Conformal field theory (CFT)

A conformal field theory is a quantum field theory invariant under conformal transformations. Local operators are organized into conformal families labelled by scaling dimension and spin.

A conformal cross-ratio is a dimensionless combination of point separations invariant under conformal transformations. Four generic points in dimension greater than two have two independent cross-ratios,

The scaling dimension of a local operator is its eigenvalue under dilations. Under $x\mapsto \lambda x$, a scalar operator of dimension $\Delta$ transforms by $\mathcal{O}(x)\mapsto {\lambda }^{-\Delta }\mathcal{O}(\lambda x)$.

An exactly marginal operator has scaling dimension equal to the spacetime dimension and a deformation coupling whose beta function vanishes to all orders. Varying that coupling moves along a family of conformal field theories without introducing a scale.

The conformal algebra in $d$ Euclidean dimensions is generated by translations $P_a$, rotations $M_{ab}$, dilations $D$, and special conformal transformations $K_a$.

A conformal primary operator is annihilated at the origin by every special conformal generator. Acting with translation generators produces its conformal descendants.

A scalar primary operator transforms without a spin rotation and is characterized by its scaling dimension. Conformal symmetry fixes its two- and three-point functions up to constants, while a four-point function can depend on conformal cross-ratios.

A conformal descendant is obtained by applying one or more translation generators to a conformal primary operator. Its scaling dimension exceeds that of the primary by a nonnegative integer.

Positivity of descendant norms constrains the scaling dimension of a primary operator. For a nonidentity scalar primary in $d\ge 2$, unitarity requires $\Delta \ge (d-2)/2$.

The state–operator correspondence maps a local operator of scaling dimension $\Delta$ to a state on $S^{d-1}\times \mathbb{R}$ whose cylinder energy is $\Delta /r$ for sphere radius $r$.

For a primary operator, $T(z)\mathcal{O}(0)\sim h\mathcal{O}(0)/z^2+\partial \mathcal{O}(0)/z$. Taking the residue against $z^{n+1}$ gives the weight for $n=0$, the derivative descendant for $n=-1$, and zero for every $n>0$. Conversely the absence of positive-mode actions excludes higher poles in this operator product and gives the primary transformation law. The antiholomorphic argument is identical. Thus primary states and primary operators correspond under radial quantization.

If the only light field in AdS4 is the graviton, the dual CFT3 spectrum begins with the vacuum at dimension zero, the spin-two stress tensor at dimension three, and its non-null level-one descendants of spins two and three at dimension four.