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

TL;DR

This paper surveys cohomological physics, highlighting its development through homological algebra extensions like the Batalin-Vilkovisky approach, which integrates Lie algebra cohomology with the Koszul-Tate resolution for quantizing gauge theories.

Contribution

It presents a revisionist view of the Batalin-Vilkovisky machinery as a novel reconstruction of homological algebra with innovative ideas.

Findings

Connection between Lie algebra cohomology and the Koszul-Tate resolution.

Application of cohomological methods to gauge theory quantization.

Development of the Batalin-Vilkovisky approach for string field theory.

Abstract

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.