Primary field

Primary fields in a D-dimensional conformal field theory were introduced in 1969 by Mack and Salam{{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 }} where they were called interpolating fields. They were then studied by Ferrara, Gatto, and Grillo{{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

}} who called them irreducible conformal tensors, and by 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

}} who called them lowest weights. Polyakov{{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 }} 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{{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

}} in the context of two-dimensional conformal field theory. This terminology is now used both for D=2 and D>2.

In $d>2$ dimensions conformal primary fields can be defined in two equivalent ways. {{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 }}

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

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

A conformal primary field $\backslash hat\{\backslash phi\}^M\_\{\backslash rho\}(x)$, in the $\backslash rho$ representation of the Lorentz group and with conformal dimension $\backslash Delta$, transforms under a conformal transformation $\backslash eta\_\{\backslash mu\; \backslash nu\}\backslash mapsto\; \backslash Omega^2(x)\backslash eta\_\{\backslash mu\; \backslash nu\}$ as

:$\backslash hat\{\backslash phi\text{'}\}^M\_\{\backslash rho\}(x\text{'})=\backslash Omega^\{\backslash Delta\}(x)\backslash mathcal\{D\}\{\backslash left[R(x)\backslash right]^M\}\_\{N\}\backslash hat\{\backslash phi\}^N\_\{\backslash rho\}(x)$

where $\{R^\{\backslash mu\}\}\_\{\backslash nu\}(x)=\backslash Omega^\{-1\}(x)\backslash frac\{\backslash partial\; x^\{\backslash mu\}\}\{\backslash partial\; x\text{'}^\{\backslash nu\}\}$ and $\backslash mathcal\{D\}\{\backslash left[R(x)\backslash right]^M\}\_\{N\}$ implements the action of $R$ in the $SO(d-1,1)$ representation of $\backslash hat\{\backslash phi\}^\{M\}\_\{\backslash rho\}(x)$.

In two dimensions, conformal field theories are invariant under an infinite dimensional Virasoro algebra with generators n>0, which are the lowering generators. Descendants are obtained from the primaries by acting with $L\_n,\; \backslash bar\{L\}\_n$ with n<0.

The Virasoro algebra has a finite dimensional subalgebra generated by $L\_n,\; \backslash bar\{L\}\_n,\; -1\backslash le\; n\backslash le\; 1$. Operators annihilated by $L\_1,\; \backslash bar\{L\}\_1$ 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 D>2 dimensional case.

Source:{{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 |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}}

In $D\backslash le\; 6$ 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 $D>2$ dimensions, superconformal primaries are annihilated by $K\_\backslash mu$ 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.

In unitary (super)conformal field theories, dimensions of primary operators satisfy lower bounds called the unitarity bounds.{{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

}}{{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

}} 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{{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 }} and by Mack.

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Category:Conformal field theory