A first AdS/CFT dictionary from QFT and GR

Suppose a solution of a gravitational field equation could calculate a strongly interacting quantum correlation function. What data would we have to prescribe, and which part of the solution would be the answer? We will solve one such boundary problem completely. The key distinction is familiar from response theory: an externally specified source and the response it induces are different quantities.

Prepare with conformal fixed points, sources and currents and universal observables. Basic GR supplies the metric and covariant wave equation. The worked example has a three-dimensional boundary and a four-dimensional Euclidean bulk. It is a linear scalar probe on a fixed geometry, with no asserted gauge group or supersymmetry. A particular supersymmetric duality requires much more data than this probe.

The extra coordinate and the boundary

Anti-de Sitter space, abbreviated AdS, is the maximally symmetric Lorentzian spacetime of negative curvature. Its Poincaré coordinates have

$$ds^2=\frac{L^2}{z^2}\left(dz^2+\eta_{ij}dx^i dx^j\right), \qquad z>0,\qquad i=0,\ldots,d-1. \tag{X02.1}$$

Here $L$ is a length, the curvature radius, and $x^i,z$ have length units. We use the GR curvature convention in which a round sphere has positive scalar curvature: $R_{\rm GR}=-d(d+1)/L^2$ for this metric. The target curved-SUSY paper uses the opposite curvature convention, so its $R_P=-R_{\rm GR}$. No supersymmetry equation is being inferred from that sign.

For Euclidean calculations replace $\eta_{ij}$ by $\delta_{ij}$. The resulting hyperbolic space is often called Euclidean AdS. Lorentzian AdS has isometry group $SO(d,2)$, locally the conformal group of $d$-dimensional Minkowski space; the Euclidean counterpart is $SO(d+1,1)$. The Lorentzian coordinate chart above covers a patch, not all of global AdS.

The surface $z=0$ is at infinite proper radial distance: $\int_\epsilon^{z_0}L\,dz/z$ diverges as $\epsilon\to0$. Nevertheless the rescaled metric $(z/L)^2ds^2$ has a finite boundary metric. Rescaling by another function with the same simple zero changes that metric by a Weyl factor. This is a conformal boundary, carrying a conformal class rather than a preferred metric. This construction is explicit in Witten (2.1)–(2.4).

The simultaneous transformation $(z,x)\mapsto(az,ax)$ leaves (X02.1) unchanged. Thus moving outward toward small $z$ accompanies shorter boundary length scales. The solution below makes this relation concrete: a boundary Fourier mode of momentum magnitude $p$ penetrates appreciably only to $z$ of order $p^{-1}$. This is a useful scale interpretation of the radial coordinate, not an identification of a boundary point with a unique bulk point.

Turn a boundary condition into a source

For a proposed dual pair, the dictionary relates the bulk partition function with prescribed asymptotic fields to the boundary generating functional. For one scalar operator $\mathcal O$, choose

$$\frac{Z[J]}{Z[0]} =\left\langle\exp\!\left(\int d^dx\,J\mathcal O\right)\right\rangle_0, \qquad S_E[J]=S_E[0]-\int d^dx\,J\mathcal O. \tag{X02.2}$$

This fixes the source sign. If a coupling $j$ is instead added to the action as $+\int j\mathcal O$, then $j=-J$. The earlier convention $\mathcal W=-\log Z$ remains valid; here it is convenient to differentiate $\log Z$, so $\delta\log Z/\delta J=\langle\mathcal O\rangle_J$.

At a dominant classical bulk saddle, the proposed relation becomes

$$\log\frac{Z[J]}{Z[0]} \simeq-S_{\rm ren,cl}[J] +S_{\rm ren,cl}[0]. \tag{X02.3}$$

Here cl means the action is evaluated on a classical solution. The bulk field solves its equation with the prescribed source and an interior condition. The subscript “ren” means that divergent local boundary terms have been subtracted in a stated prescription. The underlying quantum correspondence is stronger than this saddle approximation; Witten (2.10)–(2.12) separates the two statements. Solving a scalar equation alone does not establish a complete dual pair.

For a scalar of bulk mass $m$, the near-boundary equation follows from $(\nabla^2-m^2)\phi=0$. With $\phi\sim z^\alpha$ and no leading boundary derivatives it gives

$$\alpha(\alpha-d)=m^2L^2,\qquad \Delta=\frac d2+\sqrt{\frac{d^2}{4}+m^2L^2}. \tag{X02.4}$$

For the ordinary Dirichlet choice used here, the two behaviors are a source $z^{d-\Delta}J$ and a response $z^\Delta B$. Under a boundary rescaling, their weights give $[J]=d-\Delta$ and $[\mathcal O]=\Delta$ in mass units, as required by (X02.2). Local derivative terms can occur between these two powers. The mass–dimension relation is Witten (2.43)–(2.44), with $L$ restored. The two roots are not automatically dimensions of two different operators: boundary conditions and the admissible quantization matter. We need only a massless field and the larger-root choice below.

Solve a massless scalar in Euclidean AdS4

Set $d=3$, $m=0$, and take a real dimensionless field with positive dimensionless normalization $\mathcal N$:

$$S[\phi]=\frac{\mathcal N}{2L^2} \int_{z\geq\epsilon}d^4x\sqrt g\, g^{MN}\partial_M\phi\partial_N\phi. \tag{X02.5}$$

The cutoff $\epsilon>0$ will be removed. This coefficient makes the action dimensionless; changing it changes the operator normalization. Write $\phi(z,x)=\int_p e^{ip\cdot x}\phi(z,p)$, where $\int_p=\int d^3p/(2\pi)^3$ and $p=|\boldsymbol p|$ when it appears without an index. For $p>0$, the wave equation is

$$z^2\phi''-2z\phi'-p^2z^2\phi=0. \tag{X02.6}$$

Its two independent solutions are $(1+pz)e^{-pz}$ and $(1-pz)e^{pz}$. The second grows in the interior. Requiring regular decay and asymptotic value $J(p)$ selects

$$\begin{aligned} \phi(z,p)&=(1+pz)e^{-pz}J(p),\\ &=J(p)-\frac{p^2}{2}J(p)z^2+B(p)z^3+O(z^4), \qquad B(p)=\frac{p^3}{3}J(p). \end{aligned} \tag{X02.7}$$

At zero momentum the regular solution is constant. The $z^2$ coefficient is fixed locally by derivatives of the source. The $z^3$ coefficient knows about the interior condition: in position space its $|p|^3$ dependence is nonlocal. Finding that coefficient is already a response calculation, but its conversion to $\langle\mathcal O\rangle$ still needs the action normalization and subtraction.

Integrate (X02.5) by parts on the solution. At the lower boundary the outward unit normal has $n^z=-\epsilon/L$. Its sign gives

$$\begin{aligned} S_{\rm reg} &=-\frac{\mathcal N}{2\epsilon^2} \int_p\phi(\epsilon,p)\partial_z\phi(\epsilon,-p)\\ &=\frac{\mathcal N}{2}\int_p J(p)J(-p) \left(\frac{p^2}{\epsilon}-p^3+O(\epsilon)\right). \end{aligned} \tag{X02.8}$$

The divergence is a polynomial in momentum, hence local in the source. With $\gamma$ the induced metric at $z=\epsilon$, subtract it using

$$S_{\rm ct}=-\frac{\mathcal N}{2L} \int_{z=\epsilon}d^3x\sqrt\gamma\, \gamma^{ij}\partial_i\phi\partial_j\phi. \tag{X02.9}$$

Indeed $S_{\rm ct}=-(\mathcal N/2\epsilon)\int_p p^2\phi(\epsilon,p)\phi(\epsilon,-p)$. Substitution of the exact solution is especially useful: the sum is exactly $-(\mathcal N/2)\int_p p^3(1+p\epsilon)e^{-2p\epsilon}J(p)J(-p)$. Therefore

$$S_{\rm ren}[J]=\lim_{\epsilon\to0}(S_{\rm reg}+S_{\rm ct}) =-\frac{\mathcal N}{2}\int_p p^3J(p)J(-p). \tag{X02.10}$$

Although the unrenormalized bulk integral is positive, its finite part after subtraction need not be positive. Differentiating its negative as in (X02.3) gives

$$\boxed{\langle\mathcal O(p)\rangle_J =\mathcal N p^3J(p)=3\mathcal N B(p),\qquad G(p)=\mathcal N|p|^3.} \tag{X02.11}$$

Here $\langle\mathcal O(p)\mathcal O(q)\rangle_c=(2\pi)^3\delta^3(p+q)G(p)$ at zero source. This is the quadratic probe result, up to local contact terms. In a full theory bulk interactions, backreaction and quantum corrections require their own analysis. The dimensions check: $\Delta=3$, the position-space source is dimensionless, and the position-space operator has mass dimension three.

Check the response without its Fourier solution

The same boundary problem can be solved by convolution with

$$K(z,r)=\frac4{\pi^2}\frac{z^3}{(z^2+r^2)^3},\qquad \phi(z,x)=\int d^3y\,K(z,|x-y|)J(y). \tag{X02.12}$$

Direct differentiation verifies its wave equation. With $r=zu$, the radial integral $\int_0^\infty4\pi r^2K(z,r)dr=1$ fixes its coefficient; as $z\to0$ its support concentrates at $r=0$. It therefore approaches a delta function and reproduces the source. For $x\ne y$, its $z^3$ coefficient is $4/(\pi^2|x-y|^6)$. Multiplying by the response factor $3\mathcal N$ gives

$$\langle\mathcal O(x)\mathcal O(y)\rangle_c =\frac{12\mathcal N}{\pi^2|x-y|^6},\qquad x\ne y. \tag{X02.13}$$

This has the expected scalar conformal form and a positive separated-point coefficient. Its coincident singularity requires distributional extension, which is where local contact terms enter. A finite local subtraction changes a polynomial in $p^2$, not the nonanalytic $|p|^3$ term or (X02.13).

For a smooth worked source, take $J(x)=e^{-|x|^2/a^2}$ with width $a$. At the center, the convolution yields $\phi=0.94950233,0.55393691,0.13456658$ at $z/a=0.15,0.7,2$. A separate momentum integral using $J(p)=\pi^{3/2}a^3e^{-a^2p^2/4}$ and (X02.7) gives the same numbers. The profile becomes smoother and smaller with depth; it is not simply equal to the boundary value on a vertical line. Both integrals can be checked directly, without assuming a field-theory dual.

What a compact factor and a brane mean

A bulk geometry often contains an additional compact space $Y$. Expanding a higher-dimensional field in eigenfunctions of $-\nabla_Y^2$ gives fields in AdS. For a direct product with compact radius $R_Y$, an eigenvalue $\kappa/R_Y^2$ shifts a minimally coupled scalar mass to $m_{\rm eff}^2=m_0^2+\kappa/R_Y^2$. Each mode then has its own mass–dimension relation. “Compact” does not mean that all these modes can be discarded: if $R_Y$ is comparable to $L$, their masses are comparable to the AdS curvature scale. Fluxes and mixing can modify the simple scalar example.

A $p$-brane is an object extended in $p$ spatial directions, with a $(p+1)$-dimensional worldvolume. It can support a low-energy field theory and also source a higher-dimensional geometry. Maldacena’s construction compares appropriate decoupling and near-horizon descriptions of such systems; the introduction and (2.1)–(2.3) provide the starting example. A brane is not an extra boundary coordinate, and the compact factor is not an extra field-theory spacetime dimension.

The radius and metric alone do not determine gauge rank or supersymmetry. Those require the theory, flux integers, global data and spinor equations. This general scalar calculation supplies no such dictionary and assumes no preserved supercharge.

When can classical gravity be used?

Three distinct approximations have entered. First, a classical bulk saddle needs suppressed quantum fluctuations, commonly measured by $G_{d+1}/L^{d-1}\ll1$ with the effective Newton constant $G_{d+1}$. Second, a two-derivative gravity description needs small curvature in units of the scale of omitted higher-derivative physics, such as the string length. Third, our linear probe needs sufficiently weak sources and negligible backreaction. The applicable conditions depend on the full compactification and observable.

Large gauge-group size can suppress bulk quantum effects without by itself suppressing string-scale corrections. Nor does classical bulk gravity mean weakly coupled boundary fields; Maldacena’s discussion on printed pp. 4–5 separates these limits in its example. Conversely, a protected field-theory answer may be exact outside a classical-gravity regime. One must compare the same observable in a common regime, with matched source normalization and contact prescription.

Exercises

1. Give the compact direction a momentum

Consider a massless scalar on the direct product of Euclidean AdS4 of radius $L$ and a circle of radius $R$. The circle coordinate $\theta$ has period $2\pi$. Find the AdS mass of its mode $e^{in\theta}$. For $R=L$ and $n=2$, find the larger-root operator dimension and source dimension. Does choosing this mode specify a gauge rank or preserved supersymmetry?

Hint 1

The circle contribution to the Laplacian is $R^{-2}\partial_\theta^2$. Separate the wave equation before applying the AdS relation.

Hint 2

The source and operator dimensions must sum to three. The integer Fourier label is not a color number.

Solution

Writing $\Phi(z,x,\theta)=\phi_n(z,x)e^{in\theta}$ gives $(\nabla_{\rm AdS}^2-n^2/R^2)\phi_n=0$, hence $m_n^2=n^2/R^2$. At the specified values $m_n^2L^2=4$ and $\Delta(\Delta-3)=4$ has roots $4,-1$. The chosen Dirichlet prescription uses $\Delta=4$, so $[J]=3-4=-1$. This is a source for an irrelevant operator, treated infinitesimally in the probe calculation; a finite interacting deformation would require additional ultraviolet care. The zero circle mode instead has $\Delta=3$, checking the worked example. No gauge group or supersymmetry has been specified by separating this scalar equation.

2. Keep the source and change the interior condition

For $p>0$, consider

$$\phi_\alpha(z,p)=\big[(1-\alpha)(1+pz)e^{-pz} +\alpha(1-pz)e^{pz}\big]J(p).$$

Find its source and $z^3$ response coefficients. Which value of $\alpha$ meets the regularity condition? Explain why a boundary source alone is insufficient to specify the answer.

Hint 1

Both independent solutions approach one at $z=0$. Their cubic Taylor coefficients have opposite signs.

Hint 2

A small nonzero coefficient still multiplies an exponentially growing function at large $z$.

Solution

Expansion gives $\phi_\alpha=J-p^2Jz^2/2+(1-2\alpha)p^3Jz^3/3+O(z^4)$. Every $\alpha$ has the same source and local quadratic coefficient, but a different response. Interior regularity requires $\alpha=0$. A nonzero choice does not compute the stated vacuum response; its growing contribution also invalidates dropping the interior surface term in the action. In Lorentzian problems, the state and causal prescription play corresponding roles in choosing a solution. Merely observing the correct boundary falloff does not settle them.

3. Change a local finite subtraction

Add $S_{\rm fin}=c\int d^3x\,\partial_iJ\partial_iJ/2$ to the renormalized action, with constant $c$ of mass dimension one. Find the change in $G(p)$ and its position-space support. Can it change the coefficient in (X02.13)?

Hint 1

The generating functional is the negative of the renormalized action at this order. In Fourier space the added term is quadratic in $J$ with kernel $cp^2$.

Hint 2

The Fourier transform of $p^2$ is $-\partial^2\delta^3(x-y)$. Distinguish a contact distribution from a function at noncoincident points.

Solution

The change is $\delta G(p)=-cp^2$, hence $\delta\langle\mathcal O(x)\mathcal O(y)\rangle_c=c\partial_x^2\delta^3(x-y)$. It has support only at coincidence and cannot change (X02.13). The dimensionful coefficient introduces a contact-term scale into this prescription; it does not change the separated-point conformal power law. Absorbing the nonanalytic $|p|^3$ term would instead need a nonlocal source functional, which is not an allowed local counterterm. This distinguishes a normalization fixed by the bulk action from a removable local ambiguity.

The essential calculation is now complete: choose the boundary source and interior condition, solve for the bulk field, subtract local divergences, and differentiate the finite action. A theory-specific AdS/CFT dictionary must additionally determine which bulk field represents which operator, the coefficient $\mathcal N$, the compact geometry and the parameters controlling the approximation.

A first AdS5 times S5 dictionary supplies those data for four-dimensional N=4 Yang–Mills. Derive its coupling, radius and Newton coefficient, then test the separate conditions needed for a classical bulk calculation.

For three-dimensional Chern–Simons matter, the ABJM holographic dictionary supplies the quotient geometry and two gravity regimes. Its finite sphere action is a new observable to compare using the boundary subtraction taught here.