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

@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},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {37},
     number = {4},
     year = {1987},
     pages = {99-137},
     doi = {10.5802/aif.1113},
     zbl = {0625.58041},
     mrnumber = {89i:17006},
     language = {en},
     url = {http://www.numdam.org/item/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. http://www.numdam.org/item/AIF_1987__37_4_99_0/

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