This paper is a detailed introduction to the differential geometric and cohomological framework underlying B.R.S. (Becchi-Rouet-Stora) - transformations and anomalies of gauge fields. Section 1 describes the de Rham complex \(\Lambda^*\) of differential forms on a principal bundle P with values in the symmetric algebra over the Lie algebra L of the structure group G and the commuting actions of G and the gauge group \({\mathcal G}\) on this complex \(\Lambda^*\). Section 2 describes the subcomplex \(A^*\) of G-invariant elements in \(\Lambda^*\). Section 3 describes the complex of alternating multilinear differential operators on the Lie algebra \({\mathcal L}\) of the gauge group \({\mathcal G}\) with values in the smooth functions on the base manifold.
Section 4 describes the complex of alternating multilinear differential operators on the gauge Lie algebra \({\mathcal L}\) with values in \(A^*\). It admits a canonical (tautologic) algebraic connection, the ghost field of the physical literature, leading to the “B.R.S.”-relations, treated in section 5. Then this double complex is changed to have coefficients in the space of all local nonlinear operators on the space of connections with values in the reals, given by integrals of forms over chains, in section 6. The cohomology of this last complex contains then finally the anomalies.
All possible anomalies can be classified via an extended Chern-Weil- homomorphism following M. Dubois-Violette, C. M. Viallet and M. Talon [Commun. Math. Phys. 102, 105-122 (1985; Zbl 0604.58055)]. An account of this is given in the last section. Two appendices, on graded differential algebras and on algebraic connections in graded differential Lie algebras, conclude the paper. This paper suffers from a somewhat topheavy not very systematic notation. The reader is advised to look also at the paper of M. Dubois-Violette [J. Geom. Phys. 3, 525-565 (1986; Zbl 0627.53064)].