Kontsevich product and gauge invariance

Abstract

We analyze the question of $U_{\star }(1)$ gauge invariance in a flat non-commutative space where the parameter of non-commutativity, ${\theta }^{\mu \nu }(x)$, is a local function satisfying Jacobi identity (and thereby leading to an associative Kontsevich product). We show that in this case, both gauge transformations as well as the definitions of covariant derivatives have to modify so as to have a gauge invariant action. We work out the gauge invariant actions for the matter fields in the fundamental and the adjoint representations up to order ${\theta }^2$ while we discuss the gauge invariant Maxwell theory up to order $\theta$. We show that despite the modifications in the gauge transformations, the covariant derivative and the field strength, Seiberg-Witten map continues to hold for this theory. In this theory, translations do not form a subgroup of the gauge transformations (unlike in the case when ${\theta }^{\mu \nu }$ is a constant) which is reflected in the stress tensor not being conserved.

arXiv comment7 pages

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Cited by in corpus (10)

• Noncommutative Gravity
• Monopole and Berry Phase in Momentum Space in Noncommutative Quantum Mechanics
• Non-commutative gauge theory on D-branes in Melvin Universes
• On Supergravity Solutions of Branes in Melvin Universes
• Coordinate noncommutativity in strong non-uniform magnetic fields
• Noncommutative theories and general coordinate transformations
• Effects of wave propagation in canonical Poisson gauge theory under an external magnetic field
• Field Theory with Coordinate Dependent Noncommutativity
• Seiberg-Witten Map with Lorentz-Invariance and Gauge-Covariant Star Product
• Planar field theories with space-dependent noncommutativity