Quantum entanglement measures from Hyperscaling violating geometries with finite radial cut off at general d, $\theta$ from the emergent global symmetry

Abstract

The quantum entanglement measures for $T{\overline{T}}$ deformed field theory on boundary, deformation coefficient $\mu$, with dual bulk geometry with finite radial cutoff $\rho_c$, for entangling region is single or disjoint intervals on the boundary, of length l is expected to give global description of these measures over the complete parameter-regime of $(l, \mu)$ or on 2D $(l,\rho_c)$ plane, because it is solvable irrelevant deformation. Here, to find quantum-measures through RT prescription, from Hyperscaling violating bulk geometry with finite radial cut off, we found mathematically it is impossible, to obtain such global form, since the turning point $\rho_0 (l,\rho_c)$, neither in its exact or in any perturbative form, is solvable globally, can describe these quantum measures at most locally over some specific regime in 2D $(l,\rho_c)$ plane! However, to find such global form, we found, on application of RT formalism, a global symmetry structure, from the considered geometry emerges, over 2D parameter-space, irrespective of d, $\theta$, which alongwith global b.c and other consistency conditions, fix $\rho_0(l,\rho_c)$ globally, exactly in $l >> \rho_c $ and $\rho_c >> l$ regime and as some interpolating expressions, very close to the exact one in other regime. Some of the quantum entanglement measures with this $\rho_0 (l,\rho_c)$, with our intuitively predicted behaviour for them in the deformed theory, derived and shown, behaving accordingly. The impact of this emergent symmetry on these quantum-measures is discussed, the possible space time origin of this symmetry is explored, although the later aspect is subjected to a proper and detailed study.