We consider superconformal and supersymmetric field theories on four-dimensional Lorentzian curved space-times, and their five-dimensional holographic duals. As in the Euclidean signature case, preserved supersymmetry for a superconformal theory is equivalent to the existence of a charged conformal Killing spinor. Differently from the Euclidean case, we show that the existence of such spinors is equivalent to the existence of a null conformal Killing vector. For a supersymmetric field theory with an R-symmetry, this vector field is further restricted to be Killing. We demonstrate how these results agree with the existing classification of supersymmetric solutions of minimal gauged supergravity in five dimensions.
We study chaos in non-Lorentzian field theories, specifically Galilean and Carrollian conformal field theories in two dimensions. In a large central charge limit, we find that the Lyapunov exponent saturates the bound on chaos, conjectured originally for relativistic field theories. We recover the same Lyapunov exponent holographically by a shock-wave calculation in three-dimensional flat space cosmologies, providing further evidence for flat space holography.
We study N <= 2 superconformal and supersymmetric theories on Lorentzian threemanifolds with a view toward holographic applications, in particular to BPS black hole solutions. As in the Euclidean case, preserved supersymmetry for asymptotically locally AdS solutions implies the existence of a (charged) "conformal Killing spinor" on the boundary. We find that such spinors exist whenever there is a conformal Killing vector which is null or timelike. We match these results with expectations from supersymmetric four-dimensional asymptotically AdS black holes. In particular, BPS bulk solutions in g…
M-theory on $AdS_7 \times S^4$ admits a description where the $AdS_7$ factor is constructed as a timelike Hopf fibration over a non-compact three dimensional complex projective space $\tilde{\mathbb{CP}}^3$. We consider the worldvolume theory for M5-branes at a fixed $\tilde{\mathbb{CP}}^3$ radius which, after reduction along the timelike fibre, is given by an $Ω$-deformed Yang-Mills theory with eight supercharges. Taking the radius to infinity then induces a classical RG flow. We construct the fixed point action which has an enhanced 24 supercharges and which can be understood as the $(2,0)$ …
We consider Lorentzian correlation functions in theories with non-relativistic Schrodinger symmetry. We employ the method developed by Skenderis and van Rees in which the contour in complex time defining a given correlation function is associated holographically with the gluing together of Euclidean and Lorentzian patches of spacetimes. This formalism extends appropriately to geometries with Schrodinger isometry.
De Boer et. al. have found an asymptotic equivalence between the Hamilton-Jacobi equations for supergravity in (d+1)-dimensional asymptotic anti-de Sitter space, and the Callan-Symanzik equations for the dual d-dimensional perturbed conformal field theory. We discuss this correspondence in Lorentzian signature. We construct a gravitational dual of the generating function of correlation functions between initial and final states, in accordance with the construction of Marolf, and find a class of states for which the result has a classical supergravity limit. We show how the data specifying the …
We investigate the structure of two point functions for the QFT dual to an asymptotically Lorentzian AdS-wormhole. The bulk geometry is a solution of 5-dimensional second order Einstein Gauss Bonnet gravity and causally connects two asymptotically AdS space times. We revisit the GKPW prescription for computing two-point correlation functions for dual QFT operators O in Lorentzian signature and we propose to express the bulk fields in terms of the independent boundary values phi_0^\pm at each of the two asymptotic AdS regions, along the way we exhibit how the ambiguity of normalizable modes in …
Some recent literature has examined the holographic-induced cosmology of a brane universe in the background of an anti-de Sitter-black hole geometry. In this regard, curved-brane scenarios have begun to receive considerable attention. Our current interest is in a formal discrepancy that exists between two such works by Padilla (hep-th/0111247) and Youm (hep-th/0111276). In particular, these authors have incorporated different values for a conformal factor that is used to relate the thermodynamics of the relevant (AdS bulk and CFT brane) spacetimes. After a more general review, we clarify this …
We consider a holographic model of strongly interacting plasma with a gravitational anomaly. In this model, we compute parity-odd responses of the system at finite temperature and chemical potential to external electromagnetic and gravitational fields. Working within the linearized fluid/gravity duality, we performed the calculation up to the third order in gradient expansion. Besides reproducing the chiral magnetic(CME) and vortical (CVE) effects we also obtain gradient corrections to the CME and CVE due to the gravitational anomaly. Additionally, we find energy-momentum and current responses…
We study superconformal and supersymmetric theories on Euclidean four- and three-manifolds with a view toward holographic applications. Preserved supersymmetry for asymptotically locally AdS solutions implies the existence of a (charged) "conformal Killing spinor" on the boundary. We study the geometry behind the existence of such spinors. We show in particular that, in dimension four, they exist on any complex manifold. This implies that a superconformal theory has at least one supercharge on any such space, if we allow for a background field (in general complex) for the R-symmetry. We also s…
Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；Universitat de Barcelona（巴塞罗那大学）；Inamori Research Institute for Science（稻盛科学研究所）
We study space-time symmetries in scalar quantum field theory (including interacting theories) on static space-times. We first consider Euclidean quantum field theory on a static Riemannian manifold, and show that the isometry group is generated by one-parameter subgroups which have either self-adjoint or unitary quantizations. We analytically continue the self-adjoint semigroups to one-parameter unitary groups, and thus construct a unitary representation of the isometry group of the associated Lorentzian manifold. The method is illustrated for the example of hyperbolic space, whose Lorentzian…
This paper focuses on the semiclassical behavior of the spinfoam quantum gravity in 4 dimensions. There has been long-standing confusion, known as the flatness problem, about whether the curved geometry exists in the semiclassical regime of the spinfoam amplitude. The confusion is resolved by the present work. By numerical computations, we explicitly find curved Regge geometries from the large-$j$ Lorentzian Engle-Pereira-Rovelli-Livine (EPRL) spinfoam amplitudes on triangulations. These curved geometries are with small deficit angles and relate to the complex critical points of the amplitude.…
M/string theory on noncompact, negatively curved, cosets which generalize $AdS_{D+1}=SO(D,2)/SO(D,1)$ is considered. Holographic descriptions in terms of a conformal field theory on the boundary of the spacetime are proposed. Examples include $SU(2,1)/U(2)$, which is a Euclidean signature (4,0) space with no supersymmetry, and $SO(2,2)/SO(2)$ and $SO(3,2)/SO(3)$, which are Lorentzian signature (4,1) and (6,1) spaces with eight supersymmetries. Qualitatively new features arise due to the degenerate nature of the conformal boundary metric.
We construct two-dimensional quantum states associated to four-dimensional linearized rotating self-dual black holes in $(2,2)$ signature Klein space. The states are comprised of global conformal primaries circulating on the celestial torus, the Kleinian analog of the celestial sphere. By introducing a generalized tower of Goldstone operators we identify the states as coherent exponentiations carrying an infinite tower of ${\rm w}_{1+\infty}$ charges or soft hair. We relate our results to recent approaches to black hole scattering, including a connection to Wilson lines, $\mathcal{S}$-matrix r…
We study the consequences of unbroken rigid supersymmetry of four-dimensional field theories placed on curved manifolds. We show that in Lorentzian signature the background vector field coupling to the R-current is determined by the Weyl tensor of the background metric. In Euclidean signature, the same holds if two supercharges of opposite R-charge are preserved, otherwise the (anti-)self-dual part of the vector field-strength is fixed by the Weyl tensor. As a result of this relation, the trace and R-current anomalies of superconformal field theories simplify, with the trace anomaly becoming p…
These notes are an expanded version of lectures given at the Croatian School on Black Holes at Trpanj, June 21-25, 2010. The aim is to provide a practical introduction to quantum field theory in curved spacetime and related black hole physics, with AdS/CFT as the loose motivation.
The one-loop quantum corrections to the free energy associated with scalar field in a higher dimensional static curved space-time is investigated making use of the conformal transformation method. For a space-time with bifurcate horizon, horizon divergences are accounted for choosing the Planck length as natural cutoff. The leading term in the high temperature quantum correction satifies holographically the "area law", like the tree level Bekenstein-Hawking term. Furthermore it is stressed that only for the asymptotically AdS black holes one may have a microscopic interpretation of the entropy…
The essential features of a quantum group deformation of classical symmetries of General Relativity in the case with non-vanishing cosmological constant $Λ$ are presented. We fully describe (anti-)de Sitter non-commutative spacetimes and curved momentum spaces in (1+1) and (2+1) dimensions arising from the $κ$-deformed quantum group symmetries. These non-commutative spacetimes are introduced semiclassically by means of a canonical Poisson structure, the Sklyanin bracket, depending on the classical $r$-matrix defining the $κ$-deformation, while curved momentum spaces are defined as orbits gener…
String propagation on D-dimensional curved backgrounds with Lorentzian signature is formulated as a geometrical problem of embedding surfaces. When the spatial part of the background corresponds to a general WZW model for a compact group, the classical dynamics of the physical degrees of freedom is governed by the coset conformal field theory SO(D-1)/SO(D-2), which is universal irrespective of the particular WZW model. The same holds for string propagation on D-dimensional flat space. The integration of the corresponding Gauss-Codazzi equations requires the introduction of (non-Abelian) parafe…
We present a mathematical framework which underlies the connection between information theory and the bulk spacetime in the AdS$_3$/CFT$_2$ correspondence. A key concept is kinematic space: an auxiliary Lorentzian geometry whose metric is defined in terms of conditional mutual informations and which organizes the entanglement pattern of a CFT state. When the field theory has a holographic dual obeying the Ryu-Takayanagi proposal, kinematic space has a direct geometric meaning: it is the space of bulk geodesics studied in integral geometry. Lengths of bulk curves are computed by kinematic volum…
We review a few rigorous and partly unpublished results on the regularisation of the stress-energy in quantum field theory on curved spacetimes: 1) the symmetry of the Hadamard/Seeley-DeWitt coefficients in smooth Riemannian and Lorentzian spacetimes 2) the equivalence of the local $ζ$-function and the Hadamard-point-splitting procedure in smooth static spacetimes 3) the equivalence of the DeWitt-Schwinger- and the Hadamard-point-splitting procedure in smooth Riemannian and Lorentzian spacetimes.
We consider the holographic prescription problem in a (Lorentzian) AdS background, deriving from first principles the explicit formulas that relate the field at infinity with the field in the bulk. In contrast with the previous studies of the "real-time" holography problem, our derivation uses purely classical arguments that involve causality, as in the usual treatment of the holographic prescription problem in Wick-rotated spaces of Euclidean signature. We show that there is a unique propagator that preserves causality and see that this provides a simple picture of the relationship between th…
As was recently pointed out by Cadoni, a certain class of two-dimensional gravitational theories will exhibit (black hole) thermodynamic behavior that is reminiscent of a free field theory. In the current letter, a direct correspondence is established between these two-dimensional models and the strongly curved regime of (arbitrary-dimensional) anti-de Sitter gravity. On this basis, we go on to speculatively argue that two-dimensional gravity may ultimatley be utilized for identifying and perhaps even understanding holographic dualities.
We study N=2 superconformal theories on Euclidean and Lorentzian four-manifolds with a view toward applications to holography and localization. The conditions for supersymmetry are equivalent to a set of differential constraints including a "generalised" conformal Killing spinor equation depending on various background fields. We solve these equations in the general case and give very explicit expressions for the auxiliary fields that we need to turn on to preserve some supersymmetry. As opposed to what has been observed for the N=1 case, the conditions for unbroken supersymmetry turn out to b…
We construct explicit examples of non-relativistic supersymmetric field theories on curved Newton-Cartan three-manifolds. These results are obtained by performing a null reduction of four-dimensional supersymmetric field theories on Lorentzian manifolds and the Killing spinor equations that their supersymmetry parameters obey. This gives rise to a set of algebraic and differential Killing spinor equations that are obeyed by the supersymmetry parameters of the resulting three-dimensional non-relativistic field theories. We derive necessary and sufficient conditions that determine whether a Newt…
We study holography for asymptotically AdS spaces with an arbitrary genus compact Riemann surface as the conformal boundary. Such spaces can be constructed from the Euclidean AdS_3 by discrete identifications; the discrete groups one uses are the so-called classical Schottky groups. As we show, the spaces so constructed have an appealing interpretation of ``analytic continuations'' of the known Lorentzian signature black hole solutions; it is one of the motivations for our generalization of the holography to this case. We use the semi-classical approximation to the gravity path integral, and c…
In 2+1 dimensions at finite temperature, spontaneous symmetry breaking of global symmetries is precluded by large thermal fluctuations of the order parameter. The holographic correspondence implies that analogous effects must also occur in 3+1 dimensional theories with gauged symmetries in certain curved spacetimes with horizon. By performing a one loop computation in the background of a holographic superconductor, we show that bulk quantum fluctuations wash out the classical order parameter at sufficiently large distance scales. The low temperature phase is seen to exhibit algebraic long rang…
Holonomy Reconstruction and Character Varieties of Lorentzian Curved Tetrahedra
Institute for Theoretical Sciences, and Departments of Physics and Astronomy, Westlake University（西湖大学理论科学研究所及物理与天文系）；Westlake Institute for Advanced Study（西湖高等研究院）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
We consider gauge/gravity correspondence between maximally supersymmetric Yang-Mills theory in ($p+1$) dimensions and superstring theory on the near-horizon limit of the D$p$-brane solution. The string-frame metric is AdS$_{p+1}\times S^{8-p}$ times a Weyl factor, and there is no conformal symmetry except for $p=3$. In a previous paper by one of the present authors, the free-field result of gauge theory has been reproduced from string theory for a particular operator which has angular momentum along $S^{8-p}$. In this paper, we extend this result to operators which have angular momenta along A…
We use holography in order to study the entropy of thermal CFTs on (1+1)-dimensional curved backgrounds that contain horizons. Starting from the metric of the BTZ black hole, we perform explicit coordinate transformations that set the boundary metric in de Sitter or black-hole form. For a de Sitter boundary, the dual picture describes a CFT at a temperature different from that of the cosmological horizon. We determine minimal surfaces that allow us to compute the entanglement entropy of a boundary region, as well as the temperature affecting the energy associated with a probe quark on the boun…
For Euclidean quantum field theories, Holland and Hollands have shown operator product expansion (OPE) coefficients satisfy "flow equations": For interaction parameter $λ$, the partial derivative of any OPE coefficient with respect to $λ$ is given by an integral over Euclidean space of a sum of products of other OPE coefficients. In this paper, we generalize these results for flat Euclidean space to curved Lorentzian spacetimes in the context of the solvable "toy model" of massive Klein-Gordon scalar field theory, with $m^2$ viewed as the "self-interaction parameter". Even in Minkowski spaceti…
We obtain the action for a curved superconformal abelian M5 brane with the background R-symmetry gauge field turned on. We then restrict ourselves to superconformal M5 brane on a sphere times flat Minkowski space. We choose R-symmetry SO(1,4) instead of SO(5), which enables us to partially twist on Minkowski space and replace it by some curved Lorentzian manifold. We obtain M5 brane actions on $M_{1,1} \times S^4$ and $M_{1,2} \times S^3$ where actions and all fields, including the background gauge field, are real. Dimensional reduction along time gives real 5d SYM actions with nonabelian gene…
We show that the Lorentzian Snyder models, together with their Galilei and Carroll limiting cases, can be rigorously constructed through the projective geometry description of Lorentzian, Galilean and Carrollian spaces with nonvanishing constant curvature. The projective coordinates of such curved spaces take the role of momenta, while translation generators over the same spaces are identified with noncommutative spacetime coordinates. In this way, one obtains a deformed phase space algebra, which fully characterizes the Snyder model and is invariant under boosts and rotations of the relevant …
We construct a symplectic isomorphism, h, from classical Klein Gordon solutions of mass m on (d+1)-dimensional Lorentzian Anti de Sitter space (equipped with the usual symplectic form) to a certain symplectic space of functions on its conformal boundary (only) for all integer and half-integer Delta (= d/2 + (1/2)(d^2 + 4m^2)^{1/2}). h induces a large family of new examples of Rehren's algebraic holography in which the net of local quantum Klein Gordon algebras in AdS is seen to map to a suitably defined net of local algebras for the (generalized free) scalar conformal field with anomalous dime…
In this paper, we study the 6d Little String Theory (LST) (the decoupled theory on the worldvolume of $N$ NS5-branes) on curved manifolds, by using its holographic duality to Type II string theory in asymptotically linear dilaton backgrounds. We focus on backgrounds with a large number of Killing vectors (namely, products of maximally symmetric spaces), without requiring supersymmetry (we do not turn on any background fields except the metric). LST is non-local so it is not obvious which spaces it can be defined on; we show that holography implies that the theory cannot be put on negatively cu…
We propose a codimension two holography between a gravitational theory on a $d+1$ dimensional wedge spacetime and a $d-1$ dimensional CFT which lives on the corner of the wedge. Formulating this as a generalization of AdS/CFT, we explain how to compute the free energy, entanglement entropy and correlation functions of the dual CFTs from gravity. In this wedge holography, the holographic entanglement entropy is computed by a double minimization procedure. Especially, for a four dimensional gravity ($d=3$), we obtain a two dimensional CFT and the holographic entanglement entropy perfectly reprod…
We study baryons as Skyrmions in holographic QCD with D4/D8/D8-bar multi-D brane system in type IIA superstring theory, and also in the non-linear sigma model with hidden local symmetry (HLS). Comparing these two models, we find that the extra-dimension and its nontrivial curvature can largely change the role of (axial) vector mesons for baryons in four-dimensional space-time. In the HLS approach, the rho-meson field as a massive Yang-Mills field has a singular configuration in Skyrmion, which gives a strong repulsion for the baryon as a stabilizer. When a_1 meson is added in this approach, th…
We show that for an eikonal limit of gravity in a space-time of any dimension with a non-vanishing cosmological constant, the Einstein -- Hilbert action reduces to a boundary action. This boundary action describes the interaction of shock-waves up to the point of evolution at which the forward light-cone of a collision meets the boundary of the space-time. The conclusions are quite general and in particular generalize the previous work of E. and H. Verlinde. The role of the off-diagonal Einstein action in removing the bulk part of the action is emphasised. We discuss the sense in which our res…
We investigate a new approach to holography in asymptotically AdS spacetimes, in which time rather than space is the emergent dimension. By making a sufficiently large T^2-deformation of a Euclidean CFT, we define a holographic theory that lives on Cauchy slices of the Lorentzian bulk. (More generally, for an arbitrary Hamiltonian constraint equation that closes, we show how to obtain it by an irrelevant deformation from a CFT with suitable anomalies.) The partition function of this theory defines a natural map between the bulk canonical quantum gravity theory Hilbert space, and the Hilbert sp…
This paper concerns the motion of a relativistic string in a curved space-time. As a general framework, we first analyze relativistic string equations, i.e., the basic equations for the motion of a one-dimensional extended object in a curved enveloping space-time $(\mathscr{N}, \tilde g)$, which is a general Lorentzian manifold, and then investigate some interesting properties enjoyed by these equations. Based on this, under suitable assumptions we prove the global existence of smooth solutions of the Cauchy problem for relativistic string equations in the curved space-time $(\mathscr{N}, \til…
We calculate the relevant Spencer cohomology of the minimal Poincaré superalgebra in 5 spacetime dimensions and use it to define Killing spinors via a connection on the spinor bundle of a 5-dimensional lorentzian spin manifold. We give a definition of bosonic backgrounds in terms of this data. By imposing constraints on the curvature of the spinor connection, we recover the field equations of minimal (ungauged) 5-dimensional supergravity, but also find a set of field equations for an $\mathfrak{sp}(1)$-valued one-form which we interpret as the bosonic data of a class of rigid supersymmetric th…
A simple procedure is given to construct curved, non-self-dual (complexified) Kaehler metrics on space-time in terms of deformations of holomorphic quadric surfaces in flat twistor space. Imposing Lorentzian reality conditions, the Schwarzschild, Kerr, and Plebanski-Demianski space-times (among others) are derived as examples of the construction.
In this thesis we study the AdS3 Wess-Zumino-Novikov-Witten model. We compute the Operator Product Expansion of primary fields as well as their images under the spectral flow automorphism in all sectors of the model by considering it as a Wick rotation of the H3+ coset model. We argue that the symmetries of the affine algebra require a truncation which establishes the closure of the fusion rules on the Hilbert space of the theory. These results are then used to discuss the factorization of four point functions by applying the bootstrap approach. We also study the modular properties of the mode…
We present an algorithm for constructing the Wilson operator product expansion (OPE) for perturbative interacting quantum field theory in general Lorentzian curved spacetimes, to arbitrary orders in perturbation theory. The remainder in this expansion is shown to go to zero at short distances in the sense of expectation values in arbitrary Hadamard states. We also establish a number of general properties of the OPE coefficients: (a) they only depend (locally and covariantly) upon the spacetime metric and coupling constants, (b) they satisfy an associativity property, (c) they satisfy a renorma…
We consider a general Kaluza-Klein reduction of a truncated Lovelock theory. We find necessary geometric conditions for the reduction to be consistent. The resulting lower-dimensional theory is a higher derivative scalar-tensor theory, depends on a single real parameter and yields second-order field equations. Due to the presence of higher-derivative terms, the theory has multiple applications in modifications of Einstein gravity (Galileon/Horndesky theory) and holography (Einstein-Maxwell-Dilaton theories). We find and analyze charged black hole solutions with planar or curved horizons, both …
