Local cohomology in gauge theories, BRST transformations and anomalies

Abstract

We introduce a geometric framework needed for a mathematical understanding of the BRST symmetries and chiral anomalies in gauge field theories. We define the BRST bicomplex in terms of local cohomology using differential forms on the infinite jet bundle and consider variational aspects of the problem in this cohomological context. The adjoint representation of the structure group induces a representation of the infinite dimensional Lie algebra g of infinitesimal gauge transformations on the space of local differential forms, with respect to which the BRST bicomplex is defined using the Chevalley-Eilenberg construction. The induced coboundary operator of the associated cohomology H ∗ loc ( g ) is the BRST operator s. With this we derive the classical BRST transformations of the vector potential A and the ghost field η as sA = dη+[A, η] , and sη = - 1 2 [η, η] . Moreover the ghost field η is identified with the canonical Maurer-Cartan form of the infinite dimensional Lie group G of gauge transformations. We give a homotopy formula on the BRST bicomplex and with the introduction of Chern-Simon type forms we derive the associated descent equations and show that the non-Abelian anomalies, which satisfy the Wess-Zumino consistency condition, represent cohomology classes in H 1 loc ( g ) .