Papers

Strong subadditivity (SSA) on a common light cone gives the Casini-Huerta entropic proof of the three-dimensional $F$-theorem. We ask whether holographic entropy inequalities beyond SSA similarly constrain renormalization group flows for which every intermediate theory admits a semiclassical holographic description. For small, disjoint deformations of a common light-cone region, we show that the second-order response of a broad class of balanced holographic inequalities depends only on pairwise correlations already controlled by SSA. A six-party example shows that the full finite inequality nevertheless contains genuinely multipartite information, so its disappearance is a limitation of the second-order expansion rather than of the inequality itself. Two natural finite constructions do not recover the missing information. We nevertheless find two ways in which information beyond SSA survives. A five-party inequality bounds the rate at which a conditional correlation grows as one region is enlarged. Separately, a continuum limit of the odd-cyclic inequalities gives a constraint on the angular shape dependence of entanglement entropy, and Lorentz symmetry relates this constraint to radial evolution. Thus holographic entropy inequalities beyond SSA do constrain entanglement along RG flows, although we do not obtain a second universal analogue of the $F$-function.

Recently, some research has emerged that used the Tsallis statistics in Quantum Chromodynamics (QCD). In Tsallis statistics, the Tsallis parameter $\mathfrak{q}$ reflects long-range interactions, non-Markovian memory, and multifractal boundary effects, linking directly to temperature variance in fluctuating systems, which vanish in the Boltzmann-Gibbs limit. This is our motivation to investigate the thermodynamical variables of a hot and dense system. Within a non-extensive color QCD MIT-Bag Model and using the Tsallis distributions to describe the thermal response functions related to the equation of state. The study focuses on the deconfinement phase transition from a hadronic gas to a quark-gluon plasma in order to analyze the behavior of thermodynamical quantities of the system, such as the energy density, the pressure and the interaction measure at finite volume under Tsallis statistics. Furthermore, the effect of the $\mathfrak{q}$-parameter on the phase transition point is investigated. We find that the transition point is significantly affected by the $\mathfrak{q}$-parameter in the framework of non-extensive statistics.

$N$-body simulations by M.~Milosavljevic and D.~Merritt \cite{Milosavljevic:2001vi} suggest that for supermassive black hole binaries in their expected nearly circular orbits, before gravitational wave emission dominates, the inspiral rate causes the dimensionless orbital relative velocity squared, $(v/c)^2$, to increase nearly linearly with time at a rate that is approximately the inverse of 210 Gyr (about 15 times the present age of the universe), nearly independently of the mass of the binary. This by itself would not lead to coalescence by the present time, but when it is augmented by gravitational radiation, it can lead to coalescence for equal-mass black hole binaries up to about $1.4\times 10^{14} M_\odot$, which might then be conjectured to be a very crude upper limit to the mass of black holes in our universe at the present time. This also suggests that the gravitational wave strain from coalescing supermassive black hole binaries should decrease for wave periods greater than about 100\,000 years.

The $n$-point single-minus graviton amplitudes are nonzero in a special kinematic region that preserves some supersymmetry when embedded in supergravity. We compute these amplitudes within the BFSS matrix theory by relating them to an index problem considered by A. Sen. More precisely, the single-minus amplitudes are obtained from an index that counts BPS states in the Coulomb branch of four-dimensional ${\cal N}=4$ super Yang-Mills. We explicitly check this relation in a specific kinematic region by computing the index using BPS wall-crossing formulae and matching it to the gravity result. We further employ BPS wall-crossing to show that the index satisfies a tower of $w_{1+\infty}$ soft theorems in any kinematic regime, again matching gravity. Finally, we show that for certain kinematics, the single-minus amplitudes can be realized by scattering a graviton off a plane wave.

We study the dynamics of pure-gauge SU(N) through the time evolution of a spatial Wilson loop. We compute its out-of-time-order correlator at weak 't Hooft coupling and large N, and propose a framework for the strong-coupling description. We find that operators in pure-gauge Yang-Mills grow only when the group is non-abelian and d>2, and conjecture that pure gauge theories are chaotic only when non-abelian at d>2. Even at weak coupling the correlator's exponent inherits the IR sensitivity of the gluon self-energy imaginary part, which rests on the non-perturbative magnetic scale. We then explore the possibility that the loop's own size supplies that scale, since the loop operator is blind to any momenta softer than 1/2R. With the cutoff imposed that way the exponent changes sign at a critical diameter: smaller loops scramble, larger ones do not, and the scrambling time diverges at the crossing. The transition follows from that cutoff. Further investigation, in particular a first-principles treatment of the thermal decay width, is required to establish whether it is physical.

Tree-level partial-wave unitarity of the open bosonic Veneziano amplitude follows indirectly from the no-ghost theorem, but a direct proof from the beta function has been missing at the critical dimension. We prove nonnegativity at every mass level and spin throughout the sharp range 3 < D <= 26, using a Bessel recurrence that reduces the infinite sign problem to a manifestly positive bulk and a finite exact boundary.

We investigate the impact of scalar-tensor theories on the 21cm brightness temperature. We consider a specific model in Horndeski scalar-tensor theory in which dark matter and dark energy in the framework of shift-symmetric generalized Galileon theories was described and an effective unified cosmic fluid was obtained. We calculate the Hubble parameter of the model and we plot the curve of the evolution of the 21-cm temperature in a specific range of the redshift parameter $z$ and compare it to the curve corresponding to $\Lambda$CDM paradigm.

Three dimensional compact U(1) gauge theory supports a finite density of monopoles at a fixed lattice gauge coupling. After showing evidence for the expected dependence on the lattice coupling of the density of monopoles, we study the low lying spectrum of a two flavors of two-component Wilson fermions as a function of the bare Wilson mass. The Wilson-Dirac operator that realizes the two flavors are hermitian conjugate of each other to restore parity invariance and the commutativity of the two operators expected in the typical continuum limit is not the case in the presence of monopoles for all values of the lattice gauge coupling. This results in an anomalous behavior where we see clear evidence for the presence of very small eigenvalues for a wide range of a particular sign of the bare Wilson mass. These eigenmodes are typically in one-to-one correspondence with the number of monopoles as defined by the gauge field and the spectrum associated with these modes are separated by a gap from the bulk in the asymptotic regime of the lattice gauge coupling. The eigenvalues come in pairs around zero and the eigenvalues of one sign couple to the monopoles while the modes of the opposite sign couple to the anti-monopole implying flavor symmetry breaking.

Conformal invariance is expected to be an emergent property of many systems in their critical regime. However, approximation schemes generically spoil this property. This is in particular the case of the derivative expansion, a widely used approximation scheme in the framework of the functional renormalization group. In this article, we consider Ward identities associated with conformal invariance in the 3-d Ising universality class with truncations at order 4 (next-to-next-to-leading order) in the derivative expansion, with $Z_2$ invariant composite operators. Our results confirm that the regulating functions which yield a small breaking of conformal invariance also present a small sensitivity of the universal critical exponents with the choice of this regulating function. We also show that, in the vicinity of regulator-parameter values for which the conformal constraints are best satisfied, the breaking of conformal invariance reduces as the order of the derivative expansion is increased, providing a new indication of the convergence of this approximation scheme.

We study tropical open strings arising from the analytically continued action of tropological sigma models in the presence of a constant Kalb-Ramond background. We show that the background field separates the theory into a generic sector and a critical sector. In the generic sector, the canonical structure produces a kinematical nonlocal bracket between the transverse and leafwise target-space coordinates. At the critical value of the Kalb-Ramond field, the Legendre map degenerates and the theory develops a primary constraint, requiring quantization by the Dirac-Bergmann procedure. This critical sector has a finite reduced phase-space structure and differs from the naive limit of the generic theory.

We prove three results about refined $SU(N)$ VW invariants on Hirzebruch surfaces. First, we crucially simplify the modular anomaly equation satisfied by the generating series of these invariants by expressing it in terms of complementary generalized error functions, which requires extending these error functions to their singular loci. Second, we evaluate the variation of the modular completions of the generating series upon change of polarization. Finally, a representation of the generating functions in terms of indefinite theta series allows us to compute VW invariants at the walls of marginal stability, and to express them in terms of the invariants on the two sides of the wall and those of lower ranks. All the results exhibit a universal structure suggesting their wider applicability.

Recent advances in cosmological measurements such as DESI and ACT may indicate an increase in the preferred value of the spectral tilt $n_s$ such that it disfavors rather popular models such as the Higgs or Starobinski inflation models. We argue that it actually means that the data is now sensitive enough to quantum corrections beyond simple tree-level models. Resurrecting the old theme of Coleman-Weinberg effective potential, we analyze the predictions of such models, as well as likelihood analysis, leading to a more established and interesting predictive framework.

Equivariant localization in the $\Omega$-background is a powerful technique for the evaluation of the partition function of $\mathcal{N}=2$ supersymmetric field theory on a toric four-manifold. For the complex projective plane $\mathbb{CP}^2$, we apply equivariant localization to supersymmetric Yang-Mills theories with gauge groups $SU(2)$ and $SO(3)$ with $N_f$ massive hypermultiplets in the fundamental representation, and the $\mathcal{N}=2^*$ theory with the hypermultiplet in the adjoint representation. We evaluate equivariant correlation functions, which provide an equivariant extension of intersection numbers of moduli spaces of instantons on $\mathbb{CP}^2$, such as Euler numbers and Segre invariants. In the non-equivariant limit, we compare our results with the evaluation using low energy field theory and integration over the $u$-plane. Using this comparison, we identify a specific overall factor, which captures the difference between a partition function for gauge group $U(2)$ and $SU(2)$ or $SO(3)$ on a compact four-manifold. We also discuss our results in the context of $S$-duality of the $N_f=4$ and $\mathcal{N}=2^*$ theory, which includes the triality automorphism group of the flavor group for $N_f=4$.

Anti-ultralocality refers to the growth of spatial gradient terms relative to velocity terms in the coupled Einstein–scalar field equations. It is a characteristic feature of decelerated expansion before the onset of inflation. Previous numerical relativity studies have shown that anti-ultralocality prevents the onset of inflation in models with power-law inflaton potentials. In this paper, we show that models with plateau-shaped inflaton potentials, which are considered to be the simplest way to generate a tensor-to-scalar ratio below current observational upper limits, are especially vulnerable to anti-ultralocality effects. The reasons are the flatness of the plateau and the energy density gap of $\sim 10$ orders of magnitude between the Planck density and the plateau potential energy. To study the problem, we develop a protocol for assessing the viability of inflationary models in general, and we apply it to a plateau potential using a previously validated numerical relativity code. We find that, starting from generic initial conditions, the growth of gradient terms in the Einstein equations relative to non-gradient terms either prevents inflation from lasting for enough $e$-folds or triggers a phase of quantum runaway. We show that the fine-tuning of initial conditions necessary to avoid these issues becomes more severe as the energy scale of inflation is made smaller, disfavoring common approaches for reducing the tensor-to-scalar ratio.

The supermoduli stack ${\mathfrak M}_g$, parametrizing smooth unpunctured super Riemann surfaces of genus $g$, is known to be non-projected for $g \ge 5$. We extend this result to the even-spin component of ${\mathfrak M}_4$. This is done by exhibiting an appropriate compact curve in the spin moduli space ${\mathcal S\mathcal M}_4^{+}$, and computing the restriction to it of the class obstructing the projectedness of ${\mathfrak M}_4^{+}$. This restriction turns out to be non-zero, proving that ${\mathfrak M}_4^{+}$ is non-projected.

Gravitational radiation from the early Universe offers a powerful window into physics beyond the Standard Model (SM). In this paper, we investigate the low-frequency tail of gravitational radiation produced by the decay of massive particles. The energy spectrum in Minkowski space-time, $dE/d\omega$, exhibits two distinct regimes: a flat bremsstrahlung/memory spectrum above a characteristic cutoff frequency and a suppressed quadratic frequency scaling below it. The typical rate of momentum change of the decay products determines the cutoff frequency. We calculate $dE/d\omega$ in two independent but equivalent ways: using Weinberg's soft graviton theorem and by solving the equations of motion. This further establishes the connection between Weinberg's soft graviton theorem and the gravitational memory formula, which we generalize to incorporate the effects of final-state scattering. We then compute the cosmological stochastic gravitational energy spectrum $h^2\Omega_{\rm h}$ generated by many particle decays in the early Universe. We demonstrate that $h^2\Omega_{\rm h}$ exhibits linear scaling with frequency above the cutoff frequency, transitioning to cubic scaling below it. Our findings reveal a suppression of the low-frequency signal compared to a naive linear extrapolation, with implications for detection prospects. Our results also reveal the connection between bremsstrahlung and memory in the context of cosmological stochastic gravitational energy spectra from the early Universe.

We establish a uniqueness theorem in four-dimensional spacetime, demonstrating that a nonlinear spinor system naturally constrains the background geometry to the Kastor-Traschen spacetime, which describes dynamical, cosmic multi-black-holes. By analyzing a nonlinear field equation where the super-covariant derivative of a Dirac spinor is sourced by intrinsic fermion currents, we evaluate its integrability under two physical requirements: a phase-locking condition enforcing a real hermitian product $\overline{\psi}_R \psi_L$, and a pure-electric source condition. We show that the field equation uniquely compels the spinor to be an eigenvector of the super-covariant Dirac operator, with the eigenvalue dynamically identified as the Hubble constant of the de Sitter background. These results provide a novel geometric mechanism wherein both the cosmological constant and dynamical black hole configurations spontaneously emerge from fermion condensates.

We study CFT$_3$ realizations and celestial representations of the $\mathcal{L}_{\Lambda}w_{1+\infty}$ algebra. Free scalar and Dirac theories yield the same one-particle light-ray action. This action extends to the critical $O(N)$ model at $N=\infty$. At large but finite $N$, we use the light-ray algebra under its closure assumptions and check prescribed iterated limits of the frequency kernel. In the dimension-completed celestial hard-graviton representation, a closed raising subalgebra annihilates $G_3^+$ exactly. We construct two nonzero singular descendants in its induced module and prove that both vanish in the celestial realization. They implement the finite-spin condition and massless spin-two shortening of the generated highest-weight sector. The nonzero-$\Lambda$ quadratic Casimir has eigenvalue three throughout the hard family. The null descendants and low-spin Ward identities yield ordered differential–difference equations for celestial correlators.

Scalar-induced gravitational waves (SIGWs) probe primordial curvature perturbations beyond cosmic microwave background scales, while primordial black holes (PBHs) probe the rare tail of the same statistics. We ask whether SIGW morphology can reveal scale-dependent primordial non-Gaussianity. We adopt the internal-leg separable kernel, $\mathcal F_{\rm NL}(\mathbf{k};\mathbf q,\mathbf{k}-\mathbf q)=\bar f_{\rm NL}f(q)f(\lVert\mathbf{k}-\mathbf q\rVert)$, and compute Gaussian, reducible, and connected contributions through $\mathcal O(\bar f_{\rm NL}^{4})$ in the second-order tensor-source approximation. We compare persistent power-law running with a UV-matched tanh weight that saturates at high scalar momentum. Across narrow, finite-width, asymmetric, and multi-slope scalar spectra, running produces more than amplitude renormalization. Power-law running shifts the scalar support sampled by non-Gaussian convolutions, generating peak displacement, asymmetric shoulders, and persistent ultraviolet deformations. The tanh template instead produces a transition-localized modification and approaches a momentum-independent ultraviolet plateau. Analytical estimates of the perturbative hierarchy, weighted slopes, log-normal saddle displacement, peak shifts, and infrared scaling explain the trends and the enhanced sensitivity of the $\mathcal O(\bar f_{\rm NL}^{4})$ sector. For localized sources with finite weighted moments, smooth running preserves the leading $k^3$ infrared behavior up to logarithmic corrections. Since the PBH mass scales as $M\propto k^{-2}$, preferential high-wavenumber weighting conditionally favors lower PBH masses in a narrow-support interpretation. Peak position, spectral curvature, shoulders, and ultraviolet slopes are diagnostics for PTAs, LISA, and third-generation observations, providing finite-order templates for primordial interactions on otherwise inaccessible scales.

Recent advancements in direct detection (DD) experiments stimulate the investigation of the interactions of dark matter (DM) with nucleons and nuclei. In the framework of Composite Higgs Models (CHMs) the lightest Dirac composite particle (LDCP) can be stable composing a significant fraction \xi of the observed DM relic abundance. We consider the elastic scattering of the LDCP on nucleons as well as on xenon (Xe) and argon (Ar) nuclei within the CHMs in which the LDCP magnetic moment, its mass and its coupling to the Higgs doublet are suppressed by an approximate U(1) symmetry. The LDCP with non-zero magnetic dipole moment can result in a substantial enhancement of the differential event rate in the DD experiments at low recoil energies of nuclei. Assuming \xi\ge 0.1, we identify the region of the parameter space where such enhancement may be potentially observable. In addition we specify some observables that can be useful in discriminating between the DM fermions with non-zero magnetic moment and other types of dark matter particles which don't have similar electromagnetic properties.

Fidelity susceptibility is a useful probe of quantum phase transitions and quantum metrology, but its direct evaluation on a quantum device is challenging. By integrating Krylov-subspace methods with a resolvent-based reformulation, we present a method for certifying fidelity susceptibility using randomized single-copy measurements across repeated ground-state preparations. The resulting approximations form monotone lower bounds to the exact fidelity susceptibility and converge geometrically. We also present applications to metrology and linear static susceptibilities.

Using the string/field theory correspondence as a tool, we study the dependence of the Hagedorn temperature of strongly coupled, planar gauge theories on angular velocities and chemical potentials for $U(1)$ global currents. Our results are obtained from an interplay of world-sheet semiclassical quantization methods and target space low energy effective ones. The Hagedorn temperature is given as an expansion in the inverse (large) 't Hooft coupling limit. Working to quadratic order in the fluctuations, the world-sheet analysis provides the leading, the next-to-leading and part of the next-to-next-to leading order terms. In particular, it captures the NNLO $\log 2$ terms which are due to the contributions of non-zero modes to the zero-point energy. These modes are not accounted for by the effective approach, within which they have to be included by hand using suitable educated guesses. On the other hand, the effective methods allow to compute in a relatively easy way the missing NNLO pieces (and even further subleading corrections), which could be accounted for from the world-sheet approach only working at quartic (or higher) order in the fluctuations. We present general results and several examples in various dimensions, including both confining models and conformal field theories on spheres (such as $\mathcal N=4$ SYM and ABJM) dual to string theories on global $AdS$ spaces.

In this work we introduce exact cosmological solutions in General Relativity in (3+1) dimensions which take into account both the usual fluid component as well as a superfluid component describing superfluid Pionic vortices. Such superfluid vortices are source of inhomogeneities which can be treated exactly, no perturbation theory is required. These analytic solutions are relevant to obtain a non-perturbative description of inhomogeneities in cosmology. We discuss the peculiar effects of these hadronic inhomogeneities on the propagation of electromagnetic fields on such cosmological backgrounds analyzing possible observable effects. Finally, the properties that the effective superfluid satisfy for such solutions to exist are discussed, together with possible extensions of this family of solutions.

We extend the earlier analysis of cosmic censorship and the Weak Gravity Conjecture (WGC) in Einstein–Maxwell-dilaton (EMD) theory by performing a complete higher-order Sorce–Wald gedankenexperiment on the static charged Gibbons–Maeda–Garfinkle–Horowitz–Strominger (GMG) black hole. In the probe limit, an extremal GMG black hole can apparently be overcharged by a test particle, seemingly violating the weak cosmic censorship conjecture (WCCC), which is nothing but first-order variational identity in the Iyer–Wald formalism. This window can be closed by either incorporating backreaction via the Hoop Conjecture or further including second-order variational identities. In this paper, we explicitly compute third-order variational identities. The extremality parameter eta remains non-negative to the third perturbative orders confirming WCCC protection, regardless of the apparent failure of all-order generalization, model-independent proof of Lu, et al.. We further incorporate the Swampland Distance Conjecture (SDC): the resulting infinite tower of light charged states discharges any would-be overcharged configuration on a timescale tau much less than 2M, exponentially faster than a horizon can be destroyed, reinforcing WCCC in dynamical regimes and strongly relaxing the original WGC charge-to-mass bound.

We first present an introduction to the differential geometry of noncommutative algebras that carry a representation of a triangular Hopf algebra. Their noncommutativity is encoded in the universal R-matrix. The differential geometry is canonically constructed from these data. We then develop a new approach to the Bianchi identities for curvature and torsion of arbitrary connections (not necessarily bimodule connections). Using the Cartan calculus for connections, we prove the equivalence of their global formulations in terms of exterior forms and tensor fields. In particular, we obtain the noncommutative analogues of the first and second Bianchi identities familiar from general relativity.

Quantum simulators based on Bose-Einstein condensates (BECs) provide a powerful platform to study relativistic quantum-field phenomena in the lab and in particular emulate relativistic analog particle detectors. However, such detectors typically couple to excitations beyond the acoustic phonon regime, so that the nonlinear Bogoliubov dispersion introduces corrections that cause the effective relativistic description to break down. We propose to overcome this limitation using an intrinsically momentum-selective Unruh-DeWitt detector model in which a tweezer-trapped atom couples to the condensate through a multimode optical cavity. Although the atom is effectively pointlike, the cavity-mediated interaction kernel causes it to sample density fluctuations over a controllable finite region of the BEC equivalent to a finite-size detector. The effective detector size is tunable via the cavity mode structure determining a controllable momentum cutoff that filters out nonphononic excitations. We analytically calculate the response of such a detector for uniform linear acceleration and uniform circular motion and show that this momentum selectivity can enhance the signal at experimentally accessible accelerations. Compatible with existing cavity-QED platforms our scheme thus provides a versatile setting for analog quantum-field measurements or relativistic quantum-information protocols.

In this work, we introduce a systematic study for studying the scalar propagator and tadpole self-energy by considering an arbitrary parameter $\sigma$ that allows for a path integral description in real-time formalism (RTF). The closed time path formalism (CTP) and Thermofield Dynamics (TFD) are two popular choices for the parameter $\sigma$ in the Feynman rules. We have constructed a scalar propagator and a tadpole self-energy in two different bases in the momentum space as well as the mixed space. The results show that the diagonal components of self-energy in both spaces for the 1/2 basis are the same in both approaches within RTF, whereas the other off-diagonal components of self-energy are different because they depend on the path parameter. On the other hand, the diagonal components of self-energy in both spaces for the new basis are not the same in both approaches within RTF, whereas the off-diagonal components of self-energy are vanishing. That means the new basis allows one to reduce the components for the quantities studied, like self-energy or other.

We examine the tidal response of Kerr–Anti-de Sitter black holes using the Teukolsky formalism in a vacuum background. We consider massless perturbations with spin weights $s = 0,\ \pm\frac{1}{2},\ \pm1,\ \pm\frac{3}{2},\ \pm2$, corresponding to scalar, fermionic, electromagnetic, and gravitational perturbations, respectively. We determine the Tidal Love number using the near-horizon approximation. For asymptotically flat black holes in general relativity, the conservative tidal response vanishes identically. In contrast, we find that the presence of a negative cosmological constant parameter in the Kerr–Anti-de Sitter black holes leads to a non-trivial tidal response, containing both conservative and dissipative contributions. We systematically study the static and dynamical tidal responses of Kerr–Anti-de Sitter black holes. In particular, we examine the behaviour of the conservative and dissipative response coefficients and investigate their dependence on the black-hole rotation parameter and the AdS curvature scale.

Using the effective Lagrangian for the low-energy/temperature of photon-photon interaction and the lowest-order photon self-energy is calculated in the Real Time Formalism (RTF) for an arbitrary path specified by the $\sigma$-parameter within the new basis. The causal Green's functions (without chemical potential) for the scalar field are evaluated to derive the usual thermal propagators in the mixed space. It is shown that the symmetric propagator does not depend on $\sigma-$ parameter. Furthermore, the photon self-energy is used to calculate some electromagnetic properties, such as dielectric tensor and velocity of light from photon self-energy in the mixed space that greatly simplifies calculations in RTF. Their time dependence is investigated, and a comparison between our results and those obtained by other models is discussed.

We reconstruct the primordial power spectrum (PPS) across the full range of cosmological scales accessible to Cosmic Microwave Background (CMB) observations. Using the Modified Richardson–Lucy algorithm, we perform free-form reconstructions through deconvolution of Planck PR3 and PR4, Atacama Cosmology Telescope (ACT) DR6, and South Pole Telescope 3G (SPT-3G) D1 data. Across different regularization schemes, we find no evidence for significant deviations from a power-law primordial spectrum. The reconstructed spectra show a strong correlation between Planck and ACT, even at the level of localized features over their overlapping range, $0.07 < k < 0.14\,\mathrm{Mpc}^{-1}$, demonstrating consistency between the two observations. Using a complementary parametric Bayesian reconstruction, we find that the previously discussed preference for a blueward tilt in the ACT data at small scales is preferred only at the $1\sigma$ level.