AdS/CFT Isometry Mapping

1. The Symmetry of the "C": The Conformal Group

First, let's characterize the symmetry group of a d-dimensional Conformal Field Theory. A conformal transformation is a coordinate transformation $x^\mu \to x'^\mu$ that preserves angles, which is equivalent to rescaling the metric by a position-dependent factor: $g'{\mu\nu}(x') = \Omega(x) g{\mu\nu}(x)$.

In a d-dimensional flat spacetime, these transformations are generated by a finite set of vector fields (for $d > 2$). Your background in QFT has likely made you familiar with the Poincaré group, which is a subgroup of the conformal group. The full set of generators is:

• Translations ($P_\mu$): $d$ generators.
• Lorentz Transformations ($M_{\mu\nu}$): $d(d-1)/2$ generators. These two form the Poincaré group.
• Dilatations ($D$): 1 generator, responsible for scale transformations $x^\mu \to \lambda x^\mu$.
• Special Conformal Transformations ($K_\mu$): $d$ generators. These can be thought of as an inversion, followed by a translation, followed by another inversion.

The total number of generators is $d + \frac{d(d-1)}{2} + 1 + d = \frac{(d+2)(d+1)}{2}$. This is precisely the dimension of the Lie group $SO(d,2)$ for a Lorentzian signature metric, or $SO(d+1,1)$ for a Euclidean metric. The commutation relations of these generators indeed form the Lie algebra $\mathfrak{so}(d,2)$.

The key takeaway is that the symmetry group of a d-dimensional CFT is the conformal group, which is isomorphic to $SO(d,2)$.

2. The Symmetry of the "AdS": The Isometry Group of AdS Space

Now, let's turn our attention to the gravity side of the duality. The symmetry of a spacetime is described by its isometry group—the set of transformations that leave the metric invariant. These are generated by Killing vectors. What is the isometry group of $(d+1)$-dimensional Anti-de Sitter space?

While we can write down the metric for AdS and find its Killing vectors directly, there is a much more elegant and powerful way to reveal its symmetry, which will also prove essential for connecting it to the conformal group. We can define $AdS_{d+1}$ as a geometric surface (a hyperboloid) embedded in a flat $(d+2)$-dimensional space.

Consider a flat space $\mathbb{R}^{d,2}$ with coordinates $X^A = (X^{-1}, X^0, X^1, ..., X^{d-1}, X^d)$ and a metric with signature $(-,+,\dots,+, -)$:

Note: Conventions for the signature vary. Here we use the one from resource LINK.

Within this flat ambient space, $AdS_{d+1}$ is defined as the hyperboloid given by the equation:

where $R$ is the radius of the AdS space.

The isometries of $AdS_{d+1}$ are the transformations of the ambient space $\mathbb{R}^{d,2}$ that leave this equation invariant. The transformations that preserve the quadratic form $X^A X_A$ are, by definition, the generalized Lorentz group $O(d,2)$. Thus, the isometry group of $AdS_{d+1}$ is $SO(d,2)$ (restricting to the connected component).

We have now arrived at a remarkable result:

• Symmetry of d-dim CFT: Conformal Group $\cong SO(d,2)$
• Symmetry of (d+1)-dim AdS: Isometry Group $\cong SO(d,2)$

The symmetry groups match perfectly!

3. The Manifest Isomorphism: The Embedding Space Formalism

The fact that the two groups are isomorphic is powerful, but we can do even better. We can show that they are, in a deep sense, the same group acting in different ways on the bulk and boundary. The embedding space formalism makes this relationship manifest.

You've already seen how AdS space can be embedded in $\mathbb{R}^{d,2}$. It turns out we can also represent the d-dimensional spacetime of the CFT within this same ambient space.

We do this by identifying the d-dimensional Minkowski space with the projective null cone in $\mathbb{R}^{d,2}$. The null cone is the set of points satisfying:

To get a d-dimensional space, we remove the ambiguity of scaling along the light rays by identifying all points on the same ray from the origin, $X^A \sim \lambda X^A$. This is called a projective space.

The magic is that the linear $SO(d,2)$ transformations acting on the embedding space coordinates $X^A$ induce precisely the (non-linear) conformal transformations on the d-dimensional physical spacetime coordinates.

This formalism provides a unified picture:

• We have a single ambient space, $\mathbb{R}^{d,2}$, with the symmetry group $SO(d,2)$.
• Anti-de Sitter space ($AdS_{d+1}$) is the hyperboloid $X^A X_A = -R^2$ within this space. The $SO(d,2)$ transformations act as isometries on it.
• The CFT spacetime can be identified with the conformal boundary of AdS. In the embedding formalism, this boundary is reached as the coordinates on the hyperboloid go to infinity. This boundary is projectively equivalent to the null cone $X^A X_A = 0$. The $SO(d,2)$ transformations act as conformal transformations on it.

CAPTION: A visualization of the embedding space for AdS_3. The ambient space is R^{2,2}. AdS_3 is the hyperboloid, and its boundary is the cylinder at infinity. A 2D CFT can be defined on this boundary. The SO(2,2) isometries of the bulk act as conformal transformations on the boundary.

This beautiful geometric unity is the first and most fundamental piece of evidence for the AdS/CFT correspondence. The isometries of the bulk spacetime are precisely the conformal symmetries of the boundary field theory.

Conclusion

Today we performed a foundational consistency check of the AdS/CFT correspondence by matching the symmetries on both sides of the duality.

Key Takeaways:

• The global symmetry group of a d-dimensional CFT is the conformal group, which is isomorphic to $SO(d,2)$.
• The symmetry group of a (d+1)-dimensional Anti-de Sitter space is its isometry group, which is also $SO(d,2)$.
• The embedding space formalism makes this isomorphism manifest. Both AdS space (as a hyperboloid) and the CFT spacetime (as a projective null cone) can be embedded within the same flat $\mathbb{R}^{d,2}$ space. The $SO(d,2)$ transformations of this ambient space act as isometries in the bulk and as conformal transformations on the boundary.

This matching of symmetries is far from a coincidence; it is a deep statement about the geometry of holography. The boundary theory inherits its symmetries directly from the isometries of the bulk spacetime.

Next Lesson Preview:

With the correspondence now grounded in a shared symmetry structure, we can build a more detailed dictionary. In the next lesson, we will explore the operator-field correspondence, which provides a precise map between operators in the CFT and fields propagating in the AdS bulk. This will be our first step toward using the duality for concrete calculations, showing how the mass of a bulk field is determined by the scaling dimension of its dual boundary operator.