Conformal Symmetry and Representations

Conformal symmetry is more restrictive than scale symmetry because it constrains angles, causal cones, local operators, and their descendants at once. A sound conformal-symmetry claim must therefore pass several distinct tests: the transformation must exist on the stated spacetime domain, the generators must realize the correct real form, local operators must form the asserted modules, and any unitarity or shortening conclusion must follow from a positive inner product with a declared adjoint. This chapter builds those tests before correlation functions and crossing symmetry enter.

Helpful background. Lie Groups, Lie Algebras, the Exponential Map, and the Adjoint Action supplies the group–algebra distinction. Local Composite-Operator Insertions supplies renormalized local operators. Lorentz-Field and Poincaré-Particle Representations supplies Lorentz and Spin representations. Multiplets, Invariants, and Selection Rules supplies the general language of symmetry multiplets.

Enter this chapter

The natural route begins with spacetime and ends with the hypotheses under which scale symmetry can be enhanced:

Immediate question	Start with	Result to carry forward
Which finite maps are conformal, and where are they defined?	Conformal Geometry, Maps, and Compactification	A signature- and patch-aware action of conformal maps
How does the conformal Killing equation produce the generators?	The Conformal Algebra and Its Generators	A fixed commutator and Hermiticity convention for $P$, $M$, $D$, and $K$
How are local operators organized?	Primaries, Descendants, and Conformal Multiplets	Conformal families, descendant levels, reducibility, and null states
How are spin and tensor indices encoded?	Spin and Tensor Representations	Spin-cover labels and calculational polarization variables
What does positivity force?	Unitarity Bounds and Null States	Dimension bounds and the null equations at saturation
How are currents and the stress tensor normalized?	Conserved Currents and the Stress Tensor	Conserved multiplets, improvements, Ward-normalized OPE data, $C_J$, and $C_T$
How can a whole module be counted?	Characters and Conformal Multiplet Counting	Long and short characters with null submodules removed
Does scale invariance imply conformal invariance?	Scale versus Conformal Invariance	A hypothesis-by-hypothesis enhancement test and counterexample diagnosis

Readers preparing for the bootstrap should follow the order shown. Readers checking a proposed conformal theory can instead enter at the first failed condition: global geometry, algebraic closure, representation labels, positivity, conservation, or enhancement.

Conventions that connect the pages

Lorentzian spacetime uses the site’s mostly-minus metric $(+,-,\ldots,-)$; Euclidean space uses a positive-definite metric. In $d>2$, the local conformal Lie algebra is

$$\mathfrak{so}(d+1,1) \quad\text{in Euclidean signature}, \qquad \mathfrak{so}(d,2) \quad\text{in Lorentzian signature}.$$

The group acting globally depends additionally on connected component, discrete quotients, spacetime compactification, and—when spinors occur—the chosen Spin cover. Equality of Lie algebras alone does not settle these global questions.

For calculations in Euclidean radial quantization, the chapter uses the common differential-generator convention

$$[D,P_\mu]=P_\mu, \qquad [D,K_\mu]=-K_\mu, \qquad [K_\mu,P_\nu]=2\delta_{\mu\nu}D-2M_{\mu\nu}.$$

These generators are not all Hermitian: radial conjugation gives $D^\dagger=D$, $P_\mu^\dagger=K_\mu$, and $M_{\mu\nu}^\dagger=-M_{\mu\nu}$. Multiplying generators by appropriate factors of $i$ converts to a Hermitian-charge convention and inserts corresponding factors of $i$ into the structure constants. The pages never combine formulas from the two conventions without translating them. This distinction is central to the positivity derivations in Simmons-Duffin 2017, §§ 3.1–3.2.

Four logically different obligations

Obligation	Question that must be answered	Typical failure
Local geometry	Does the conformal Killing equation hold on the chosen patch?	A formula is singular on a null cone or denominator hypersurface
Global action	Does the local map extend to the stated compactification and component of the group?	Inversion or a special conformal transformation crosses infinity without a second patch
Representation theory	Which real form and Spin-cover representation acts on each operator, and which descendants are quotiented?	A Lorentz irrep is confused with a Euclidean rotation irrep, or a null state is counted twice
Positivity and dynamics	Is there a positive form with the required adjoint, and do Ward identities or field equations justify shortening?	A nonunitary module is subjected to a unitarity bound, or conservation is inferred from dimension without its hypotheses

These obligations interact but are not interchangeable. A locally conformal coordinate change need not be a globally defined symmetry. Algebraic reducibility does not by itself imply that a state has zero norm. Conversely, a vanishing descendant in a unitary module has both an algebraic consequence—the quotient is shorter—and a local consequence such as a conservation or free-field equation.

From geometry to bootstrap data

At the origin, a primary operator obeys $[K_\mu,\mathcal O(0)]=0$. Translations generate descendants, so a conformal family has the schematic form

$$\mathcal O(0), \quad P_\mu\mathcal O(0), \quad P_{\mu_1}P_{\mu_2}\mathcal O(0), \quad\ldots.$$

The rotation representation and scaling dimension of the primary label the generic module. Positivity of descendant norms bounds the scaling dimension. At saturation a descendant becomes null, and quotienting its submodule produces a short representation. For a symmetric traceless primary of spin $\ell\geq1$ in a reflection-positive $d\geq3$ CFT,

$$\Delta\geq \ell+d-2,$$

with saturation giving the conservation equation. A nonidentity scalar instead obeys $\Delta\geq(d-2)/2$, and saturation gives a level-two free-field null. The hypotheses and the distinct scalar derivation are developed on the unitarity page; they are not consequences of scale invariance alone. The classification and its bootstrap use are reviewed in Poland, Rychkov, and Vichi 2019, §§ III.B and III.E.

Currents and the stress tensor are the most important short multiplets because their conservation equations encode global-symmetry and spacetime Ward identities. Their two-point normalizations $C_J$ and $C_T$, together with Ward-fixed OPE coefficients, become numerical bootstrap data. Characters then count the remaining descendants after the null submodule has been removed.

Boundaries and handoffs

This chapter assumes, rather than redevelops, the general theory of Lie groups and Lorentz representations in Mathematical Methods, local composite operators in Foundations of Quantum Field Theory, Noether currents and anomalies in Symmetry and Gauge Structure, and fixed-point machinery in Renormalization and Effective Field Theory. Theorem-first reconstruction, conformal nets, and exceptional analytic hypotheses remain with Mathematical QFT. This chapter’s task is to assemble the conformal specialization with enough convention data that later calculations are reproducible.

Radial Quantization and State–Operator Correspondence constructs the cylinder Hilbert space, reflection-positive inner product, and Gram matrices. Correlators, OPE, and Conformal Blocks turns representation labels into correlators. From the Local OPE to Conformal Data makes conformal families into an expansion, and Crossing Equations and Positivity supplies the consistency equations. Questions about exceptional dimensions, nonunitary theories, and theorem-strength global claims must retain the hypotheses stated on the individual pages.

Review the chapter

Explain why the formula $x'^\mu=(x^\mu-b^\mu x^2)/(1-2b\mathbin{\cdot}x+b^2x^2)$ does not by itself define a global symmetry of an uncompactified spacetime.

Solution

The denominator can vanish, so the image reaches conformal infinity and leaves the original coordinate patch. In Lorentzian signature the singular set and causal domains also depend on null directions. A global claim therefore needs a compactification, a choice of connected component or discrete extension, and—if spinors transform—a lift to a Spin cover.

Explain why $\Delta=\ell+d-2$ is not enough, without further assumptions, to prove that a spin-$\ell$ operator is conserved.

Solution

The conclusion uses a positive-energy, reflection-positive conformal representation with radial adjoint $P^\dagger=K$. Positivity turns saturation of the descendant-norm inequality into a null state, and the state–operator correspondence turns that null state into a local conservation equation. In a nonunitary or logarithmic representation the same dimension need not imply a vanishing descendant.

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