arXiv subjects
Publications and source records attributed to F. Brandt.
The formulation of Seiberg-Witten maps from the point of view of consistent deformations of gauge theories in the context of the Batalin-Vilkovisky antifield formalism is reviewed. Some additional remarks on noncommutative Yang-Mills theory are made.
The Wess-Zumino consistency condition for four-dimensional Einstein gravity is investigated in the space of local forms involving the fields, the ghosts, the antifields and their derivatives. Its general solution is constructed for all values of the form degree and of the ghost number. It is shown in particular that the antifields (= sources for the BRST variations) can occur only through cohomologically trivial terms.
We establish general theorems on the cohomology $H^*(s|d)$ of the BRST differential modulo the spacetime exterior derivative, acting in the algebra of local $p$-forms depending on the fields and the antifields (=sources for the BRST variations). It is shown that $H^{-k}(s|d)$ is isomorphic to $H_k(δ|d)$ in negative ghost degree $-k\ (k>0)$, where $δ$ is the Koszul-Tate differential associated with the stationary surface. The cohomological group $H_1(δ|d)$ in form degree $n$ is proved to be isomorphic to the space of constants of the motion, thereby providing a cohomological reformulation of Noether theorem. More generally, the group $H_k(δ|d)$ in form degree $n$ is isomorphic to the space of $n-k$ forms that are closed when the equations of motion hold. The groups $H_k(δ|d)$ $(k>2)$ are shown to vanish for standard irreducible gauge theories. The group $H_2(δ|d)$ is then calculated explicitly for electromagnetism, Yang-Mills models and Einstein gravity. The invariance of the groups $H^{k}(s|d)$ under the introduction of non minimal variables and of auxiliary
Yang-Mills models with compact gauge group coupled to matter fields are considered. The general tools developed in a companion paper are applied to compute the local cohomology of the BRST differential $s$ modulo the exterior spacetime derivative $d$ for all values of the ghost number, in the space of polynomials in the fields, the ghosts, the antifields (=sources for the BRST variations) and their derivatives. New solutions to the consistency conditions $sa+db=0$ depending non trivially on the antifields are exhibited. For a semi-simple gauge group, however, these new solutions arise only at ghost number two or higher. Thus at ghost number zero or one, the inclusion of the antifields does not bring in new solutions to the consistency condition $sa+db=0$ besides the already known ones. The analysis does not use power counting and is purely cohomological. It can be easily extended to more general actions containing higher derivatives of the curvature, or Chern-Simons terms.
It is shown that generally the consistency equation for anomalies of quantum field theories has solutions which depend nontrivially on the sources of the (generalized) BRS-transformations of the fields. Explicit previously unknown examples of such solutions are given for Yang-Mills and super Yang-Mills theories.