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The authors study Weyl symmetry from a cohomological point of view in theories including gravity in order to determine 0- and 1-cocycles (Weyl invariants and Weyl anomalies). For pedagogical reasons they rederive known results in four dimensions. Then they solve the same problem in six dimensions. The relation of Weyl anomalies to cocycles of the diffeomorphisms are determined: they can be obtained from each other by subtracting a suitable counterterm from the action. Finally it is shown by an example that such a subtraction corresponds to changing the regularisation scheme.

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We show that ghosts in gauge theories can be interpreted as Maurer-Cartan forms in the infinite dimensional group ^ of gauge transformations. We examine the cohomology of the Lie algebra of ^ and identify the coboundary operator with the BRS operator. We describe the anomalous terms encountered in the renormalization of gauge theories (triangle anomalies) as elements of these cohomology groups.

Abstract:We discuss BRST and anti-BRST transformations for an Abelian antisymmetric gauge field in 4D and find that, in order for them to anticommute, we have to impose a condition on the auxiliary fields. This condition is similar to the Curci-Ferrari condition for the 4D non-Abelian 1-form gauge theories and represents a consistency requirement. We interpret it as a signal that our Abelian 2-form gauge field theory is based on gerbes. To support this interpretation we discuss, in particular, the case of the 1-gerbe for our present field theory and write the relevant equations and symmetry transformations for 2-gerbes.

Motivated by the search for possible CP violating terms in the trace of the energy-momentum tensor in theories coupled to gravity we revisit the problem of trace anomalies in chiral theories. We recalculate the latter and ascertain that in the trace of the energy-momentum tensor of theories with chiral fermions at one-loop the Pontryagin density appears with an imaginary coefficient. We argue that this may break unitarity, in which case the trace anomaly has to be used as a selective criterion for theories, analogous to the chiral anomalies in gauge theories. We analyze some remarkable consequences of this fact, that seem to have been overlooked in the literature.