Deformation quantization; Homological perturbation theory; Non-commutative Poisson bracket; Poisson orbifold; Mathematical Physics; Physics and Astronomy (all); Geometry and Topology; General Physics and Astronomy
Abstract :
[en] Whenever a given Poisson manifold is equipped with discrete symmetries the corresponding algebra of invariant functions or the algebra of functions twisted by the symmetry group can have new deformations, which are not captured by Kontsevich Formality. We consider the simplest example of this situation: R2 with the reflection symmetry Z2. The usual quantization leads to the Weyl algebra. While Weyl algebra is rigid, the algebra of even or twisted by Z2 functions has one more deformation, which was identified by Wigner and is related to Feigin's glλ and to fuzzy sphere. With the help of homological perturbation theory we obtain explicit formula for the deformed product, the first order of which can be extracted from Shoikhet–Tsygan–Kontsevich formality.
Research center :
AGIF - Algèbre, Géométrie et Interactions fondamentales
Disciplines :
Physics
Author, co-author :
Sharapov, Alexey; Physics Faculty, Tomsk State University, Tomsk, Russian Federation
Skvortsov, Evgeny ; Université de Mons - UMONS ; Lebedev Institute of Physics, Moscow, Russian Federation
Sukhanov, Arseny; Moscow Institute of Physics and Technology, Dolgoprudnyi, Russian Federation
Language :
English
Title :
Deformation quantization of the simplest Poisson orbifold
S827 - Physique de l'Univers, Champs et Gravitation
Research institute :
R150 - Institut de Recherche sur les Systèmes Complexes
Funders :
ERC - European Research Council F.R.S.-FNRS - Fonds de la Recherche Scientifique
Funding text :
We would like to thank Xiang Tang for a very useful correspondence. The work of E.S. was partially supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 101002551 ) and by the Fonds De La Recherche Scientifique - FNRS under Grant No. F.4544.21 . The results of Appendix A on HPT for Poisson orbifolds were obtained under exclusive support of the Ministry of Science and Higher Education of the Russian Federation (project No. FSWM-2020-0033 ). A. Sh. gratefully acknowledges the financial support of the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS” . The work of A.S. was supported by the Russian Science Foundation grant 18-72-10123 in association with the Lebedev Physical Institute.
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