Geometrie quantisation on (infinite dimensional) graded symplectic manifolds is elaborated for a restricted class of phase spaces. The formalism indudes the treatment of femionic field theories. The chiral anomaly (U(1)-anomaly) as well as the non abelian (covariant) anomaly of D-dimensional non Abelian gauge theories is calculated in this framework. Thus previous work of the present authors is generalised.
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