Super boson-fermion correspondence

Kac, Victor G.; Leur, W. Van De

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 $\tilde{g}{l}_{1|1}$. 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 $\tilde{g}{l}_{1|1}$ et de quelques repréentations intéressantes de $\tilde{g}{l}_{1|1}\left(\mathbf{C}\right)$ 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 $\tilde{g}{l}_{1|1}$. 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 $\tilde{g}{l}_{1|1}$ and of some interesting representations of $\tilde{g}{l}_{\infty |\infty }\left(\mathbf{C}\right)$, and also derive some new combinatorial identities. We hope that this construction will provide representation theoretical framework for hierarchies of super soliton equations.

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