Papers

This primer offers an first introduction to Celestial Holography, a rapidly evolving framework aiming to describe quantum gravity in asymptotically flat spacetimes. We start from the established foundation of the AdS/CFT correspondence before venturing into the wilds of Minkowski space. The journey covers the essential Bondi-Sachs formalism and the geometry of null infinity ($\mathscr{I}^\pm$), the discovery of the infinite-dimensional BMS asymptotic symmetry group. We explain the core holographic map given by Mellin transform that recasts four-dimensional scattering amplitudes as correlation functions of a conjectured 2D Celestial Conformal Field Theory (CCFT) on the celestial sphere. The presentation is made intuitive but the mathematics is still precise (hopefully!). It is designed to equip newcomers with both the conceptual landscape and the technical vocabulary to explore this exciting frontier.

We study the recently discussed $2+1$-dimensional Hull-type twisted $\mathcal{N}=2$ Carroll superalgebra. This structure admits both electric- and magnetic-type realizations at the level of the supersymmetry transformations. At the Lagrangian level, magnetic Carroll theories describe fields that propagate in space while remaining constrained in time, and vice versa for electric theories. We first realize this symmetry algebra through the dimensional reduction of a $2+2$-dimensional $\mathcal{N}=(1,1)$ parent algebra, originally discussed in the context of self-dual supergravity. We further find that a consistent reduction of this higher-dimensional algebra admits infinite-dimensional lifts to left- and right-chiral super-$\mathrm{BMS}_4$-type algebras, which can be combined by restoring the $R$-symmetry, leading to the type I-I magnetic super-$\mathrm{BMS}_4$ algebra. We then consistently construct magnetic off-shell scalar and Abelian Yang–Mills theories and comment on on-shell theories. Finally, in the appendix, we explain how an ordinary supersymmetric theory can be analytically continued to its twisted counterpart.

We introduce a non-commutative Laplace transform between functionals on path space and formal series in a tensor algebra, under which a natural convolution of path functionals becomes an algebraic product of series. Applying it to a random unitary matrix-valued path development - the quantum path signature - we show that the governing planar loop equations take a non-commutative spectral form. We then extend the loop equations to a $1/N$ genus expansion, organised by topological recursion, and obtain a hierarchy of integral equations on path space for the corrections.

Cloud-of-strings (CS) geometries provide an effective description of one-dimensional string distributions. However, their central singularity cannot be removed through the standard regular black holes (RBH) mechanism based on an effective mass function, since the string sector contributes independently to the ultraviolet structure of the spacetime. Motivated by this observation, we investigate whether string-supported black holes can be consistently regularized while preserving the CS asymptotics and admitting a physically meaningful string-fluid interpretation. Using the gravitational decoupling method, we construct a RBH supported by an effective anisotropic string fluid. We show that the string sector deforms the de Sitter core, modifies the local topology of the spacelike slices, and introduces a longer-range correction dominating the usual Hayward/LQG term. The geometry admits non-extremal and extremal RBH, as well as a regular horizonless compact object. Moreover, the string parameter qualitatively modifies the thermodynamic evolution by shifting the Davies phase transition and the size of the black-hole remnant. Finally, the scalar quasinormal-mode spectrum exhibits systematic changes in both the oscillation frequencies and damping rates. These results show that regularizing string supported black holes is a physically distinct problem, with the matter sector governing the ultraviolet structure, thermodynamics, and dynamical response of the spacetime.

We propose a new method for computing $N$-point tree-level open string amplitudes, inspired by the field-theoretical perturbiner framework and bypassing world-sheet moduli integration. The construction rests on a weak associativity condition for a point-split, on-shell product of string vertices, introducing a binary-tree prescription with all physical channels manifest. The output is a multi-parametric series representation of the partial amplitude that is amenable to mass-level truncation and, therefore, numerical implementation. Besides the Veneziano amplitude, automatically represented by a dual-channel series with one free parameter, we present its five-point generalization, involving two free parameters and matched against known results in the literature. Higher multiplicities are obtained via the same systematic branching, with no separate input.

Universal hallmarks of quantum chaos—such as the ramp and plateau in the spectral form factor—and the ramp's gravitational duals involving wormholes are widely interpreted as consequences of spectral or ensemble averaging. In this paper, following an earlier proposal of~\cite{Liu25c}, we develop an alternative approach: these phenomena arise as macroscopic smooth structures hidden within erratic microscopic data, which can be isolated through a smooth filter projection. Using the semiclassical Gutzwiller trace formula as a paradigmatic example, we illustrate how many features characteristic of random matrix models—including the ramp, the plateau, the spectral curve, and single-eigenvalue instantons—can be derived in the semiclassical limit without invoking ensemble or explicit spectral averages. We postulate the existence of a minimal Gutzwiller-like structure in the large-$N$ limit of holographic systems and explore its consequences. Beyond deriving the ramp and the plateau, this Gutzwiller-like structure predicts universal rapid macroscopic oscillations in the density of states and the possible existence of hyper-instantons, both of which involve double exponentials in $1/N^2$. On the gravity side, we demonstrate how spacetime wormholes enable the construction of emergent hyper-non-perturbative objects—such as baby-universe and wormhole condensates—which yield double exponential effects in $G_N$. This mirrors the postulated boundary Gutzwiller-like structure and provides a dual gravitational derivation of the universal rapid macroscopic oscillations in the density of states and the spectral plateau.

We compute the energy and angular momentum radiated by compact binaries through gravitational waves, scalar and vector fields, using the field-theoretic approach. We observe that each emitted graviton carries away an angular momentum of $\dot{J}/\dot {N}\simeq 2\hbar$, not only for elliptical orbits (as has already been pointed out by Page [arXiv:2409.00305]) but also for hyperbolic orbits. We extend the analysis to scalar and vector radiation and find that the angular momentum carried by each emitted quantum of radiation is approximately $\hbar$, i.e., $\dot{J}/\dot{N}\simeq \hbar$ for both types of radiation. We demonstrate that the radiated energy and angular momentum of binary systems can be determined from the evolution of the orbital angular frequency and the eccentricity of the quasi-Kepler orbit. However, a gauge-invariant decomposition of the angular momentum carried by the emitted particles into the spin and orbital components is not possible.

We study the BRST quantization of the null string after completing its local gauge symmetry by Carroll-Weyl transformations. The resulting worldsheet theory possesses three first-class constraints, $C_1 = P^2$, $C_2 = P \cdot X'$, and $C_3 = P \cdot X$, whose modes realize a Weyl-BMS algebra. The additional Carroll-Weyl constraint qualitatively changes the quantum gauge complex: its scalar $s$-ghost is intrinsically coupled to the BMS $bc$-ghost sector, and the anomaly analysis involves three independent cocycles rather than a single Virasoro-type central charge. Starting from the gauge-fixed action, we derive the complete Faddeev-Popov complex, construct the matter and ghost currents and the BRST charge, and evaluate their equal-time operator products in the flipped, equivalently highest-weight, representation. The matter and ghost anomaly coefficients are $(c_{LL},c_{LS},c_{SS})_{\mathrm{matter}}=(2D,-D,-D)$ and $(c_{LL},c_{LS},c_{SS})_{\mathrm{ghost}}=(-54,6,4)$. Because the corresponding central terms multiply linearly independent ghost bilinears in $Q_B^2$, BRST nilpotency requires the three conditions $D=27$, $D=6$, and $D=4$, respectively. These conditions are mutually incompatible. Consequently, there is no target-space dimension in which the minimal flat Carroll-Weyl matter-plus-ghost complex is anomaly-free in the highest-weight representation. The familiar $D=26$ condition of the ILST null string is recovered only after truncation to the two-constraint BMS subsector, which defines a different quantum gauge complex.

We present a fermionic lattice Hamiltonian that exactly realizes the $\mathrm{U(1)}_\mathrm{L}\times \mathrm{U(1)}_\mathrm{R}$ global symmetry and the associated chiral anomalies of a massless Dirac fermion in 1+1 dimensions. The construction couples Villain bosons to a Kitaev chain of Majorana fermions, where the bosonic fields are essential for evading Nielsen-Ninomiya-type no-go theorems. For two copies of our model, we realize the anomaly-free $\mathrm{U(1)}_{3450}$ global symmetry and construct symmetric boundary conditions. We analytically demonstrate symmetric mass generation by mapping the symmetry-preserving six-fermion interactions to fermion bilinear terms using fermionic T-duality. Finally, by gauging general anomaly-free global symmetries, we obtain a broad class of lattice chiral gauge theories. As nontrivial applications, we compute the mass spectra of the lattice Schwinger model and the 3450 gauge theory, finding agreement with the corresponding continuum results.

Thermal fluctuations are generally expected to destroy order and restore symmetries at sufficiently high temperatures. Recently, however, a family of scalar theories in $2+1$ dimensions was shown to exhibit $\mathbb Z_2$ symmetry breaking that persists to arbitrarily high temperatures using a variety of approaches, including the $\epsilon$-expansion, the FRG, and large-$N$ techniques. Although the similarities among these theories suggest that they belong to the same universality class, this connection has not been established explicitly. In this work, we fill this gap. Using large-$N$ methods, we analytically study the biconical vector model in a range of spacetime dimensions, including $2+1$, at leading and next-to-leading order in the $1/N$ expansion. We determine the RG flow, fixed-point structure, and the CFT data of the infrared theory, reproducing and extending previous results and thereby unifying the apparently distinct constructions within a common analytic framework. We derive the effective potential, establish its stable minima at zero and finite temperature, and demonstrate spontaneous $\mathbb{Z}_2$ symmetry breaking at arbitrarily high temperatures for large finite $N$.

We develop a partial-wave formalism providing the missing spin-resolved framework for multipositivity among massless planar amplitudes. We present a systematic procedure for constructing kinematic charts realizing the complex-forward limit. Applying it to obtain the four-, five-, and six-point charts, we show how to derive the one-to-two and one-to-three wavefunctions and construct the associated partial waves. The resulting formalism places higher-point multipositivity on the same footing as conventional partial-wave analyses of two-to-two scattering.

We study the behaviour of the topological defect lines of the $k^{\rm th}$ ${\cal N}=2$ minimal models that are preserved by the least relevant perturbation to first order. It is usually believed that these defects should then also define symmetries of the IR theory, which for the usual "massless” flow should be the $(k-2)^{\rm nd}$ ${\cal N}=2$ minimal model. Using CFT arguments we show that this is not possible. We also reproduce this result using matrix factorisation techniques: while the corresponding B-type defects can be adjusted to first order in the deformation, there is an obstruction at second order, which is associated with a supersymmetry anomaly. By contrast, for the associated massive integrable flow, which corresponds to a Chebyshev deformation of the superpotential, all of these defects can be consistently deformed, and they indeed define symmetries of the massive IR theory.

We investigate Krylov spread complexity along holographic renormalisation-group flows using the proposal that its growth rate is captured by the proper radial momentum of an in-falling massive probe. We focus on the second time derivative of the complexity ${U}$, which is determined locally by the redshift and radial metric functions of the dual geometry. For Lorentz-invariant domain-wall flows preserving the spacetime dimension, we derive a new relation between ${U}$ and the covariant central charge $c_{\text{cov}}$. The decrease of the covariant central function towards the infrared is accompanied by a monotonic increase of the complexity acceleration. For the top-down Dp-brane family, we obtain a universal relation. We then examine flows across dimensions, including a twisted compactification from four to two dimensions and from six to four dimensions. In these examples $c_{\rm cov}$ and ${U }$ are co-monotonic, in sharp contrast with the inverse correlation characteristic of fixed-dimensional flows. We argue that this reversal reflects the reorganisation, rather than simple depletion, of degrees of freedom into lower-dimensional sectors under compactification. Our results identify complexity acceleration as a sensitive geometric diagnostic connecting information spreading, holographic central functions and RG evolution.

Analytic calculations of zero- and single-scale quantities in perturbative quantum field theory result into special numbers and functions, the first of which have been revealed during the last decades. These are generalizations of the polylogarithm in form of Kummer-Poincaré iterative integrals over special alphabets and extensions thereof.With growing order in the coupling constant, the polylogarithms, Nielsen integrals, the iterated integrals over linear denominator terms, cyclotomic letters, letters induced by quadratic forms, square-root valued letters, and more general functions contribute. For the nested sums we consider nested harmonic sums, generalized harmonic sums, nested sums implied by quadratic forms, cyclotomic harmonic sums, and nested sums containing central binomials. We construct the $q$-extensions of these special functions and of the nested sums, which are associated to them by the series expansion at $x=0$, and their Mellin transform in the $q$-free case. These functions are expected to play a role in perturbative calculations in the case of $q$-deformed commutation relations. For the simpler function spaces closed form solutions are presented. For more involved alphabets we present the algorithmic steps leading to the $q$-extension for the individual cases. We also derive the determining differential and difference equations of these higher transcendental functions. The $q$-extended special functions are quite different form the corresponding $\mu$-extended functions.

At large $N$, two coupled Sachdev-Ye-Kitaev models realize an eternal traversable wormhole with a discrete spectrum. We show that at finite $N$ the spectrum organizes into clusters labeled by operator size. Low-size clusters evolve into the conformal towers and define a weak-ergodicity-breaking subspace responsible for the long-lived wormhole revival dynamics. Moreover, the competition between size energy and size entropy provides a microscopic interpretation of the wormhole-black hole transition. These results reveal operator size as the bridge between the finite-$N$ spectrum and the emergent gravitational physics at large $N$.

We show that the category of twisted/untwisted representations of the Heisenberg conformal net $\text{Heis}$ is a continuous Tambara-Yamagami category for the group $\mathbb{R}$. We compute the associators and the $\mathbb{Z}/2$-crossed braiding. By taking a $\mathbb{Z}/2$-equivariantization, we obtain an explicit computation of the braided continuous tensor category of representations of the fixed-points conformal net $\text{Heis}^{\mathbb{Z}/2}$. This provides the first explicit computation of a category of representations of a conformal net containing irreducible representations whose tensor product is a direct integral of irreducible representations.

Imaginary Liouville theory was recently proposed as a path integral construction of a non-rational conformal field theory (CFT) with central charge less than one (Usciati et al. 2026). In the present work, we establish Ward identities and Belavin-Polyakov-Zamolodchikov differential equations in this framework, assuming a precise conjecture on the decay of the Laplace transform of Imaginary Gaussian multiplicative chaos. These results provide a first step towards a rigorous implementation of the conformal bootstrap program for conformal field theories with central charge lower than one. This work is the first to investigate the intrinsic properties of Imaginary Liouville theory beyond exact computations.

We present a class of cosmological scenarios in which a preferred spatial direction in the matter sector coexists with an exactly homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker (FLRW) geometry. The Cosmological Principle is realized on shell, with the vector-field equations enforcing a vanishing momentum density and suitable interactions eliminating the anisotropic stress. Consequently, the construction yields an entire FLRW branch without tuning the matter-field initial conditions. The preferred direction reemerges in perturbations, producing direction-dependent propagation and mixing among scalar, vector, and tensor modes already at linear order. In particular, the mixing opens a linear channel through which perturbations in the scalar sector can in principle source gravitational waves. Our construction thus reveals a new route by which preferred-direction physics can leave observable cosmological signatures while remaining hidden in the background geometry.

We study the relationship between quantum error correction, confinement, and lattice Yang-Mills theory. We first formulate decoding for finite Abelian homological codes in terms of higher form gauge fields. For positive local noise, the logical classes are topological sectors of a Nishimori ensemble, and the optimal decoding error is determined by the relative weights of the nontrivial sectors. We derive Fourier relations between logical probabilities, disorder operators, and information in the channel environment, and we give contour and fractional moment criteria for a threshold. We then study a four dimensional $\Z_N$ memory and its possible relation to confining $\PSU(N)$ vacua. Finally, we define a finite curvature center sheet model coupled to Wilson $\SU(N)$ link variables. In this model the conditional logical probabilities are center twisted Yang-Mills partition functions. A strong coupling expansion gives the leading effective interaction for the syndrome and shows that local syndrome correlations can decay even when the global sheet sectors are mixed. We also show that the likelihood for a separated pair of syndrome worldlines is the center monopole correlator. Its decay determines a transfer matrix mass. This distinguishes the suppression of global flux sectors from the local spectral information needed to discuss a mass gap.

We ask when the sphere free energy of a quantum field theory, after subtraction of its local counterterms in a fixed renormalization prescription, decreases monotonically along the renormalization-group (RG) flow. The counterterms fix the required zeros of the subtraction polynomial, but positivity of the spectral data alone does not determine monotonicity. For Gaussian determinants, complete monotonicity of a rescaled integrated heat trace is sufficient. This condition proves monotonicity for Robin boundary flows on every sphere $S^p$, $p\ge2$. A related identity proves monotonicity for massive Dirac fields on odd spheres $S^d$, $d\ge3$. Conversely, under the stated endpoint and decay assumptions, an approach to the infrared value from below excludes every endpoint-preserving finite polynomial subtraction. For the double-trace families considered here, driven by $\mathcal O^2$ on spherical surface defects and, at leading order in large $N$, on $S^3$, a monotone subtraction exists exactly when the dimension of $\mathcal O$ is at least $1/2$ and $(3-\sqrt{37/13})/2$, respectively. Below these thresholds no subtraction is monotone, although the spectra remain positive. The same determinants describe double-trace boundary flows of a free massive scalar in AdS$_3$ and AdS$_4$, where the classifications carry over. Finally, with contact terms fixed, the leading infrared sphere coefficient of a flow from a three-dimensional conformal field theory to a gapped phase is proportional to a difference of nonnegative stress-tensor spectral moments. In the massive critical $O(N)$ model this difference vanishes at leading order and is negative at order $N^0$.

We study subleading asymptotic charges of a U(1) gauge field coupled to a charged massless scalar field in Minkowski spacetime. We consider a phase space for which the scalar-field initial data on null infinity have compact support, while the gauge-field initial data admit an asymptotic expansion with inverse-power tails in retarded (advanced) time, which we refer to as initial-data tails. We identify three finite charges at the null infinities which are conserved under scattering. One is a renormalized version of the standard subleading soft charge. The second is a logarithmic charge which is the difference of the coefficients of the leading order tails at the future and past ends of null infinity. The third is a new charge that probes the average of the tail coefficients. We conjecture that the third charge is associated with a new class of large gauge transformations which diverge linearly in retarded time.

We investigate the phase diagram of the (2+1)-dimensional Gross-Neveu-Yukawa (GNY) model at finite temperature, density, and magnetic field beyond mean-field, using the Functional Renormalization Group (FRG) in the local potential approximation. Large magnetic fields result in magnetic catalysis, a dimensional reduction of the system, and enhancement of chiral symmetry breaking. We employ a hydrodynamical algorithm to solve the FRG flow equation for the effective potential, which allows for go further into the infrared region than with previously used methods. We find that the chiral condensate exhibits non-trivial behavior in various regions of the phase diagram: several first-order phase transitions and de Haas – van Alphen oscillations at small magnetic field and large chemical potential, as well as a critical endpoint which shifts to higher temperature with increasing magnetic field.

We formulate the loop equations of pure $4d, N=1$ super Yang–Mills (SYM) theory in a finite geometric form. The usual equal-point loop derivatives are singular because the order of gauge-field insertions is lost when contour points coincide. Our path-ordered operator calculus (POOC) keeps the insertions in distinct ordered slots, forms the graded commutators and Jacobi combinations, and only then takes the coincidence limit. This removes the spurious kinematical singularities without introducing a cutoff; the genuine ultraviolet contact remains a separate physical distribution. We apply POOC to the exact Itoyama–Takashino superloop hierarchy and construct a Lorentzian supersymmetric Hodge-dual (SHD) surface functional. Its Hodge-resolved area derivative is annihilated by the local chiral and anti-chiral loop operators. Exact additivity under surface-geodesic sewing shows that the exponential of this zero mode multiplies any solution of the complete finite-$N$ hierarchy without changing the equations. For planar contours the SHD functional is the geometric area. In particular, for a long rectangular Wilson loop, $W[C_{T,L}]\sim\exp[-i\sigma_kLT]$ and $E_k(L)=\sigma_kL>0$. Thus the construction gives an exact nonperturbative confining area-law factor in $ N=1$ SYM. After Euclidean continuation, the same SHD factor is an exact fixed point of the deterministic zero-noise SYM gradient flow. Its interpretation as a dynamically selected equilibrium still requires stability in the long-flow-time, large-volume, and zero-noise limits, but this does not affect the exact zero-mode and finite-$N$ dressing theorems. The complete planar Wilson-loop solution, including the undressed fluctuation factor and excitation spectrum, will be developed in the next papers of this series.

Coarse-graining should retain the response to the perturbations and observations of interest. We study this requirement for systems defined by generating operations and observation maps. Equality of all allowed future outputs defines an observation-sufficient quotient; fibre consistency gives differentiable descent for bounded surjective linear maps between Banach spaces. Linearising the state equation and the observation together produces a response whose kernel, source, readout and direct term transform consistently under coarse-graining. We derive an exact comparison identity valid also for noninvertible coarse maps. Eliminating residual states yields a memory representation and, under decay bounds, controlled finite-memory approximations. For isolated critical modes, analytic Fredholm theory gives the pole and visibility conditions; reciprocal realisations with a positive crossing matrix have positive finite-rank residues. An arithmetic merge and a reflecting heat protocol illustrate these results by explicit calculations. Gaussian averaging and quantum-field-theory channels provide further realisations, including the identification of response poles with vacuum spectral atoms under matched spectral hypotheses.

The quality of the QCD axion can be protected by a gauge symmetry. This work considers a gauged chiral $U(1)_{B-L}$. The vacuum expectation values of two complex scalars break $U(1)_{B-L}$. One linear combination of their two phase degrees of freedom is eaten by the $U(1)_{B-L}$ gauge boson, while the other becomes the QCD axion. Since the PQ symmetry is accidental, there is no ambiguity in the PQ charge assignment. High axion quality implies a large anomalous coupling between the axion and the $U(1)_{B-L}$ gauge boson (or gauged Majoron). A `quality floor', namely a minimal coupling determined by the axion quality, arises for the coupling between the axion and SM fermions, extending the parameter space of QCD axions toward improved testability.

We present a novel approach to the superintegrability of collective Hamiltonians invariant under a Hamiltonian action of a connected semisimple compact Lie group, $G$, on a symplectic manifold, $M$. By exploiting a Hamiltonian torus action that goes back to Guillemin and Sternberg [GS,1983], we demonstrate that the functional dimensions of $\mathfrak{H} := \mathcal{J}^*(C^\infty(\mathfrak{g}^*)^G)$, where $\mathcal{J}: M \to \mathfrak{g}^*$ is the momentum map of the $G$ action, and its centralizer $\mathfrak{F}$ in $C^\infty(M)$ satisfy the equality $\mathrm{ddim}(\mathfrak{H}) + \mathrm{ddim}(\mathfrak{F}) = \mathrm{dim}(M)$. Together with a non-triviality condition, this ensures that the Abelian Poisson algebra $\mathfrak{H}\subset C^\infty(M)$ represents a superintegrable system, and it also follows that the momentum map of the GS torus action yields action variables for the system. Our work provides a new insight into collective superintegrability complementing earlier results of Bolsinov and Jovanović.

We investigate the holographic dual of the extended bulk thermodynamics of dyonic dilaton AdS black holes by allowing both the cosmological constant and Newton's gravitational constant to vary. In the dual CFT thermodynamics, the central charge $C$ and chemical potential $\mu$ as its conjugate enter the thermodynamic relations, in addition to $(\tilde{T},\tilde{S})$, $(\tilde{Q},\tilde{\Phi})$, $(\tilde{P},\tilde{\Psi})$, $(\tilde{\mathcal{P}},\tilde{V})$. We consider sixteen different ensembles related to those five pairs of thermodynamic variables which we separate into fixed volume and pressure ensembles. Due to the symmetry on ($\tilde{Q},\tilde{P}$), we only need to analyze five ensembles each in fixed volume and pressure ensembles. In the fixed volume ensembles, it is found that the fixed ($\tilde{Q},\tilde{P},\tilde{V},\mu$) and ($\tilde{\Phi},\tilde{P},\tilde{V},\mu$) exhibit a zeroth-order phase transition. Interestingly, the fixed ($\tilde{\Phi},\tilde{\Psi},\tilde{V},C$) ensemble shows the zeroth- and first-order phase transitions at above and below critical potential $\tilde{\Upsilon}_c$, respectively. In the fixed pressure ensembles, it is found that richer structures appear where the sign of $\mu$ heavily influences phase space for several ensembles. For fixed ($\tilde{Q},\tilde{P},\tilde{\mathcal{P}},\mu$) and ($\tilde{\Phi},\tilde{P},\tilde{\mathcal{P}},\mu$), they show the zeroth-, first-, and second-order phase transitions. For the fixed ($\tilde{\Phi},\tilde{\Psi},\tilde{\mathcal{P}},\mu$) ensemble, there is only a zeroth-order phase transition occurs. The other ensembles which we do not mention eventually do not show any phase transition. These findings provide deeper insight into how bulk gravitational variations, particularly regarding the cosmological and gravitational constants, translate to exact critical phenomena and richer phase structures within CFT.

We propose a quantum dominant energy condition (QDEC) for the stress tensor in the context of quantum field theory in curved spacetimes. A rigorous proof is given for the case of Rindler wedges, and a heuristic discussion about possible generalizations to more general geometric setups, including curved spacetime, is provided. In Minkowski spacetime, we establish a connection with state recovery bounds. We illustrate the QDEC for coherent states of the free scalar field, where it turns out to be related to the ordinary DEC for the stress tensor of the classical solution that defines the coherent state.

We investigate how the gluon condensate modifies the thermal spectral functions and melting patterns of heavy vector mesons, specifically charmonium and bottomonium, using an improved soft-wall AdS/QCD model. The framework is extended to finite temperature via a dilaton black hole geometry that consistently incorporates the backreaction from the gluon condensate. We numerically track the evolution of spectral resonance peaks as functions of both temperature and gluon condensate strength. Our calculations reveal that increasing temperature systematically broadens and suppresses spectral peaks, signaling in-medium dissociation. In contrast, a stronger gluon condensate considerably mitigates peak broadening and enhances the spectral weight of both ground and excited states. This behavior indicates that the gluon condensate hinders thermal dissociation, thereby stabilizing heavy quarkonia within the quark-gluon plasma. These findings are consistent with existing studies and provide new holographic evidence for the stabilising role of the gluon condensate from the perspective of thermal spectral functions.

Cosmological collider signals are usually discussed for isolated massive particles, whose exchange produces characteristic logarithmic oscillations in primordial correlators. In this work, we study how this signal is modified when the exchanged states form a continuous mass spectrum. We first develop a spectral representation for inflationary correlators mediated by a continuum field. In the soft limit, the non-analytic part of the seed function is expressed as a Fourier–Laplace transform of the spectral weight with respect to logarithmic momentum variables. This representation shows that a continuum superposes clock signals with different frequencies and can dephase the sharp oscillation associated with a single massive particle. We then realize this mechanism in an RS2-like inflationary braneworld, where the inflaton is localized on a de Sitter brane in a five-dimensional AdS bulk. The tensor sector contains a localized massless graviton and a continuum of Kaluza-Klein gravitons starting at \(m=3H/2\). We derive the KK wavefunctions and identify the brane spectral weight fixed by the continuum wavefunction on the brane. Applying this weight to the inflaton four-point function, we find a smooth seed function rather than a persistent logarithmic clock oscillation. This behavior follows from the fact that the continuum starts at zero clock frequency, while the KK spectral weight vanishes near threshold. Our results provide a concrete higher-dimensional example in which deviations from the standard cosmological collider signal encode the structure of a continuum high-energy spectrum.