The geometry of BRS transformations (Q1122174)

In 1976 \textit{C. Becchi}, \textit{A. Rouet} and \textit{R. Stora} [Renormalization of gauge theories, Ann. Phys. 98, 287-321 (1976)] noticed that in gauge field theories the effective Lagrangian, which is no longer gauge invariant, is still invariant under a new class of transformations now called BRS transformations \[ sA=d\eta +[A,\eta],\quad s\eta =-[\eta,\eta] \] where A is the potential field (connection one form) and \(\eta\) is the ghost field. We show how these BRS transformations can be interpreted as purely differential geometric objects. We define a general BRS cohomology \(H^{q,p}\) of the infinite dimensional Lie algebra \({\mathfrak g}\) of infinitesimal gauge transformations with respect to an induced representation. As a special case, namely with respect to the adjoint representation, we obtain the classical BRS transformations as coboundary operator \(s: {\mathbb{C}}^{q,p}\to {\mathbb{C}}^{q+1,p}\) of this complex. The Wess-Zumino consistency condition is expressed as \(s^ 2=0\), while the ghost field \(\eta\) is interpreted as the canonical Maurer-Cartan form on the infinite dimensional Lie group G of gauge transformations.