Deforming a two dimensional conformal field theory on one side of a trivial defect line gives rise to a defect separating the original theory from its deformation. The Casimir force between these defects and other defect lines or boundaries is used to construct flows on bulk moduli spaces of CFTs. It turns out, that these flows are constant reparametrizations of gradient flows of the g-functions of the chosen defect or boundary condition. The special flows associated to supersymmetric boundary conditions in N=(2,2) superconformal field theories agree with the attractor flows studied in the context of black holes in N=2 supergravity.
We study the attractor flow and near horizon geometry of two-charge small black holes in heterotic string theory. The Hessian of Sen's entropy function with respect to the moduli fields has standard attractor properties and shows the interesting factorization at the attractor fixed points. We notice that the stability conditions are preserved under arbitrary $α^{\prime}$-corrections to the black hole solutions.
In this note, we use the first order formalism for attractor flows in N=2 SUGRA extremal black hole backgrounds to establish a formal correspondence between the RG flow of moduli in the underlying N=(2,2) SCFT and bulk attractor flows. Starting from a study of moduli flow trajectories in the CFT, we derive a potential which generates the aforesaid flow. This potential is shown to be a symplectic invariant with the same form as the black hole potential which drives the attractor flows in the bulk. We use these results to make comments on the non-renormalization of extremal black hole entropy an…
In this paper we investigate the attractor mechanism in the five dimensional low energy supergravity theory corresponding to M-theory compactified on a Calabi-Yau threefold $CY_3$. Using very special geometry, we derive the general first-order attractor flow equations for BPS and non-BPS solutions in five-dimensional Gibbons-Hawking spaces. Especially, considering the supersymmetric solution, we obtain the first-order flow equations for supersymmetric (multi)black rings. We also solve the flow equations and discuss some properties of the solutions of flow equations.
We propose a program for counting microstates of four-dimensional BPS black holes in N >= 2 supergravities with symmetric-space valued scalars by exploiting the symmetries of timelike reduction to three dimensions. Inspired by the equivalence between the four dimensional attractor flow and geodesic flow on the three-dimensional scalar manifold, we radially quantize stationary, spherically symmetric BPS geometries. Connections between the topological string amplitude, attractor wave function, the Ooguri-Strominger-Vafa conjecture and the theory of automorphic forms suggest that black hole degen…
We discuss the non-equilibrium attractors of systems undergoing Gubser flow within kinetic theory by means of nonlinear dynamical systems. We obtain the attractors of anisotropic hydrodynamics, Israel-Stewart (IS) and transient fluid (DNMR) theories. These attractors are non-planar and the basin of attraction is three dimensional. We compare the asymptotic attractors of each hydrodynamic model with the one obtained from the exact Gubser solution of the Boltzmann equation within the relaxation time approximation. Anisotropic hydrodynamics matches, up to high numerical accuracy, the attractor of…
Following the same treatment of Bellucci et.al., we obtain the hitherto unknown general solutions of the radial attractor flow equations for extremal black holes, both for non-BPS with non-vanishing and vanishing central charge Z for the so-called st^2 model, the minimal rank-2 N=2 symmetric supergravity in d=4 space-time dimensions. We also make useful comparisons with results that already exist in literature,and introduce the fake supergravity (first-order) formalism to be used in our analysis. An analysis of the BPS bound all along the non-BPS attractor flows and of the marginal stability o…
We investigate the existence of supersymmetric static dyonic black holes with spherical horizon in the context of N= 2 U(1) gauged supergravity in four dimensions. We analyze the conditions for their existence and provide the general first-order flow equations driving the scalar fields and the metric warp factors from the asymptotic AdS4 geometry to the horizon. We work in a general duality-symmetric setup, which allows to describe both electric and magnetic gaugings. We also discuss the attractor mechanism and the issue of moduli (de-)stabilization.
We classify the stability region, marginal stability walls (MS) and split attractor flows for two-center extremal black holes in four-dimensional N=2 supergravity minimally coupled to n vector multiplets. It is found that two-center (continuous) charge orbits, classified by four duality invariants, either support a stability region ending on a MS wall or on an anti-marginal stability (AMS) wall, but not both. Therefore, the scalar manifold never contains both walls. Moreover, the BPS mass of the black hole composite (in its stability region) never vanishes in the scalar manifold. For these rea…
We extend the discussion of the "Kerr/CFT correspondence" and its recent developments to the more general gauge/gravity correspondence in the full extremal black hole space-time of the bulk by using a technique of the holographic renormalization group (RG) flow. It is conjectured that the extremal black hole space-time is holographically dual to the chiral two dimensional field theory. Our example is a typical four dimensional Reissner-Nordstrom black hole, a system in which the M5-brane is wrapped on four cycles of Calabi-Yau threefold. In five dimensional supergravity view point this near ho…
We investigate the attractor mechanism in gauged supergravity in the presence of higher derivatives terms. In particular, we discuss the attractor behaviour of static black hole horizons in anti-de Sitter spacetime by using the effective potential approach as well as Sen's entropy function formalism. We use the holographic techniques to interpret the moduli flow as an RG flow towards the IR attractor horizon. We find that the holographic c-function obeys the expected properties and point out some subtleties in understanding attractors in AdS.
We present a simple example of a supersymmetric attractor mechanism in the purely open string context of D-branes embedded in curved space-time. Our example involves a class of D3-branes embedded in the 2-charge D1-D5 background of type IIB whose worldvolume contains a 2-sphere. Turning on worldvolume fluxes, these branes carry induced (p,q) string charges. Supersymmetric configurations display a flow of the open string moduli towards an attractor solution independent of their asymptotics. The equations governing this mechanism closely resemble the attractor flow equations for supersymmetric b…
We investigate the spectrum of type IIA BPS D-branes on the quintic from a four dimensional supergravity perspective and the associated split attractor flow picture. We obtain some very concrete properties of the (quantum corrected) spectrum, mainly based on an extensive numerical analysis, and to a lesser extent on exact results in the large radius approximation. We predict the presence and absence of some charges in the BPS spectrum in various regions of moduli space, including the precise location of the lines of marginal stability and the corresponding decay products. We explain how the ge…
We derive extremal black hole solutions for a variety of four dimensional models which, after Kaluza-Klein reduction, admit a description in terms of 3D gravity coupled to a sigma model with symmetric target space. The solutions are in correspondence with certain nilpotent generators of the isometry group. In particular, we provide the exact solution for a non-BPS black hole with generic charges and asymptotic moduli in N=2 supergravity coupled to one vector multiplet. Multi-centered solutions can also be generated with this technique. It is shown that the non-supersymmetric solutions lack the…
We perform a systematic search for static solutions in different sectors of 5d $N=8$ supergravities with compact and non-compact gauged R-symmetry groups, finding new and listing already known backgrounds. Due to the variety of possible gauge groups and resulting scalar potentials, the maximally symmetric vacua we encounter in these theories can be Minkowski, de Sitter, or anti-de Sitter. There exist BPS and non-BPS near-horizon geometries and full solutions with all these three types of asymptotics, corresponding to black holes, branes, strings, rings, and other black objects with more exotic…
We study flows of non-supersymmetric attractor black holes in the context of gauge/gravity correspondence. As our bulk theory, we consider the Einstein-Maxwell-Dilaton system with a single dilaton field coupled to two Maxwell fields and make a relevant deformation by adding a bare potential to the dilaton field. We find two types of extremal black hole solutions with attractor mechanism: The one smooth at the horizon and the other non-smooth. We show from both bulk and boundary theory perspective that the former is thermodynamically unstable, while the latter is stable.
We present a simple and yet rigorous derivation of the flow equations for the supersymmetric black-hole solutions of all 4-dimensional supergravities based on the recently found general form of all those solutions.
We review the construction of multi-centered black hole solutions through dimensional reduction over time. This method does not rely on Killing spinor equations or gradient flow equations, but on solving the second order field equations in terms of harmonic functions. The black hole attractor equations are obtained directly from the field equations.
We consider causal higher order theories of relativistic viscous hydrodynamics in the limit of one-dimensional boost-invariant expansion and study the associated dynamical attractor. We obtain evolution equations for the inverse Reynolds number as a function of Knudsen number. The solutions of these equations exhibit attractor behavior which we analyze in terms of Lyapunov exponents using several different techniques. We compare the attractors of the second-order Müller-Israel-Stewart (MIS), transient DNMR, and third-order theories with the exact solution of the Boltzmann equation in the relax…
We show that topological defects in the language of Landau-Ginzburg models carry information about the RG flow between the non-compact orbifolds $\mathbb{C}/\mathbb{Z}_d$. We show that such defects correctly implement the bulk-induced RG flow on the boundary.
We study the effect of bulk perturbations of N=(2,2) superconformal minimal models on topological defects. In particular, symmetries and more general topological defects which survive the flow to the IR are identified. Our method is to consider the topological subsector and make use of the Landau-Ginzburg formulation to describe RG flows and topological defects in terms of matrix factorizations.
Using the properties of defect lines, we study boundary renormalisation group flows. We find that when there exists a flow between maximally symmetric boundary conditions "a" and "b" then there also exists a boundary flow between "c x a" and "c x b" where "x" denotes the fusion product. We also discuss applications of this simple observation.
The nonequilibrium evolution in a boost-invariant Bjorken flow of a hybrid viscous fluid model containing two interacting components with different viscosities, such that they represent strongly and weakly self-coupled sectors, is shown to be characterized by a hydrodynamic attractor which has an early-time behavior that is reminiscent of the so-called bottom-up thermalization scenario in heavy-ion collisions. The hydrodynamization times for the two sectors can differ strongly, with details depending on the curve realized on the two-dimensional attractor surface, which might account for differ…
Late-time attractors in relativistic spin hydrodynamics in Gubser flow
University of Science and Technology of China（中国科学技术大学）；The University of Tokyo（东京大学）；Institute of Modern Physics, Chinese Academy of Sciences（中国科学院近代物理研究所）
We consider defect operators in scalar field theories in dimensions $d=4-ε$ and $d=6-ε$ with self-interactions given by a general marginal potential. In a double scaling limit, where the bulk couplings go to zero and the defect couplings go to infinity, the bulk theory becomes classical and the quantum defect theory can be solved order by order in perturbation theory. We compute the defect $β$ functions to two loops and study the Renormalization Group flows. The defect fixed points can move and merge, leading to fixed point annihilation; and they exhibit a remarkable factorization property whe…
Perturbative expansions, such as the well-known gradient series and the recently proposed slow-roll expansion, have been recently used to investigate the emergence of hydrodynamic behavior in systems undergoing Bjorken flow. In this paper we determine for the first time the large order behavior of these perturbative expansions in relativistic hydrodynamics in the case of Gubser flow. While both series diverge, the slow-roll series can provide a much better overall description of the system's dynamics than the gradient expansion when both series are truncated at low orders. The truncated slow-r…
Extending the quantum effective approach of Son and Nicolis and incorporating dissipation, we develop a MIS formalism for describing a superfluid out of equilibrium by including the Goldstone boson and the condensate together with the hydrodynamic modes as the effective degrees of freedom. We find that the evolution of the superfluid undergoing Bjorken flow is governed by the conventional hydrodynamic attractor with unbroken symmetry and an even number of novel non-dissipative fixed points with broken symmetry. If the initial temperature is super-critical, then the condensate becomes exponenti…
The non-equilibrium attractors of systems undergoing Gubser flow within relativistic kinetic theory are studied. In doing so we employ well-established methods of nonlinear dynamical systems which rely on finding the fixed points, investigating the structure of the flow diagrams of the evolution equations, and characterizing the basin of attraction using a Lyapunov function near the stable fixed points. We obtain the attractors of anisotropic hydrodynamics, Israel-Stewart (IS) and transient fluid (DNMR) theories and show that they are indeed non-planar and the basin of attraction is essentiall…
There is mounting evidence suggesting that relativistic hydrodynamics becomes relevant for the physics of quark-gluon plasma as the result of nonhydrodynamic modes decaying to an attractor apparent even when the system is far from local equilibrium. Here we determine this attractor for Bjorken flow in N=4 supersymmetric Yang-Mills theory using Borel summation of the gradient expansion of the expectation value of the energy momentum tensor. By comparing the result to numerical simulations of the flow based on the AdS/CFT correspondence we show that it provides an accurate and unambiguous approx…
We investigate the implications of the general frame approach for conformal Bjorken flow beyond the earlier studies. We show that the power series solution at late times is not unique and is accompanied by an exact solution of the form $1/τ$, which becomes unphysical if taken on shell. In contrast to the Müller-Israel-Stewart formalism, a matching between $\NFSYM$ results and the hydro expansion is only possible up to the first order, which gives rise to $η/s=1/4π$. Matching the results to the next order gives rise to causality/stability-violating values. Furthermore, we show that the pressure…
The appearance of a generalized (or Borcherds-) Kac-Moody algebra in the spectrum of BPS dyons in N=4, d=4 string theory is elucidated. From the low-energy supergravity analysis, we identify its root lattice as the lattice of the T-duality invariants of the dyonic charges, the symmetry group of the root system as the extended S-duality group PGL(2,Z) of the theory, and the walls of Weyl chambers as the walls of marginal stability for the relevant two-centered solutions. This leads to an interpretation for the Weyl group as the group of wall-crossing, or the group of discrete attractor flows. F…
Motivated by the interpretation of the Ooguri-Strominger-Vafa conjecture as a holographic correspondence in the mini-superspace approximation, we study the radial quantization of stationary, spherically symmetric black holes in four dimensions. A key ingredient is the classical equivalence between the radial evolution equation and geodesic motion of a fiducial particle on the moduli space M^*_3 of the three-dimensional theory after reduction along the time direction. In the case of N=2 supergravity, M^*_3 is a para-quaternionic-Kahler manifold; in this case, we show that BPS black holes corres…
We consider line defects in d-dimensional Conformal Field Theories (CFTs). The ambient CFT places nontrivial constraints on Renormalization Group (RG) flows on such line defects. We show that the flow on line defects is consequently irreversible and furthermore a canonical decreasing entropy function exists. This construction generalizes the g theorem to line defects in arbitrary dimensions. We demonstrate our results in a flow between Wilson loops in 4 dimensions.
In [L-Y5] (D(6): arXiv:1003.1178 [math.SG]) we introduced the notion of Azumaya $C^{\infty}$-manifolds with a fundamental module and morphisms therefrom to a complex manifold. In the current sequel, we use this notion to give a prototypical definition of supersymmetric D-branes of A-type (i.e. A-branes) -- in an appropriate region of the Wilson's theory-space of string theory -- as special Lagrangian morphisms from such objects with a unitary, minimally flat connection-with-singularities. This merges Donaldson's picture of special Lagrangian submanifolds and the Polchinski-Grothendieck Ansatz …
