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It is suggested that the original, minimal Kaluza-Klein theory should be extended by adding a 5-dimensional version of the Gibbons-Hawking gravitational surface term. It is then demonstrated that the usual dimensional reduction of the newly added surface (boundary) term leads to the emergence of the famous Abelian Chern-Simons term. It is stressed that the advent of this Chern-Simons term is not merely a parametrization artefact but a real thing. Finally, the issue of finite-ranged electromagnetic interaction due to massive photons on a plane has been interpreted in terms of the violation of the local gauge invariance of this extended version of Kaluza-Klein theory., Comment: 10 pages

We explore mathematical relationship between holographic Wilsonian renormalization group(HWRG) and stochastic quantization(SQ) motivated by the similarity of the monotonicity in RG flow with Langevin dynamics of non-equilibrium thermodynamics. We look at scalar field theory in AdS space with its generic mass, self-interaction, and boundary deformation in the momentum space. Identifying the stochastic time $t$ with radial coordinate $r$ in AdS, we establish maps between the fictitious time evolution of stochastic multi point correlation function and the radial evolution of multi-trace deformation, which respectively, express the relaxation process of Langevin dynamics and holographic RG flow. We especially consider marginal multi-trace deformation on the AdS boundary which is successfully captured by a Langevin dynamics of SQ., Comment: 50+1 pages, 3 figures, typos are corrected, review section is added

In this paper, we explore the relationship between holographic Wilsonian renormalization groups and stochastic quantization in conformally coupled scalar theory in AdS$_{4}$. The relationship between these two different frameworks is firstly proposed in arXiv:1209.2242 and tested in various free theories. However, research on the theory with interactions has recently begun. In this paper, we show that the stochastic four-point function obtained by the Langevin equation is completely captured by the holographic quadruple trace deformation when the Euclidean action $S_{E}$ is given by $S_{E}=-2I_{os}$ where $I_{os}$ is the holographic on-shell action in the conformally coupled scalar theory in AdS$_{4},$ together with a condition that the stochastic fictitious time $t$ is also identified with AdS radial variable $r$. We extensively explore a case that the boundary condition on the conformal boundary is Dirichlet boundary condition, and in that case, the stochastic three-point function trivially vanishes. This agrees with that the holographic triple trace deformation vanishes when Dirichlet boundary condition is applied on the conformal boundary., Comment: 17+1 pages

We explore the mathematical relationship between holographic Wilsonian renormalization group(HWRG) and stochastic quantization(SQ) of scalar field theory with its generic mass, self-interaction and $n$-multiple-trace deformation on the $d$-dimensional conformal boundary defined in AdS$_{d+1}$ spacetime. We understand that once we define our Euclidean action, $S_E$ as $S_E\equiv -2S_B$, then the stochastic process will reconstruct the holographic Wilsonian renormalization group data via solving Langevin equation and computing stochastic correlation functions. The $S_B$ is given by $S_B=S_{\rm ct}+S_{\rm def}$, where $S_{\rm ct}$ is the boundary counter term and $S_{\rm def}$ is the boundary deformation which gives a boundary condition. In our study, we choose the boundary condition adding (marginal)$n$-multiple trace deformation to the holographic dual field theory. In this theory, we establish maps bewteen ficticious time, $t$ evolution of stochastic $n$-point, ($2n-2$)-point correlation functions and the (AdS)radial, $r$ evolution of $n$-multiple-trace and ($2n-2$)-multiple-trace deformations respectively once we take identifications of $r=t$ and between some of constants appearing in both sides., Comment: 41 pages and 3 figures, typos are corrected

We study holographic entanglement entropy in 5-dimensional charged black brane geometry obtained from Einstein-SU(2)Yang-Mills theory defined in asymptotically AdS space. This gravity system undergoes second order phase transition near its critical point affected by a spatial component of the Yang-Mills fields, which is normalizable mode of the solution. This is known as phase transition between isotropic and anisotropic phases. We get analytic solutions of holographic entanglement entropies by utilizing the solution of bulk spacetime geometry given in arXiv:1109.4592, where we consider subsystems defined on AdS boundary of which shapes are wide and thin slabs and a cylinder. It turns out that the entanglement entropies near the critical point shows scaling behavior such that for both of the slabs and cylinder, $\Delta_\varepsilon S\sim\left(1-\frac{T}{T_c}\right)^\beta$ and the critical exponent $\beta=1$, where $\Delta_\varepsilon S\equiv S^{iso}-S^{aniso}$, and $S^{iso}$ denotes the entanglement entropy in isotropic phase whereas $S^{aniso}$ denotes that in anisotropic phase. We suggest a quantity $O_{12}\equiv S_1-S_2$ as a new order parameter near the critical point, where $S_1$ is entanglement entropy when the slab is perpendicular to the direction of the vector order whereas $S_2$ is that when the slab is parallel to the vector order. $O_{12}=0$ in isotropic phase but in anisotropic phase, the order parameter becomes non-zero showing the same scaling behavior. Finally, we show that even near the critical point, the first law of entanglement entropy is hold. Especially, we find that the entanglement temperature for the cylinder is $\mathcal T_{cy}=\frac{c_{ent}}{a}$, where $c_{ent}=0.163004\pm0.000001$ and $a$ is the radius of the cylinder., Comment: 1+29 pages, 4 figures, typos are corrected, a few paragraphs are added in introduction and discussion section is added

Holographic model of massive scalar field with its self-interaction $\lambda\phi^n$ in AdS space is able to give a logarithmic scale dependence to marginal multi trace deformation couplings on its dual conformal field theory, where $\lambda$ is the self-interaction coupling of the scalar field, $\phi$ and $n$ is an integral number. In arXiv:1501.06664, the authors realize this feature by looking at bulk scalar solutions near AdS boundary imposing a specific boundary condition between the coefficients of non-normalizable and normalizable modes of the scalar field excitations. We study the same holographic model to see scale dependence of marginal deformations on the dual conformal field theory by employing completely different method: {\it holographic Wilsonian renormalization group}. We solve Hamilton-Jacobi equation derived from the holographic model of massive scalar with $\lambda\phi^n$ interaction and obtain the solution of marginal multi trace deformations up to the leading order in $\lambda$. It turns out that the solution of marginal multi trace deformation also presents logarithmic behavior in energy scale near UV region., Comment: 11 pages

We study a mathematical relationship between holographic Wilsonian renormalization group and stochastic quantization framework. We extend the original proposal given in arXiv:1209.2242 to interacting theories. The original proposal suggests that fictitious time(or stochastic time) evolution of stochastic 2-point correlation function will be identical to the radial evolution of the double trace operator of certain classes of holographic models, which are free theories in AdS space. We study holographic gravity models with interactions in AdS space and establish a map between the holographic renormalization flow of multi-trace operators and stochastic $n$-point functions. To give precise examples, we extensively study conformally coupled scalar theory in AdS$_6$. What we have found is that the stochastic time $t$ dependent 3-point function obtained from Langevin equation with its Euclidean action being given by $S_E=2I_{os}$ is identical to holographic renormalization group evolution of holographic triple trace operator as its energy scale $r$ changes once an identification of $t=r$ is made. $I_{os}$ is the on-shell action of holographic model of conformally coupled scalar theory at the AdS boundary. We argue that this can be fully extended to mathematical relationship between multi point functions and multi trace operators in each framework., Comment: 1+23 pages