Redirected from "local BV-cohomology".
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Context
Algebraic Quantum Field Theory
Higher geometry
Contents
Idea
Given a Lagrangian field theory with field bundle over some spacetime and local Lagrangian density , then its local BV-BRST complex (or local BRST complex, for short) is the realization of the BV-BRST complex not on local observables given by functionals on the space of field histories which are transgressions of variational differential forms on the jet bundle, but on these variational differential forms themselves (whence “local”, i.e. before transgression).
If denotes the BV-BRST differential in a BV-resolution of the restriction to the shell of the variational bicomplex with its total spacetime derivative (horizontal derivative), then the local BV-BRST cohomology is the cochain cohomology of , hence of the total complex of the double complex given by and .
Generally, considering variational differential forms up to -exact terms is the “local” incarnation of what under the integration involved in the transgression is integration by parts and it is in this way that “local BV-BRST cohomology” knows about the actual BV-BRST cohomology on local observables.
Example
Consider local coordinates on the fibers of the field bundle. The corresponding antifield coordinates are to be denoted and the BV-BRST differential takes them to the corresponding component
of the Euler-Lagrange form.
In degree the -closed elements in vanishing ghost degree are pairs consisting of an infinitesimal symmetry of the Lagrangian , regarded as an antifield density , together with a corresponding conserved Noether current :
Such pairs are -exact if on-shell the infintiesimal symmetry coincides with an infinitesimal gauge symmetry. To see this, recall:
An infinitesimal gauge symmetry of gauge parameter is a vector field on the jet bundle with components of the form
such that this is an infinitesimal symmetry of the Lagrangian in that
for all .
The corresponding antighosts are taken by the BV-BRST differential to the antifield-preimage of the term on the left:
Moreover, an on-shell vanishing infinitesimal symmetry of the Lagrangian is a vector field with components of the form
for a skew-symmetric system of smooth functions on the jet bundle.
The linear combination of such an infinitesimal gauge symmetry and an on-shell vanishing infinitesimal symmetry is -exact:
(Barnich-Brandt-Henneaux 94, p. 20)
It may be useful to organize this expression into the -bicomplex like so:
References
The general theory:
Review:
Details on the local antibracket:
Application to gravity and/or Yang-Mills theory (Einstein-Yang-Mills theory) is discussed in