Scalar Two-Point Functions

The main idea

The first real calculation in AdS/CFT should be as simple as possible, but not simpler. We take a scalar primary operator $\mathcal O$ in a $d$-dimensional CFT and its dual scalar field $\phi$ in Euclidean $\mathrm{AdS}_{d+1}$. The goal is to compute the leading large-$N$ two-point function

$$\langle \mathcal O(x)\mathcal O(y)\rangle$$

from the classical bulk action.

The answer has to take the conformal form

$$\langle \mathcal O(x)\mathcal O(y)\rangle = \frac{C_{\mathcal O}}{|x-y|^{2\Delta}},$$

where $\Delta$ is the scaling dimension of $\mathcal O$. What holography adds is a way to determine $\Delta$ from the bulk mass and $C_{\mathcal O}$ from the normalization of the bulk action.

The calculation is the cleanest illustration of the GKPW prescription:

$$Z_{\mathrm{CFT}}[\phi_{(0)}] = \left\langle \exp\!\left(\int d^d x\,\phi_{(0)}(x)\mathcal O(x)\right) \right\rangle \simeq \exp\!\left[-I_E^{\mathrm{ren}}[\phi_{\mathrm{cl}}]\right],$$

where $\phi_{\mathrm{cl}}$ solves the bulk equation of motion with boundary source $\phi_{(0)}$. Defining the connected generating functional by

$$W[\phi_{(0)}] = \log Z_{\mathrm{CFT}}[\phi_{(0)}] \simeq -I_E^{\mathrm{ren}}[\phi_{\mathrm{cl}}],$$

we extract correlators by functional differentiation:

$$\langle \mathcal O(x)\mathcal O(y)\rangle = \left. \frac{\delta^2 W}{\delta\phi_{(0)}(x)\delta\phi_{(0)}(y)} \right|_{\phi_{(0)}=0}.$$

A scalar source $\phi_{(0)}$ on the boundary determines a regular classical bulk solution through the bulk-to-boundary propagator $K_\Delta$. The renormalized on-shell action $S_{\rm os}^{\rm ren}[\phi_{(0)}]$ is then differentiated twice to obtain the separated-point correlator $\langle \mathcal O(x)\mathcal O(y)\rangle\propto |x-y|^{-2\Delta}$.

There are three morals to keep in mind from the start. First, the power law comes from the near-boundary scaling of the bulk field, not from putting conformal invariance in by hand. Second, the coefficient is not universal until the normalization of $\phi$ in the bulk action is fixed. Third, the on-shell action is divergent and must be renormalized before it is a CFT generating functional.

This page performs the calculation in Euclidean signature. Lorentzian real-time correlators require additional information about the interior or horizon boundary condition and will be treated later.

The bulk problem

Work in Euclidean Poincare AdS$_{d+1}$,

$$ds^2 = \frac{L^2}{z^2} \left(dz^2+d x^i d x^i\right), \qquad z>0, \qquad x\in \mathbb R^d.$$

The conformal boundary is at $z=0$. The bulk action for a scalar field is

$$I_E[\phi] = \frac{\mathcal N_\phi}{2} \int_{z>\epsilon} d^{d+1}X\sqrt g \left(g^{ab}\partial_a\phi\partial_b\phi+m^2\phi^2\right) + I_{\mathrm{ct}}[\phi;\epsilon].$$

Here $\mathcal N_\phi$ is an overall normalization. In a top-down compactification it is determined by the ten- or eleven-dimensional action and the normalization of the Kaluza-Klein mode. In a bottom-up model it is a parameter. The cutoff $z=\epsilon$ regulates the infinite volume near the boundary, and $I_{\mathrm{ct}}$ contains local counterterms on that cutoff surface.

The equation of motion is

$$\left(\nabla^2-m^2\right)\phi=0.$$

With the Poincare metric this becomes

$$z^{d+1}\partial_z\left(z^{1-d}\partial_z\phi\right) +z^2\partial_i\partial_i\phi -m^2L^2\phi=0.$$

Near $z=0$, the $x$-derivatives are subleading compared with radial derivatives. Trying $\phi\sim z^\alpha$ gives

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

It is convenient to define

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

Then

$$m^2L^2=\Delta(\Delta-d),$$

and the two independent near-boundary behaviors are

$$z^{d-\Delta}=z^{\frac d2-\nu}, \qquad z^\Delta=z^{\frac d2+\nu}.$$

For standard quantization, the coefficient of the slower falloff is the source and the coefficient of the faster falloff is related to the expectation value:

$$\phi(z,x) = z^{d-\Delta}\left[\phi_{(0)}(x)+\cdots\right] + z^\Delta\left[A(x)+\cdots\right].$$

The dots denote derivative corrections determined locally by $\phi_{(0)}$, and sometimes logarithmic terms. At separated points, the nonlocal part of $A(x)$ is the part that matters for the two-point function.

The bulk-to-boundary propagator

The regular Euclidean solution with boundary source $\phi_{(0)}$ can be written as

$$\phi(z,x) = \int d^d y\,K_\Delta(z,x;y)\phi_{(0)}(y),$$

where the scalar bulk-to-boundary propagator is

$$K_\Delta(z,x;y) = C_\Delta \left( \frac{z}{z^2+|x-y|^2} \right)^\Delta,$$

with

$$C_\Delta = \frac{\Gamma(\Delta)}{\pi^{d/2}\Gamma(\Delta-d/2)} = \frac{\Gamma(\Delta)}{\pi^{d/2}\Gamma(\nu)}.$$

This normalization is chosen so that

$$\lim_{z\to0}z^{\Delta-d}K_\Delta(z,x;y) = \delta^{(d)}(x-y)$$

as a distribution. The expression looks as though it vanishes like $z^\Delta$ at fixed $x\ne y$, but near $x=y$ it becomes sharply peaked. This is the familiar approximate-identity mechanism behind the delta function.

The constant $C_\Delta$ follows from the integral

$$\int d^d u\,\frac{1}{(1+u^2)^\Delta} = \pi^{d/2}\frac{\Gamma(\Delta-d/2)}{\Gamma(\Delta)}, \qquad \Delta>\frac d2.$$

Indeed,

$$\int d^d x\,z^{\Delta-d}K_\Delta(z,x;y) = C_\Delta \int d^d x\, \frac{z^{2\Delta-d}}{(z^2+|x-y|^2)^\Delta} =1.$$

The propagator also makes conformal covariance manifest. Under the scaling

$$z\to \lambda z, \qquad x\to \lambda x, \qquad y\to \lambda y,$$

one has

$$K_\Delta(\lambda z,\lambda x;\lambda y) = \lambda^{-\Delta}K_\Delta(z,x;y),$$

which is precisely the scaling expected for a boundary operator of dimension $\Delta$.

The on-shell action is a boundary term

For a solution of the equation of motion, the bulk action reduces to a boundary term plus counterterms. Integrating by parts gives

$$I_E^{\mathrm{bare}}[\phi_{\mathrm{cl}}] = \frac{\mathcal N_\phi}{2} \int_{z=\epsilon}d^d x\sqrt\gamma\,\phi\,n^a\partial_a\phi.$$

At the cutoff surface, the induced metric is

$$\gamma_{ij}=\frac{L^2}{\epsilon^2}\delta_{ij}, \qquad \sqrt\gamma=\frac{L^d}{\epsilon^d}.$$

For the region $z\ge \epsilon$, the outward-pointing unit normal points toward decreasing $z$:

$$n^a\partial_a=-\frac{z}{L}\partial_z.$$

Thus

$$I_E^{\mathrm{bare}}[\phi_{\mathrm{cl}}] = -\frac{\mathcal N_\phi L^{d-1}}{2} \int d^d x\,\epsilon^{1-d}\phi(\epsilon,x)\partial_z\phi(\epsilon,x).$$

Substituting the near-boundary expansion produces divergent local terms. The leading divergence is removed by a counterterm of the schematic form

$$I_{\mathrm{ct}} = \frac{\mathcal N_\phi}{2} \int_{z=\epsilon}d^d x\sqrt\gamma\, \frac{d-\Delta}{L}\phi^2+ \text{derivative counterterms}.$$

The derivative counterterms are local functions of $\phi$ and the cutoff metric $\gamma_{ij}$. They are essential for a fully finite variational principle, but they only change contact terms in the separated-point two-point function.

After renormalization, the variation of the on-shell action has the universal form

$$\delta I_E^{\mathrm{ren}} = -\mathcal N_\phi L^{d-1}(2\Delta-d) \int d^d x\, A(x)\,\delta\phi_{(0)}(x) + \text{local terms}.$$

Since $W=-I_E^{\mathrm{ren}}$, the CFT one-point function in the source background is

$$\langle\mathcal O(x)\rangle_{\phi_{(0)}} = \frac{\delta W}{\delta\phi_{(0)}(x)} = \mathcal N_\phi L^{d-1}(2\Delta-d)A(x) + \text{local terms}.$$

The word “local” is doing important work here. Local terms are polynomials in derivatives acting on the source. After differentiating again, they produce contact terms supported at $x=y$. They do not affect the separated-point correlator.

Extracting the two-point function

The bulk-to-boundary solution has the near-boundary expansion

$$K_\Delta(z,x;y) = z^{d-\Delta}\delta^{(d)}(x-y) + z^\Delta\frac{C_\Delta}{|x-y|^{2\Delta}} + \text{local derivative terms},$$

where the first term is meant distributionally. Therefore the nonlocal part of $A(x)$ is

$$A(x) = C_\Delta \int d^d y\, \frac{\phi_{(0)}(y)}{|x-y|^{2\Delta}} + \text{local terms}.$$

Differentiating the one-point function gives the Euclidean two-point function at separated points:

$$\boxed{ \langle\mathcal O(x)\mathcal O(y)\rangle = \mathcal N_\phi L^{d-1}(2\Delta-d) \frac{\Gamma(\Delta)}{\pi^{d/2}\Gamma(\Delta-d/2)} \frac{1}{|x-y|^{2\Delta}} }$$

for $x\ne y$, in the normalization of the scalar action written above.

This is the first “full-stack” AdS/CFT computation in the course:

$$\text{bulk mass} \longrightarrow \Delta, \qquad \text{regular Dirichlet solution} \longrightarrow A[\phi_{(0)}], \qquad \text{renormalized on-shell action} \longrightarrow \langle\mathcal O\mathcal O\rangle.$$

The power $|x-y|^{-2\Delta}$ is fixed by conformal invariance. The coefficient is dynamical, because it knows the normalization $\mathcal N_\phi L^{d-1}$ inherited from the bulk theory.

A useful shorthand is

$$C_{\mathcal O} = \mathcal N_\phi L^{d-1}(2\Delta-d)C_\Delta.$$

However, $C_{\mathcal O}$ is not invariant under the rescaling

$$\mathcal O\to a\mathcal O, \qquad \phi_{(0)}\to a^{-1}\phi_{(0)}.$$

Only after the source normalization is fixed by a physical convention, for example by embedding the scalar into a known supergravity mode dual to a specified single-trace operator, does the absolute coefficient become meaningful.

Momentum-space viewpoint

The same calculation is often cleaner in momentum space. Write

$$\phi(z,x) = \int\frac{d^d p}{(2\pi)^d} e^{ip\cdot x}\phi_p(z).$$

The radial equation becomes

$$z^2\phi_p''-(d-1)z\phi_p'-p^2z^2\phi_p-m^2L^2\phi_p=0.$$

The solution regular in the Euclidean interior is

$$\phi_p(z) = \phi_{(0)}(p) \frac{p^\nu}{2^{\nu-1}\Gamma(\nu)} z^{d/2}K_\nu(pz),$$

where $K_\nu$ is the modified Bessel function. The normalization is chosen so that the leading near-boundary behavior is

$$\phi_p(z) \sim z^{d-\Delta}\phi_{(0)}(p).$$

For noninteger $\nu$, the small-argument expansion gives

$$z^{d/2}K_\nu(pz) = 2^{\nu-1}\Gamma(\nu)p^{-\nu}z^{d/2-\nu} + 2^{-\nu-1}\Gamma(-\nu)p^\nu z^{d/2+\nu} + \cdots.$$

Thus

$$A(p) = \frac{\Gamma(-\nu)}{2^{2\nu}\Gamma(\nu)}p^{2\nu}\phi_{(0)}(p) + \text{local analytic terms}.$$

The momentum-space two-point function has the nonlocal structure

$$\langle\mathcal O(p)\mathcal O(-p)\rangle \propto p^{2\nu} = p^{2\Delta-d},$$

again up to convention-dependent constants and local analytic terms. This is the Fourier transform of $|x|^{-2\Delta}$.

When $\nu$ is an integer, the expansion of $K_\nu$ contains logarithms. The nonlocal momentum-space term becomes schematically

$$p^{2\nu}\log p^2.$$

This is not a pathology. It is the momentum-space avatar of the distribution $|x|^{-2\Delta}$ when $\Delta-d/2$ is an integer, together with the conformal anomaly and scheme-dependent contact terms.

Example: a massless scalar in AdS$_5$

For AdS$_5$/CFT$_4$, take $d=4$. A massless scalar has

$$m^2L^2=0, \qquad \Delta(\Delta-4)=0.$$

The standard quantization root is

$$\Delta=4.$$

The leading behavior is

$$\phi(z,x)=\phi_{(0)}(x)+z^4A(x)+\cdots,$$

where logarithmic terms may appear in the complete expansion when derivatives of the source are included. The separated-point two-point function scales as

$$\langle\mathcal O(x)\mathcal O(0)\rangle \propto \frac{1}{|x|^8}.$$

This is the correct scaling for a marginal scalar operator in four dimensions, such as a protected operator in the same supergravity sector as the dilaton. In the canonical $\mathrm{AdS}_5\times S^5$ example, the overall coefficient is proportional to $N^2$ before one rescales the operator to unit two-point normalization, because the classical five-dimensional action carries an overall factor

$$\frac{L^3}{G_5}\sim N^2.$$

That $N^2$ is not an accident; it is the number of adjoint degrees of freedom in the large-$N$ gauge theory and the inverse strength of bulk quantum gravity.

Example: the BF-window ambiguity in AdS$_4$

Consider AdS$_4$, so $d=3$, and a scalar with

$$m^2L^2=-2.$$

Then

$$\Delta(\Delta-3)=-2, \qquad \Delta=1\quad\text{or}\quad \Delta=2.$$

Both choices are allowed because the mass lies in the Breitenlohner-Freedman window

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

In standard quantization, the dual operator has $\Delta=2$ and

$$\langle\mathcal O(x)\mathcal O(0)\rangle \propto \frac{1}{|x|^4}.$$

In alternate quantization, the dual operator has $\Delta=1$ and

$$\langle\widetilde{\mathcal O}(x)\widetilde{\mathcal O}(0)\rangle \propto \frac{1}{|x|^2}.$$

The same bulk mass therefore does not always specify the CFT by itself. One must also specify the boundary condition, or equivalently which mode is treated as the source.

Why the calculation is classical at large $N$

The result above comes from the quadratic classical action. In the large-$N$ expansion, this is the leading connected two-point function of a single-trace operator, in the normalization where the source couples to an unnormalized single-trace operator.

Schematically,

$$I_{\mathrm{bulk}} \sim \frac{L^{d-1}}{G_{d+1}} \int d^{d+1}X\,\mathcal L \sim N^2\int d^{d+1}X\,\mathcal L$$

for matrix large-$N$ theories with Einstein-like duals. Thus

$$\langle\mathcal O\mathcal O\rangle \sim N^2$$

for unnormalized single-trace operators. If instead one defines a unit-normalized operator

$$\widehat{\mathcal O}=\frac{1}{N}\mathcal O,$$

then

$$\langle\widehat{\mathcal O}\widehat{\mathcal O}\rangle \sim N^0.$$

Bulk loop corrections are suppressed by powers of $G_{d+1}/L^{d-1}\sim1/N^2$. Stringy corrections modify the bulk action and propagator at finite ‘t Hooft coupling, and are suppressed only when the AdS radius is large compared with the string length.

Relation to the geodesic approximation

For a heavy operator, $\Delta\gg1$, the bulk-to-boundary propagator can be written as an exponential:

$$K_\Delta(z,x;y) = C_\Delta\exp\!\left[-\Delta\log\left(\frac{z^2+|x-y|^2}{z}\right)\right].$$

At large $\Delta$, the path integral for the bulk field is dominated by a worldline saddle. The two-point function is then approximated by

$$\langle\mathcal O(x)\mathcal O(y)\rangle \sim \exp\!\left[-m\ell_{\mathrm{ren}}(x,y)\right],$$

where $\ell_{\mathrm{ren}}(x,y)$ is the renormalized length of the spacelike geodesic connecting the two boundary points. This reproduces

$$\ell_{\mathrm{ren}}(x,y) \sim 2L\log\frac{|x-y|}{\epsilon}, \qquad mL\simeq\Delta,$$

and hence

$$\langle\mathcal O(x)\mathcal O(y)\rangle \sim |x-y|^{-2\Delta}.$$

The geodesic calculation is useful intuition, but the scalar field calculation is more general. It works for arbitrary $\Delta$ above the unitarity and stability bounds, fixes the normalization, and gives the correct contact-term structure after renormalization.

What is universal, and what is not

The scalar two-point function teaches a useful separation between universal structure and convention-dependent data.

Quantity	Status	Bulk origin
The power $	x-y	^{-2\Delta}$
The relation $m^2L^2=\Delta(\Delta-d)$	Universal for scalars	Near-boundary radial equation
The coefficient $C_{\mathcal O}$	Normalization-dependent	Overall kinetic term and field/operator normalization
Contact terms	Scheme-dependent	Local counterterms at $z=\epsilon$
Alternate quantization	Boundary-condition-dependent	Choice of source mode in the BF window
Lorentzian pole prescription	State- and contour-dependent	Interior or horizon boundary condition

This is a theme throughout holographic correlator calculations. The bulk geometry fixes much of the answer, but a precise CFT observable requires precise boundary conditions, precise normalization, and precise renormalization.

Common mistakes

Mistake 1: treating $K_\Delta$ as if it simply vanishes near the boundary. At fixed $x\ne y$, $K_\Delta$ goes like $z^\Delta$, but distributionally it contains the source term $z^{d-\Delta}\delta^{(d)}(x-y)$.

Mistake 2: forgetting the counterterms. The bare on-shell action diverges. Even if one only wants separated-point correlators, the correct finite answer is defined by the renormalized variational problem.

Mistake 3: confusing the two roots for $\Delta$. The equation $m^2L^2=\Delta(\Delta-d)$ has two roots. In standard quantization $\Delta=d/2+\nu$. Alternate quantization is available only in the appropriate BF window.

Mistake 4: dropping the overall bulk normalization. The coefficient of the two-point function is not fixed by the mass alone. It depends on $\mathcal N_\phi L^{d-1}$ and on how the boundary operator is normalized.

Mistake 5: using Euclidean regularity as a real-time prescription. Euclidean regularity is enough for Euclidean correlators. Retarded, advanced, Feynman, and thermal correlators require Lorentzian contour and interior boundary data.

Exercises

Exercise 1: The scalar wave equation in Poincare AdS

Starting from

$$ds^2=\frac{L^2}{z^2}\left(dz^2+d x^i d x^i\right),$$

show that the scalar equation $(\nabla^2-m^2)\phi=0$ becomes

$$z^{d+1}\partial_z\left(z^{1-d}\partial_z\phi\right) +z^2\partial_i\partial_i\phi -m^2L^2\phi=0.$$

Then insert $\phi\sim z^\alpha$ near the boundary and derive

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

Solution

For the metric,

$$\sqrt g=\left(\frac Lz\right)^{d+1}, \qquad g^{zz}=g^{ij}=\frac{z^2}{L^2}\delta^{ij}.$$

The Laplacian is

$$\nabla^2\phi = \frac{1}{\sqrt g}\partial_a\left(\sqrt g\,g^{ab}\partial_b\phi\right).$$

The radial part is

$$\frac{1}{\sqrt g}\partial_z\left(\sqrt g\,g^{zz}\partial_z\phi\right) = \frac{z^{d+1}}{L^2}\partial_z\left(z^{1-d}\partial_z\phi\right),$$

while the boundary derivative part is

$$\frac{z^2}{L^2}\partial_i\partial_i\phi.$$

Multiplying $(\nabla^2-m^2)\phi=0$ by $L^2$ gives the desired equation. Near $z=0$, take $\phi\sim z^\alpha$ and ignore the $x$-derivative term. Then

$$z^{d+1}\partial_z\left(z^{1-d}\partial_z z^\alpha\right) = \alpha(\alpha-d)z^\alpha.$$

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

Exercise 2: Normalizing the bulk-to-boundary propagator

Show that

$$K_\Delta(z,x;y) = C_\Delta \left(\frac{z}{z^2+|x-y|^2}\right)^\Delta$$

obeys

$$\lim_{z\to0}z^{\Delta-d}K_\Delta(z,x;y)=\delta^{(d)}(x-y)$$

provided

$$C_\Delta=\frac{\Gamma(\Delta)}{\pi^{d/2}\Gamma(\Delta-d/2)}.$$

Solution

Test the distribution against a smooth function $f(x)$:

$$\int d^d x\,z^{\Delta-d}K_\Delta(z,x;y)f(x) = C_\Delta\int d^d x\, \frac{z^{2\Delta-d}}{(z^2+|x-y|^2)^\Delta}f(x).$$

Set $x=y+zu$. Then $d^d x=z^d d^d u$, so

$$\frac{z^{2\Delta-d}}{(z^2+|x-y|^2)^\Delta}d^d x = \frac{d^d u}{(1+u^2)^\Delta}.$$

As $z\to0$, $f(y+zu)\to f(y)$, and the limit is

$$C_\Delta f(y) \int d^d u\,\frac{1}{(1+u^2)^\Delta}.$$

Using

$$\int d^d u\,\frac{1}{(1+u^2)^\Delta} = \pi^{d/2}\frac{\Gamma(\Delta-d/2)}{\Gamma(\Delta)},$$

the limit equals $f(y)$ precisely for the stated $C_\Delta$.

Exercise 3: Extracting the two-point coefficient

Assume that the nonlocal part of the near-boundary coefficient is

$$A(x)=C_\Delta\int d^d y\,\frac{\phi_{(0)}(y)}{|x-y|^{2\Delta}}.$$

Using

$$\langle\mathcal O(x)\rangle_{\phi_{(0)}} = \mathcal N_\phi L^{d-1}(2\Delta-d)A(x),$$

derive the separated-point two-point function.

Solution

Differentiate the one-point function with respect to the source:

$$\langle\mathcal O(x)\mathcal O(y)\rangle = \left. \frac{\delta\langle\mathcal O(x)\rangle_{\phi_{(0)}}}{\delta\phi_{(0)}(y)} \right|_{\phi_{(0)}=0}.$$

Since

$$\frac{\delta A(x)}{\delta\phi_{(0)}(y)} = \frac{C_\Delta}{|x-y|^{2\Delta}}$$

away from coincident points, one obtains

$$\langle\mathcal O(x)\mathcal O(y)\rangle = \mathcal N_\phi L^{d-1}(2\Delta-d)C_\Delta \frac{1}{|x-y|^{2\Delta}}, \qquad x\ne y.$$

Substituting $C_\Delta$ gives the boxed result in the main text.

Exercise 4: Momentum-space nonanalyticity

For noninteger $\nu$, use the expansion

$$K_\nu(q) = 2^{\nu-1}\Gamma(\nu)q^{-\nu} +2^{-\nu-1}\Gamma(-\nu)q^\nu+ \text{analytic corrections}$$

to show that the nonlocal part of the momentum-space two-point function scales as

$$\langle\mathcal O(p)\mathcal O(-p)\rangle \propto p^{2\nu}=p^{2\Delta-d}.$$

Why do integer values of $\nu$ require special care?

Solution

The regular solution normalized to the source is

$$\phi_p(z) = \phi_{(0)}(p) \frac{p^\nu}{2^{\nu-1}\Gamma(\nu)} z^{d/2}K_\nu(pz).$$

Using the expansion of $K_\nu$, the leading term is

$$z^{d/2}\frac{p^\nu}{2^{\nu-1}\Gamma(\nu)} \left[2^{\nu-1}\Gamma(\nu)(pz)^{-\nu}\right] = z^{d/2-\nu}=z^{d-\Delta}.$$

The subleading nonlocal term is

$$z^{d/2}\frac{p^\nu}{2^{\nu-1}\Gamma(\nu)} \left[2^{-\nu-1}\Gamma(-\nu)(pz)^\nu\right] = \frac{\Gamma(-\nu)}{2^{2\nu}\Gamma(\nu)}p^{2\nu}z^{d/2+\nu}.$$

Thus $A(p)\propto p^{2\nu}\phi_{(0)}(p)$, and the two-point function is proportional to $p^{2\nu}=p^{2\Delta-d}$, up to local terms and normalization constants.

When $\nu$ is an integer, logarithms appear in the Bessel expansion. The nonlocal term is then of the form $p^{2\nu}\log p^2$, while polynomial terms in $p^2$ are contact terms that depend on the renormalization scheme.

Exercise 5: Alternate quantization in the BF window

Let a scalar in AdS$_{d+1}$ have

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

Show that both

$$\Delta_+=\frac d2+\nu, \qquad \Delta_- = \frac d2-\nu$$

satisfy the scalar mass-dimension relation. What changes in the two-point function when one uses alternate quantization?

Solution

Both roots satisfy

$$\Delta(\Delta-d)=m^2L^2$$

because they are the two roots of the same quadratic equation. In standard quantization, the source is the coefficient of $z^{d-\Delta_+}=z^{\Delta_-}$ and the operator has dimension $\Delta_+$. The two-point function scales as

$$\langle\mathcal O_+(x)\mathcal O_+(0)\rangle \propto \frac{1}{|x|^{2\Delta_+}}.$$

In alternate quantization, the role of source and response is exchanged by a Legendre-transform-type boundary condition. The dual operator has dimension $\Delta_-$, and the two-point function scales as

$$\langle\mathcal O_-(x)\mathcal O_-(0)\rangle \propto \frac{1}{|x|^{2\Delta_-}}.$$

The bulk differential equation is the same, but the boundary condition and therefore the CFT are different.

Further reading

• S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, “Gauge Theory Correlators from Non-Critical String Theory”.
• E. Witten, “Anti De Sitter Space and Holography”.
• D. Z. Freedman, S. D. Mathur, A. Matusis, and L. Rastelli, “Correlation Functions in the CFT$_d$/AdS$_{d+1}$ Correspondence”.
• E. D’Hoker and D. Z. Freedman, “Supersymmetric Gauge Theories and the AdS/CFT Correspondence”.
• K. Skenderis, “Lecture Notes on Holographic Renormalization”.