The (secret?) homological algebra of the Batalin-Vilkovisky approach

This is a survey of `Cohomological Physics', a phrase that first appeared in the context of anomalies in gauge theory. Differential forms were implicit in physics at least as far back as Gauss (1833) (cf. his electro-magnetic definition of the linking number), and more visibly in Dirac's magnetic monopole (1931). The magnetic charge was given by the first Chern number. Thus were characteristic classes (and by implication the cohomology of Lie algebras and of Lie groups) introduced into physics. The `ghosts' introduced by Fade'ev and Popov were incorporated into what came to be known as BRST cohomology. Later the ghosts were reinterpreted as generators of the Chevalley-Eilenberg complex for Lie algebra cohomology. Cohomological physics also makes use of group theoretic cohomology, algebraic deformation theory and especially a novel extension of homological algebra, combining Lie algebra cohomology with the Koszul-Tate resolution, the major emphasis of the talk. This synergistic combination of both kinds of cohomology appeared in the Batalin-Fradkin-Vilkovisky approach to the cohomological reduction of constrained Poisson algebras. An analogous `odd' version was developed in the Batalin-Vilkovisky approach to quantizing particle Lagrangians and Lagrangians of string field theory. A revisionist view of the Batalin- Vilkovisky machinery recognizes parts of it as a reconstruction of homological algebra with some powerful new ideas undreamt of in that discipline.