Primary field
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. History of the concept. 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, , 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. Conformal field theory in "D">2 spacetime dimensions. The lowering generators of the conformal algebra in "D">2 dimensions  are the special conformal transformation generators formula_1. Primary operators inserted at formula_2 are annihilated by these generators: formula_3. The descendants are obtained by acting on the primaries with the translation generators formula_4; these are just the derivatives of the primaries. Conformal field theory in "D"2 dimensions. In two dimensions, conformal field theories are invariant under an infinite dimensional Virasoro algebra with generators formula_5. Primaries are defined as the operators annihilated by all formula_6 with "n">0, which are the lowering generators. Descendants are obtained from the primaries by acting with formula_6 with "n"<0. The Virasoro algebra has a finite dimensional subalgebra  generated by formula_8. Operators annihilated by formula_9 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. Superconformal field theory. In formula_10 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 formula_11 dimensions, superconformal primaries are annihilated by  formula_1 and by the fermionic generators formula_13 (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 formula_14 on the superconformal primary. There exist also special "chiral" superconformal primary operators, which are primary operators annihilated by some combination of the supercharges. In formula_15 dimensions, superconformal field theories are invariant under super Virasoro algebras, which include infinitely many fermionic operators. Superconformal primaries are annihilated by all lowering operators, bosonic and fermionic. Unitarity bounds. In unitary (super)conformal field theories, dimensions of primary operators satisfy lower bounds called  the unitarity bounds. 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.