# Definition of concepts: AdS/CFT Gauge Quantum Gravity
Understanding digital AdS/CFT Gauge Quantum Gravity requires a deep dive into several important mathematical equations.
## Essential Formulas
1. **AdS Space Formula**
The Anti-de Sitter (AdS) space formula defines the geometry of a negatively curved space. The metric is represented as:
```
ds^2 = L^2 / z^2 (dz^2 + dx^2_1 + ... + dx^2_d)
```
where `L` is the AdS radius, `z` is the holographic coordinate, and `d` is the number of spatial dimensions.
2. **Schwarzschild-AdS Metric**
The Schwarzschild-AdS metric describes a black hole in AdS space. The formula is given by:
```
ds^2 = -(1 - mG / r^(d-2) + r^2 / L^2) dt^2 + dr^2 / (1 - mG / r^(d-2) + r^2 / L^2) + r^2 dΩ^2_(d-1)
```
where `m` is the mass of the black hole, `G` is the gravitational constant, `r` is the radial coordinate, and `dΩ^2_(d-1)` represents the metric of the unit (d-1)-sphere.
3. **Holographic Entanglement Entropy Formula**
The holographic entanglement entropy formula connects the entanglement entropy of a holographic CFT with the area of a minimal surface in the dual AdS space. The formula is:
```
S_EE = Area(Γ_A) / 4G_N
```
where `Γ_A` is the minimal surface in AdS whose boundary coincides with the entangling surface of the CFT, and `G_N` is the Newton's gravitational constant in the bulk.
4. **N-Point Correlation Functions**
The n-point correlation functions in the CFT are related to the n-point functions of the dual AdS fields. An example of the two-point correlation function is:
```
<O(x) O(y)> = C / |x - y|^(2Δ)
```
where `C` is a constant, `x` and `y` are the points in the CFT, and `Δ` is the scaling dimension of the operator `O`.
5. **Wilson Loop**
The Wilson loop is an important observable in gauge theories. In the context of AdS/CFT, the expectation value of the Wilson loop in the CFT is related to the minimal area of a surface in AdS:
```
<W(C)> = exp(-Area(Σ)/4πα')
```
where `W(C)` is the Wilson loop in the CFT, `Σ` is the minimal area of the surface in AdS whose boundary coincides with the loop `C`, and `α'` is the string tension.
6. **Holographic Renormalization**
The process of holographic renormalization is used to compute finite quantities in the CFT from the divergent quantities in the bulk. One example is the computation of the expectation value of the energy-momentum tensor:
```
<T^μν> = lim(z->0) z^d (K^μν - h^μν K)
```
where `K^μν` is the extrinsic curvature, `h^μν` is the induced metric on the boundary, and `K` is the trace of the extrinsic curvature.
7. **Thermodynamics and Phase Transitions**
The thermodynamics and phase transitions of a holographic CFT can be studied using the thermodynamic properties of the dual AdS black holes. For example, the Hawking-Page phase transition between a thermal AdS and a Schwarzschild-AdS black hole is given by:
```
S_Hawking-Page = (Area(A)) / 4G_N
```
where `Area(A)` is the area of the horizon of the black hole, and `G_N` is the Newton's gravitational constant in the bulk.
These equations provide a solid foundation for further investigation into the fascinating world of digital AdS/CFT Gauge Quantum Gravity. By understanding and applying these formulas, researchers can gain valuable insights into the intricate connection between quantum gravity and quantum field theories.
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