Abstract
We study holographically non-local observables in field theories at finite temperature and in the large $d$ limit. These include the Wilson loop, the entanglement entropy, as well as an extension to various dual extremal surfaces of arbitrary codimension. The large $d$ limit creates a localized potential in the near horizon regime resulting in a simplification of the analysis for the non-local observables, while at the same time retaining their qualitative physical properties. Moreover, we study the monotonicity of the coefficient $\alpha$ of the entanglement's area term, the so called area theorem. We find that the difference between the UV and IR of the $\alpha$-values, normalized with the thermal entropy, converges at large $d$ to a constant value which is obtained analytically. Therefore, the large $d$ limit may be used as tool for the study and (in)validation of the renormalization group monotonicity theorems. All the expectation values of the observables under study show rapid convergence to certain values as $d$ increases. The extrapolation of the large $d$ limit to low and intermediate dimensions shows good quantitative agreement with the numerical analysis of the observables.
Highlights
The nonlocal observables have played an important role in the study of theoretical physics over the years
We rely on the fact that the bulk geometry is well defined at any dimension and we will assume that the dictionary between extremal surfaces and nonlocal observables is valid at any d
We have considered the limit of large number of dimensions d, both analytically and numerically
Summary
The nonlocal observables have played an important role in the study of theoretical physics over the years. The entanglement entropy as well as the holographic functions α and c have not been expressed in an exact closed form in the presence of scales Their monotonicity is guaranteed only for theories with Lorentz symmetry and it is an interesting task to analytically obtain them and study their properties along the different RG flows. We find that the difference of the UV and the IR contributions to the function α converges in the large d limit to a certain number, simplifying tremendously the computation This number reveals the violation of the area theorem in the theories under discussion, as expected, due to the breaking of Lorentz invariance. We observe that the extrapolation of the large d limit to low dimensions shows a good quantitative agreement with the numerical analysis on the observables
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