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A representation of general translation-invariant star products ? in the
algebra of M(C) = limNï¿½??ï¿½?? MN (C) is introduced which results in the Moyal-Weyl-Wigner
quantization. It provides a matrix model for general translation-invariant 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 translation-invariant noncommutative quantum field theories based on
the achievements for the commutative fields, and is comparable to the Seiberg-Witten
map for the Moyal case. Employing the Chern-Weil 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 translation-invariant
noncommutative Yang-Mills theory. Thereby we study the mentioned Yang-Mills 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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