Geodesic
The $6j$-symbols for the $SL(2,\mathbb C)$ group
Teoretičeskaâ i matematičeskaâ fizika, Tome 198 (2019) no. 1, pp. 32-53
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We study $6j$-symbols or Racah coefficients for the tensor products of infinite-dimensional unitary principal series representations of the group $SL(2,\mathbb C)$. Using the Feynman diagram technique, we reproduce the results of Ismagilov in constructing these symbols (up to a slight difference associated with equivalent representations). The resulting $6j$-symbols are expressed either as a triple integral over complex plane or as an infinite bilateral sum of integrals of the Mellin–Barnes type.
Mots-clés : $3j$-symbol, $6j$-symbol
Keywords: Feynman diagram, $SL(2,\mathbb C)$ group, hypergeometric integral. 
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S. È. Derkachev; V. P. Spiridonov. The $6j$-symbols for the $SL(2,\mathbb C)$ group. Teoretičeskaâ i matematičeskaâ fizika, Tome 198 (2019) no. 1, pp. 32-53. http://geodesic.mathdoc.fr/item/TMF_2019_198_1_a2/

[1] G. Racah, “Theory of complex spectra. II”, Phys. Rev., 62:9–10 (1942), 438–462 | DOI

[2] G. E. Andrews, R. Askey, R. Roy, Special Functions, v. 71, Encyclopedia of Mathematics and its Applications, Cambridge Univ. Press, Cambridge, 1999 | DOI | MR

[3] D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, Kvantovaya teoriya uglovogo momenta, Nauka, L., 1975 | DOI | MR

[4] I. M. Gelfand, M. A. Naimark, Unitarnye predstavleniya klassicheskikh grupp, Tr. MIAN SSSR, 36, Izd-vo AN SSSR, M.–L., 1950 | MR | Zbl

[5] L. N. Lipatov, “Zatravochnyi pomeron v kvantovoi khromodinamike”, ZhETF, 90:5 (1986), 1536–1552

[6] L. N. Lipatov, “High-energy asymptotics of multicolor QCD and two-dimensional conformal field theories”, Phys. Lett. B, 309:3–4 (1993), 394–396 | DOI

[7] L. D. Faddeev, G. P. Korchemsky, “High-energy QCD as a completely integrable model”, Phys. Lett. B, 342:1–4 (1995), 311–322, arXiv: hep-th/9404173 | DOI

[8] L. N. Lipatov, “Asimptotika mnogotsvetnoi KKhD pri bolshikh energiyakh i tochno reshaemye spinovye modeli”, Pisma v ZhETF, 59:9 (1994), 571–574

[9] M. A. Naimark, “Razlozhenie tenzornogo proizvedeniya neprivodimykh predstavlenii sobstvennoi gruppy Lorentsa na neprivodimye predstavleniya. Chast I. Sluchai tenzornogo proizvedeniya predstavlenii osnovnoi serii”, Tr. MMO, 8 (1959), 121–153 | MR | Zbl

[10] R. S. Ismagilov, “Ob operatorakh Raka”, Funkts. analiz i ego pril., 40:3 (2006), 69–72 | DOI | DOI | MR | Zbl

[11] R. S. Ismagilov, “Ob operatorakh Raká dlya predstavlenii osnovnoi serii gruppy $\mathrm{SL}(2,\mathbb C)$”, Matem. sb., 198:3 (2007), 77–90 | DOI | DOI

[12] W. Groenevelt, “Wilson function transforms related to Racah coefficients”, Acta Appl. Math., 91:2 (2006), 133–191 | DOI | MR

[13] S. E. Derkachëv, A. N. Manashov, “Obschee reshenie uravneniya Yanga–Bakstera s gruppoi simmetrii $\mathrm{SL}(\mathrm n,\mathbb C)$”, Algebra i analiz, 21:4 (2009), 1–94 | DOI

[14] L. D. Faddeev, “Modular double of a quantum group”, Conférence Moshé Flato 1999. Quantization, Deformations, and Symmetries, v. I, Mathematical Physics Studies, 21, eds. G. Dito, D. Sternheimer, Kluwer, Dordrecht, 2000, 149–156 | MR

[15] B. Ponsot, J. Teschner, “Clebsch–Gordan and Racah–Wigner coefficients for a continuous series of representations of $U_q(sl(2,\mathbb{R}))$”, Commun. Math. Phys., 224:3 (2001), 613–655 | DOI | MR

[16] M. Pawelkiewicz, V. Schomerus, P. Suchanek, “The universal Racah–Wigner symbol for $U_q(osp(1|2))$”, JHEP, 04 (2014), 079, 25 pp., arXiv: 1307.6866 | DOI

[17] S. É. Derkachov, A. N. Manashov, “Spin chains and Gustafson's integrals”, J. Phys. A: Math. Theor., 50:29 (2017), 294006, 20 pp. | DOI | MR

[18] S. É. Derkachov, A. N. Manashov, P. A. Valinevich, “Gustafson integrals for $SL(2,\mathbb{C})$ spin magnet”, J. Phys. A: Math. Theor., 50:29 (2017), 294007, 12 pp. | DOI | MR

[19] I. M. Gelfand, M. I. Graev, N. Ya. Vilenkin, Obobschennye funktsii, v. 5, Integralnaya geometriya i svyazannye c nei voprosy teorii predstavlenii, Fizmatlit, M., 1962 | MR | Zbl

[20] D. Chicherin, S. E. Derkachov, V. P. Spiridonov, “From principal series to finite-dimensional solutions of the Yang–Baxter equation”, SIGMA, 12 (2016), 028, 34 pp. | DOI | MR

[21] A. A. Belavin, A. M. Polyakov, A. B. Zamolodchikov, “Infinite conformal symmetry in two-dimensional quantum field theory”, Nucl. Phys. B, 241:2 (1984), 333–380 | DOI | MR

[22] S. É. Derkachov, L. D. Faddeev, “$3j$-symbol for the modular double of $SL_q(2,\mathbb{R})$ revisited”, J. Phys.: Conf. Ser., 532:1 (2014), 012005, 14 pp. | DOI

[23] L. Hadasz, M. Pawelkiewicz, V. Schomerus, “Self-dual continuous series of representations for $U_q(sl(2))$ and $U_q(osp(1|2))$”, JHEP, 10 (2014), 091, 29 pp. | DOI | MR

[24] S. G. Gorishnii, A. P. Isaev, “Ob odnom podkhode k vychisleniyu mnogopetlevykh bezmassovykh feinmanovskikh integralov”, TMF, 62:3 (1985), 345–358 | DOI

[25] V. P. Spiridonov, “Ocherki teorii ellipticheskikh gipergeometricheskikh funktsii”, UMN, 63:3(381) (2008), 3–72 | DOI | DOI | MR | Zbl