AdS isometries and holographic symmetry

The isometry algebra of Lorentzian $AdS_{d+1}$ is the conformal algebra $\mathfrak{so}(2,d)$ of a $d$-dimensional Lorentzian boundary. This matches symmetry representations. It does not, by itself, establish a holographic duality or determine a boundary theory’s spectrum and interactions. The global spacetime, group cover, field equation and boundary condition are all part of a concrete calculation.

Here $d\geq3$, the AdS radius is $L>0$, and the bulk metric has one timelike direction with mostly-plus signature. The scalar examples concern a free linearized field obeying $(\Box-m^2)\phi=0$ on fixed AdS, with $m^2$ including any constant curvature-coupling contribution. They are not formulas for arbitrary spin or interacting bulk theories. The CFT guide supplies the primary and descendant conventions.

The hyperboloid and its time cover

Embed the quadric in $\mathbb R^{2,d}$:

$$-X_{-1}^2-X_0^2+\sum_{i=1}^{d}X_i^2=-L^2.$$

Its connected matrix isometry group is $SO_0(2,d)$. Parametrize it by

$$X_{-1}=L\cosh\rho\cos\tau,\qquad X_0=L\cosh\rho\sin\tau,\qquad X_i=L\sinh\rho\,n_i, \qquad \mathbf n\in S^{d-1}.$$

The induced metric is

$$ds^2=L^2\left(-\cosh^2\rho\,d\tau^2+d\rho^2 +\sinh^2\rho\,d\Omega_{d-1}^2\right).$$

On the embedded hyperboloid $\tau$ has period $2\pi$, giving topology $S^1\times\mathbb R^d$ and closed timelike curves. In the usual physical AdS spacetime one instead takes $\tau\in\mathbb R$: this is the universal cover of the space, with topology $\mathbb R\times\mathbb R^d$. The construction and metric are Aharony et al., Large N Field Theories, String Theory and Gravity, §2.2, equations (2.20)–(2.25).

The connected matrix isometries lift to a covering group acting on the unwrapped space. A $2\pi$ time translation is now a nontrivial isometry, not the identity. When discussing spinor states one must also retain the appropriate spin action. Unwrapping the time circle of the spacetime and taking a spin double cover of the isometry group are different operations; the word “Spin” does not guarantee a universal group cover in indefinite signature.

At $\rho\to\infty$, rescaling the metric gives the boundary conformal metric $-d\tau^2+d\Omega_{d-1}^2$. Thus the unwrapped boundary is the Lorentzian cylinder $\mathbb R\times S^{d-1}$. A Minkowski boundary coordinate patch covers only part of it. After Euclidean continuation, AdS becomes hyperbolic space with connected isometry group $SO_0(1,d+1)$, a different real form; see Aharony et al., §2.2, equations (2.26)–(2.29).

The AdS4 and AdS5 matrix groups

The low-dimensional identifications in the atlas are actual maps, not just coincidences of algebra dimensions:

Bulk dimension	Connected matrix group	Connected spin double cover	Kernel of the displayed cover
$AdS_4$	$SO_0(2,3)$	$Sp(4,\mathbb R)$	$\{\pm I_4\}$
$AdS_5$	$SO_0(2,4)$	$SU(2,2)$	$\{\pm I_4\}$

Here $Sp(4,\mathbb R)$ acts on four real dimensions; it is not compact $Sp(2)=USp(4)$. Its five-dimensional orthogonal representation comes from the primitive part of $\Lambda^2\mathbb R^4$. For $SU(2,2)$, a real six-dimensional subspace of $\Lambda^2\mathbb C^4$ carries the orthogonal form. In both cases $-I_4$ acts by $+I$ on an exterior square, explaining the covering kernel. These constructions are Garrett, §§2.8 and 2.11, pp. 7 and 9–10.

The maximal compact subgroups are $U(2)$ and $S(U(2)\times U(2))$, respectively; each has fundamental group $\mathbb Z$. Thus neither matrix spin group is simply connected. For example, the determinant of the first $U(2)$ block maps $S(U(2)\times U(2))$ onto $U(1)$ with simply connected kernel $SU(2)\times SU(2)$, yielding $\pi_1=\mathbb Z$. Passing from each group to its maximal compact subgroup preserves this fundamental group by the global decomposition; compare Knapp, Chapter VI, §4 and Theorem 6.46, pp. 314–318. The relevant Cartan-retraction facts and maps are recorded at Sp(4,R), SU(2,2), the AdS4 spin map and the AdS5 spin map.

Do not replace $SU(2,2)$ by compact $SU(4)$ in a unitarity question just because both complexify to type $A_3$. Their defining fours preserve Hermitian forms of different signatures. An indefinite preserved form is not the positive Hilbert inner product of a physical state representation.

Bulk field, normal mode and boundary primary

A local bulk scalar has one field component. The positive-frequency solutions of its wave equation, after a suitable boundary condition is chosen, form an infinite-dimensional one-particle Hilbert space. A positive-energy irreducible module is labeled by a lowest dimensionless energy $E_0$ and a compact spatial-spin representation; for a scalar the latter is trivial. On a cylinder of radius $L$, energies relative to the vacuum satisfy $E_0=L E_{\mathrm{physical}}$.

The conformal identification is $E_0=\Delta$ for the lowest state. For the usual free scalar quantization, the global normal modes have

$$\omega_{n\ell}=\Delta+2n+\ell, \qquad n,\ell\in\mathbb Z_{\geq0},$$

where $\omega$ is conjugate to dimensionless $\tau$ and $\ell$ labels spherical harmonics on $S^{d-1}$. The states form the primary and its descendants. This is a free-particle statement with the selected reflecting boundary condition, not a claim that interacting multiparticle energies are additive. The construction using the conformal Casimir and global modes is Penedones, TASI Lectures on AdS/CFT, §3.1, equations (61)–(70).

Derive the scalar mass–dimension relation

Near a Poincaré boundary, use

$$ds^2=\frac{L^2}{z^2} \left(dz^2+\eta_{\mu\nu}dx^\mu dx^\nu\right), \qquad z>0, \qquad \eta=\operatorname{diag}(-,+,\ldots,+).$$

The scalar wave operator follows from $\Box\phi=|g|^{-1/2}\partial_A(|g|^{1/2}g^{AB}\partial_B\phi)$:

$$L^2\Box\phi= \left[z^2\partial_z^2-(d-1)z\partial_z+z^2\Box_\eta\right]\phi.$$

For a leading falloff $\phi\sim z^\alpha f(x)$ with smooth boundary variation, the boundary-derivative term is two powers higher in $z$. The indicial equation is therefore

$$\boxed{m^2L^2=\alpha(\alpha-d),\qquad \Delta_\pm=\frac d2\pm\nu,\qquad \nu=\sqrt{\frac{d^2}{4}+m^2L^2}.}$$

In the standard quantization the boundary primary dimension is $\Delta_+$. Its source falloff has exponent $d-\Delta_+=\Delta_-$; the other independent falloff has exponent $\Delta_+$. A falloff exponent is therefore not, without specifying the quantization, an operator dimension. The relation and the two roots are Klebanov–Witten, AdS/CFT Correspondence and Symmetry Breaking, §2, equations (2.1)–(2.2), with their unit AdS radius restored here.

Real exponents require the Breitenlohner–Freedman inequality $m^2L^2\geq-d^2/4$. For a free scalar this is the usual lower stability bound with admissible AdS boundary conditions; it is not a stability test for an interacting potential or an arbitrary mixed boundary condition. In the strict window

$$-\frac{d^2}{4}<m^2L^2<-\frac{d^2}{4}+1 \qquad(0<\nu<1),$$

both standard and alternate AdS-invariant quantizations are available, with dimensions $\Delta_+$ and $\Delta_-$. The alternate choice exchanges the source/response roles. See Klebanov–Witten, §2, equation (2.3) and the discussion following it. The strict inequalities avoid the coincident-root logarithms at $\nu=0$ and the separate endpoint analysis at $\nu=1$.

A scalar in AdS4

Choose $d=3$ and $m^2L^2=-2$. Then

$$\nu=\frac12,\qquad \Delta_-=1,\qquad\Delta_+=2, \qquad -\frac94<-2<-\frac54.$$

Both roots satisfy $\Delta(\Delta-3)=-2$ and the three-dimensional nonidentity scalar bound $\Delta\geq1/2$. The leading expansion is $\phi\sim a(x)z+b(x)z^2$. Standard quantization has a dimension-two operator; alternate quantization has a dimension-one operator. The mass and isometry algebra are identical, but the boundary theories and source assignments differ. Under the standard free boundary condition, the first global mode has $\omega=2$; its vector descendants have $\omega=3$.

For contrast, a massless scalar in $AdS_5$ has $d=4$ and roots $0,4$. It lies outside the two-quantization window and the standard dimension is $4$. The zero root is the source exponent, not a second propagating identity operator. This is one reason that solving the quadratic equation alone does not finish the representation problem.

What matching symmetry establishes

The equality of algebras organizes a possible correspondence between bulk states and boundary operators. A claim of AdS/CFT requires more: the theories, boundary conditions, spectra, interactions and observable dictionary must match. In a weakly coupled bulk approximation, one-particle fields can often be associated with suitable single-trace primaries; this interpretation uses additional large-$N$ and dynamical assumptions. Penedones separates QFT on fixed AdS in §3.2 from gravity and the correspondence in §§3.3–4.

The site records selected real groups and maps in its spacetime collection. Its compact highest-weight calculator is useful for internal compact or rotation labels, with conventions fixed. It does not classify positive-energy AdS modules, solve a bulk boundary-value problem, or test holographic duality. Return to the spacetime learning path to connect these distinctions with Lorentz fields, particle states and local operators.