Comptes Rendus
no. 5
Lie algebras/Topology
Dirac families for loop groups as matrix factorizations
[Familles d'opérateurs de Dirac pour les groupes de lacets et factorisations en matrices]

Présenté par : Michèle Vergne

Daniel S. Freed1 ; Constantin Teleman2
1 UT Austin, Mathematics Department, RLM 8.100, 2515 Speedway C1200, Austin, TX 78712, USA
2 UC Berkeley, Mathematics Department, 970 Evans Hall #3840, Berkeley, CA 94720, USA
Comptes Rendus. Mathématique, Volume 353 (2015) no. 5, pp. 415-419.

On identifie la catégorie des représentations intégrables de plus bas poids du groupe de lacets LG d'un groupe de Lie compact G avec la catégorie des complexes de Fredholm tordus, courbés et équivariants pour conjugaison sur le groupe G : plus précisément, les factorisations en matrices d'un potentiel provenant de la rotation des lacets dans LG. Cette construction relève l'isomorphisme de K-groupes de [3–5] en une équivalence de catégories. La construction fait appel aux familles d'opérateurs de Dirac.

We identify the category of integrable lowest-weight representations of the loop group LG of a compact Lie group G with the category of twisted, conjugation-equivariant curved Fredholm complexes on the group G: namely, the twisted, equivariant matrix factorizations of a super-potential built from the loop rotation action on LG. This lifts the isomorphism of K-groups of [3–5] to an equivalence of categories. The construction uses families of Dirac operators.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.02.011
Daniel S. Freed 1 ; Constantin Teleman 2

1 UT Austin, Mathematics Department, RLM 8.100, 2515 Speedway C1200, Austin, TX 78712, USA
2 UC Berkeley, Mathematics Department, 970 Evans Hall #3840, Berkeley, CA 94720, USA 
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Daniel S. Freed; Constantin Teleman. Dirac families for loop groups as matrix factorizations. Comptes Rendus. Mathématique, Volume 353 (2015) no. 5, pp. 415-419. doi : 10.1016/j.crma.2015.02.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.02.011/

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