Relevant Deformations and Domain Walls

The main idea

A conformal field theory is a fixed point of the renormalization group. To make it flow, we perturb it. The simplest and most controlled deformation is a relevant scalar operator,

$$S_{\mathrm{QFT}} = S_{\mathrm{CFT}} + \int d^d x\, \lambda\,\mathcal O(x), \qquad \Delta < d.$$

The coupling has mass dimension

$$[\lambda]=d-\Delta>0,$$

so the deformation becomes more important at long distances. In ordinary field-theory language, it changes the theory away from the UV CFT and asks what the infrared physics becomes: another CFT, a mass gap, confinement, spontaneous symmetry breaking, or something more exotic.

In holography, the same operation has a beautifully geometric avatar. The operator $\mathcal O$ is dual to a bulk scalar field $\phi$. Turning on the coupling $\lambda$ means imposing a non-normalizable boundary condition for $\phi$. The scalar profile backreacts on the metric. If the deformation preserves $d$-dimensional Poincaré invariance, the bulk solution takes the form of a domain wall,

$$ds^2 = dr^2+e^{2A(r)}\eta_{\mu\nu}dx^\mu dx^\nu, \qquad \phi=\phi(r).$$

Near the boundary, the metric approaches AdS and the dual theory approaches the UV CFT. Deeper in the bulk, the changing scalar and warp factor describe the RG evolution of the deformed theory.

A relevant deformation $\lambda\mathcal O$ of the UV CFT is implemented holographically by a scalar boundary condition. The bulk solution is a domain wall whose warp factor $A(r)$ sets the approximate energy scale $\mu\sim e^{A(r)}$, while the scalar profile $\phi^i(r)$ encodes running couplings and expectation values. The IR may be another AdS region, a smooth cap, a horizon, or a physically acceptable singular endpoint.

This page introduces the basic technology. Later pages will use the same setup to derive holographic $c$-theorems, model confinement, diagnose mass gaps, and add flavor degrees of freedom.

CFT deformations and dimensionless couplings

Start with a UV CFT in $d$ dimensions. A scalar primary operator $\mathcal O$ has scaling dimension $\Delta$, so under a scale transformation $x\to b x$,

$$\mathcal O(x)\to b^{-\Delta}\mathcal O(b^{-1}x).$$

The deformation

$$\delta S = \int d^d x\,\lambda\mathcal O$$

is dimensionless only if $[\lambda]=d-\Delta$. Therefore:

Type of operator	Condition	Coupling dimension	UV behavior
relevant	$\Delta<d$	positive	grows toward the IR
marginal	$\Delta=d$	zero	requires higher-order beta function
irrelevant	$\Delta>d$	negative	suppressed in the IR but dangerous in the UV

For a relevant deformation, a convenient dimensionless coupling at energy scale $\mu$ is

$$g(\mu)=\lambda\,\mu^{\Delta-d}.$$

At leading order near the UV fixed point,

$$\mu\frac{d g}{d\mu} = (\Delta-d)g+O(g^2).$$

Because $\Delta-d<0$, $g(\mu)$ grows as $\mu$ is lowered. This is the field-theory origin of the bulk intuition that a small UV scalar boundary condition can become large in the interior.

A source deformation is not the same thing as choosing a different state. The deformation changes the Lagrangian or Hamiltonian. A state changes expectation values while leaving the theory fixed. In holography this distinction becomes the difference between the two leading near-boundary modes of the bulk scalar.

The bulk model

The minimal bulk system for a holographic RG flow is Einstein gravity coupled to scalar fields. For one scalar, use

$$S = \frac{1}{2\kappa_{d+1}^2} \int d^{d+1}x\sqrt{-g} \left[ R-\frac12(\partial\phi)^2-V(\phi) \right] +S_{\mathrm{GH}}+S_{\mathrm{ct}}.$$

Here $S_{\mathrm{GH}}$ is the Gibbons-Hawking term and $S_{\mathrm{ct}}$ contains local counterterms needed to define finite one-point functions. For several scalars,

$$-\frac12(\partial\phi)^2 \quad\longrightarrow\quad -\frac12G_{IJ}(\phi)\partial_a\phi^I\partial^a\phi^J.$$

A CFT fixed point corresponds to an AdS critical point of the scalar potential:

$$\left.\frac{dV}{d\phi}\right|_{\phi_*}=0, \qquad V(\phi_*)=-\frac{d(d-1)}{L_*^2}.$$

The AdS radius $L_*$ determines the scale of CFT data such as the stress-tensor two-point coefficient,

$$C_T\sim \frac{L_*^{d-1}}{G_{d+1}},$$

up to a dimension-dependent normalization. If the same potential has two AdS critical points, a domain wall may interpolate between a UV CFT and an IR CFT. If the flow instead ends at a cap or a singularity, the field theory may have a mass gap, confinement, or another nonconformal infrared behavior.

Near a UV critical point, expand

$$V(\phi) = -\frac{d(d-1)}{L_{\mathrm{UV}}^2} +\frac12m^2\phi^2+\frac{1}{3!}v_3\phi^3+\frac{1}{4!}v_4\phi^4+\cdots.$$

The mass of $\phi$ is related to the dimension of the dual operator by

$$m^2L_{\mathrm{UV}}^2=\Delta(\Delta-d).$$

Thus a relevant deformation corresponds to a scalar whose mass lies in the range associated with $\Delta<d$. The mass may be negative without indicating an instability, as long as it obeys the Breitenlohner-Freedman bound,

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

Domain-wall equations

For a Poincaré-invariant RG flow, take

$$ds^2=dr^2+e^{2A(r)}\eta_{\mu\nu}dx^\mu dx^\nu, \qquad \phi=\phi(r).$$

The coordinate $r$ is chosen so that the UV boundary lies at $r\to +\infty$. Pure AdS is recovered when

$$A(r)=\frac{r}{L}, \qquad \phi=\phi_*.$$

For one scalar with canonical kinetic term, the equations of motion reduce to

$$\phi''+dA'\phi'-\frac{dV}{d\phi}=0,$$ $$A''=-\frac{1}{2(d-1)}(\phi')^2,$$

and the Hamiltonian constraint

$$(A')^2 = \frac{1}{d(d-1)} \left( \frac12(\phi')^2-V(\phi) \right).$$

The second equation is crucial:

$$A''\leq 0.$$

It says that, in this simple two-derivative model with positive scalar kinetic energy, the warp factor is concave as a function of the radial coordinate. This is the geometric seed of holographic monotonicity. The next page will turn it into a $c$-theorem.

The scalar equation has an intuitive interpretation. The term $dA'\phi'$ acts like friction in the radial evolution. Close to the UV boundary, $A'\simeq 1/L_{\mathrm{UV}}$, so the linearized equation is a damped equation whose two independent solutions are precisely the source and response modes.

The near-boundary dictionary

It is often clearer to use Fefferman-Graham coordinate $z$, related near the UV boundary to $r$ by

$$z\sim e^{-r/L_{\mathrm{UV}}}.$$

The metric then has the asymptotic form

$$ds^2 \simeq \frac{L_{\mathrm{UV}}^2}{z^2} \left(dz^2+\eta_{\mu\nu}dx^\mu dx^\nu\right), \qquad z\to 0.$$

The scalar behaves as

$$\phi(z) = z^{d-\Delta}\phi_{(s)} +z^{\Delta}\phi_{(v)} +\cdots,$$

when $\Delta\neq d/2$ and no logarithmic degeneracy is present. In terms of $r$,

$$\phi(r) = \phi_{(s)}e^{-(d-\Delta)r/L_{\mathrm{UV}}} + \phi_{(v)}e^{-\Delta r/L_{\mathrm{UV}}} + \cdots.$$

In standard quantization:

$$\phi_{(s)}\quad\leftrightarrow\quad \text{source }\lambda,$$ $$\phi_{(v)}\quad\leftrightarrow\quad \text{response, related to }\langle\mathcal O\rangle.$$

The precise relation between $\phi_{(v)}$ and $\langle\mathcal O\rangle$ is not simply a universal numerical constant. It can receive local contributions from counterterms, nonlinear terms in the asymptotic expansion, and conformal anomaly terms. Schematically,

$$\langle\mathcal O\rangle = \frac{1}{2\kappa_{d+1}^2} \left[ (2\Delta-d)\phi_{(v)}+\text{local terms in sources} \right].$$

For separated-point correlators the local terms often do not matter. For one-point functions, Ward identities, and thermodynamics, they matter enormously.

Source flows versus vev flows

A common trap is to call every scalar domain wall an RG flow triggered by a relevant operator. This is too quick.

There are two qualitatively different possibilities:

Boundary data	Field-theory interpretation	Typical name
$\phi_{(s)}\neq 0$	explicit deformation of the UV CFT	source flow
$\phi_{(s)}=0$, $\phi_{(v)}\neq 0$	nonzero expectation value in a state or vacuum	vev flow

A source flow changes the theory. A vev flow changes the vacuum of the same theory. This distinction is familiar in QFT: adding a mass term is not the same as choosing a vacuum with $\langle\mathcal O\rangle\neq0$.

The distinction is also visible in the trace Ward identity. For a deformed flat-space CFT, one expects schematically

$$\langle T^\mu{}_{\mu}\rangle = \sum_I \beta^I\langle\mathcal O_I\rangle +\mathcal A,$$

where $\mathcal A$ denotes possible anomaly terms. Near a fixed point with dimensionful sources, this is often written, up to sign conventions, as

$$\langle T^\mu{}_{\mu}\rangle \sim \sum_I (d-\Delta_I)\lambda^I\langle\mathcal O_I\rangle +\mathcal A.$$

If the source vanishes but the expectation value is nonzero, conformal invariance may be spontaneously rather than explicitly broken. Then the Ward identity and the spectrum can look very different. Coulomb-branch flows of $\mathcal N=4$ SYM are standard examples: the bulk is not simply dual to adding a relevant term to the Lagrangian.

How the radial direction becomes an RG scale

For the domain-wall metric,

$$ds^2=dr^2+e^{2A(r)}\eta_{\mu\nu}dx^\mu dx^\nu,$$

a natural local energy scale is

$$\mu(r)\sim e^{A(r)}.$$

This is not a literal equality with a unique scheme-independent RG scale. A radial redefinition changes how one labels slices. What is robust is the asymptotic statement: near AdS,

$$A(r)\simeq \frac{r}{L}, \qquad \mu\sim e^{r/L}\sim \frac{1}{z}.$$

Given scalar profiles $\phi^I(r)$, one can define holographic beta functions by differentiating with respect to $A$:

$$\beta^I(\phi) = \frac{d\phi^I}{d\log\mu} \simeq \frac{d\phi^I}{dA} = \frac{(\phi^I)'}{A'}.$$

This formula is useful, but it should be read with care. The bulk scalar value $\phi^I(r)$ is not always identical to a renormalized field-theory coupling in a canonical scheme. Field redefinitions in the bulk correspond to coupling redefinitions in the boundary theory. The invariant content is not the coordinate value of $\phi^I$ at a particular radial position, but the complete renormalized generating functional and its Ward identities.

Still, the formula captures the central intuition. A holographic RG flow is a radial evolution problem in which the boundary conditions at large $r$ specify UV data and the regularity or admissibility condition in the interior selects physical IR data.

First-order flow and superpotentials

Many important domain walls can be written in first-order form. Suppose there is a function $W(\phi)$ such that

$$V(\phi) = \frac12\left(\frac{dW}{d\phi}\right)^2 - \frac{d}{4(d-1)}W(\phi)^2.$$

Then the second-order equations are solved by

$$\phi'=\frac{dW}{d\phi}, \qquad A'=-\frac{1}{2(d-1)}W(\phi),$$

with the sign convention chosen so that $W<0$ for an AdS solution with $A'>0$.

For several scalars,

$$V = \frac12G^{IJ}\partial_IW\partial_JW - \frac{d}{4(d-1)}W^2,$$ $$(\phi^I)'=G^{IJ}\partial_JW, \qquad A'=-\frac{W}{2(d-1)}.$$

Then the holographic beta functions become

$$\beta^I = \frac{(\phi^I)'}{A'} = -2(d-1)G^{IJ}\frac{\partial_J W}{W}.$$

In supersymmetric flows, $W$ is usually a genuine superpotential determined by the supersymmetry variations of the gauged supergravity. In non-supersymmetric flows, a function $W$ may still exist locally; then it is often called a fake superpotential. The first-order form is powerful because it turns a boundary-value problem into a gradient-flow-like system, but it does not eliminate the need for UV holographic renormalization or IR regularity conditions.

What happens in the infrared?

The IR endpoint determines the physical character of the deformed theory. Common possibilities include:

Bulk endpoint	Boundary interpretation
another AdS region	flow to an IR CFT
smooth cap	mass gap; often confinement-like behavior
black-brane horizon	finite-temperature plasma state
acceptable singularity	possible zero-temperature IR phase, often requiring uplift or extra criteria
unacceptable singularity	likely unphysical solution or incomplete truncation

If the flow reaches another AdS critical point $\phi_{\mathrm{IR}}$, then

$$V(\phi_{\mathrm{IR}})=-\frac{d(d-1)}{L_{\mathrm{IR}}^2}.$$

The IR CFT has operator dimensions determined by the Hessian of $V$ at $\phi_{\mathrm{IR}}$,

$$m_{{\mathrm{IR}},I}^2L_{\mathrm{IR}}^2 = \Delta_{{\mathrm{IR}},I}(\Delta_{{\mathrm{IR}},I}-d).$$

Notice the subtlety: the same bulk scalar can correspond to one operator dimension near the UV fixed point and another near the IR fixed point. Operator identities along an RG flow are not usually as simple as tracking one scalar label. The scalar labels fields in a bulk effective theory; the CFT interpretation depends on the fixed point around which one linearizes.

If the flow ends at a singularity, one must be cautious. Some singularities are artifacts of a lower-dimensional truncation and become smooth after uplifting to ten or eleven dimensions. Some are physically meaningful endpoints of brane distributions. Others are pathological. A widely used practical criterion is that a physically acceptable singularity should admit a sensible finite-temperature deformation, or equivalently should satisfy appropriate boundedness conditions on the scalar potential along the solution. This is a useful diagnostic, not a magic theorem.

Holographic renormalization along flows

A domain wall dual to a relevant deformation is usually asymptotically AdS in the UV. Therefore the divergences of the on-shell action are still controlled by the near-boundary expansion. One introduces a cutoff surface $r=r_c$ or $z=\epsilon$, evaluates the regulated action, adds local covariant counterterms, and removes the cutoff.

The renormalized one-point functions are obtained by variation with respect to the boundary sources:

$$\langle \mathcal O_I\rangle = \frac{1}{\sqrt{g_{(0)}}} \frac{\delta S_{\mathrm{ren}}}{\delta \phi^I_{(s)}}.$$

For the stress tensor,

$$\langle T^{\mu\nu}\rangle = \frac{2}{\sqrt{g_{(0)}}} \frac{\delta S_{\mathrm{ren}}}{\delta g^{(0)}_{\mu\nu}}.$$

The Ward identities express the remaining bulk gauge redundancies. For scalar sources on a curved boundary, they take the schematic form

$$\nabla_\mu\langle T^{\mu}{}_{\nu}\rangle = \sum_I \langle\mathcal O_I\rangle\nabla_\nu\phi^I_{(s)},$$

and

$$\langle T^\mu{}_{\mu}\rangle = \sum_I (d-\Delta_I)\phi^I_{(s)}\langle\mathcal O_I\rangle +\mathcal A +\text{scheme-dependent local terms}.$$

For constant sources in flat space, the diffeomorphism Ward identity reduces to stress-tensor conservation. The trace Ward identity is the interesting one: it says the stress tensor is no longer traceless because the theory is no longer conformal.

This is one of the cleanest ways to see the difference between a conformal state and a deformed theory. A thermal state of a CFT on flat space still has a traceless stress tensor when the theory is exactly conformal, up to anomalies on curved backgrounds. A relevant deformation changes the operator identity for the trace.

A minimal computational workflow

Given a proposed holographic RG flow, the disciplined calculation goes as follows.

Step	Bulk task	Boundary interpretation
1	Specify $V(\phi)$ and the scalar kinetic metric	choose operator content and possible fixed points
2	Find UV AdS asymptotics	identify the UV CFT and operator dimensions
3	Choose source data $\phi_{(s)}$	specify the deformation $\lambda\mathcal O$
4	Impose IR regularity or admissibility	select the physical vacuum or phase
5	Add counterterms and vary $S_{\mathrm{ren}}$	compute $\langle\mathcal O\rangle$ and $\langle T_{\mu\nu}\rangle$
6	Study fluctuations around the background	compute spectra, correlators, transport, or stability

The most important practical point is that a domain-wall background alone is not the full answer. The background gives one-point functions and thermodynamics. Two-point functions require linearized fluctuations around the background. Spectra require normalizability and IR boundary conditions. Wilson loops require string worldsheets in the background. Entanglement requires extremal surfaces. The domain wall is the geometry on which many other observables are computed.

Example: a flow between two fixed points

Suppose $V(\phi)$ has two extrema:

$$V'(\phi_{\mathrm{UV}})=0, \qquad V'(\phi_{\mathrm{IR}})=0,$$

with

$$V(\phi_{\mathrm{UV}})=-\frac{d(d-1)}{L_{\mathrm{UV}}^2}, \qquad V(\phi_{\mathrm{IR}})=-\frac{d(d-1)}{L_{\mathrm{IR}}^2}.$$

A domain wall interpolating between them has

$$A(r)\sim \frac{r}{L_{\mathrm{UV}}} \quad (r\to +\infty),$$

and

$$A(r)\sim \frac{r}{L_{\mathrm{IR}}} \quad (r\to -\infty),$$

after choosing the IR coordinate orientation appropriately. Near each endpoint the geometry is approximately AdS, but the radius can differ. Since the number of degrees of freedom scales like $L^{d-1}/G_{d+1}$, the ratio

$$\frac{C_{T,\mathrm{IR}}}{C_{T,\mathrm{UV}}} \sim \left(\frac{L_{\mathrm{IR}}}{L_{\mathrm{UV}}}\right)^{d-1}$$

measures how many degrees of freedom remain at the IR fixed point. Under suitable energy conditions, holographic monotonicity implies the IR value should not exceed the UV value. This is the geometric version of the intuition that RG flow integrates out degrees of freedom.

Real top-down examples are richer. Supersymmetric mass deformations of $\mathcal N=4$ SYM lead to scalar profiles in five-dimensional gauged supergravity. Some flows approach nontrivial IR fixed points. Others describe confining or singular endpoints. The important lesson for this course is not the details of a particular potential, but the method: relevant deformations are boundary conditions; RG evolution is encoded in radial profiles; IR physics is encoded in the interior endpoint.

What is universal, and what is model-dependent?

The following statements are robust in standard two-derivative Einstein-scalar holography:

• a relevant scalar source changes the UV CFT action;
• the scalar near-boundary expansion separates source and response data;
• Poincaré-invariant deformations are described by domain-wall geometries;
• the radial scale is approximately an RG scale near AdS regions;
• positive scalar kinetic energy gives $A''\leq0$;
• holographic renormalization gives finite one-point functions and Ward identities.

The following statements are model-dependent:

• whether the IR endpoint is a CFT, a gap, confinement, or a singularity;
• whether a singularity is acceptable;
• the detailed beta functions away from fixed points;
• the precise spectrum of glueball-like modes;
• whether a bottom-up potential has a controlled string-theory embedding;
• how closely a phenomenological model resembles QCD.

The difference matters. Holographic RG flows are powerful because they organize strongly coupled dynamics geometrically. They are dangerous when the geometry is treated as if every feature automatically had a known UV-complete field-theory interpretation.

Common mistakes

Mistake 1: equating the radial coordinate with a unique RG scale. The relation $\mu\sim e^A$ is useful, especially near AdS regions, but it is not a scheme-independent local identity.

Mistake 2: calling every scalar profile a source deformation. The source mode and vev mode have different field-theory meanings. Many famous domain walls are vacuum or Coulomb-branch flows rather than explicit relevant deformations.

Mistake 3: forgetting backreaction. A scalar profile generically changes the metric. Treating it as a probe is a separate approximation and usually misses stress-tensor one-point functions and central-charge flow.

Mistake 4: trusting singular interiors automatically. A singular domain wall is not automatically wrong, but it is not automatically right either. One must check regularity, uplift, finite-temperature deformations, fluctuation spectra, and physical observables.

Mistake 5: reading the potential too literally. A bottom-up scalar potential is a model for a sector of RG physics. Without a top-down construction or a sharply defined boundary QFT, it is not a first-principles Lagrangian derivation.

Exercises

Exercise 1: Relevant coupling and RG growth

Let $\mathcal O$ be a scalar operator of dimension $\Delta<d$, and define the dimensionless coupling

$$g(\mu)=\lambda\mu^{\Delta-d}.$$

Show that the leading UV beta function is

$$\mu\frac{dg}{d\mu}=(\Delta-d)g.$$

Explain why this means the deformation grows toward the IR.

Solution

Since $\lambda$ is a fixed dimensionful parameter at leading order,

$$\mu\frac{dg}{d\mu} = \mu\frac{d}{d\mu}\left(\lambda\mu^{\Delta-d}\right) =(\Delta-d)\lambda\mu^{\Delta-d} =(\Delta-d)g.$$

For a relevant operator, $\Delta-d<0$. Therefore $g(\mu)$ decreases as $\mu$ is increased toward the UV and increases as $\mu$ is lowered toward the IR. Equivalently, if $\ell\sim1/\mu$ is a length scale, then $g\sim\lambda\ell^{d-\Delta}$ grows at large distances.

Exercise 2: Deriving the domain-wall equations

For the action

$$S =\frac{1}{2\kappa_{d+1}^2}\int d^{d+1}x\sqrt{-g} \left[R-\frac12(\partial\phi)^2-V(\phi)\right],$$

use the ansatz

$$ds^2=dr^2+e^{2A(r)}\eta_{\mu\nu}dx^\mu dx^\nu, \qquad \phi=\phi(r),$$

to show that

$$A''=-\frac{1}{2(d-1)}(\phi')^2.$$

Solution

The Einstein equations can be written as

$$R_{ab}=\frac12\partial_a\phi\partial_b\phi+\frac{1}{d-1}g_{ab}V(\phi).$$

For the domain-wall metric,

$$R_{rr}=-d\left(A''+(A')^2\right),$$

and

$$R_{\mu\nu}=-\left(A''+d(A')^2\right)g_{\mu\nu}.$$

The $\mu\nu$ equation gives

$$-\left(A''+d(A')^2\right)=\frac{V}{d-1}.$$

The $rr$ equation gives

$$-d\left(A''+(A')^2\right)=\frac12(\phi')^2+\frac{V}{d-1}.$$

Substituting the first relation into the second eliminates $V$ and yields

$$-(d-1)A''=\frac12(\phi')^2,$$

so

$$A''=-\frac{1}{2(d-1)}(\phi')^2.$$

Exercise 3: Source and vev in $r$ coordinates

In Fefferman-Graham coordinate $z$, a scalar behaves as

$$\phi(z)=z^{d-\Delta}\phi_{(s)}+z^\Delta\phi_{(v)}+\cdots.$$

Using $z=e^{-r/L}$ near the boundary, rewrite this in terms of $r$ and identify which term is the source in standard quantization.

Solution

Substituting $z=e^{-r/L}$ gives

$$z^{d-\Delta}=e^{-(d-\Delta)r/L}, \qquad z^{\Delta}=e^{-\Delta r/L}.$$

Therefore

$$\phi(r) = \phi_{(s)}e^{-(d-\Delta)r/L} + \phi_{(v)}e^{-\Delta r/L} +\cdots.$$

In standard quantization, $\phi_{(s)}$ is the source for the operator $\mathcal O$, while $\phi_{(v)}$ is related to $\langle\mathcal O\rangle$ after holographic renormalization.

Exercise 4: First-order flow implies the constraint

Assume

$$V(\phi) = \frac12(W')^2-\frac{d}{4(d-1)}W^2,$$

and

$$\phi'=W', \qquad A'=-\frac{W}{2(d-1)}.$$

Show that the Hamiltonian constraint

$$(A')^2 = \frac{1}{d(d-1)}\left(\frac12(\phi')^2-V\right)$$

is automatically satisfied.

Solution

Using the first-order equations,

$$(A')^2=\frac{W^2}{4(d-1)^2}, \qquad \frac12(\phi')^2=\frac12(W')^2.$$

The right-hand side of the Hamiltonian constraint is

$$\frac{1}{d(d-1)} \left[ \frac12(W')^2- \left( \frac12(W')^2-\frac{d}{4(d-1)}W^2 \right) \right].$$

The $W'$ terms cancel, leaving

$$\frac{1}{d(d-1)}\frac{d}{4(d-1)}W^2 = \frac{W^2}{4(d-1)^2} = (A')^2.$$

Thus the constraint is automatically satisfied.

Exercise 5: The trace Ward identity for a constant source

Consider a flat-space CFT deformed by a constant source $\lambda$ for an operator $\mathcal O$ of dimension $\Delta$. Ignoring anomalies and scheme-dependent contact terms, explain why dimensional analysis suggests

$$\langle T^\mu{}_{\mu}\rangle \propto (d-\Delta)\lambda\langle\mathcal O\rangle.$$

Solution

Under a Weyl rescaling, the source must transform with weight $d-\Delta$ so that

$$\int d^d x\sqrt g\,\lambda\mathcal O$$

remains dimensionless. The trace of the stress tensor measures the response of the renormalized generating functional to a Weyl rescaling. Therefore the explicit breaking caused by the source is proportional to the Weyl weight of the source, namely $d-\Delta$, times the source and the conjugate operator expectation value.

The precise sign depends on whether the deformation is written with $+\lambda\mathcal O$ or $-\lambda\mathcal O$ in the action and on Euclidean versus Lorentzian conventions. The invariant lesson is that the trace is controlled by beta functions or dimensionful sources times their conjugate operators.

Further reading

• D. Z. Freedman, S. S. Gubser, K. Pilch, and N. P. Warner, Renormalization Group Flows from Holography—Supersymmetry and a c-Theorem.
• J. de Boer, E. Verlinde, and H. Verlinde, On the Holographic Renormalization Group.
• M. Bianchi, D. Z. Freedman, and K. Skenderis, Holographic Renormalization.
• M. Bianchi, O. DeWolfe, D. Z. Freedman, and K. Pilch, Anatomy of Two Holographic Renormalization Group Flows.
• K. Skenderis, Lecture Notes on Holographic Renormalization.
• S. S. Gubser, Curvature Singularities: The Good, the Bad, and the Naked.