Analytic integrability for holographic duals with Joverline{T} deformations

Dibakar Roychowdhury

doi:10.1007/jhep09(2020)053

Dibakar Roychowdhury

Open Access

https://doi.org/10.1007/jhep09(2020)053

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Abstract

We probe warped BTZ ×S3 geometry with various string solitons and explore the classical integrability criteria of the associated phase space configurations using Kovacic’s algorithm. We consider consistent truncation of the parent sigma model into one dimension and obtain the corresponding normal variational equations (NVE). Two specific examples have been considered where the sigma model is reduced over the subspace of the full target space geometry. In both examples, NVEs are found to possess Liouvillian form of solutions which ensures the classical integrability of the associated phase space dynamics. We address similar issues for the finite temperature counterpart of the duality, where we analyse the classical phase space of the string soliton probing warped BTZ black string geometry. Our analysis reveals a clear compatibility between normal variational equations and the rules set by the Kovacic’s criteria. This ensures the classical integrability of the parent sigma model for the finite temperature extension of the duality conjecture.

Highlights

JHEP09(2020)053 where we set the AdS length scale, l = 1
We show analytic integrability for classical strings probing warped BT Z × S3 geometries both for the zero as well as finite temperature examples
We reduce the parent sigma model over different subspaces of the full target space geometry and obtain the corresponding dynamical equations those are consistent with various other physical constraints of the sigma model

Summary

Analysis and results

Where (τ, σ) are the so called world-sheet coordinates, XM s are the target space coordinates and (α )−1 measures the tension of the string soliton. The invariant plane [28] in this reduced phase space may be obtained by setting, θ = θ = 0 which trivially solves (2.22) and is an allowed space of solution. In order to obtain NVE in a consistent way, we set r = r0 = constant which chooses the invariant plane [28] in the phase space as, {r = r0, Jr ∼ r = 0}. Substituting this into (2.49) and (2.50) we find, φ = χ = 0. The above analysis ensures the classical integrability of strings over warped BT Z × S1

Sigma models on warped BTZ black string

Summary and final remarks