The Conformal Algebra and Its Generators

In flat dimension $d>2$, the conformal Killing equation has only a finite-dimensional solution space. Its constants become the generators of translations $P_\mu$, rotations or Lorentz transformations $M_{\mu\nu}$, dilatations $D$, and special conformal transformations $K_\mu$. Their commutators form $\mathfrak{so}(d+1,1)$ in Euclidean signature and $\mathfrak{so}(d,2)$ in Lorentzian signature. Signs and factors of $i$ depend on whether one writes differential actions or Hermitian charges, so this page fixes one convention and gives the translation rule.

Required background. Conformal Geometry, Maps, and Compactification supplies the conformal Killing equation, signature, and global-domain qualifications. Lie Groups, Lie Algebras, the Exponential Map, and the Adjoint Action supplies Lie brackets, exponentiation, and global-form distinctions. Helpful background. Classical Symmetries, Currents, and the Stress Tensor supplies Noether charges and their action on fields.

Solving the conformal Killing equation

With a constant Euclidean or Lorentzian metric $g_{\mu\nu}$, an infinitesimal conformal vector field obeys

$$\partial_\mu v_\nu+\partial_\nu v_\mu =\frac{2}{d}g_{\mu\nu}\,\partial_\rho v^\rho.$$

Define $\sigma=(1/d)\partial\mathbin{\cdot}v$. Differentiating the equation and permuting its three indices gives

$$\partial_\mu\partial_\nu v_\rho =g_{\mu\rho}\partial_\nu\sigma +g_{\nu\rho}\partial_\mu\sigma -g_{\mu\nu}\partial_\rho\sigma.$$

Taking $\partial^\rho$ and using $\partial^\rho v_\rho=d\sigma$ yields

$$(d-2)\partial_\mu\partial_\nu\sigma +g_{\mu\nu}\partial^2\sigma=0.$$

Its trace gives $2(d-1)\partial^2\sigma=0$. For $d>2$, therefore, $\partial_\mu\partial_\nu\sigma=0$: $\sigma$ is affine and $v^\mu$ is at most quadratic. Integrating gives

$$v^\mu(x) =a^\mu+\omega^\mu{}_{\nu}x^\nu +\lambda x^\mu +2(b\mathbin{\cdot}x)x^\mu-b^\mu x^2, \qquad \omega_{\mu\nu}=-\omega_{\nu\mu}.$$

There are $d$ parameters $a^\mu$, $d(d-1)/2$ parameters $\omega^{\mu\nu}$, one $\lambda$, and $d$ parameters $b^\mu$, for a total $(d+1)(d+2)/2$. This equals the dimension of $\mathfrak{so}(d+1,1)$ and $\mathfrak{so}(d,2)$. The step multiplying $\partial_\mu\partial_\nu\sigma$ by $d-2$ is exactly why the argument does not apply in $d=2$. In $d=1$ every local reparametrization is conformal at the metric level; in $d=2$ holomorphic and antiholomorphic vector fields give infinite-dimensional local algebras. The finite set above remains the global conformal subalgebra on the compactified geometry Simmons-Duffin 2017, § 2.1.

One commutator convention

For the remainder of this chapter, $P_\mu$, $M_{\mu\nu}=-M_{\nu\mu}$, $D$, and $K_\mu$ denote the Euclidean radial-quantization generators with no explicit factors of $i$. Replacing $\delta_{\mu\nu}$ below by the mostly-minus $\eta_{\mu\nu}$ gives the corresponding Lorentzian real-form brackets:

$$\begin{aligned} [M_{\mu\nu},M_{\rho\sigma}] ={}&g_{\nu\rho}M_{\mu\sigma} -g_{\mu\rho}M_{\nu\sigma} +g_{\nu\sigma}M_{\rho\mu} -g_{\mu\sigma}M_{\rho\nu},\\ [M_{\mu\nu},P_\rho] ={}&g_{\nu\rho}P_\mu-g_{\mu\rho}P_\nu,\\ [M_{\mu\nu},K_\rho] ={}&g_{\nu\rho}K_\mu-g_{\mu\rho}K_\nu,\\ [D,P_\mu]={}&P_\mu, \qquad [D,K_\mu]=-K_\mu,\\ [K_\mu,P_\nu] ={}&2g_{\mu\nu}D-2M_{\mu\nu}. \end{aligned}$$

All other independent brackets vanish. This algebra and its action on local operators are summarized independently in Poland, Rychkov, and Vichi 2019, §§ III.A–III.B. Radial conjugation is

$$D^\dagger=D, \qquad P_\mu^\dagger=K_\mu, \qquad M_{\mu\nu}^\dagger=-M_{\mu\nu}.$$

Thus $M_{\mu\nu}$ is represented anti-Hermitian in a unitary finite-dimensional rotation representation, while $P$ and $K$ are adjoints of one another rather than separately Hermitian. If instead $\widehat Q$ denotes a Hermitian Lorentzian Noether charge, its action is conventionally $\delta\mathcal O=i[\epsilon\widehat Q,\mathcal O]$ and its algebra contains explicit $i$ multiplying the same real structure constants. Dropping those $i$‘s without also changing the generator definitions reverses signs in later norm calculations. The convention and radial adjoint above agree with Simmons-Duffin 2017, Eqs. (32)–(37) and §§ 7.1–7.2.

Embedding the algebra

Introduce antisymmetric generators $J_{AB}$ in an auxiliary space with two extra coordinates and metric $G_{AB}$ of signature $(d+1,1)$ or $(d,2)$. They obey

$$[J_{AB},J_{CD}] =G_{BC}J_{AD}-G_{AC}J_{BD} +G_{BD}J_{CA}-G_{AD}J_{CB}.$$

In a null basis $(+,-,\mu)$ with $G_{+-}=G_{-+}=1$, $G_{++}=G_{--}=0$, and $G_{\mu\nu}=g_{\mu\nu}$, choose

$$M_{\mu\nu}=J_{\mu\nu}, \qquad D=J_{+-}, \qquad P_\mu=J_{+\mu}, \qquad K_\mu=2J_{-\mu}.$$

With the displayed metric, these definitions reproduce the preceding brackets directly. Rescaling $J_{+\mu}$ and $J_{-\mu}$ inversely changes the displayed factors but not the algebra. The invariant statement is that $P$ and $K$ occupy opposite null directions while $D$ is the boost in their plane. This construction explains the real forms but does not choose a global quotient or Spin cover.

The geometric relation is summarized below. Inspect how the same complexified algebra acquires different real forms and global domains.

The generators act linearly in an embedding space and nonlinearly in an affine spacetime patch. Euclidean signature gives $\mathfrak{so}(d+1,1)$; Lorentzian signature gives $\mathfrak{so}(d,2)$. Compactification resolves coordinate infinities, but the connected group, discrete quotient, universal cover, and Spin lift remain separate choices. The diagram is schematic.

Feature	Euclidean realization	Lorentzian realization
Metric used in brackets	Positive $\delta_{\mu\nu}$	Mostly-minus $\eta_{\mu\nu}$
Local conformal real form	$\mathfrak{so}(d+1,1)$	$\mathfrak{so}(d,2)$
Natural compactified base	$S^d$	$(S^{d-1}\times S^1)/\mathbb Z_2$ or its universal cover
Positivity convention used later	Reflection-positive radial quantization, $P^\dagger=K$	Positive-energy Hilbert space after a declared continuation and charge convention
Information not fixed by the algebra	Orientation component, quotient, Spin cover	Time orientation, quotient, time cover, Spin cover
Scale symmetry alone	Contains $P$, $M$, and $D$ but does not imply $K$	Same; enhancement requires a stress-tensor and virial-current argument

Action on a primary scalar

Let $\mathcal O(0)$ be a scalar primary of dimension $\Delta$:

$$[D,\mathcal O(0)]=\Delta\mathcal O(0), \qquad [M_{\mu\nu},\mathcal O(0)]=0, \qquad [K_\mu,\mathcal O(0)]=0.$$

Translations define $\mathcal O(x)=e^{x\cdot P}\mathcal O(0)e^{-x\cdot P}$. Repeated use of the Baker–Campbell–Hausdorff formula terminates because $[P_\rho,[P_\sigma,K_\mu]]$ is proportional to $P$ and the next commutator vanishes. The result is

$$[K_\mu,\mathcal O(x)] =\left(2x_\mu x\mathbin{\cdot}\partial -x^2\partial_\mu+2\Delta x_\mu\right)\mathcal O(x).$$

For a spinning primary one adds $-2x^\nu S_{\mu\nu}$ in this convention. At $x=0$ the right-hand side vanishes, as required. Acting with $D$ on the same expression gives weight $-1$ for $K_\mu$, while commuting it with translations reproduces $[K_\mu,P_\nu]=2g_{\mu\nu}D-2M_{\mu\nu}$. This is a concrete closure check rather than an appeal to the name of the group.

Exponentiating the scalar action gives the finite special conformal law

$$\mathcal O'(x') =\bigl(1-2b\mathbin{\cdot}x+b^2x^2\bigr)^\Delta \mathcal O(x)$$

on a Euclidean patch where the denominator has fixed positive sign. If $\Delta$ is not an integer, crossing a negative or complex value requires a branch prescription; in Lorentzian signature it must be correlated with the causal boundary value. The algebraic exponential does not remove the domain qualifications established on the geometry page.

A bounded calculation can be used for checking commutators and small descendant Gram matrices after the dimension, signature, generator convention, and representation matrices have been supplied. Its output is evidence for those declared inputs, not a replacement for the analytic derivation above.

Primaries, Descendants, and Conformal Multiplets uses this grading to build local-operator modules. The separate Radial Quantization and the State–Operator Correspondence constructs the positive form and explains why the radial adjoint is physical rather than a formal involution.

Common pitfalls

Changing signature by replacing one symbol. The metric substitution translates local brackets, but the adjoint, positive-energy condition, compactification, and global group also change. State all of them before transferring a positivity result.

Calling every generator Hermitian. That is inconsistent with the no-$i$ Euclidean convention used here. Either keep the radial adjoint above or convert every bracket and field action to Hermitian charges.

Using the finite-dimensional derivation in two dimensions. The inference $\partial_\mu\partial_\nu\sigma=0$ divided by $d-2$. Two-dimensional local conformal transformations require a separate complex-analytic treatment.

Exercises

Count the conformal generators in $d>2$ and verify that the result equals $\dim\mathfrak{so}(d+2)$.

Solution

There are $d$ translations, $d(d-1)/2$ rotations or boosts, one dilatation, and $d$ special conformal generators. Their sum is

$$2d+1+\frac{d(d-1)}2=\frac{(d+1)(d+2)}2,$$

which is the number of independent antisymmetric pairs in $d+2$ dimensions.

Use the algebra to show that $P_\mu\lvert\mathcal O\rangle$ has scaling dimension $\Delta+1$ when $D\lvert\mathcal O\rangle=\Delta\lvert\mathcal O\rangle$.

Solution

From $[D,P_\mu]=P_\mu$,

$$D P_\mu\lvert\mathcal O\rangle =P_\mu D\lvert\mathcal O\rangle+[D,P_\mu]\lvert\mathcal O\rangle =(\Delta+1)P_\mu\lvert\mathcal O\rangle.$$

References

• Poland, David, Slava Rychkov, and Alessandro Vichi. “The Conformal Bootstrap: Theory, Numerical Techniques, and Applications.” Reviews of Modern Physics 91 (2019): 015002. DOI; Open PDF
• Simmons-Duffin, David. “TASI Lectures on the Conformal Bootstrap.” In New Frontiers in Fields and Strings, 1–74. Singapore: World Scientific, 2017. DOI; Open PDF