Boson-fermion correspondence — 2 articles found.

Comments Related to Infinite Wedge Representations

C. R. Math. Rep. Acad. Sci. Canada Vol. 39 (1) 2017, pp. 13-35
Vol.39 (1) 2017
Nathan Grieve Details
(Received: 2016-06-30 , Revised: 2016-11-07 )
(Received: 2016-06-30 , Revised: 2016-11-07 )

Nathan Grieve,Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB,Canada E3B 5A3; e-mail:


We study the infinite wedge representation and show how it is related to the universal central extension of \(g[t,t^{-1}]\), the loop algebra of a complex semi-simple Lie algebra \(g\). We also give an elementary proof of the boson-fermion correspondence. Our approach to proving this result is based on a combinatorial construction combined with an application of the Murnaghan-Nakayama rule.

Nous étudions l’algèbre extérieure en dimension infinie et montrons comment elle est reliée à l’extension centrale universelle de \(g[t,\!t^{-1}]\), l’algèbre de lacets sur une algèbre de Lie \(g\) semi-simple complexe. De plus, nous donnons une preuve élémentaire de la correspondance boson-fermion. Pour ce faire, nous utilisons une construction combinatoire, ainsi que la règle de Murnaghan-Nakayama.

Keywords: Boson-fermion correspondence, Infinite wedge representation, Murnaghan-Nakayama rule

AMS Subject Classification: Symmetric functions, Completely integrable systems; integrability tests; bi-Hamiltonian structures; hierarchies (KdV; KP; Toda; etc.) 05E05, 37K10

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A geometric boson-fermion correspondence

C. R. Math. Rep. Acad. Sci. Canada Vol. 28 (3) 2006, pp. 65–84
Vol.28 (3) 2006
Alistair Savage Details
(Received: 2006-09-07 )
(Received: 2006-09-07 )

Alistair Savage, University of Ottawa, Ottawa, Ontario K1N 6N5; email:


The fixed points of a natural torus action on the Hilbert schemes of points in \(\C^2\) are quiver varieties of type \(A_\infty\). The equivariant cohomology of the Hilbert schemes and quiver varieties can be given the structure of bosonic and fermionic Fock spaces respectively. Then the localization theorem, which relates the equivariant cohomology of a space with that of its fixed point set, yields a geometric realization of the important boson-fermion correspondence.

Les points fixes d’une action canonique d’un tore sur le schéma de Hilbert de \(\C^2\) sont des variétés de quiver de type \(A_\infty\). On peut donner la cohomologie équivariante des schémas de Hilbert et des variétés de quiver la structure des éspaces de Fock fermionique et bosonique, respectivement. Alors, la théorème de localisation, qui lie la cohomologie équivariante d’une éspace avec la cohomologie équivariante de son ensemble des point fixes, nous permet de donner une réalisation géométrique de la correspondance bosonique-fermionique.

Keywords: Boson-fermion correspondence, Hilbert schemes, affine Lie algebras, equivariant cohomology, quiver varieties, quivers, vertex algebras

AMS Subject Classification: Parametrization (Chow and Hilbert schemes) 14C05

PDF(click to download): A geometric boson-fermion correspondence

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