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.

     author = {Kac, Victor G. and Leur, W. Van De},
     title = {Super boson-fermion correspondence},
     journal = {Annales de l'Institut Fourier},
     pages = {99--137},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {37},
     number = {4},
     year = {1987},
     doi = {10.5802/aif.1113},
     zbl = {0625.58041},
     mrnumber = {89i:17006},
     language = {en},
     url = {}
AU  - Kac, Victor G.
AU  - Leur, W. Van De
TI  - Super boson-fermion correspondence
JO  - Annales de l'Institut Fourier
PY  - 1987
SP  - 99
EP  - 137
VL  - 37
IS  - 4
PB  - Institut Fourier
PP  - Grenoble
UR  -
DO  - 10.5802/aif.1113
LA  - en
ID  - AIF_1987__37_4_99_0
ER  - 
%0 Journal Article
%A Kac, Victor G.
%A Leur, W. Van De
%T Super boson-fermion correspondence
%J Annales de l'Institut Fourier
%D 1987
%P 99-137
%V 37
%N 4
%I Institut Fourier
%C Grenoble
%R 10.5802/aif.1113
%G en
%F AIF_1987__37_4_99_0
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.

[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

Cited by Sources: