Localized quench in 1 + 1 conformal field theory

We study the boundary dual of AdS3 spacetime with a point particle. Particles in AdS3 generate topological defects, which allows to formulate the geodesic image method for boundary correlators. We propose the generalization of the geodesic recipe to arbitrary time intervals in case of the bulk spacetime deformed by a point particle.


Introduction
The AdS/CFT -correspondence [1] is the correspondence between (d + 1) dimensional gravity of the AdS spacetime and d -dimensional quantum field theory on the boundary of the space.The useful and important tool in holographic calculations is the geodesic approximation [2].This approximation allows to get in a simple way the correlation function.
The geodesic approximation directly relates correlation functions of the boundary QFT to geometry of the bulk spacetime.However, the geodesic prescription is valid only either for Euclidean spacetimes, or for spacelike-separated points in the Lorentzian case.Timelike geodesics in asymptotically AdS spacetimes cannot reach the boundary, therefore the timelike region is unavailable to the prescription, unless there is an analytical continuation from the original Euclidean form.Because of this, the information carried by large-time dynamics, or even real-time correlators in general, cannot be obtained from the geodesics approximation in general spacetimes.In [3] a non-trivial Euclidean continuation was constructed for the Vaidya spacetime.Another method that was used in [3,4] is making use of discontinuous timelike geodesics which go through Poincare horizon.
In our work the prescription for calculation two point correlation function in geodesic approximation for the case when the particle deforms the AdS 3 spacetime is presented.Moving particle in the bulk realizes a quantum quench in the boundary theory.Here the formula for correlation function in geodesic approximation has the form: (a,b) . ( We sum over all geodesics that connect a and b points.In the work [5] it was shown that geodesic approximation gives the exact answer (i.e.coincides with the answer from the [6,7]) for the correlation function if the AdS 3 spacetime is orbifold and does not reproduce exact answer otherwise.We generalize the prescription to timelike-separated points by utilizing a set of auxiliary geodesics.
The AdS 3 is maximally symmetric solution of Einstein equations with the negative cosmological constant.In the embedding coordinates the AdS spacetime is a hyperboloid.If we parametrize the embedding coordinates by global coordinates (t, χ, φ) the metric of AdS 3 spacetime can be written as follows: where t is the time coordinate, χ is the radial coordinate and φ is the angular coordinate with period 2π.The AdS 3 conformal boundary corresponds to χ → ∞.

AdS 3 spacetime with static particle
We briefly discuss the Deser-Jackiw solution of Einstein equations with delta function in the RHS and cosmological constant equals to −1 [8,9].It describes a static particle in the center of the AdS 3 spacetime.Solution of these equations has the form: The metric is the same as the metric for empty spacetime but an angle of the space has different ranges.
Here A is the parameter associated with the mass parameter: where G is the Newtonian constant and μ is the mass of the particle.In the Fig. 1A we show the point particle in the center of the AdS 3 spacetime.This particle cuts out the wedge and faces of the wedge must be glued.

AdS 3 spacetime with moving particle
In this section we explain how the moving particle can be obtained in the AdS 3 spacetime.In [10] it was shown that using the group language for description the AdS 3 spacetime with a particle we can boost this particle by Lorentz transformation and get the spacetime with a moving particle in a simple way.The particle moves periodically with the T = 2π period.Massive particle does not reach the boundary.
Consider the action of the isometry in the AdS 3 spacetime.The * -transformation is the isometry corresponding to the identification of the faces of the wedge.In the case of moving particle the isometry transformation is as follows [10]: (5) where B ξ and F ξ are defined as We are interested in the boundary theory so in the formulae (5) we take a limit of radial coordinate (i.e.χ → ∞) and get the action of isometry on the boundary (b means the boundary): In the Fig. 1B we illustrate the AdS 3 spacetime with a moving massive particle.With the expression for the induced boundary identification at hand, one can now formulate the geodesic images prescription for spacelike geodesics, which was explained in [10].

Holographic image method and correlation function
Consider the case of the empty AdS 3 spacetime.For the calculation of two point correlation function we use the geodesics approximation [2].In this approximation one defines the correlator: where a and b are boundary points with coordinates (φ a , t a ) and (φ b , t b ).If the interval between a and b points is spacelike then the L ren is the renormalized length of the geodesic connecting these points [10].In case of timelike interval between a and b points we have to use the the prescription from the work [11].
The length of geodesic between two points in the AdS 3 is: QUARKS-2016 then the correlation function (10) has the form: In case of spacetime deformed by particle we have to change the general formulae for correlation function (12) and use the follows: where G Δ,AdS (φ, t) is given by ( 12).The presence of the Φ-factors in summands is related to the change of the causal relation between two points on the boundary under the isometry transformation (9).
Here the subscript l.s. in the LHS means a living space of the boundary of the AdS 3 with static or moving defects and (φ a , t a ) are coordinates of the image points obtained by n-times applications of the isometry * and #transformations respectively.We can use the #-transformation as an inverse * -transformation as follows: (φ # a , t # a ) * = (φ a , t a ).
In case of deformed AdS spacetime for calculation of correlation function we have to use the extra factors: where Factors Z are related to renormalizations [10].Geodesic in (13) connecting points a * , a * * ,... and b are image geodesics.Length of each image geodesic equals to the length of a winding geodesic between a and b points [10].
The images prescription for the correlator can be generalized to the case when the bulk spacetime is deformed by several particles.

Conclusion
We have calculated the two point correlation function in holographic approach via geodesic approximation.We have considered the case when geometry of AdS 3 is deformed by moving and static point massive particles.The whole Virasoro symmetry remains unchanged only in case when the mass parameter equals to 2π/n, where n is integer (see Fig. 2) but in the general case the symmetry is broken by the conical defect and this is manifest in the fact that correlation function in the geodesic approximation has the singularities in ϕ dependence (see Fig. 3).

Figure 1 .
Figure 1. A. AdS 3 with static massive particle in the center.Particle cats out the wedge from the spacetime.B. AdS 3 with moving massive particle.Here the mass parameter α = π/8 and the boost parameter ξ = 1.

Figure 2 .Figure 3 .
Figure 2. Density plot of inverse correlation function as a function of G −1 (t, ϕ) is presented.In both of the cases the massive moving particle with the mass parameter 2π/n where n is integer deforms the AdS 3 spacetime.Parameter values are: ξ = 0, α = 3π/2.In the left plot the massive particle is static.In the right plot the massive particle is moving.Parameter values are: ξ = 0.6, α = 3π/2 )