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Paper IPM / M / 16514  


Abstract:  
A representation of general translationinvariant star products ? in the
algebra of M(C) = limNï¿½??ï¿½?? MN (C) is introduced which results in the MoyalWeylWigner
quantization. It provides a matrix model for general translationinvariant noncommutative
quantum field theories in terms of the noncommutative calculus on differential graded
algebras. Upon this machinery a cohomology theory, the so called ?cohomology, with
groups Hk
?
(C), k ï¿½?ï¿½ 0, is worked out which provides a cohomological framework to
formulate general translationinvariant noncommutative quantum field theories based on
the achievements for the commutative fields, and is comparable to the SeibergWitten
map for the Moyal case. Employing the ChernWeil theory via the integral classes of
Hk
?
(Z) a noncommutative version of the Chern character is defined as an equivariant
form which contains topological information about the corresponding translationinvariant
noncommutative YangMills theory. Thereby we study the mentioned YangMills theories
with three types of actions of the gauge fields on the spinors, the ordinary, the inverse,
and the adjoint action, and then some exact solutions for their anomalous behaviors are
worked out via employing the homotopic correlation on the integral classes of ?cohomology.
Finally, the corresponding consistent anomalies are also derived from this topological Chern
character in the ?cohomology.
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