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.
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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, 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 theories. 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
Conformal field theory in D=2 dimensions
In two dimensions, conformal field theories are invariant under an infinite dimensional Virasoro algebra with generators
The Virasoro algebra has a finite dimensional subalgebra generated by
Superconformal field theory
In
In D>2 dimensions, superconformal primaries are annihilated by
In D=2 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.