Abstract
We introduce ABJM quantum field theory in the noncommutative spacetime by using the component formalism and show that it is mathcal{N} = 6 supersymmetric. For the U(1)κ × U(1)−κ case, we compute all one-loop 1PI two and three point functions in the Landau gauge and show that they are UV finite and have well-defined commutative limits θμν → 0, corresponding exactly to the 1PI functions of the ordinary ABJM field theory. This result also holds for all one-loop functions which are UV finite by power counting. It seems that the noncommutative quantum ABJM field theory is free from the noncommutative IR instabilities.
Highlights
ABJM field theory at the level κ was introduced in [1] to provide a holographic dual of the M theory on the AdS4 × S7/Zk, furnishing a concrete realization of the famous gauge/gravity duality conjecture [2]
It is plain that the analysis carried out for the ΠμA1AμA2ˆμ+3 tensor will apply to the tensor ΠμA1AμA2ˆμ−3 as well, so that the limit θμν → 0 of the latter is given by the corresponding Green function in the ordinary ABJM theory too
By using component formalism we have shown that the theory has an N = 6 supersymmetry
Summary
ABJM field theory at the level κ was introduced in [1] to provide a holographic dual of the M theory on the AdS4 × S7/Zk, furnishing a concrete realization of the famous gauge/gravity duality conjecture [2]. As will be shown below, by possessing six supersymmetries our noncommutative ABJM (NCABJM) action does fulfill the necessary condition to become dual to the superstring/supergravity theory on the deformed background constructed in [19] Another important aim of this paper is to check on the quantum level, whether the limit θμν → 0 of the noncommutative ABJM theory restores back the ordinary/commutative ABJM theory introduced in [1]. Values of some integrals contributing to a certain Feynman diagram — see appendix C.2, for example — remains bounded as one approaches θμν = 0 point, but the θμν → 0 limit does not exist Putting it all together, we conclude that it is far from clear that the limit θμν → 0 of the 1PI Green functions in the noncommutative formulation of the ABJM quantum field theory are the corresponding functions in the commutative ABJM quantum field theory. Remaining appendices are needed for properly understanding of the main text
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