Geodesic
Quantization of the $N{=}2$ supersymmetric $\text{KdV}$ hierarchy
Teoretičeskaâ i matematičeskaâ fizika, Tome 147 (2006) no. 2, pp. 303-314 Cet article a éte moissonné depuis la source Math-Net.Ru

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We continue the study of the quantization of supersymmetric integrable KdV hierarchies. We consider the $N{=}2$ KdV model based on the $sl^{(1)}(2\,|\,1)$ affine algebra but with a new algebraic construction for the $L$-operator, different from the standard Drinfeld–Sokolov reduction. We construct the quantum monodromy matrix satisfying a special version of the reflection equation and show that in the classical limit, this object precisely gives the monodromy matrix of the $N{=}2$ supersymmetric KdV system. We also show that at both the classical and the quantum levels, the trace of the monodromy matrix {(}transfer matrix{\rm)} is invariant under two supersymmetry transformations and the zero mode of the associated $U(1)$ current.
Keywords: superconformal field theory, quantum superalgebras, supersymmetric KdV equation, supersymmetric integrable systems
Mots-clés : quantization. 
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     title = {Quantization of the $N{=}2$ supersymmetric $\text{KdV}$ hierarchy},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {303--314},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2006_147_2_a6/}
}
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A. M. Zeitlin. Quantization of the $N{=}2$ supersymmetric $\text{KdV}$ hierarchy. Teoretičeskaâ i matematičeskaâ fizika, Tome 147 (2006) no. 2, pp. 303-314. http://geodesic.mathdoc.fr/item/TMF_2006_147_2_a6/

[1] P. P. Kulish, A. M. Zeitlin, Phys. Lett. B, 581 (2004), 125 ; ; П. П. Кулиш, А. М. Цейтлин, ТМФ, 142 (2005), 252 ; hep-th/0312159hep-th/0501018 | DOI | MR | Zbl | DOI | MR | Zbl

[2] P. P. Kulish, A. M. Zeitlin, Phys. Lett. B, 597 (2004), 229 ; ; Nucl. Phys. B, 709 (2005), 578 ; hep-th/0407154hep-th/0501019 | DOI | MR | Zbl | DOI | MR | Zbl

[3] P. P. Kulish, A. M. Zeitlin, Nucl. Phys. B, 720 (2005), 289 ; hep-th/0506027 | DOI | MR | Zbl

[4] V. V. Bazhanov, S. L. Lukyanov, A. B. Zamolodchikov, Commun. Math. Phys., 177 (1996), 381 ; 190 (1997), 247 ; 200 (1999), 297 | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[5] F. Delduc, L. Gallot, Supersymmetric Drinfeld–Sokolov reduction, solv-int/9802013 | MR

[6] T. Inami, H. Kanno, Nucl. Phys. B, 359 (1991), 201 | DOI | MR | Zbl

[7] L. A. Takhtadzhyan, L. D. Faddeev, Gamiltonov podkhod v teorii solitonov, Nauka, M., 1986 | MR | Zbl

[8] V. V. Bazhanov, A. G. Shadrikov, TMF, 73 (1988), 402 | MR

[9] S. M. Khoroshkin, V. N. Tolstoy, Twisting of quantum (super)algebras. Connection of Drinfeld's and Cartan–Weyl realizations for quantum affine algebras, hep-th/9404036 | MR

[10] B. Feigin, E. Frenkel, “Integrals of motion and quantum groups”, Integrable Systems and Quantum Groups (Montecatini Terme, Italy, June 14–22, 1993), Lect. Notes Math., 1620, eds. M. Francaviglia, S. Greco, Springer, Berlin, 1996, 349 ; P. Bouwknegt, J. McCarthy, K. Pilch, hep-th/9110007 | DOI | MR | Zbl

[11] S. Klevtsov, On vertex operator construction of quantum affine algebras, hep-th/0110148

[12] V. V. Bazhanov, A. N. Hibberd, S. M. Khoroshkin, Nucl. Phys. B, 622 (2002), 475 | DOI | MR | Zbl

[13] L. D. Faddeev, “How the algebraic Bethe ansatz works for integrable models”, Quantum symmetries/Symétries quantiques, Proc. of the Les Houches Summer School, Session LXIV (Les Houches, France, August 1 – September 8, 1995), eds. A. Connes, K. Gawedzki, J. Zinn-Justin, Elsevier, Amsterdam, 1998, 149 ; ; P. P. Kulish, E. K. Sklyanin, “Quantum spectral transform method: recent development”, Integrable Quantum Field Theories, Lect. Notes Phys., 151, eds. J. Hietarinta, C. Montonen, Springer, Berlin, 1982, 61 hep-th/9605187 | MR | DOI | MR

[14] P. P. Kulish, E. K. Sklyanin, J. Phys. A, 25 (1992), 5963 ; P. P. Kulish, R. Sasaki, Prog. Theor. Phys., 89 (1993), 741 | DOI | MR | Zbl | DOI | MR

[15] W. Lerche, C. Vafa, N. P. Warner, Nucl. Phys. B, 324 (1989), 427 | DOI | MR