AdS/CFT Correspondence
The AdS/CFT correspondence, also known as the gauge/gravity duality, is a conjectured relationship between a gravitational theory in Anti-de Sitter space (AdS) and a conformal field theory (CFT) defined on its boundary. The correspondence states that certain observables in the gravitational theory are equivalent to correlation functions in the CFT.
While there is no single set of formulas that encapsulates the entire AdS/CFT correspondence, I can provide you with some key formulas and relations that are often used in the context of this duality:
1. AdS metric:
The metric for the five-dimensional Anti-de Sitter space (AdS5) can be written as:
ds^2 = (R^2 / z^2) (dz^2 + dx^2 + dy^2 - dt^2)
where R is the AdS radius, (x, y, t) are the boundary coordinates, and z is the radial coordinate.
2. Boundary-to-bulk relation:
The correspondence relates bulk fields in AdS to boundary operators in the CFT. In particular, for a bulk scalar field φ(z, x, y, t), the relation is given by:
φ(z, x, y, t) = z^Δ O(x, y, t)
where O(x, y, t) is the dual operator in the CFT and Δ is the scaling dimension of the operator.
3. AdS Energy-Momentum Tensor:
The energy-momentum tensor (Tμν) in the CFT corresponds to the bulk metric perturbations in the AdS gravitational theory. The precise relation involves the metric fluctuations hμν and the AdS radius R:
Tμν(x) = lim(z→0) ((1 / 16πG5) (hμν(z, x) - (1 / 2) h(z, x) gμν(z, x)))
where G5 is the five-dimensional Newton's constant.
4. Entanglement entropy:
The entanglement entropy of a region in the CFT can be related to the area of a minimal surface in the bulk AdS. For a boundary region A, the entanglement entropy S_A is given by:
S_A = (Area of minimal surface in AdS) / (4G_N)
where G_N is the Newton's constant in the bulk theory.
It's important to note that the AdS/CFT correspondence is a highly intricate and active area of research, and the formulas and techniques used can vary depending on the specific context and details of the theories involved. The formulas above provide a glimpse into some of the key relations used in this duality.
