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

Abstract

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 $H_k(\delta |d)$ in negative ghost degree $-k\ (k>0)$, where $\delta$ is the Koszul-Tate differential associated with the stationary surface. The cohomological group $H_1(\delta |d)$ 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 $H_k(\delta|d)$ in form degree $n$ is isomorphic to the space of $n-k$ forms that are closed when the equations of motion hold. The groups $H_k(\delta|d)$ $(k>2)$ are shown to vanish for standard irreducible gauge theories. The group $H_2(\delta|d)$ 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 auxiliaryComment: 48 pages LaTeX file, ULB-PMIF-94/06 NIKEF-H 94-13 (minor changes in section 10

E 7(7) symmetry in perturbatively quantised $ \mathcal{N} = 8 $ supergravity

“…The existence of an anomaly therefore requires that the cohomology of the linearised Slavnov-Taylor operator F → (Σ, F) on the set of local functionals {F} of the fields is non-trivial. This cohomology is equivalent to the cohomology H 1 (δ g ) of the differential operator δ g which generates the non-linear e 7(7) action on the set of local functionals of the fields identified modulo the equations of motion [35]. As we already pointed out, the property that A is a local functional is known as the quantum action principle [14].…”

Interactions for a collection of spin-two fields intermediated by a massless p-form

“…We recall that the local cohomology H (δ|d) is completely trivial in both strictly positive antighost and pure ghost numbers (for instance, see Theorem 5.4 from [27] and also [28]). Using the fact that the Cauchy order of the free theory under study is equal to two, the general results from [27] and [28], according to which the local cohomology of the Koszul-Tate differential in pure ghost number zero is trivial in antighost numbers strictly greater than its Cauchy order, ensure that…”

“…Using the fact that the Cauchy order of the free theory under study is equal to two, the general results from [27] and [28], according to which the local cohomology of the Koszul-Tate differential in pure ghost number zero is trivial in antighost numbers strictly greater than its Cauchy order, ensure that…”

“…We employ a systematic approach to the construction of interactions in gauge theories [19,20,21], based on the cohomological reformulation of Lagrangian BRST symmetry [22,23,24,25,26]. In this approach interactions result from the analysis of consistent deformations of the generator of the Lagrangian BRST symmetry (known as the solution of the master equation) by means of specific cohomological techniques, relying on local BRST cohomology [27,28]. The emerging deformations, and hence also the interactions, are constructed under the general hypotheses of locality, smoothness in the coupling constant, Poincaré invariance, Lorentz covariance, and derivative order assumption.…”

Space-time locality in theSp(2)-symmetric Lagrangian formalism

“…For the standard master equation of the BV formalism the existence of a local solution, the structure of renormalization, anomalies and related questions have been discussed in literature (see [12], [13], [14] and references therein). The method for solving the locality problem is based on considering instead of equations for functionals, equations for corresponding functions on spaces, which have the fields and their finite order derivatives in an arbitrary spacetime point as coordinates(jet-spaces [15]).…”