Derivations of the Moyal Algebra and Noncommutative Gauge Theories

• Дата: 2009
• Автор: Wallet, Jean-Christophe
• Формат: Стаття
• Мова: English
• Опубліковано: Інститут математики НАН України 2009
• Назва видання: Symmetry, Integrability and Geometry: Methods and Applications
• Онлайн доступ: http://dspace.nbuv.gov.ua/handle/123456789/149248
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Цитувати: Derivations of the Moyal Algebra and Noncommutative Gauge Theories / Jean-Christophe Wallet // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 52 назв. — англ.

The differential calculus based on the derivations of an associative algebra underlies most of the noncommutative field theories considered so far. We review the essential properties of this framework and the main features of noncommutative connections in the case of non graded associative unital algebras with involution. We extend this framework to the case of Z₂-graded unital involutive algebras. We show, in the case of the Moyal algebra or some related Z₂-graded version of it, that the derivation based differential calculus is a suitable framework to construct Yang-Mills-Higgs type models on Moyal (or related) algebras, the covariant coordinates having in particular a natural interpretation as Higgs fields. We also exhibit, in one situation, a link between the renormalisable NC φ4-model with harmonic term and a gauge theory model. Some possible consequences of this are briefly discussed.