Super boson-fermion correspondence
Annales de l'Institut Fourier, Volume 37 (1987) no. 4, pp. 99-137.

We establish a super boson-fermion correspondence, generalizing the classical boson-fermion correspondence in 2-dimensional quantum field theory. A new feature of the theory is the essential non-commutativity of bosonic fields. The superbosonic fields obtained by the super bosonization procedure from super fermionic fields form the affine superalgebra g ˜l 1|1 . The converse, super fermionization procedure, requires introduction of the super vertex operators. As applications, we give vertex operator constructions of all degenerate highest weight representations of g ˜l 1|1 and of some interesting representations of g ˜l | (C), and also derive some new combinatorial identities. We hope that this construction will provide representation theoretical framework for hierarchies of super soliton equations.

Nous établissons une correspondance super boson-fermion, généralisant la correspondance analogue dans la théorie des champs quantiques 2-dimensionnels. Un nouvel élément dans cette théorie est la non-commutativité essentielle des champs bosoniques. Les champs superbosoniques obtenus par la procédure de superbonisation des champs superfermioniques constituent la superalgèbre affine g ˜l 1|1 . La procédure converse de superfermionisation exige l’introduction des superopérateurs de sommet.

Comme applications, nous donnons la construction de toutes les représentations dégénérées ayant un plus haut poids de g ˜l 1|1 et de quelques repréentations intéressantes de g ˜l 1|1 (C) à l’aide des superopérateurs de sommet. Aussi nous dérivons quelques nouvelles identités combinatoires. Nous espérons que cette construction fournira un cadre de théorie des représentations pour les hiérarchies des équations supersolitoniques.

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     title = {Super boson-fermion correspondence},
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Kac, Victor G.; Leur, W. Van De. Super boson-fermion correspondence. Annales de l'Institut Fourier, Volume 37 (1987) no. 4, pp. 99-137. doi : 10.5802/aif.1113. http://www.numdam.org/articles/10.5802/aif.1113/

[1] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, Operator approach to the Kadomstev-Petviashvili equation. Transformation groups for soliton equations III, J. Phys. Soc. Japan, 50 (1981), 3806-3812. | MR | Zbl

[2] E. Date, M. Jimbo, M. Kashiwara, T. Miwa, Transformation groups for soliton equations, in : Proceedings of RIMS Symposium, M. Jimbo and T. Miwa, eds., World Scientific, 1983, 34-120. | MR | Zbl

[3] J. C. Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Math., 750 Springer-Verlag, 1979. | MR | Zbl

[4] M. Jimbo and T. Miwa, Solitons and infinite dimensional Lie algebras, Publ. RIMS, 19 (1983), 943-1001. | MR | Zbl

[5] V. G. Kac, Infinite-dimensional Lie algebras and Dedekinds η-function, Funkt. Anal. i ego Prilozh, 8 (1974), No. 1, 77-78. English translation : Funct. Anal. Appl., 8 (1974), 68-70. | Zbl

[6] V. G. Kac, Lie superalgebras, Advances in Math., 26, No. 1 (1977), 8-96. | MR | Zbl

[7] V. G. Kac, Infinite-dimensional algebras, Dedekind's η-function, classical Möbius function and the very strange formula, Advances in Math., 30 (1978), 85-136. | MR | Zbl

[8] V. G. Kac, Representations of classical Lie superalgebras, Lecture Notes in Mathematics, 676 (1978), 597-626. | MR | Zbl

[9] V. G. Kac, Contravariant form for infinite dimensional Lie algebras and superalgebras, Lecture Notes in Physics, 94 (1979), 441-445. | Zbl

[10] V. G. Kac, Infinite Dimensional Lie Algebras. Progress in Mathematics, 44, Birkhäuser, Boston, 1983. Second edition, Cambridge University Press, 1985. | MR | Zbl

[11] V. G. Kac, Highest weight representations of conformal current algebras, Symposium on Topological and Geometric and methods in Field theory. Espoo, Finland, World Scientific (1986), 3-16. | Zbl

[12] V. G. Kac, D. A. Kazhdan, Structure of representations with highest weight of infinite dimensional Lie algebras, Advances in Math., 34 (1979), 97-108. | MR | Zbl

[13] V. G. Kac, D. Peterson, Lectures on the infinite wedge representation and the MKP hierarchy. Séminaire de Math. Supérieures, Les Presses de L'Université de Montréal, 102 (1986), 141-186. | MR | Zbl

[14] J. W. Van De Leur, Contragredient Lie superalgebras of finite growth, Thesis Utrecht, May 1986. | Zbl

[15] Ju. I. Manin and A. O. Radul, A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy, Comm. Math. Phys., 98 (1985), 65-77. | MR | Zbl

B. Kupershmidt, Odd and even Poisson brackets in dynamical systems, Lett. Math. Phys., 9 (1985), 323-330. | MR | Zbl

[16] M. Sato, Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds, RIMS Kokyuroku, 439 (1981), 30-46. | Zbl

[17] T. Shiota, Characterization of Jacobian varieties in terms of soliton equations, Invent. Math., 83 (1986), 333-382. | MR | Zbl

[18] E. Arbarello and C. De Concini, On a set of equations characterising Riemann matrices, Ann. Math., 120 (1984), 119-140. | MR | Zbl

[19] M. Mulase, Cohomological structure in soliton equation and Jacobian varieties, J. Diff. Geom., 19 (1984), 403-430. | MR | Zbl

[20] T. H. R. Skyrme, Kinks and the Dirac equation, J. Math. Physics, 12 (1971), 1735-1743.

[21] K. Ueno, H. Yamada, A supersymmetric extension of infinite-dimensional Lie algebras, RIMS-Kokyuroku, 554 (1955), 91-101.

[22] K. Ueno and H. Yamada, A supersymmetric extension of nonlinear integrable systems. Symposium on Topological and Geometric methods in Field theory. Espoo, Finland, World Scientific (1986), 59-72. | MR | Zbl

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