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

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.

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.

@article{AIF_1987__37_4_99_0,
     author = {Kac, Victor G. and Leur, W. Van De},
     title = {Super boson-fermion correspondence},
     journal = {Annales de l'Institut Fourier},
     pages = {99--137},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {37},
     number = {4},
     year = {1987},
     doi = {10.5802/aif.1113},
     mrnumber = {89i:17006},
     zbl = {0625.58041},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_1987__37_4_99_0/}
}
Kac, Victor G.; Leur, W. Van De. Super boson-fermion correspondence. Annales de l'Institut Fourier, Tome 37 (1987) no. 4, pp. 99-137. doi : 10.5802/aif.1113. https://aif.centre-mersenne.org/item/AIF_1987__37_4_99_0/

[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 84m:58119c | Zbl 0571.35099

[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 86a:58093 | Zbl 0571.35098

[3] J. C. Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Math., 750 Springer-Verlag, 1979. | MR 81m:17011 | Zbl 0426.17001

[4] M. Jimbo and T. Miwa, Solitons and infinite dimensional Lie algebras, Publ. RIMS, 19 (1983), 943-1001. | MR 85i:58060 | Zbl 0557.35091

[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 0299.17005

[6] V. G. Kac, Lie superalgebras, Advances in Math., 26, No. 1 (1977), 8-96. | MR 58 #5803 | Zbl 0366.17012

[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 83a:17014a | Zbl 0391.17010

[8] V. G. Kac, Representations of classical Lie superalgebras, Lecture Notes in Mathematics, 676 (1978), 597-626. | MR 80f:17006 | Zbl 0388.17002

[9] V. G. Kac, Contravariant form for infinite dimensional Lie algebras and superalgebras, Lecture Notes in Physics, 94 (1979), 441-445. | Zbl 0574.17002

[10] V. G. Kac, Infinite Dimensional Lie Algebras. Progress in Mathematics, 44, Birkhäuser, Boston, 1983. Second edition, Cambridge University Press, 1985. | MR 86h:17015 | Zbl 0537.17001

[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 0637.17012

[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 81d:17004 | Zbl 0427.17011

[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 88f:17021 | Zbl 0638.35070

[14] J. W. Van De Leur, Contragredient Lie superalgebras of finite growth, Thesis Utrecht, May 1986. | Zbl 0685.17013

[15] Ju. I. Manin and A. O. Radul, A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy, Comm. Math. Phys., 98 (1985), 65-77. | MR 87b:58038 | Zbl 0607.35075

B. Kupershmidt, Odd and even Poisson brackets in dynamical systems, Lett. Math. Phys., 9 (1985), 323-330. | MR 86h:58057 | Zbl 0585.58020

[16] M. Sato, Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds, RIMS Kokyuroku, 439 (1981), 30-46. | Zbl 0507.58029

[17] T. Shiota, Characterization of Jacobian varieties in terms of soliton equations, Invent. Math., 83 (1986), 333-382. | MR 87j:14047 | Zbl 0621.35097

[18] E. Arbarello and C. De Concini, On a set of equations characterising Riemann matrices, Ann. Math., 120 (1984), 119-140. | MR 86a:14025 | Zbl 0551.14016

[19] M. Mulase, Cohomological structure in soliton equation and Jacobian varieties, J. Diff. Geom., 19 (1984), 403-430. | MR 86f:14016 | Zbl 0559.35076

[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 91c:58053 | Zbl 0653.35076