Deformation quantization for Heisenberg supergroup

Pierre Bieliavsky (1) , Axel de Goursac (1) , Gijs M. Tuynman (2)

1 UCL IRMP - Institut de Recherche en Mathématiques et Physique
2 LPP - Laboratoire Paul Painlevé - UMR 8524

We construct a non-formal deformation machinery for the actions of the Heisenberg supergroup analogue to the one developed by M. Rieffel for the actions of R^d . However, the method used here differs from Rieffel's one: we obtain a Universal Deformation Formula for the actions of R^{m|n} as a byproduct of Weyl ordered Kirillov's orbit method adapted to the graded setting. To do so, we have to introduce the notion of C* -superalgebra, which is compatible with the deformation, and which can be seen as corresponding to noncommutative superspaces. We also use this construction to interpret the renormalizability of a noncommutative quantum field theory.

Mots clés: Deformation quantization; Heisenberg supergroup; Harmonic analysis; Fréchet functional spaces; Graded operator algebras; Supermanifolds; Renormalization

Domaines: Algèbres quantiques [math.QA]; Physique mathématique [math-ph]; Géométrie différentielle [math.DG]; Algèbres d'opérateurs [math.OA]

Pierre Bieliavsky, Axel de Goursac, Gijs M. Tuynman. Deformation quantization for Heisenberg supergroup. Journal of Functional Analysis, 2012, 263 (3), pp.549-603. ⟨10.1016/j.jfa.2012.05.002⟩. ⟨hal-00707817⟩