Bulk Fields and Boundary Operators

Mass alone does not finish a bulk-to-boundary dictionary. For a scalar, the radial equation first supplies candidate weights; a boundary condition then chooses the source, and positivity plus CFT unitarity decide whether the choice is admissible. Spin adds a Lorentz representation, on-shell constraints, and sometimes gauge shortening. This page derives that chain for a free scalar on AdS$_{d+1}$ and then gives the corresponding symmetric-spin test.

Required background. Anti-de Sitter geometry supplies the Poincaré metric. Primaries, descendants, and conformal multiplets supplies the boundary representations. Helpful background. Unitarity bounds and null states tests the candidate dimension, while timelike AdS boundary conditions supplies the self-adjointness framework.

The scalar radial equation

In the inherited Lorentzian convention, Poincaré AdS has

$$\mathrm ds^2=\frac{L^2}{z^2} \left(\eta_{ij}\,\mathrm dx^i\mathrm dx^j-\mathrm dz^2\right), \qquad z>0,$$

with the conformal boundary at $z=0$. Consider a real scalar with positive kinetic normalization $\mathcal N_\phi>0$,

$$S=\frac{\mathcal N_\phi}{2} \int\mathrm d^{d+1}x\sqrt{\lvert g\rvert} \left(g^{MN}\partial_M\phi\partial_N\phi-m^2\phi^2\right).$$

Here $m^2$ is the complete mass in the linearized Klein–Gordon operator about the chosen AdS vacuum; any quadratic curvature coupling has already been absorbed into it. Since $\sqrt{\lvert g\rvert}=L^{d+1}z^{-(d+1)}$, the equation $(\Box+m^2)\phi=0$ becomes

$$\left[ -z^2\partial_z^2+(d-1)z\partial_z +z^2\Box_\eta+m^2L^2 \right]\phi=0.$$

For smooth boundary data, or for a Fourier mode at fixed boundary momentum $k_i$, the $z^2\Box_\eta$ term is suppressed as $z\to0$. Substituting an indicial ansatz $\phi=z^\delta f(x)+\cdots$ therefore gives

$$\delta(\delta-d)=m^2L^2.$$

The two candidate weights are

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

This is the scalar mass–dimension relation Witten 1998, §2.5, especially eqs. (2.35) and (2.43)–(2.44), Open PDF. Representation-theoretically, the same equation matches the quadratic Casimir of the AdS mode to that of a scalar conformal primary.

Indicial roots and the BF threshold

Reality of the two powers requires

$$m^2L^2\geq-\frac{d^2}{4}.$$

The same value is the Breitenlohner–Freedman stability threshold, but the logic matters. The indicial calculation only finds real roots. Positive-energy stability is a global statement about the spatial operator together with its domain: suitable standard boundary conditions are stable at and above the threshold, whereas a bad mixed boundary condition can still be unstable. Below the threshold, the usual positive-energy scalar quantization fails Breitenlohner and Freedman 1982, pp. 259–268 and Ishibashi and Wald 2004, §§3.2–3.3, especially theorem 3.2, Open PDF.

For generic nonresonant $\nu>0$, the two independent asymptotic coefficients appear as

$$\phi(z,x) =z^{\Delta_-} \left[\alpha(x)+z^2\alpha_{(2)}(x)+\cdots\right] +z^{\Delta_+}\left[\beta(x)+\cdots\right].$$

The coefficients before $\beta$ are determined locally from $\alpha$ by the radial recursion. For a free scalar with this even-power recursion, a positive integer $\nu$ produces a resonance: the slow series reaches $z^{\Delta_+}$ and a source-local term $z^{\Delta_+}\log(z\mu)$ can appear. At the BF endpoint $\nu=0$, the roots themselves coalesce and the independent pair instead begins as

$$\phi=z^{d/2} \left[\alpha_{\log}(x)\log(z\mu)+\beta(x)+\cdots\right].$$

Thus the shorthand $z^{\Delta_-}\alpha+z^{\Delta_+}\beta$ must not be used blindly at an endpoint or resonance de Haro, Skenderis, and Solodukhin 2001, §5.1, especially eqs. (5.5)–(5.11), Open PDF.

Normalizability, flux, and the alternate window

After complexifying the solution space, the ordinary Klein–Gordon inner product on a constant-time slice has the near-boundary scaling

$$(\phi_1,\phi_2)_{\mathrm{KG}} \sim i\mathcal N_\phi L^{d-1} \int_0^\epsilon \mathrm dz\,z^{1-d} \phi_1^*\!\stackrel{\leftrightarrow}{\partial_t}\!\phi_2.$$

For a mode with regular time dependence, the fast falloff gives $\int_0^\epsilon\mathrm dz\,z^{1+2\nu}$, while the slow falloff gives

$$\int_0^\epsilon\mathrm dz\,z^{1-2\nu}.$$

The slow mode therefore has finite ordinary Klein–Gordon norm precisely when $\nu<1$. Independently, for $d>2$ the scalar CFT unitarity bound gives

$$\Delta_-\geq\frac{d-2}{2} \quad\Longleftrightarrow\quad \nu\leq1.$$

The strict elementary alternate-quantization window is consequently

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

Inside it, the boundary symplectic flux is proportional to

$$\Omega_{\partial M}(\delta_1\phi,\delta_2\phi) \propto 2\nu\mathcal N_\phi L^{d-1} \int_{\partial M}\!\mathrm d^d x\, \left(\delta_1\alpha\,\delta_2\beta -\delta_2\alpha\,\delta_1\beta\right).$$

Fixing $\alpha$ gives standard quantization and operator dimension $\Delta_+$. Fixing $\beta$ after the appropriate Legendre transform gives alternate quantization and dimension $\Delta_-$. More general relations between $\alpha$ and $\beta$ must conserve flux and give a positive self-adjoint dynamics; normalizability alone is not enough Klebanov and Witten 1999, §2.1, especially eqs. (2.1)–(2.20), Open PDF.

The five regimes are compared below. On a narrow screen, scroll horizontally without shrinking the text.

Ordinary scalar quantization by mass regime

Regime	Radial behavior	Standard branch	Alternate branch	Decisive issue
Below the BF bound	Complex powers and logarithmic oscillation	No ordinary positive-energy branch	No	The global spatial operator is unbounded below under the usual assumptions.
ν = 0	A repeated power with a logarithmic partner	Yes, with the BF-endpoint prescription	No second distinct dimension	The logarithmic coefficient and boundary scale require separate treatment.
0 < ν < 1	Two distinct ordinary-KG-normalizable powers	Yes, with dimension Δ+	Yes, with dimension Δ−	Flux conservation and positivity still restrict the boundary condition.
ν = 1	A resonant slow power at the scalar unitarity bound	Yes; general sources require logarithmic renormalization	Not as an ordinary bulk scalar	The slow ordinary KG norm diverges logarithmically; a unitary endpoint needs an exceptional singleton limit.
ν > 1	Two real powers	Yes, for an admissible stable boundary condition	No ordinary unitary branch	The slow norm diverges and Δ− violates the scalar bound for d > 2.

At $\nu=0$, the two roots do not define two distinct conformal quantizations. In the natural conformal prescription, $J_{\mathrm{BF}}=\alpha_{\log}$ up to a declared rescaling, source-free modes obey $\alpha_{\log}=0$, and $\beta$ is the response modulo local, $\mu$-dependent shifts. Other self-adjoint relations can introduce a scale, and the $\pm2\nu$ one-point formulas below cannot simply be evaluated at $\nu=0$ Klebanov and Witten 1999, §2.1, printed p. 11, especially eq. (2.21), Open PDF.

At $\nu=1$ and $d>2$, $\Delta_-=(d-2)/2$ saturates the scalar bound and must behave as a free field with a null descendant. A unitary construction is an exceptional singleton limit—in which the physical module reduces to the boundary free-field degrees of freedom—not the usual alternate scalar quantization Ohl and Uhlemann 2012, §II, especially eqs. (12)–(26), Open PDF. Specialized counterterm-renormalized Neumann constructions outside the open window alter the norm and introduce additional infrared, null-state, or ghost questions; they are not licensed by the ordinary norm test above Andrade and Marolf 2012, §2 and appendix A, Open PDF.

Source, response, and one-point normalization

For a nonresonant standard quantization, set the Euclidean source convention

$$Z_{\mathrm{CFT}}[J_+]=e^{-S_{E,\mathrm{ren}}[\alpha]}, \qquad J_+=\alpha.$$

With the bulk normalization displayed above, holographic renormalization gives

$$\left\langle\mathcal O_+\right\rangle_\alpha =2\nu\mathcal N_\phi L^{d-1}\beta +\mathcal F_{\mathrm{local}}[\alpha].$$

Within $0<\nu<1$, after the corresponding Legendre transform, choosing $J_-=\beta$ gives

$$\left\langle\mathcal O_-\right\rangle_\beta =-2\nu\mathcal N_\phi L^{d-1}\alpha +\widetilde{\mathcal F}_{\mathrm{local}}[\beta].$$

The source normalization and $\mathcal N_\phi$ fix the separated-point normalization. Finite local counterterms shift the displayed local terms and contact terms, not that nonlocal normalization. At resonance, the split between the power coefficient and source-local logarithm also depends on the renormalization scale; at $\nu=0$, the formulas require a rescaled logarithmic-source convention de Haro, Skenderis, and Solodukhin 2001, §5.1, eqs. (5.10)–(5.11), Open PDF.

The word “response” also has a signature-dependent qualification. In a specified Euclidean saddle, interior regularity can determine $\beta$ nonlocally from $\alpha$. In Lorentzian signature, the source alone does not specify a solution: normalizable initial-state data and any horizon prescription remain to be supplied.

Symmetric spin and the meaning of mass

Now let $s\geq1$ and $d\geq3$. A totally symmetric bulk field is paired with a boundary primary in the symmetric-traceless spin-$s$ representation. Define $m_s^2$ to be the flat-limit or Stueckelberg–Fierz–Pauli mass deformation, normalized to vanish at the massless Fronsdal gauge point. Then

$$m_s^2L^2=(\Delta+s-2)(\Delta-s-d+2),$$

so

$$\Delta_\pm=\frac d2\pm \sqrt{\left(s+\frac d2-2\right)^2+m_s^2L^2}.$$

In Metsaev’s notation the symbol $d$ is the bulk dimension, so his $d$ equals $d+1$ here, and his lowest energy $E_0$ is the boundary dimension $\Delta$. With that translation, the convention and its relation to the lowest AdS energy follow from Metsaev 2004, §5, especially eqs. (5.66), (5.69), and (5.74), printed pp. 8–9, Open PDF. For a unitary symmetric-traceless primary,

$$\Delta\geq s+d-2.$$

Thus $m_s^2>0$ selects a long unitary multiplet on the $\Delta_+$ branch. At $m_s^2=0$,

$$\Delta_+=s+d-2, \qquad \Delta_-=2-s.$$

The first root saturates the bound: bulk gauge redundancy removes longitudinal polarizations, and the boundary divergence descendant becomes null, so the operator is conserved. The second root is normally the source or shadow weight, not a second unitary local-current quantization. If $m_s^2<0$, roots can remain real while $\Delta_+<s+d-2$; reality alone therefore does not produce an ordinary unitary spin-$s$ primary Costa, Penedones, Poland, and Rychkov 2011, §5.1, especially eqs. (5.1)–(5.8), Open PDF.

Mass notation for spin is especially easy to mishandle. Propagator literature may instead call the transverse-traceless covariant-Laplacian eigenvalue $\kappa_s^2$, defined after Euclidean continuation by $D^2h=\kappa_s^2h$. That convention obeys

$$\kappa_s^2L^2=\Delta(\Delta-d)-s =m_s^2L^2+(s-2)(s+d-2)-s.$$

For a vector, $\kappa_1^2L^2=m_A^2L^2-d$; for spin two, $\kappa_2^2L^2=m_{\mathrm{FP}}^2L^2-2$. A Maxwell field and a graviton therefore have $m_s^2=0$ even though their tensor Laplacians retain curvature shifts Costa, Gonçalves, and Penedones 2014, §3, eqs. (22)–(27) and (34)–(37), Open PDF.

For $m_s^2>0$, transversality and tracelessness are massive on-shell constraints, not gauge choices. Full massless Fronsdal gauge redundancy appears at $m_s^2=0$. Isolated negative mass values can instead have a partial gauge symmetry; in AdS their partially shortened boundary modules are not ordinary unitary representations. Antisymmetric forms and mixed Young symmetries carry different Lorentz representations and require their own Casimir and gauge-complex analysis; Spinning Fields, Forms, and Mixed-Symmetry Operators develops those cases.

Checks that catch wrong dictionaries

For $d=4$ and a scalar with $m^2L^2=-3$,

$$\nu=1, \qquad (\Delta_-,\Delta_+)=(1,3).$$

The standard $\Delta=3$ branch is admissible, but a general source is resonant and requires the $z^3\log(z\mu)$ term and a logarithmic counterterm. The $\Delta=1$ root is not licensed as an ordinary alternate bulk-scalar quantization.

Now give a spin-two field the same numerical Fierz–Pauli mass, $m_{\mathrm{FP}}^2L^2=-3$. Its algebraic roots are again $1$ and $3$, but the spin-two unitarity bound is $\Delta\geq4$. This value is the partially massless spin-two point in AdS$_5$: it has a partial gauge symmetry, yet its $\Delta=3$ boundary module is nonunitary. Neither root defines an ordinary unitary spin-two primary. This comparison isolates the central lesson: the quadratic equation supplies candidates, while the representation, gauge complex, and norm decide admissibility.

As a shortening check, $m_A^2=0$ gives vector roots $d-1$ and $1$, of which the unitary operator is a conserved current with $\Delta=d-1$. Likewise, $m_{\mathrm{FP}}^2=0$ gives spin-two roots $d$ and $0$, and the unitary root is the stress tensor.

What survives when a test fails

Below the BF threshold. The wave equation still has formal complex indicial roots, but no ordinary positive-energy unitary scalar dictionary survives under the standard assumptions.

Outside the alternate window. The slow real root still solves the indicial equation. What fails is the stronger claim that it defines an ordinary finite-Klein–Gordon-norm, unitary alternate quantization.

For a wrong spin representation. A real $\Delta$ can still be below the appropriate spin-dependent bound. The surviving information is only the Casimir root in the declared mass convention, not an admissible local operator.

Interactions can shift physical AdS masses and scaling dimensions and can mix fields with identical quantum numbers. The scalar formula uses the complete physical mass in the Klein–Gordon operator. The symmetric-spin formula instead uses the physical flat-limit or Stueckelberg–Fierz–Pauli deformation normalized to vanish at the massless point. With those definitions the displayed equations are exact representation relations; a bare Lagrangian coefficient need not already be either quantity.

Exercises

Starting from the Poincaré metric, compute $g^{MN}$ and $\sqrt{\lvert g\rvert}$ and recover the displayed radial Klein–Gordon equation. Why must the boundary momentum be held fixed in the indicial limit?

Solution

The inverse components are $g^{ij}=z^2\eta^{ij}/L^2$ and $g^{zz}=-z^2/L^2$, while $\sqrt{\lvert g\rvert}=L^{d+1}z^{-(d+1)}$. Substitution into $\Box\phi=\lvert g\rvert^{-1/2}\partial_M(\sqrt{\lvert g\rvert}g^{MN}\partial_N\phi)$ gives

$$\Box\phi=\frac1{L^2} \left[-z^2\partial_z^2+(d-1)z\partial_z+z^2\Box_\eta\right]\phi.$$

For a Fourier mode, the last term scales as $z^2k^2\phi$. It is subleading only when $k$ remains finite as $z\to0$; taking $k\sim z^{-1}$ would not be the fixed-data asymptotic problem used to define an indicial exponent.

For a scalar in AdS$_4$, take $d=3$ and $m^2L^2=-2$. Find both dimensions, test the slow Klein–Gordon norm, and identify the source in each quantization.

Solution

Here $\nu=\sqrt{9/4-2}=1/2$, so $(\Delta_-,\Delta_+)=(1,2)$. The slow-mode radial norm scales as $\int_0^\epsilon\mathrm dz\,z^{1-2\nu}=\int_0^\epsilon\mathrm dz$, which is finite. Standard quantization fixes $J_+=\alpha$ and assigns dimension $2$; alternate quantization fixes $J_-=\beta$ after the Legendre transform and assigns dimension $1$. In either case the complete boundary condition must also conserve flux and give positive dynamics.

In $d=4$, compare a scalar and a symmetric spin-two field when their respective declared mass parameters both obey $m^2L^2=-3$. Why does the same quadratic pair not imply the same admissibility?

Solution

Both equations give roots $1$ and $3$. For the scalar, $\Delta=3$ is an admissible standard-quantization dimension, while $\nu=1$ excludes the ordinary alternate branch. For spin two, the unitary bound is $\Delta\geq4$, so neither root is an ordinary unitary spin-two primary. The scalar BF analysis and the spin-two representation bound answer different questions.

Controlled handoffs

Asymptotically Locally AdS Boundary Data develops the full radial recursion and resonant logarithms. The GKPW Generating-Functional Dictionary derives correlators from renormalized source variation. Boundary Conditions, Alternate Quantization, and Deformations treats flux-preserving mixed conditions, Legendre transforms, and multi-trace flows. Spinning Fields, Forms, and Mixed-Symmetry Operators owns the detailed higher-spin, form, and mixed-symmetry extensions.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Andrade, Tomás, and Donald Marolf. “AdS/CFT beyond the Unitarity Bound.” Journal of High Energy Physics 2012, 049 (2012). DOI. Open PDF.
• Breitenlohner, Peter, and Daniel Z. Freedman. “Stability in Gauged Extended Supergravity.” Annals of Physics 144 (1982): 249–281. DOI.
• Costa, Miguel S., Vasco Gonçalves, and João Penedones. “Spinning AdS Propagators.” Journal of High Energy Physics 2014, 064 (2014). DOI. Open PDF.
• Costa, Miguel S., João Penedones, David Poland, and Slava Rychkov. “Spinning Conformal Correlators.” Journal of High Energy Physics 2011, 071 (2011). DOI. Open PDF.
• de Haro, Sebastian, Kostas Skenderis, and Sergey N. Solodukhin. “Holographic Reconstruction of Spacetime and Renormalization in the AdS/CFT Correspondence.” Communications in Mathematical Physics 217 (2001): 595–622. DOI. Open PDF.
• Ishibashi, Akihiro, and Robert M. Wald. “Dynamics in Non-Globally-Hyperbolic Static Spacetimes III: Anti-de Sitter Spacetime.” Classical and Quantum Gravity 21 (2004): 2981–3014. DOI. Open PDF.
• Klebanov, Igor R., and Edward Witten. “AdS/CFT Correspondence and Symmetry Breaking.” Nuclear Physics B 556 (1999): 89–114. DOI. Open PDF.
• Metsaev, R. R. “Massive Totally Symmetric Fields in AdS(d).” Physics Letters B 590 (2004): 95–104. DOI. Open PDF.
• Ohl, Thorsten, and Christoph F. Uhlemann. “Saturating the Unitarity Bound in AdS/CFT$_{\mathrm{AdS}}$.” Journal of High Energy Physics 2012, 161 (2012). DOI. Open PDF.
• Witten, Edward. “Anti-de Sitter Space and Holography.” Advances in Theoretical and Mathematical Physics 2 (1998): 253–291. DOI. Open PDF.