# Primary field

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{{Short description|Type of local operator in conformal field theory}}
In [theoretical physics](/source/theoretical_physics), a '''primary field''', also called a '''primary operator''', or simply a '''primary''', is a local operator in a [conformal field theory](/source/conformal_field_theory) which is annihilated by the part of the [conformal algebra](/source/conformal_algebra) consisting of the lowering generators. From the [representation theory](/source/representation_theory) point of view, a primary is the lowest dimension operator in a given [representation](/source/representation_(mathematics)) of the [conformal algebra](/source/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](/source/Abdus_Salam)<ref>{{Cite journal     
| doi = 10.1016/0003-4916(69)90278-4
| issn = 0003-4916
| volume = 53
| issue = 1
| pages = 174–202
| author = G Mack
|author2=Abdus Salam
| title = Finite-component field representations of the conformal group
| journal = Annals of Physics
| date=1969
|bibcode = 1969AnPhy..53..174M }}</ref> where they were called ''interpolating fields''.  They were then studied by Ferrara, [Gatto](/source/%3Ait%3ARaoul_Gatto), and Grillo<ref>{{Cite book
| publisher = Springer-Verlag
| isbn = 9783540062165
| last = Ferrara
| first = Sergio
|author2=Raoul Gatto |author3=A. F. Grillo
 | title = Conformal Algebra in Space-Time and Operator Product Expansion
| date = 1973
}}</ref> who called them ''irreducible conformal tensors'', and by Mack<ref name=Mack>{{Cite journal
| volume = 55
| issue = 1
| pages = 1–28
| author = G. Mack
| title = All unitary ray representations of the conformal group SU(2, 2) with positive energy
| journal = Communications in Mathematical Physics
| accessdate = 2013-12-05
| date = 1977
| url = http://projecteuclid.org/euclid.cmp/1103900926
| doi=10.1007/bf01613145
| s2cid = 119941999
}}</ref> who called them ''lowest weights''. Polyakov<ref>{{Cite journal
| issn = 1063-7761
| volume = 39
| pages = 10
| last = Polyakov
| first = A. M.
| title = Non-Hamiltonian approach to conformal quantum field theory
| journal = Soviet Journal of Experimental and Theoretical Physics
| date = 1974
|bibcode = 1974JETP...39...10P }}</ref> 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<ref>{{Cite journal
| doi = 10.1016/0550-3213(84)90052-X
| issn = 0550-3213
| volume = 241
| issue = 2
| pages = 333–380
| last = Belavin
| first = A.A. |author2=A.M. Polyakov |author3=A.B. Zamolodchikov
| title = Infinite conformal symmetry in two-dimensional quantum field theory
| journal = Nuclear Physics B
| date=1984
|bibcode = 1984NuPhB.241..333B | url = https://cds.cern.ch/record/152341
}}</ref> in the context of [two-dimensional conformal field theory](/source/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==

In <math>d>2</math>  dimensions conformal primary fields can be defined in two equivalent ways. <ref>{{cite journal |last1=Campos Delgado|first1=Ruben |title=On the equivalence of two definitions of conformal primary fields in d > 2 dimensions |journal=Eur. Phys. J. Plus |year=2022 |volume=137 |issue=9 |page=1038 |doi=10.1140/epjp/s13360-022-03228-y|s2cid=252258885 |arxiv=2112.01837 }}</ref>
=== First definition ===
Let <math>\hat{D}</math> be the generator of dilations and let <math>\hat{K}_{\mu}</math>  be the generator of special conformal transformations. A conformal primary field <math>\hat{\phi}^M_{\rho}(x)</math> , in the <math>\rho</math> representation of the [Lorentz group](/source/Lorentz_group) and with conformal dimension <math>\Delta</math>  satisfies the following conditions at <math>x=0</math> : 
#<math>\left[\hat{D},\hat{\phi}^M_{\rho}(0)\right]=-i\Delta\hat{\phi}^M_{\rho}(0)</math>;
#<math>\left[\hat{K}_{\mu},\hat{\phi}^M_{\rho}(0)\right]=0</math>.
=== Second definition ===
A conformal primary field <math>\hat{\phi}^M_{\rho}(x)</math>, in the <math>\rho</math> representation of the [Lorentz group](/source/Lorentz_group) and with conformal dimension <math>\Delta</math>, transforms under a conformal transformation <math>\eta_{\mu \nu}\mapsto \Omega^2(x)\eta_{\mu \nu}</math> as
:<math>\hat{\phi'}^M_{\rho}(x')=\Omega^{\Delta}(x)\mathcal{D}{\left[R(x)\right]^M}_{N}\hat{\phi}^N_{\rho}(x)</math>
where  <math>{R^{\mu}}_{\nu}(x)=\Omega^{-1}(x)\frac{\partial x^{\mu}}{\partial x'^{\nu}}</math> and <math>\mathcal{D}{\left[R(x)\right]^M}_{N}</math> implements the action of <math>R</math> in the  <math>SO(d-1,1)</math> representation of <math>\hat{\phi}^{M}_{\rho}(x)</math>.

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

In two dimensions, conformal field theories are invariant under an infinite dimensional [Virasoro algebra](/source/Virasoro_algebra) with generators <math>L_n, \bar{L}_n, -\infty<n<\infty</math>. Primaries are defined as the operators annihilated by all <math>L_n, \bar{L}_n</math> with ''n''>0, which are the lowering generators. Descendants are obtained from the primaries by acting with <math>L_n, \bar{L}_n</math> with ''n''<0.

The Virasoro algebra has a finite dimensional subalgebra generated by <math>L_n, \bar{L}_n, -1\le n\le 1</math>. Operators annihilated by <math>L_1, \bar{L}_1</math> 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==
Source:<ref name=MAGOO>{{Cite journal| doi = 10.1016/S0370-1573(99)00083-6| issn = 0370-1573| volume = 323| issue = 3–4| pages = 183–386| last = Aharony| first = Ofer|author2=Steven S. Gubser |author3=Juan Maldacena |author-link3=Juan Maldacena |author4=Hirosi Ooguri |author5=Yaron Oz| title = Large N field theories, string theory and gravity|journal = Physics Reports| accessdate = 2013-12-05| year = 2000| url = http://inspirehep.net/record/499969?ln=en|arxiv = hep-th/9905111|bibcode = 2000PhR...323..183A | s2cid = 119101855}}</ref>

In <math>D\le 6</math> dimensions, conformal algebra allows graded extensions containing fermionic generators. [Quantum field theories](/source/Quantum_field_theory) invariant with respect to such extended algebras are called superconformal. In superconformal field theories, one considers superconformal primary operators.

In <math>D>2</math> dimensions, superconformal primaries are annihilated by <math>K_\mu</math> and by the fermionic generators <math>S</math> (one for each [supersymmetry](/source/supersymmetry) generator). Generally, each superconformal primary representations will include several primaries of the conformal algebra, which arise by acting with the supercharges <math>Q</math> on the superconformal primary. There exist also special ''chiral'' superconformal primary operators, which are primary operators annihilated by some combination of the supercharges.<ref name=MAGOO/>

In <math>D=2</math> dimensions, superconformal field theories are invariant under [super Virasoro algebra](/source/super_Virasoro_algebra)s, 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.<ref>{{Cite journal
| volume = 2
| pages = 781–846
| last = Minwalla
| first = Shiraz
| title = Restrictions imposed by superconformal invariance on quantum field theories
| journal = Adv. Theor. Math. Phys.
| accessdate = 2013-12-05
| date = 1997
| url = http://inspirehep.net/record/452061?ln=en
| arxiv = hep-th/9712074
}}</ref><ref>{{Cite journal
| doi = 10.1016/j.physletb.2008.03.020
| issn = 0370-2693
| volume = 662
| issue = 4
| pages = 367–374
| last = Grinstein
| first = Benjamin
|author2=Kenneth Intriligator |author3=Ira Z. Rothstein
 | title = Comments on unparticles
| journal = Physics Letters B
| accessdate = 2013-12-05
| year = 2008
| url = http://inspirehep.net/record/776996?ln=en
|arxiv = 0801.1140 |bibcode = 2008PhLB..662..367G | s2cid = 5240874
}}</ref> 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<ref>{{Cite journal
| doi = 10.1103/PhysRevD.9.3564
| issn = 0556-2821
| volume = 9
| issue = 12
| pages = 3564–3565
| last = Ferrara
| first = S.
|author2=R. Gatto |author3=A. Grillo
 | title = Positivity restriction on anomalous dimensions
| journal = Physical Review D
| accessdate = 2013-12-05
| date = 1974
| url = http://inspirehep.net/record/89113?ln=en
|bibcode = 1974PhRvD...9.3564F }}</ref> and by Mack.<ref name=Mack/>

==References==
{{Reflist}}

{{DEFAULTSORT:Primary Field}}
Category:Conformal field theory

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Adapted from the Wikipedia article [Primary field](https://en.wikipedia.org/wiki/Primary_field) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Primary_field?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.