We discuss some geometrical issues related to spacelike minimal surfaces in $AdS_m$ with null polygonal boundaries at conformal infinity. In particular for $AdS_4$, two holomorphic input functions for the Pohlmeyer reduced system are identified. This system contains two coupled differential equations for two functions $α(z,\bar z)$ and $β(z,\bar z)$, related to curvature and torsion of the surface. Furthermore, we conjecture that, for a polynomial choice of the two holomorphic functions, the relative positions of their zeros encode the conformal invariant data of the boundary null $2n$-gon.
We completely classify Friedmann-Lemaître-Robertson-Walker solutions with spatial curvature $K=0,\pm 1$ and equation of state $p=wρ$, according to their conformal structure, singularities and trapping horizons. We do not assume any energy conditions and allow $ρ< 0$, thereby going beyond the usual well-known solutions. For each spatial curvature, there is an initial spacelike big-bang singularity for $w>-1/3$ and $ρ>0$, while no big-bang singularity for $w<-1$ and $ρ>0$. For $K=0$ or $-1$, $-1<w<-1/3$ and $ρ>0$, there is an initial null big-bang singularity. For each spatial curvature, there i…
We write down Crofton formulas--expressions that compute lengths of spacelike curves in asymptotically AdS$_3$ geometries as integrals over kinematic space--which apply when the curve and/or the background spacetime is time-dependent. Relative to their static predecessor, the time-dependent Crofton formulas display several new features, whose origin is the local null rotation symmetry of the bulk geometry. In pure AdS$_3$ where null rotations are global symmetries, the Crofton formulas simplify and become integrals over the null planes, which intersect the bulk curve.
This thesis is devoted to some particular aspects of integrability in $4d$ SUSY gauge theories. Taking advantage of the integrable structures emergent in the theory, non-local observables such as null polygonal Wilson loops are studied in $4d$ planar $\mathcal{N}=4$ Super Yang-Mills. Their duality with the $4d$ gluon scattering amplitudes makes the analysis even more interesting. The so-called \emph{Pentagon approach}, an application of the Operator Product Expansion (OPE) method to the null polygonal Wilsol loops, makes possible a non-perturbative evaluation of these objects. They are recast …
We present a new application of the Regge-Teitelboim method for treating symmetries which are defined asymptotically. It may be regarded as complementary to the one in their original 1974 paper. The formulation is based on replacing an asymptotic plane by the two--sheeted ``hourglass" shaped surface obtained by joining smoothly an incoming hyperboloid with an outgoing one. The hyperboloids have a fixed radius, and as one moves the center of the hourglass along the time axis one covers the whole of spacetime. The motivation is to study radiation, and the hourglass is well suited to the task bec…
We consider excited states in a CFT, obtained by applying a weak unitary perturbation to the vacuum. The perturbation is generated by the integral of a local operator $J^{(n)}$ of modular weight $n$ over a spacelike surface passing through $x = 0$. For $\vert n \vert \geq 2$ the modular Hamiltonian associated with a division of space at $x = 0$ picks up an endpoint contribution, sensitive to the details of the perturbation (including the shape of the spacelike surface) at $x = 0$. The endpoint contribution is a sum of light-ray moments of the perturbing operator $J^{(n)}$ and its descendants. …
We present a complete study of the geodesics around naked singularities in AdS$_3$, the three-dimensional anti-de Sitter spacetime. These stationary spacetimes, characterized by two conserved charges --mass and angular momentum--, are obtained through identifications along spacelike Killing vectors with a fixed point. They are interpreted as massive spinning point particles, and can be viewed as three-dimensional analogues of cosmic strings in four spacetime dimensions. The geodesic equations are completely integrated and the solutions are expressed in terms of elementary functions. We classif…
We initiate an exploration of the conformal bootstrap for $n>4$ point correlation functions. Here we bootstrap correlation functions of the lightest scalar gauge invariant operators in planar non-abelian conformal gauge theories as their locations approach the cusps of a null polygon. For that we consider consistency of the OPE in the so-called snowflake channel with respect to cyclicity transformations which leave the null configuration invariant. For general non-abelian gauge theories this allows us to strongly constrain the OPE structure constants of up to three large spin $J_j$ operators (…
The Einstein-Hilbert Lagrangian has no well-defined variational derivative with respect to the metric. This issue has to be tackled by adding a suitable surface term to the action, which is a peculiar feature of gravity. We also know that null surfaces in spacetime exhibit (observer-dependent) thermodynamic features. This suggests a possible thermodynamic interpretation of the boundary term when the boundary is a null surface. For timelike/spacelike surfaces it is easy to construct the boundary term but there are some subtleties in the case of the null surface. The correct form of boundary ter…
We work out the map between null polygonal hexagonal Wilson loops and spinning three point functions in large $N$ conformal gauge theories by mapping the variables describing the two different physical quantities and by working out the precise normalization factors entering this duality. By fixing all the kinematics we open the ground for a precise study of the dynamics underlying these dualities -- most notably through integrability in the case of planar maximally supersymmetric Yang-Mills theory.
In this article we study the geodesic motion of test particles and light in the five-dimensional Myers-Perry-anti de Sitter spacetime. We derive the equations of motion and present their solutions in terms of the Weierstraß $\wp$-, $σ$- and $ζ$-functions. With the help of parametric diagrams and effective potentials we analyze the geodesic motion and give a list of all possible orbit types for timelike, null and spacelike geodesics. We plot examples of the orbits and take a look at the photon region in five dimensions. Furthermore we study spacelike geodesics and their relation to the AdS/CFT …
