Primary field

History of the concept

Primary fields in a D-dimensional conformal field theory were introduced in 1969 by Mack and Salam{{Cite journal

The modern terms primary fields and descendants were introduced by Belavin, Polyakov and Zamolodchikov{{Cite journal

Conformal field theory in ''D''>2 spacetime dimensions

In d2 dimensions conformal primary fields can be defined in two equivalent ways.

First definition

Let \hat{D} be the generator of dilations and let \hat{K}{\mu} be the generator of special conformal transformations. A conformal primary field \hat{\phi}^M{\rho}(x) , in the \rho representation of the Lorentz group and with conformal dimension \Delta satisfies the following conditions at x=0 :

1. \left[\hat{D},\hat{\phi}^M_{\rho}(0)\right]=-i\Delta\hat{\phi}^M_{\rho}(0);
2. \left[\hat{K}{\mu},\hat{\phi}^M{\rho}(0)\right]=0.

Second definition

A conformal primary field \hat{\phi}^M_{\rho}(x), in the \rho representation of the Lorentz group and with conformal dimension \Delta, transforms under a conformal transformation \eta_{\mu \nu}\mapsto \Omega^2(x)\eta_{\mu \nu} as :\hat{\phi'}^M_{\rho}(x')=\Omega^{\Delta}(x)\mathcal{D}{\left[R(x)\right]^M}{N}\hat{\phi}^N{\rho}(x) where {R^{\mu}}{\nu}(x)=\Omega^{-1}(x)\frac{\partial x^{\mu}}{\partial x'^{\nu}} and \mathcal{D}{\left[R(x)\right]^M}{N} implements the action of R in the SO(d-1,1) representation of \hat{\phi}^{M}_{\rho}(x).

Conformal field theory in ''D''{{=}}2 dimensions

In two dimensions, conformal field theories are invariant under an infinite dimensional Virasoro algebra with generators L_n, \bar{L}_n, -\infty. Primaries are defined as the operators annihilated by all L_n, \bar{L}_n with n0, which are the lowering generators. Descendants are obtained from the primaries by acting with L_n, \bar{L}_n with n

The Virasoro algebra has a finite dimensional subalgebra generated by L_n, \bar{L}_n, -1\le n\le 1. Operators annihilated by L_1, \bar{L}_1 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 D2 dimensional case.

Superconformal field theory

Source:

In D\le 6 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 D2 dimensions, superconformal primaries are annihilated by K_\mu and by the fermionic generators S (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 Q on the superconformal primary. There exist also special chiral superconformal primary operators, which are primary operators annihilated by some combination of the supercharges.

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.{{Cite journal

References

References

1. (2022). "On the equivalence of two definitions of conformal primary fields in d > 2 dimensions". Eur. Phys. J. Plus.
2. Aharony, Ofer. (2000). "Large N field theories, string theory and gravity". Physics Reports.