Noncommutative Induced Gauge Theories on Moyal Spaces

Abstract

Noncommutative field theories on Moyal spaces can be conveniently handled within a framework of noncommutative geometry. Several renormalisable matter field theories that are now identified are briefly reviewed. The construction of renormalisable gauge theories on these noncommutative Moyal spaces, which remains so far a challenging problem, is then closely examined. The computation in 4-D of the one-loop effective gauge theory generated from the integration over a scalar field appearing in a renormalisable theory minimally coupled to an external gauge potential is presented. The gauge invariant effective action is found to involve, beyond the expected noncommutative version of the pure Yang-Mills action, additional terms that may be interpreted as the gauge theory counterpart of the harmonic term, which for the noncommutative ${\varphi }^4$-theory on Moyal space ensures renormalisability. A class of possible candidates for renormalisable gauge theory actions defined on Moyal space is presented and discussed.

24 pages, 6 figures. Talk given at the "International Conference on Noncommutative Geometry and Physics", April 2007, Orsay (France). References updated. To appear in J. Phys. Conf. Ser

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• $\kappa$-Poincaré invariant quantum field theories with KMS weight
• Gauge theories on quantum spaces
• Noncommutative gauge theories on ${\mathbb{R}}_{\theta }^2$ as matrix models
• $\kappa$-Poincaré invariant orientable field theories at 1-loop
• Gauge theories on $\kappa$-Minkowski spaces: Twist and modular operators
• Exact Partition Functions for Gauge Theories on ${\mathbb{R}}_{\lambda }^3$
• Noncommutative gauge theories on ${\mathbb{R}}_{\lambda }^3$: Perturbatively finite models
• Closed star product on noncommutative ${\mathbb{R}}^3$ and scalar field dynamics
• Slavnov-Taylor identities, non-commutative gauge theories and infrared divergences
• Symmetries of noncommutative scalar field theory
• Involutive representations of coordinate algebras and quantum spaces
• Vacuum energy and the cosmological constant problem in $\kappa$-Poincaré invariant field theories
• Spectral theorem in noncommutative field theories: Jacobi dynamics
• Electromagnetic Theory with Quantum Principal Bundles