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Type of local operator in conformal field theory
In theoretical physics, a primary field, also called a primary operator, or simply a primary, is a local operator in a conformal field theory which is annihilated by the part of the conformal algebra consisting of the lowering generators. From the representation theory point of view, a primary is the lowest dimension operator in a given representation of the conformal algebra. All other operators in a representation are called descendants; they can be obtained by acting on the primary with the raising generators.

Primary fields in a D dimensional conformal field theory were introduced in 1969 by Mack and Salam where they were called interpolating fields. They were then studied by Ferrara, Gatto, and Grillo who called them irreducible conformal tensors, and by Mack who called them lowest weights. Polyakov used an equivalent definition as fields which cannot be represented as derivatives of other fields. The modern terms primary fields and descendants were introduced by Belavin, Polyakov and Zamolodchikov in the context of two dimensional conformal field theory. This terminology is now used both for D=2 and D>2.

In dimensions conformal primary fields can be defined in two equivalent ways. Campos Delgado provided a pedagogical proof of the equivalence.

In two dimensions, conformal field theories are invariant under an infinite dimensional Virasoro algebra with generators Primaries are defined as the operators annihilated by all with n>0, which are the lowering generators. Descendants are obtained from the primaries by acting with with n<0. The Virasoro algebra has a finite dimensional subalgebra generated by Operators annihilated by are called quasi primaries. Each primary field is a quasi primary, but the converse is not true; in fact each primary has infinitely many quasi primary descendants. Quasi primary fields in two dimensional conformal field theory are the direct analogues of the primary fields in the D>2 dimensional case.

In dimensions, conformal algebra allows graded extensions containing fermionic generators. Quantum field theories invariant with respect to such extended algebras are called superconformal. In superconformal field theories, one considers superconformal primary operators. In dimensions, superconformal primaries are annihilated by and by the fermionic generators (one for each supersymmetry generator). Generally, each superconformal primary representations will include several primaries of the conformal algebra, which arise by acting with the supercharges on the superconformal primary. There exist also special chiral superconformal primary operators, which are primary operators annihilated by some combination of the supercharges. Roughly, these bounds say that the dimension of an operator must be not smaller than the dimension of a similar operator in free field theory. In four dimensional conformal field theory, the unitarity bounds were first derived by Ferrara, Gatto and Grillo and by Mack.