3$ presentation (that is, a cyclic $L_\infty$ algebra with brackets vanishing above arity three) of the variational deformation theory defined by the heterotic $G_2$ superpotential near a torsion-free standard embedding, to first order in $\alpha'$. Auxiliary fields for the inverse coframe, spinor derivative, flux and induced tangent connection give a shifted-cotangent Hamiltonian of degree at most four. Eliminating the auxiliary fields recovers the physical functional. On the flat seven-torus, this functional has a nonzero quintic Taylor coefficient in local physical spinor coordinates. For $Y=S^1\times X$, with $X$ a Calabi–Yau threefold, we restrict to the product locus with horizontal gauge and gerbe data. The functional reduces to $ \frac L4\operatorname{Im}\int_X(H+i\,d\omega)\wedge\Omega, $ where $L$ is the circle length and $\Omega=e^{-2\Phi}\Psi$ is the weighted complex volume. The induced $SU(3)$ structure selects a Hermitian tangent Courant subalgebroid. Transverse isotropic lifts in its complexification encode the holomorphic connection and string equations, together with the induced tangent connection, weighted canonical line and gerbe hierarchy. Their Taylor brackets form an $L_3$ algebra whose string part agrees with the known holomorphic heterotic algebra. We then show that this holomorphic theory is obtained by characteristic Hamiltonian reduction of a finite contractible stabilization of the product variational theory, in every cochain degree. While the quotient captures the holomorphic equations, conformal balance, gauge primitivity and leading-flux admissibility are additional conditions on a real representative.

We investigate collective charge excitations in three-dimensional semimetals with dipole-like Berry-curvature (BC) profiles, focusing on vortex nodal-ring semimetals (VNRs) and Hopf semimetals (HSMs). The VNR hosts a continuous ring of dipole-like BC sources generated by its winding pseudospin texture, whereas the HSM hosts an ideal BC dipole. This allows us to examine how distinct realisations of dipole-like BC structure affect collective charge dynamics. While plasmon modes in $\mathcal{PT}$-symmetric nodal-ring semimetals (PTNRs) and their gapped cousins were studied in a companion work, arXiv:2609.35373, here we analyse the density response, Drude weights, dielectric function, and plasmon modes of VNRs and HSMs at low doping within the random-phase approximation. For VNRs, we consider untilted, in-plane tilted, and axially tilted systems. Although VNR and PTNR have identical energy dispersions, their distinct eigenstates introduce a toroidal-angle dependence in the band-overlap factor, yielding additional interband contributions for in-plane wavevectors and weak corrections to the screened plasmon frequencies. The toroidal Fermi surface also produces intrinsically anisotropic bare plasmons. In contrast, HSMs have isotropic bare plasmons, with anisotropy arising only through their interband response. Our results show that collective charge dynamics can distinguish different realisations of dipole-like BC structure and reveal pseudospin information beyond the single-particle dispersion.

Harmonic cosmology is an algebraic framework based on the $\mathfrak{su}(1,1)$ Lie algebra that underlies the effective dynamics of loop quantum cosmology and group field theory. Focusing on the multimode bosonic realization of $\mathfrak{su}(1,1)$, we construct multimode squeezed states in harmonic cosmology and analyze their semiclassical properties by computing the relative uncertainties of the volume operator at the kinematical level. In contrast to single-mode squeezed vacua, which invariably exhibit large quantum fluctuations, we find that the relative uncertainty of the volume can be made arbitrarily small when the number of modes is large, even without displacement. Our results demonstrate that a kinematical squeezed vacuum state with a large number of independent modes can behave semiclassically providing insights into the quantum nature of bouncing cosmologies and the emergence of a classical universe.

We study the radial ultraviolet dissipation quantity $c_{UV}$ defined in arXiv:2601.16888v3 and defect OPE data in bottom-up holographic interfaces. We relate $c_{UV}$ along with the defect primary spectrum and BOPE coefficients in the large defect dimension limit to the optical length $\rho$ in the holographic bulk. In addition, we prove a new holographic bound for conformal interfaces relating entanglement entropy, energy transmission and UV dissipation: \[ c_{LR}\geq \frac{2c_{\mathrm{eff}}}{2+\pi(c/c_{\mathrm{UV}}-1)}. \] This compliments the upper bound of $c_{LR}$ by $c_\text{eff}$ proved in arXiv:2404.01515.

Cosmic birefringence, the isotropic rotation of the plane of linear polarization of photons from sources at cosmological distances, can be generated by different parity-violating mechanisms. While the net rotation angle of a single polarized source carries little information about its physical origin, sources at various redshifts $z$ can be used to distinguish between different origins. Parity-sensitive correlations of the $E$- and $B$-mode polarization fields of the cosmic microwave background measured on large and small angular scales probe sources at $z\simeq 10$ and $1100$, respectively. Thus, the detailed shape of the $EB$ power spectrum is sensitive to the redshift evolution of the birefringence source. In this paper, we calculate the $EB$ power spectrum due to a neutrino-asymmetry current, and compare it to axion dark energy and dark matter using a common Chern–Simons framework. We find that the $EB$ power spectrum due to a neutrino asymmetry is suppressed at low multipoles, $\ell\lesssim 20$, similar to the dark matter case. This can be clearly distinguished from the dark energy case, which has no such suppression. Large-scale polarization measurements, including LiteBIRD, can therefore distinguish late-time dark energy birefringence from that generated at higher redshifts by axion dark matter or neutrino asymmetry. We also find that the evolution of the neutrino density during recombination alters the shape of the $EB$ power spectrum at high multipoles. This signature can be used to distinguish between the neutrino and axion dark matter interpretations, using ground-based experiments such as the Simons Observatory.

We construct the quantized electromagnetic field and the corresponding photon propagators in a rigidly rotating cylindrical reference frame. Starting from covariant Maxwell theory, we formulate the gauge-fixed field equations and solve them in cylindrical coordinates. The vector nature of the electromagnetic field requires distinguishing orbital and intrinsic angular momentum, with the total angular momentum as the natural quantum number. We construct the two independent physical photon modes in terms of Bessel functions by solving the coupled differential equations, and provide an alternative construction based on a covariant basis of vector eigenfunctions generated by a covariantly constant tetrad. After normalizing the modes with the Maxwell inner product, we quantize the electromagnetic field and define the corresponding photon states and greater and lesser two-point functions. The physical time-ordered propagator is obtained directly from the quantized modes, with its transverse dependence encoded in polarization kernels that retain the cylindrical spatial structure and its spectral poles exhibiting the rotational energy shift. The full gauge-fixed photon propagator is constructed in two complementary ways: from the scalar Green function combined with the tensor transformation of the inertial Feynman-gauge propagator, and from the covariant vector eigensystem through its spectral decomposition and the Fock–Schwinger proper-time representation. We verify that both constructions satisfy the Green-function equation for the complete gauge-fixed Maxwell operator, providing the propagator required for covariant perturbative calculations with internal photon lines.

We study leading parity-even four-derivative corrections to the linear Christodoulou regularity criterion for a neutral, massless, minimally coupled spectator scalar in four-dimensional charged de Sitter black holes. We derive the corrected static solution, cold extremality relation, near-horizon geometry, Cauchy-horizon surface gravity, and neutral scalar quasinormal spectrum within Einstein-Maxwell effective theory. Near cold extremality, the scalar equation reduces analytically to the finite-temperature anti-de Sitter two-dimensional throat problem. We find that the dominant nonstationary spherically symmetric mode retains its cold-limit regularity ratio through first order, while higher angular sectors respond to the deformation of the relative throat radii. The finite-distance neutral-scalar Christodoulou boundary nevertheless shifts because the relevant exterior scalar spectrum and the unequal horizon surface gravities enter the regularity ratio. At smaller and intermediate cosmological scales, positive effective coefficients move the Christodoulou threshold to larger normalized separation of the Cauchy and event horizons. Toward the larger cosmological-scale part of the cold branch, the same coefficient directions move that threshold back toward the cold endpoint. The reversal accompanies a transition in the quasinormal family that supplies the spectral gap. We further show that the coefficient combination governing the fixed-charge extremal mass parameter differs generically from those governing the cold throat and the finite-distance neutral-scalar boundary. Extremality therefore does not determine a universal direction for the neutral-scalar Christodoulou-threshold correction. Our results identify cosmological scale and spectral-family competition as essential elements in relating effective gravity, black-hole extremality, and Cauchy-horizon regularity in this spectator sector.

We study the formation of $\mathcal{N}$-quark baryons during the confinement crossover of an $SU(\mathcal{N})$ gauge theory containing quarks in the fundamental representation, with masses around or below the confinement scale. We assume stepwise baryon formation through the successive fusion of pairs of quark-clusters, with color interaction rates obeying quadratic Casimir scaling. We include the complete set of reactions between $SU(\mathcal{N})$ antisymmetric clusters dressed with their specific spin-flavor multiplicities. We show that at large $\mathcal{N}$, baryon formation is hindered by a Casimir bottleneck, such that fusion of few-quark clusters in the initial steps of the baryon assembly chain is strongly outweighed by cluster destruction processes. We find that for $\mathcal{N}=12\,(18)$ the baryon-to-meson yield ratio is suppressed down to $10^{-12}\,(10^{-23})$ of its value at $\mathcal{N}=3$. We embed the mechanism in a vectorlike three-flavor model whose lightest baryon can be electrically neutral, while axial-anomaly-induced couplings and dimension-six operators induce decays of the unstable mesons and charged baryons safely before big-bang nucleosynthesis. The relic density of cosmologically stable neutral $\mathcal{N}$-baryons is thus set during confinement at values so small that, for sufficiently large $\mathcal{N}$, the baryon masses required to saturate the dark matter energy density exceed by several orders of magnitude the unitarity bound on annihilation cross sections.

We analyze two old models of black holes to see whether they realize the AdS/CFT paradigm of “multiple representations of bulk data by boundary states that differ only in the expectation value of exponentially complex operators" . We find that the explanation of the firewall paradox in these models is much simpler and corresponds to the naive way entanglement is resolved when small constituents escape from a complex chaotic quantum system. The real issue is that the quantum field theory (QFT) picture of the states near the boundary of a causal diamond is wrong. The phrase in quotes above is correct in our models because most of the states associated with any causal diamond cannot be associated with ANY states in quantum field theory.

The convergence of hydrodynamics is commonly investigated by Taylor-expanding the gapless dispersion relations $\omega(k)$ around $k=0$. We show that this diagnostic can fail arbitrarily close to the hydrodynamic limit: sound modes can collide at arbitrarily small wave number, rendering $\omega(k)$ nonanalytic, while the relationship between densities and fluxes remains analytic there. We instead formulate the gradient expansion off-shell (i.e. in the presence of external forces), directly at the level of the constitutive relations. For causal theories with relaxational spectra, we establish convergence for $|i\omega|+w|ik|<1/\tau_g$, where $w$ is the maximal information speed and $1/\tau_g$ the nonhydrodynamic gap. On the naturally Lorentzian $\{i\omega,ik\}$ plane, this region is a causal diamond, whose boundary marks the first possible encounter with nonhydrodynamic modes. Beyond it, analytic continuation reveals the nonhydrodynamic spectrum through singularities of the resummed constitutive relations. Thus, hydrodynamics is aware of the degrees of freedom beyond hydrodynamics, but this information becomes fully accessible only off-shell.

The effective action at one-loop for the quantum electrodynamics (QED) in the presence of background vectors and tensors that break the Lorentz symmetry is calculated in this work. We highlight that just the Lorentz symmetry violation through the Dirac matrices in the fermionic sector is considered since investigations in others sectors of the extended QED has been explored in recent papers. Using the proper time method, we show that some background vectors and tensors do not contribute to the effective action at one-loop and in the linear approximation. Thereby, only two background tensors contribute to the effective action at one-loop for small components of these tensors, and that a new combination of cubic and quartic non-linear electrodynamics emerge in the weak field regime.

The Island Formula for Quantum Extremal Surfaces of large AdS black holes, which are made to evaporate by coupling the CFT of the AdS/CFT correspondence to an external system, demonstrates that the space time picture can reproduce the quantum mechanically required Page curve of the CFT/bath system if a region in the interior of the black hole has become entangled with the external radiation by the Page time. In an effective quantum field theory (EFT) description in “nice slice" coordinates, this appears to imply paradoxical entanglement between degrees of freedom that have been causally disconnected since long before the Page time. We will give two arguments that all of the quantum information in the island is in fact concentrated in a tiny space-like portion of the island very close to the black hole horizon at the Page time. The first applies the covariant entropy bound\cite{fsb} (CEB) to the causal diamonds of geodesics originating on the space-like region that becomes the island, but on a time slice just before the Page time. All points on the island except those in a stretched horizon region very close to the horizon have entropy bounds that are tiny fractions of the black hole entropy. The second argument uses standard quantum statistical mechanics combined with the tensor network renormalization group regularization of CFT to argue that the wave function of a typical pure state in the microcanonical evaporating black hole always has most of its probability concentrated in a small range of wavelengths corresponding to a small range of radii in the dual tensor network/AdS slice.

We extend the modularity-based approach of arXiv:2509.02226 for the holographic thermal propagator to the physically relevant case of nonzero energy $\omega$. Via the AGT correspondence, the problem is mapped to $\mathcal{N}=2$ $SU(2)$ gauge theory with $N_f=4$ hypermultiplets. Turning on $\omega$ activates all four hypermultiplet masses, so that the full $SO(8)$ flavor symmetry constrains the low-temperature expansion, rather than the subgroup left unbroken at $\omega=0$. As a result, the building blocks of the expansion are no longer just Eisenstein series but also Jacobi theta functions. We use this fact, together with the Molien series of the resulting invariant ring and the modular anomaly equation, to fix the low-temperature expansion coefficients order by order, matching only a finite number of terms in the Zamolodchikov's $q$-recursion at each order. We obtain the low-temperature expansion of the propagator and find agreement with the result of a completely different method, the near-boundary expansion of the bulk wave equation.

We present an elementary recursive construction of Hermitian projectors on multi-quark ${\rm SU}(N)$ color states, based on the characterization of these states by their quadratic Casimir, and using the birdtrack pictorial technique.

We conjecture that the universal logarithmic coefficient for capacity of entanglement $c_{\mathrm{cap}}$ is bounded below by the corresponding entropy coefficient $c_{\mathrm{eff}}$ at conformal interfaces. We prove the bound in free boson and fermion CFTs and in holographic models with interface branes, and establish it using defect perturbation theory. Numerical calculations at interacting interfaces between the tricritical Ising and Ising CFTs further support the bound. Despite this constraint, capacity provides independent interface data. We construct interfaces with identical entropy and energy transmission coefficients but different capacity coefficients, and show that capacity can even increase as entropy decreases.

We study the entanglement structure of partially entangled thermal states in double-scaled SYK , allowing flavor degrees of freedom carried by matter insertions to be in an arbitrary mixed state. In the semiclassical limit, followed by the heavy-probe limit, our replica calculation gives a Faulkner–Lewkowycz–Maldacena formula: the boundary entropy decomposes into an `area' term (independent of the flavor state) and the bulk matter entropy in a corresponding entanglement wedge. Varying the Euclidean locations of the operator insertions produces sharp transitions in the boundary entropy, which we interpret as individual matter excitations crossing the boundary of the wedge and hence as evidence for a sharply defined entanglement wedge. To derive this result, we analyze the semiclassical behavior of general Euclidean $2n$-point functions and give a probabilistic proof of their factorization into sums over products of two-point functions. We also give a complementary derivation from exact spectral representations, using the semiclassical behavior of the quantum $6j$-symbol.