# How local fields transform in the holographic boundary

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Consider an holographic description of gravity $f:\Omega \rightarrow \partial \Omega$ such that gravitational fields and curvature in a neighbourhood $\Omega$ of 4D spacetime induce local fields on $\partial \Omega$. So if $G(\Omega)$ is a shorthand to describe all gravitational fields in the neighbourhood, then $g := f(G(\Omega))$ is a shorthand to describe all local fields in the holographic dual in $\partial \Omega$

I understand that we rarely have detailed information of the $f$ map. But usually one can ask questions about $f$ properties. In this case, I'm interested by the fact that for a given point $x$ in the spacetime, there are infinite neighbourhoods $\Omega$ such that $x \in \partial \Omega$

Question: what would you call the following property: Be $x$ a point in spacetime, and be $\Omega(x)$ a family of compact, simply connected sets such that if $\Omega \in \Omega(x)$ then $x \in \partial \Omega$. Assume that gravitational fields $G(\Omega)$ are defined everywhere on spacetime. Then the corresponding fields in the holographic dual $g:=f(G(\Omega)):\partial \Omega$ are such that if we take quantum field observable operators on $g$ and obtain vacuum expectation values of the operators around a small neighbourhood around $x$ with $x \in X_h$, then vacuum expectation values $\langle A_{g(\Omega)} \rangle_{X_h}$ of operators $A_g$ in the dual $g$ satisfy the following invariance property:

be $\Omega \in \Omega(x)$ and $\Omega' \in \Omega(x)$. Then $$\langle A_{g(\Omega)} \rangle_{X_h} = \langle A_{g(\Omega')} \rangle_{X_h}$$

for $X_h$ small enough

This property seems to be intuitively some kind of condition that the holographic dual fields also behave as local fields in the embedding space. I'm not sure if it makes physical sense to demand it, but I'm trying to understand if it is an interesting property to demand from holographic duals. Naively I would be tempted to think that such invariance suggests that the holographic dual is a real physical field, in some sense

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