Abstract
We show that natural noncommutative gauge theory models on $\mathbb{R}^3_\lambda$ can accommodate gauge invariant harmonic terms, thanks to the existence of a relationship between the center of $\mathbb{R}^3_\lambda$ and the components of the gauge invariant 1-form canonical connection. This latter object shows up naturally within the present noncommutative differential calculus. Restricting ourselves to positive actions with covariant coordinates as field variables, a suitable gauge-fixing leads to a family of matrix models with quartic interactions and kinetic operators with compact resolvent. Their perturbative behavior is then studied. We first compute the 2-point and 4-point functions at the one-loop order within a subfamily of these matrix models for which the interactions have a symmetric form. We find that the corresponding contributions are finite. We then extend this result to arbitrary order. We find that the amplitudes of the ribbon diagrams for the models of this subfamily are finite to all orders in perturbation. This result extends finally to any of the models of the whole family of matrix models obtained from the above gauge-fixing. The origin of this result is discussed. Finally, the existence of a particular model related to integrable hierarchies is indicated, for which the partition function is expressible as a product of ratios of determinants.
Highlights
Invariant classical actions can be done from suitable noncommutative differential calculi [25,26,27,28,29], the study of quantum properties is rendered difficult by technical complications stemming mainly from gauge invariance that supplement the UV/IR mixing problem inherent in NCFT on Moyal spaces
We show that natural noncommutative gauge theory models on R3λ can accommodate gauge invariant harmonic terms, thanks to the existence of a relationship between the center of R3λ and the components of the gauge invariant 1-form canonical connection
We show that natural noncommutative gauge theory models on R3λ can support gauge invariant harmonic terms, unlike the case of Moyal spaces
Summary
The algebra R3λ has been first introduced in [52] and further considered in various works [51, 53, 55]. A characterization of a natural basis has been given in [51]. Where C [x1, x2, x3, x0] is the free algebra generated by the 4 (hermitean) elements (coordinates) {xμ=1,2,3, x0} and I [R1, R2] is the two-sided ideal generated by the relations. The corresponding subfamily is related to the canonical basis of the matrix algebra M2j+1(C). J with w(j) is a center-valued weight factor to be discussed below, trj denotes the canonical trace of M2j+1(C), and Φj Ψj) an element of M2j+1(C) is defined from the expansion (2.3) of Φ by the (2j + 1) × (2j + 1) matrix Φj := (φjmn)−j≤m,n≤j Eq (2.6) defines a family of traces depending on the weight factor w(j).
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