[Familles d'opérateurs de Dirac pour les groupes de lacets et factorisations en matrices]
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.
Accepté le :
Publié le :
@article{CRMATH_2015__353_5_415_0,
author = {Freed, Daniel S. and Teleman, Constantin},
title = {Dirac families for loop groups as matrix factorizations},
journal = {Comptes Rendus. Math\'ematique},
pages = {415--419},
publisher = {Elsevier},
volume = {353},
number = {5},
year = {2015},
doi = {10.1016/j.crma.2015.02.011},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.crma.2015.02.011/}
}
TY - JOUR AU - Freed, Daniel S. AU - Teleman, Constantin TI - Dirac families for loop groups as matrix factorizations JO - Comptes Rendus. Mathématique PY - 2015 SP - 415 EP - 419 VL - 353 IS - 5 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2015.02.011/ DO - 10.1016/j.crma.2015.02.011 LA - en ID - CRMATH_2015__353_5_415_0 ER -
%0 Journal Article %A Freed, Daniel S. %A Teleman, Constantin %T Dirac families for loop groups as matrix factorizations %J Comptes Rendus. Mathématique %D 2015 %P 415-419 %V 353 %N 5 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2015.02.011/ %R 10.1016/j.crma.2015.02.011 %G en %F CRMATH_2015__353_5_415_0
Freed, Daniel S.; Teleman, Constantin. Dirac families for loop groups as matrix factorizations. Comptes Rendus. Mathématique, Tome 353 (2015) no. 5, pp. 415-419. doi : 10.1016/j.crma.2015.02.011. http://archive.numdam.org/articles/10.1016/j.crma.2015.02.011/
[1] From loop groups to 2-groups, Homology, Homotopy Appl., Volume 9 (2007), pp. 101-135
[2] Topological field theories from compact Lie groups, A Celebration of the Mathematical Legacy of Raoul Bott, CRM Proc. Lecture Notes, vol. 50, AMS, 2010, pp. 367-403
[3] Twisted K-theory and loop group representations I, J. Topol., Volume 4 (2011), pp. 737-798
[4] Twisted K-theory and loop group representations II, J. Amer. Math. Soc., Volume 26 (2013), pp. 595-644
[5] Twisted K-theory and loop group representations III, Ann. Math., Volume 174 (2011), pp. 947-1007
[6] A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups, Duke Math. J., Volume 100 (1999), pp. 447-501
[7] Multiplets of representations and Kostant's Dirac operator for equal rank loop groups, Duke Math. J., Volume 110 (2001), pp. 121-160
[8] Triangulated categories of singularities and D-branes in Landau–Ginzburg models, Tr. Mat. Inst. Steklova, Volume 246 (2004), pp. 227-248 Algebr. Geom. Metody, Svyazi i Prilozh., 240–262 (Russian) English translation in Proc. Steklov Inst. Math., 246, 2004
[9] Thom–Sebastiani and duality for matrix factorizations | arXiv
[10] Loop Groups, Oxford University Press, 1986
[11] On adding relations to homotopy groups, Ann. Math., Volume 42 (1941), pp. 409-428
Cité par Sources :
