Geodesic
Rational Conformal Field Theory in Four Dimensions
Teoretičeskaâ i matematičeskaâ fizika, Tome 132 (2002) no. 2, pp. 300-317 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We use the recently established rationality of correlation functions in a globally conformally invariant quantum field theory satisfying Wightman axioms to construct a family of solvable models in the four-dimensional Minkowski space–time. We consider the model of a neutral two-dimension scalar field $\phi$ in detail. It depends on a positive real parameter $c$, an analogue of the Virasoro central charge; for all (finite) $c$, it admits an infinite number of conserved symmetric tensor currents. The operator product algebra of $\phi$ coincides with a simpler one generated by a bilocal scalar field $V(x_1,x_2)$ of dimension 1+1. The modes of $V$ together with the unit operator span an infinite-dimensional Lie algebra $\mathfrak {L}_V$, whose vacuum (i.e. zero-energy lowest-weight) representations depend only on the central charge $c$. The Wightman positivity (i.e. unitarity of the representations of $\mathfrak {L}_V$ ) is proved equivalent to $c \in \mathbb {N}$.
Keywords: Wightman axioms, operator product expansion, bilocal fields, representations of infinite-dimensional Lie algebras. 
@article{TMF_2002_132_2_a8,
     author = {N. M. Nikolov and Ya. S. Stanev and I. T. Todorov},
     title = {Rational {Conformal} {Field} {Theory} in {Four} {Dimensions}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {300--317},
     year = {2002},
     volume = {132},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2002_132_2_a8/}
}
TY  - JOUR
AU  - N. M. Nikolov
AU  - Ya. S. Stanev
AU  - I. T. Todorov
TI  - Rational Conformal Field Theory in Four Dimensions
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2002
SP  - 300
EP  - 317
VL  - 132
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2002_132_2_a8/
LA  - ru
ID  - TMF_2002_132_2_a8
ER  - 
%0 Journal Article
%A N. M. Nikolov
%A Ya. S. Stanev
%A I. T. Todorov
%T Rational Conformal Field Theory in Four Dimensions
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2002
%P 300-317
%V 132
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2002_132_2_a8/
%G ru
%F TMF_2002_132_2_a8
N. M. Nikolov; Ya. S. Stanev; I. T. Todorov. Rational Conformal Field Theory in Four Dimensions. Teoretičeskaâ i matematičeskaâ fizika, Tome 132 (2002) no. 2, pp. 300-317. http://geodesic.mathdoc.fr/item/TMF_2002_132_2_a8/

[1] A. M. Polyakov, Pisma v ZhETF, 12 (1970), 538

[2] A. A. Migdal, Phys. Lett. B, 37 (1971), 386 | DOI | MR

[3] G. Parisi, L. Peliti, Lett. Nuovo Cimento, 2 (1971), 627 | DOI

[4] E. Schreier, Phys. Rev. D, 3 (1971), 980 | DOI

[5] B. Schroer, Lett. Nuovo Cimento, 2 (1971), 867 ; “Bjorken scaling and scale invariant quantum field theories”, Scale and Conformal Symmetry in Hadron Physics, ed. R. Gatto, Wiley, New York, 1973, 43 | DOI

[6] G. Mack, K. Symanzik, Commun. Math. Phys., 27 (1972), 247 | DOI | MR

[7] K. Symanzik, Lett. Nuovo Cimento, 3 (1972), 734 | DOI

[8] S. Ferrara, R. Gatto, A. Grillo, G. Parisi, Phys. Lett. B, 38 (1972), 333 ; Lett. Nuovo Cimento, 4 (1972), 115 ; “General consequences of conformal algebra”, Scale and Conformal Symmetry in Hadron Physics, ed. R. Gatto, Wiley, New York, 1973, 59 | DOI | DOI | MR

[9] I. T. Todorov, “Conformal invariant quantum field theory”, Strong Interaction Physics, Lect. Notes in Phys., 17, eds. W. Ruhl, A. Vancura, Springer, Berlin, 1972, 270 | DOI | MR

[10] G. Mack, I. T. Todorov, Phys. Rev. D, 8 (1973), 1764 | DOI

[11] G. Mack, “Conformal invariant quantum field theory”, Scale and Conformal Symmetry in Hadron Physics, ed. R. Gatto, Wiley, New York, 1973, 109 ; “Group theoretical approach in conformal invariant quantum field theory”, Renormalization and Invariance in Quantum Field Theory, Plenum Press, New York, 1974, 123 | MR | DOI

[12] A. M. Polyakov, ZhETF, 66 (1974), 23 | MR

[13] B. Schroer, J. A. Swieca, A. H. Volkel, Phys. Rev. D, 11 (1975), 1509 | DOI

[14] V. K. Dobrev, V. B. Petkova, S. G. Petrova, I. T. Todorov, Phys. Rev. D, 13 (1976), 887 | DOI

[15] G. Mack, Commun. Math. Phys., 53 (1977), 155 | DOI | MR

[16] I. T. Todorov, M. C. Mintchev, V. B. Petkova, Conformal Invariance in Quantum Field Theory, Scuola Normale Superiore, Pisa, 1978 | MR | Zbl

[17] I. T. Todorov, “Infinite dimensional Lie algebras in conformal QFT models”, Conformal Groups and Related Symmetries. Physical Results and Mathematical Background, Lect. Notes in Phys., 261, eds. A. O. Barut, H.-D. Doebner, Springer, Berlin, 1986, 387 | DOI | MR

[18] J. Cardy, Nucl. Phys. B, 290 (1987), 355 | DOI | MR

[19] Ya. S. Stanev, Bulg. J. Phys., 15 (1988), 93

[20] K. Lang, W. Ruhl, Nucl. Phys. B, 402 (1993), 573 | DOI | MR | Zbl

[21] H. Osborn, A. Petkou, Ann. Phys., 231 (1994), 311 | DOI | MR | Zbl

[22] E. S. Fradkin, M. Ya. Palchik, Phys. Rep., 300 (1998), 1 ; Conformal Quantum Field Theory in $D$-dimensions, Kluwer, Dordrecht, 1996 | DOI | MR | MR | Zbl

[23] F. A. Dolan, H. Osborn, Nucl. Phys. B, 599 (2001), 459 ; E-print hep-th/0011040 | DOI | MR | Zbl

[24] P. Di Francesco, P. Mathieu, D. Senechal, Conformal Field Theories, Springer, Berlin, 1996 | MR

[25] G. Aratyunov, B. Eden, A. C. Petkou, E. Sokatchev, Exceptional non-renormalization properties and OPE analysis of chiral $4$-point functions in $N = 4$ $\mathrm{SYM}_4$, E-print hep-th/0103230 | MR

[26] M. Bianchi, S. Kovacs, G. Rossi, Ya. S. Stanev, Properties of Konishi multiplet in $N=4$ SYM theory, E-print hep-th/0104016

[27] N. M. Nikolov, Ya. S. Stanev, I. T. Todorov, Four dimensional CFT models with rational correlation functions, E-print hep-th/0110230

[28] N. M. Nikolov, I. T. Todorov, Commun. Math. Phys., 218 (2001), 417 ; E-print hep-th/0009004 | DOI | MR | Zbl

[29] R. Striter, A. Vaitman, RST, spin i statistika i vse takoe, Nauka, M., 1966

[30] N. N. Bogolyubov, A. A. Logunov, A. I. Oksak, I. T. Todorov, Obschie printsipy kvantovoi teorii polya, Nauka, M., 1987 ; Н. Н. Боголюбов, А. А. Логунов, И. Т. Тодоров, Основы аксиоматического подхода в квантовой теории поля, Наука, М., 1969 | MR | MR

[31] G. Mack, Commun. Math. Phys., 55 (1977), 1 | DOI | MR | Zbl

[32] V. G. Kac, A. Radul, Commun. Math. Phys., 157 (1993), 125 ; Transform. Groups, 1:1–2 (1996), 41 | DOI | MR | DOI | MR | Zbl