Local BRST cohomology in the antifield formalism: I. General theorems

Abstract

We establish general theorems on the cohomologyH * (s/d) of the BRST differential modulo the spacetime exterior derivative, acting in the algebra of localp-forms depending on the fields and the antifields (=sources for the BRST variations). It is shown thatH −k (s/d) is isomorphic toH k (δ/d) in negative ghost degree−k (k>0), where δ is the Koszul-Tate differential associated with the stationary surface. The cohomology groupH 1 (δ/d) in form degreen is proved to be isomorphic to the space of constants of the motion, thereby providing a cohomological reformulation of Noether's theorem. More generally, the groupH k (δ/d) in form degreen is isomorphic to the space ofn−k forms that are closed when the equations of motion hold. The groupsH k (δ/d)(k>2) are shown to vanish for standard irreducible gauge theories. The groupH 2 (δ/d) is then calculated explicitly for electromagnetism, Yang-Mills models and Einstein gravity. The invariance of the groupsH k (s/d) under the introduction of non-minimal variables and of auxiliary fields is also demonstrated. In a companion paper, the general formalism is applied to the calculation ofH k (s/d) in Yang-Mills theory, which is carried out in detail for an arbitrary compact gauge group.