Primary state ($L_{n>0}|h,\tilde{h}⟩={\tilde{L}}_{n>0}|h,\tilde{h}⟩=0$)

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.