Normalizable Modes Research Articles

Coarse-graining black holes out of equilibrium with boundary observables on time slice

In black hole thermodynamics, defining coarse-grained entropy for dynamical black holes has long been a challenge, and various proposals, such as generalized entropy, have been explored. Guided by the AdS/CFT, we introduce a new definition of coarse-grained entropy for a dynamical black hole in Lorentzian Einstein gravity. On each time slice, this entropy is defined as the horizon area of an auxiliary Euclidean black hole that shares the same mass, (angular) momenta, and asymptotic normalizable matter modes with the original Lorentzian solution. The entropy is shown to satisfy a generalized first law and, through holography, the second law as well. Furthermore, by applying this thermodynamics to several Vaidya models in AdS and flat spacetime, we discover a connection between the second law and the null energy condition.

One-loop partition function of gravity with leaky boundary conditions

Leaky boundary conditions in asymptotically AdS spacetimes are relevant to discuss black hole evaporation and the evolution of the Page curve via the island formula. We explore the consequences of leaky boundary conditions on the one-loop partition function of gravity. We focus on JT gravity minimally coupled to a scalar field whose normalizable and non-normalizable modes are both turned on, allowing for leakiness through the AdS boundary. Classically, this yields a flux-balance law relating the scalar news to the time derivative of the mass. Semi-classically, we argue that the usual diffeomorphism-invariant measure is ill-defined, suggesting that the area-non-preserving diffeomorphisms are broken at one loop. We calculate the associated anomaly and its implication on the gravitational Gauss law. Finally, we generalize our arguments to higher dimensions and dS.

Holographic entanglement entropy probe on spontaneous symmetry breaking with vector order

We study holographic entanglement entropy in 5-dimensional charged black brane geometry obtained from Einstein-SU(2)Yang-Mills theory defined in asymptotically AdS space. This gravity system undergoes second order phase transition near its critical point, where a spatial component of the Yang-Mills fields appears, which is normalizable mode of the solution. This is known as phase transition between isotropic and anisotropic phases, where in anisotropic phase, SO(3)-isometry(spatial rotation) in bulk geometry is broken down to SO(2) by emergence of the spatial component of Yang-Mills fields, which corresponds to a vector order in dual field theory. We get analytic solutions of holographic entanglement entropies by utilizing the solution of bulk spacetime geometry given in arXiv:1109.4592, where we consider subsystems defined on AdS boundary of which shapes are wide and thin slabs and a cylinder. It turns out that the entanglement entropies near the critical point shows scaling behavior such that for both of the slabs and cylinder, {Delta}_{varepsilon }Ssim {left(1-frac{T}{T_c}right)}^{beta } and the critical exponent β = 1, where ∆εS ≡ Siso− Saniso, and Siso denotes the entanglement entropy in isotropic phase whereas Saniso denotes that in anisotropic phase. We suggest a quantity O12≡ S1− S2 as a new order parameter near the critical point, where S1 is entanglement entropy when the slab is perpendicular to the direction of the vector order whereas S2 is that when the slab is parallel to the vector order. O12 = 0 in isotropic phase but in anisotropic phase, the order parameter becomes non-zero showing the same scaling behavior. Finally, we show that even near the critical point, the first law of entanglement entropy is held. Especially, we find that the entanglement temperature for the cylinder is {mathcal{T}}_{cy}=frac{c_{textrm{ent}}}{a} , where cent = 0.163004 ± 0.000001 and a is the radius of the cylinder.

HKLL for the non-normalizable mode

We discuss various aspects of HKLL bulk reconstruction for the free scalar field in AdSd+1. First, we consider the spacelike reconstruction kernel for the non-normalizable mode in global coordinates. We construct it as a mode sum. In even bulk dimensions, this can be reproduced using a chordal Green’s function approach that we propose. This puts the global AdS results for the non-normalizable mode on an equal footing with results in the literature for the normalizable mode. In Poincaré AdS, we present explicit mode sum results in general even and odd dimensions for both normalizable and non-normalizable kernels. For generic scaling dimension ∆, these can be re-written in a form that matches with the global AdS results via an antipodal mapping, plus a remainder. We are not aware of a general argument in the literature for dropping these remainder terms, but we note that a slight complexification of a boundary spatial coordinate (which we call an iϵ prescription) allows us to do so in cases where ∆ is (half-) integer. Since the non-normalizable mode turns on a source in the CFT, our primary motivation for considering it is as a step towards understanding linear wave equations in general spacetimes from a holographic perspective. But when the scaling dimension ∆ is in the Breitenlohner-Freedman window, we note that the construction has some interesting features within AdS/CFT.

Gravity localization on intersecting thick braneworlds

The localization of gravity for ($5+1$)-dimensional intersecting thick braneworld models is thoroughly investigated. Departing from preliminary results for five independent proposals, from $I$ to $V$, gravity is shown to be localized in the brane. In particular, for models from $I$ to $III$, only the zero modes are analytically determined. On the other hand, for models $IV$ and $V$, massive modes are all obtained: a finite number of massive states is identified for model $IV$, while an infinite but discrete number of massive states bounded from below is exhibited by model $V$. Considering that the discreteness of the graviton modes implies that gravity does not propagate in the codimensions, the naked singularities of models $IV$ and $V$, at edges of space, are made harmless. To conclude, the phenomenological implications for sphere models, constructed out of models $III$ and $IV$, are also discussed. According to such an overall classification, if the internal space is a sphere, model $IV$ is shown to exhibit normalizable modes. More relevantly, for the sphere model assembled out of model $III$, the subtle property of reproducing a consistent Newtonian limit is identified.

On holographic Wilsonian renormalization group of massive scalar theory with its self-interactions in AdS

Holographic model of massive scalar field with its self-interaction $\lambda\phi^n$ in AdS space is able to give a logarithmic scale dependence to marginal multi trace deformation couplings on its dual conformal field theory, where $\lambda$ is the self-interaction coupling of the scalar field, $\phi$ and $n$ is an integral number. In arXiv:1501.06664, the authors realize this feature by looking at bulk scalar solutions near AdS boundary imposing a specific boundary condition between the coefficients of non-normalizable and normalizable modes of the scalar field excitations. We study the same holographic model to see scale dependence of marginal deformations on the dual conformal field theory by employing completely different method: {\it holographic Wilsonian renormalization group}. We solve Hamilton-Jacobi equation derived from the holographic model of massive scalar with $\lambda\phi^n$ interaction and obtain the solution of marginal multi trace deformations up to the leading order in $\lambda$. It turns out that the solution of marginal multi trace deformation also presents logarithmic behavior in energy scale near UV region.

De Sitter braneworld and gravitational waves

We study the braneworld theory constructed by multi-scalar fields. The model contains a smooth and infinitely large extra dimension, allowing background fields to propagate in it. We give a de Sitter solution for the four-dimensional cosmology as a good approximation to the early universe inflation. We show that the graviton has a localizable massless mode, and a series of continuous massive modes, separated by a mass gap. There could be a normalizable massive mode, depending on the background solution. The gravitational waves of the massless mode evolve as in the four-dimensional theory, while those of the massive modes evolve very differently from the massless mode.

Q-balls meet fuzzballs: non-BPS microstate geometries

We construct a three-parameter family of non-extremal microstate geometries, or “microstrata”, that are dual to states and deformations of the D1-D5 CFT. These families are non-extremal analogues of superstrata. We find these microstrata by using a Q-ball-inspired Ansatz that reduces the equations of motion to solving for eleven functions of one variable. We then solve this system both perturbatively and numerically and the results match extremely well. We find that the solutions have normal mode frequencies that depend upon the amplitudes of the excitations. We also show that, at higher order in perturbations, some of the solutions, having started with normalizable modes, develop a “non-normalizable” part, suggesting that the microstrata represent states in a perturbed form of the D1-D5 CFT. This paper is intended as a “Proof of Concept” for the Q-ball-inspired approach, and we will describe how it opens the way to many interesting follow-up calculations both in supergravity and in the dual holographic field theory.

Examining instabilities due to driven scalars in AdS

We extend the study of the non-linear perturbative theory of weakly turbulent energy cascades in AdSd+1 to include solutions of driven systems, i.e. those with time-dependent sources on the AdS boundary. This necessitates the activation of non-normalizable modes in the linear solution for the massive bulk scalar field, which couple to the metric and normalizable scalar modes. We determine analytic expressions for secular terms in the renormalization flow equations mass values {m}_{BF}^2<{m}^2le 0 , and for various driving functions. Finally, we numerically evaluate these sources for d = 4 and discuss what role these driven solutions play in the perturbative stability of AdS.

Gaussian null coordinates, near-horizon geometry and conserved charges on the horizon of extremal nondilatonic black p-branes

In this paper, we examine the emergence of conserved charges on the horizon of a particular class of extremal nondilatonic black [Formula: see text]-branes (which reduce to extremal dilatonic black holes in [Formula: see text] dimensions upon toroidal compactification) in the presence of a probe massless scalar field in the bulk. This result is achieved by writing the black [Formula: see text]-brane geometry in a Gaussian null coordinate system which allows us to get a nonsingular near-horizon geometry description. We find that the near-horizon geometry is [Formula: see text] and that the [Formula: see text] section has an internal structure which can be seen as a warped product of [Formula: see text] in Gaussian null coordinates. We show that the bulk scalar field satisfying the field equations is expanded in terms of nonnormalizable and normalizable modes, which for certain suitable quantization conditions are well-behaved at the boundary of [Formula: see text] space. Furthermore, we show that picking the normalizable modes results in the existence of conserved quantities on the horizon. We discuss the impact of these conserved quantities in the late-time regime.

Different approach to decay constants in AdS/QCD models

The holographic recipe for the calculation of decay constants is revisited. Starting from the holographic 2-point function and using the fact that normalizable bulk modes scale as $z^{\Delta-S}$, with $S$ the spin, we can obtain a consistent expression that depends on the value of the mode at the boundary, not the derivative. We apply our decay constant expression to other AdS/QCD (static and dynamic) models proving its consistency. We also demonstrated that our approach is equivalent to the usual holographic prescription.

Bulk reconstruction for spinor fields in AdS/CFT

We develop the representation of free spinor fields in the bulk of Lorentzian anti-de Sitter space in terms of smeared operators in the dual conformal field theory. To do this we expand the bulk field in a complete set of normalizable modes, work out the extrapolate dictionary for spinor fields, and show that the bulk field can be reconstructed from its near-boundary behavior. In some cases chirality and reality conditions can be imposed in the bulk. We study the action of the CFT modular Hamiltonian on bulk fermions to show that they transform with the expected spinor Lie derivative, and we calculate bulk-boundary two-point functions starting from CFT correlators.

Witten-Veneziano mechanism and pseudoscalar glueball-meson mixing in holographic QCD

We revisit the $U(1)_A$ anomaly in the holographic model of low-energy QCD by Witten, Sakai, and Sugimoto, presenting a new and direct derivation of the Witten-Veneziano mechanism for generating the mass of the $\eta'$ through an anomalous mixing of the Ramond-Ramond $C_1$ field with the singlet component of the pseudoscalar mesons. The latter turns out to have a kinetic mixing with the normalizable modes of the $C_1$ field representing pseudoscalar glueballs, yielding additional vertices for their production and their decay that dominate over those of the unmixed case considered previously in the Witten-Sakai-Sugimoto model. The leading channel is predicted to be decay into two vector mesons, followed in importance by decay into three pseudoscalar mesons. The issue of production of pseudoscalar glueballs in radiative $J/\psi$ decays and in double diffractive processes is also discussed briefly.

Quark mass correction to chiral separation effect and pseudoscalar condensate

We derived an analytic structure of the quark mass correction to chiral separation effect (CSE) in small mass regime. We confirmed this structure by a D3/D7 holographic model study in a finite density, finite magnetic field background. The quark mass correction to CSE can be related to correlators of pseudo-scalar condensate, quark number density and quark condensate in static limit. We found scaling relations of these correlators with spatial momentum in the small momentum regime. They characterize medium responses to electric field, inhomogeneous quark mass and chiral shift. Beyond the small momentum regime, we found existence of normalizable mode, which possibly leads to formation of spiral phase. The normalizable mode exists beyond a critical magnetic field, whose magnitude decreases with quark chemical potential.

Stringy horizons II

We show that the spectrum of normalizable states on a Euclidean SL(2,R)/U(1) black hole exhibits a duality between oscillator states and wound strings. This duality generalizes the FZZ correspondence, which can be thought of as an identification between a normalizable mode of dilaton gravity and a mode of the tachyon with winding number one around the Euclidean time circle. It implies that normalizable states on a large Euclidean black hole have support at widely separated scales. In particular, localized states that are extended over the cap of the cigar (the Euclidian analog of the black hole atmosphere) have a component that is localized near the tip of the cigar (the analog of the stretched horizon). As a consequence of this duality, the states exhibit a transition as a function of radial excitation level. From the perspective of a low energy probe, low lying states are naturally thought of as oscillator states in the black hole atmosphere, while at large excitation level they are naturally described as wound strings. As the excitation level increases, the size of the states first decreases and then increases. This behavior is expected to be a general feature of black hole horizons in string theory.

Mixed-symmetry fields in AdS(5), conformal fields, and AdS/CFT

Mixed-symmetry arbitrary spin massive, massless, and self-dual massive fields in AdS(5) are studied. Light-cone gauge actions for such fields leading to decoupled equations of motion are constructed. Light-cone gauge formulation of mixed-symmetry anomalous conformal currents and shadows in 4d flat space is also developed. AdS/CFT correspondence for normalizable and non-normalizable modes of mixed-symmetry AdS fields and the respective boundary mixed-symmetry anomalous conformal currents and shadows is studied. We demonstrate that the light-cone gauge action for massive mixed-symmetry AdS field evaluated on solution of the Dirichlet problem amounts to the light-cone gauge 2-point vertex of mixed-symmetry anomalous shadow. Also we show that UV divergence of the action for mixed-symmetry massive AdS field with some particular value of mass parameter evaluated on the Dirichlet problem amounts to the action of long mixed-symmetry conformal field, while UV divergence of the action for mixed-symmetry massless AdS field evaluated on the Dirichlet problem amounts to the action of short mixed-symmetry conformal field. We speculate on string theory interpretation of a model which involves short low-spin conformal fields and long higher-spin conformal fields.

Bound states in the continuum in PT-symmetric optical lattices

This Letter shows that bound states in the continuum (BIC), which are normalizable modes with energy embedded in the continuous spectrum of scattered states, exist in certain optical waveguide lattices with PT-symmetric defects. There are two distinct types of BIC modes: BIC states that exist in the broken PT phase, and correspond to exponentially localized modes with either exponentially damped or amplified optical power; and BIC modes with sub-exponential spatial localization that also can exist in the unbroken PT phase. These two types of BIC modes at the PT symmetry breaking point behave differently: in the former case, spatial localization is lost and the defect coherently radiates outgoing waves with an optical power that linearly increases with the propagation distance; in the latter case, localization is maintained and the optical power increase is quadratic.

On the stability of D7–[formula omitted] probes in near-conformal backgrounds

We investigate the perturbative stability of a nonsupersymmetric D7–D7¯ brane embedding in a particular class of type IIB backgrounds. These backgrounds are the gravitational duals of certain strongly-coupled gauge theories, that exhibit a nearly conformal regime (known as a walking regime). Previous studies in the literature have led to conflicting results as to whether the spectrum of fluctuations around the flavor D7–D7¯ embedding has a tachyonic mode or not. Here we reconsider the problem with a new analytical method and recover the previously obtained numerical results. We also point out that the earlier treatments relied on a coordinate system, in which it was not possible to take into account fluctuations of the point of confluence of the D7–D7¯ branes. Using an improved coordinate system, we confirm the presence of a normalizable tachyonic mode in this model, in agreement with the numerical calculations. Finally, we comment on the possibility of turning on worldvolume flux to stabilize the probe branes configuration.

On chiral mesons in AdS/CFT

We analyze the spectra of non-chiral and chiral bifundamental mesons arising on intersecting D7-branes in AdS 5 × S 5. In the absence of magnetic flux on the curve of intersection, the spectrum is non-chiral, and the dual gauge theory is conformal in the quenched/probe approximation. For this case we calculate the dimensions of the bifundamental mesonic operators. We then consider magnetization of the D7-branes, which deforms the dual theory by an irrelevant operator and renders the mesons chiral. The magnetic flux spoils the conformality of the dual theory, and induces a D3-brane charge that becomes large in the ultraviolet, where the non-normalizable bifundamental modes are rapidly divergent. An ultraviolet completion is therefore necessary to calculate the correlation functions in the chiral case. On the other hand, the normalizable modes are very well localized in the infrared, leading to new possibilities for local model-building on intersecting D7-branes in warped geometries.

Crystalline scaling geometries from vortex lattices

We study magnetic geometries with Lifshitz and/or hyperscaling violation exponents (both with a hard wall cutoff in the IR and a smooth black brane horizon) which have a complex scalar field which couples to the magnetic field. The complex scalar is unstable to the production of a vortex lattice in the IR. The lattice is a normalizable mode which is relevant (i.e. grows into the IR.) When one considers linearized backreaction of the lattice on the metric and gauge field, the metric forms a crystalline structure. We analyze the scaling of the free energy, thermodynamic entropy, and entanglement in the lattice phase and find that in the smeared limit, the leading order correction to thermodynamic properties due to the lattice has the scaling behavior of a theory with a hyperscaling violation exponent between 0 and 1, indicating a flow to an effectively lower-dimensional theory in the deep IR.