Primary field

A primary field in a two-dimensional conformal field theory transforms homogeneously under local conformal coordinate changes, with conformal weights $(h,\tilde{h})$. Its scaling dimension is $h+\tilde{h}$ and its two-dimensional spin is $h-\tilde{h}$.

In the holomorphic sector, its defining operator product expansion with the stress-energy tensor is $T(z)\mathcal{O}(w)\sim h\mathcal{O}(w)/(z-w{)}^2+\partial \mathcal{O}(w)/(z-w)$. The double pole encodes rescaling and the simple pole translation. Acting with negative Virasoro algebra modes generates Virasoro descendants. This separates each family into a primary starting point and symmetry-generated operators, greatly reducing the independent information needed to specify correlation functions.

A primary state is a simultaneous eigenvector of $L_0,{\tilde{L}}_0$ with eigenvalues $h,\tilde{h}$ that is annihilated by all positive modes of both Virasoro algebras. Negative modes generate descendants. The state–operator correspondence identifies it with a primary operator inserted at the origin.

For free embedding bosons, the operator product expansion of the stress tensor with ${\zeta }_{\mu }:\partial X^{\mu }{e}^{ipX}:$ has a cubic pole $-i{\alpha }^{\prime }(p\cdot \zeta ):e^{ipX}:/2$. Its vanishing is precisely the primary condition, with weight $1+{\alpha }^{\prime }{p}^2/4$. The two-derivative closed-string vertex has the two displayed transversality conditions from its holomorphic and antiholomorphic cubic poles. Being primary does not require $p^2=0$; that extra condition follows when a physical integrated vertex must have conformal weights $(1,1)$.

The conformal weights determine the response of a primary operator to holomorphic and antiholomorphic coordinate rescaling. Its scaling dimension is $\Delta =h+\tilde{h}$ and its spin is $h-\tilde{h}$.

A Virasoro descendant is obtained by applying negative Virasoro modes $L_{-n}$ to a primary state or operator. Its level is the sum of the mode numbers.