Space-time Conformal Symmetry Research Articles

Moving holographic boundaries

In this paper, we show that for one sign of the deformation coupling single-trace TT‾ deformation moves the holographic screen in Gödel universe radially inward. For the other sign of the coupling it moves the holographic screen radially outward. We (thus) argue, on general grounds, that in holography (single-trace) TT‾ deformation can be generally thought of as either moving the holographic boundary into the bulk or washing it away to infinity. We explain in what sense. In Anti-de Sitter this breaks the spacetime conformal symmetry. We further note that moving timelike holographic boundary into bulk creates (at onset) a curvature singularity. In the boundary the singularity is understood by states with imaginary energies. To define or make the bulk theory sensible we introduce an ultraviolet cutoff and thereby move the boundary into the bulk. However, it is not clear whether suitable boundary conditions that lead to a consistent string theory exist. In this paper we first obtain the Penrose limit of the single-trace TT‾ deformed string background and then perform T-duality along a space-like isometry to obtain a class of deformed Gödel universes. The string background we consider is AdS3×S3×M4. The single-trace TT‾ deformation is a particular example of the more general O(d,d) transformations.

Vanishing Love numbers of black holes in general relativity: From spacetime conformal symmetry of a two-dimensional reduced geometry

We study the underlying structure of the vanishing of the Love numbers of both Schwarzschild and Kerr black holes in terms of spacetime conformal symmetry in a unified manner for the static spin-$s$ fields. The perturbations can be reduced with the harmonic decomposition to a set of infinite static scalar fields in a two-dimensional anti--de Sitter spacetime $({\text{AdS}}_{2})$. In the reduced system, each scalar field is paired with another, implying that all multipole modes of the perturbation can be regarded as symmetric partners, which can be understood from the property of the supersymmetry algebra. The generator of the supersymmetric structure is constructed from a closed conformal Killing vector field of ${\text{AdS}}_{2}$. The associated conserved quantity allows one to show no static response, i.e., vanishing of the Love and dissipation numbers. We also discuss the vanishing Love numbers of the Kerr black hole with the nonzero dissipation numbers for the nonaxisymmetric perturbations in terms of a radial constant found in a parallel manner as the axisymmetric field case even though the interpretation for the structure is controversial. The symmetric structure corresponds to the ``ladder'' symmetry in [Hui et al., J. Cosmol. Astropart. Phys. 01 (2022) 032] although the geometrical origin is different. Our ladder operator includes the generators of hidden symmetries in previous works.

Aretakis constants and instability in general spherically symmetric extremal black hole spacetimes: Higher multipole modes, late-time tails, and geometrical meanings

We study late-time behaviors of massive scalar fields in general static and spherically symmetric extremal black hole spacetimes in arbitrary dimensions. We show the existence of conserved quantities on the extremal black hole horizons for specific mass squared and multipole modes of the scalar fields. Those quantities on the horizon are called the Aretakis constants and are constructed from the higher-order derivatives of the fields. Focusing on the region near the horizon at late times, where is well approximated by the near-horizon geometry, we show that the leading behaviors of the fields are described by power-law tails. The late-time power-law tails lead to the Atetakis instability: blowups of the transverse derivatives of the fields on the horizon. We further argue that the Aretakis constants and instability correspond to respectively constants and blowups of components of covariant derivatives of the fields at the late time in the parallelly propagated null geodesic frame along the horizon. We finally discuss the relation between the Aretakis constants and ladder operators constructed from the approximate spacetime conformal symmetry near the extremal black hole horizons.

Revisiting the Aretakis constants and instability in two-dimensional anti–de Sitter spacetimes

We discuss dynamics of massive Klein-Gordon fields in two-dimensional Anti-de Sitter spacetimes ($AdS_2$), in particular conserved quantities and non-modal instability on the future Poincar\'e horizon called, respectively, the Aretakis constants and the Aretakis instability. We find out the geometrical meaning of the Aretakis constants and instability in a parallel-transported frame along a null geodesic, i.e., some components of the higher-order covariant derivatives of the field in the parallel-transported frame are constant or unbounded at the late time, respectively. Because $AdS_2$ is maximally symmetric, any null hypersurfaces have the same geometrical properties. Thus, if we prepare parallel-transported frames along any null hypersurfaces, we can show that the same instability emerges not only on the future Poincar\'e horizon but also on any null hypersurfaces. This implies that the Aretakis instability in $AdS_2$ is the result of singular behaviors of the higher-order covariant derivatives of the fields on the whole $AdS$ infinity, rather than a blow-up on a specific null hypersurface. It is also discussed that the Aretakis constants and instability are related to the conformal Killing tensors. We further explicitly demonstrate that the Aretakis constants can be derived from ladder operators constructed from the spacetime conformal symmetry.

Mass ladder operators from spacetime conformal symmetry

Ladder operators can be useful constructs, allowing for unique insight and intuition. In fact, they have played a special role in the development of quantum mechanics and field theory. Here, we introduce a novel type of ladder operators, which map a scalar field onto another massive scalar field. We construct such operators, in arbitrary dimensions, from closed conformal Killing vector fields, eigenvectors of the Ricci tensor. As an example, we explicitly construct these objects in anti-de Sitter spacetime (AdS) and show that they exist for masses above the Breitenlohner-Freedman (BF) bound. Starting from a regular seed solution of the massive Klein-Gordon equation (KGE), mass ladder operators in AdS allow one to build a variety of regular solutions with varying boundary condition at spatial infinity. We also discuss mass ladder operator in the context of spherical harmonics, and the relation between supersymmetric quantum mechanics and so-called Aretakis constants in an extremal black hole.

Quantum 3D tensionless string in light-cone gauge

We discuss the quantization of a tensionless closed string in light-cone gauge. It is known that by using a Hamiltonian BRST scheme the tensionless p-branes have no Lorentz anomaly in any space-time dimensions and no anomaly of space-time conformal symmetry in two dimensions. In this paper, we show that tensionless 3d strings in light-cone gauge also have no anomaly of space-time conformal symmetry. We also study the spectrum of a tensionless 3d closed string.

Hamiltonian BRST quantization of the conformal string

We present a new formulation of the tensionless string ( T = 0) where the space-time conformal symmetry is manifest. Using a Hamiltonian BRST scheme we quantize this Conformal String and find that it has critical dimension D = 2. This is in keeping with our classical result that the model describes massless particles in this dimension. It is also consistent with our previous results which indicate that quantized conformally symmetric tensionless strings describe a topological phase away from D = 2. We reach our result by demanding nilpotency of the BRST charge and consistency with the Jacobi identities. The derivation is presented in two different ways: in operator language and using mode expansions. Careful attention is paid to regularization, a crucial ingredient in our calculations.

Classical and quantized tensionless strings

From the ordinary tensile string we derive a geometric action for the tensionless ( T=0) string and discuss its symmetries and field equations. The Weyl symmetry of the usual string is shown to be replaced by a global space-time conformal symmetry in the T→0 limit. We present the explicit expressions for the generators of this group in the light-cone gauge. Using these, we quantize the theory in an operator form and require the conformal symmetry to remain a symmetry of the quantum theory. Modulo details concerning zero-modes that are discussed in the paper, this leads to the stringent restriction that the physical states should be singlets under space-time diffeomorphisms, hinting at a topological theory. We present the details of the calculation that leads to this conclusion.

Realization of spacetime conformal symmetry in D=10, N=1 superspace

Starting from an investigation of D=10, N=1 superspace geometry, we identify realization of spacetime conformal symmetry on the superspace supergravity superfields. It is shown that the requirement of k -symmetry of the Green-Schwarz action is consistent with this conformal D=10, N=1 superspace supergravity theory. The relationship with the third-rank and seventh-rank field strength formulations is also clarified.

Integrability and virasoro symmetry of the noncritical baxter/ising model

The properties of a noncritical c = 1 lattice Virasoro algebra for the exactly solvable Baxter eight-vertex model are studied in the decoupling limit in which the Baxter model is equivalent to a staggered pair of Ising models. The Virasoro operator L 0 is given by the first moment of the XY spin chain density which is proportional to the logarithm of the corner transfer matrix (CTM). The eigenmodes of L 0 are explicitly constructed in terms of lattice fermion operators. They are related to the eigenmodes of the XY hamiltonian by harmonic analysis on the spectral (rapidity) torus, which arises as the CTM group manifold. At the critical point, the results reduce to a momentum space formulation of space-time conformal symmetry. Two related Virasoro algebras are constructed from the fermionic eigenmode operators of L 0. One is a generalization of the T = T c conformal algebra away from the critical point. The other represents diffeomorphisms of the real rapidity circle, which is compact due to the presence of the space-time lattice. Both are spectrum generating algebras of the corner transfer matrix. In the scaling limit, the Virasoro operators are expressed in terms of a massive, free Dirac field. For n ⩾ −1, the L n 's are given by integrals of local densities. We discuss the relation between these densities and the integrals of motion and higher spectral flows that characterize the integrability of the system.